Transcript
EFFECTS OF MEMBRANE ACTION ON THE ULTIMATE STRENGTH OF REINFORCED CONCRETE SLABS
A thesis presented for the degree of Doctor of Philosophy in Civil Engineering in the University of Canterbury, Christchurch, New Zealand.
by
D. C. HOPKINS
1969
ACKNOWLEDGEMENTS
I gratefully acknowledge the assistance that I have received during the course of this project and extend my thanks to: Professor H.J. Hopkins, Head of Department, for his general supervision and guidance. Professor R. Park for his valuable assistance and continued encouragement throughout the project and for his helpful advice during the preparation of this thesis. Members of the academic staff, particularly Dr A.J. Carr, for their assistance. Mr H.T. Watson, and members of the technical staff, particularly Messrs J. Sheard and N. Prebble, for their assistance with experimental work. The University Grants Committee for financial assistance in the form of a Postgraduate Scholarship and a research grant. Certified Concrete Limited for their assistance in making the model floor. Mrs J.M. Keoghan for typing the manuscript. Finally, I wish to thank my wife for her encouragement and assistance at all times.
SUMMARY
This thesis describes an investigation of the effects of membrane action on reinforced concrete slabs, particularly the implications of allowing for compressive membrane action in the design of slab and beam floors. An examination of the minimum reinforcement
re~uire
ments of a rectangular slab reveals that high design live loads are required before the benefits of membrane action can be fully exploited.
Studies of the effect of
c.ompression on the flexural capacity of a reinforced conc.rete section and of the effect of membrane action on a clamped circular slab with elastic lateral restraint at the circumference are undertaken.
These show that lightly
reinforced, thick slabs with high concrete strength will benefit most from compressive membrane action in practical situations, and that if the surround is flexible, tensile membrane action will be evident at the stage when the ultimate load of the slab is reached. The effects of compressive forces in the panels on the design of the supporting beams is studied.
It is shown
that some beams are required to resist considerable tension and that membrane action may have considerable effect on
the torsion induced in the edge beams.
A design method is
derived to deal with beams subject to tension. An investigation is then made of the lateral restraint provided at the edges of an interior panel by the surrounding panels, considered to be of elastic, homogeneous material. An experimental study of a quarter-scale, nine-panel slab and beam floor was conducted.
The equations derived
by Park for the ultimate strength of slabs with compressive membrane action were used to design the floor.
The membrane
action was assessed as sufficient to double the Johansen ,ultimate load of the centre panel.
A smaller enhancement
was allowed for in the centre-edge panels and none was allowed for in the corner panels.
The centre spans of all
beams were designed to carry the tension induced by the compressive membrane forces in the panels. Results of fourteen load tests on this model floor are analysed with particular reference to the effects of membrane action.
Satisfactory behaviour at service load
was observed and the floor sustained the predicted ultimate load before failure of the centre panel.
The measurement
of concrete and steel strains at critical sections revealed the presenc.e of compressive membrane forees in the centre panel and tensions in the beams that were of the order expected .
.A comparison of the volumes of steel reinforcement required in the model floor indicated that design including
compressive membrane action brings no advantage except when the additional steel that is required to resist the tensile forces induced in the beams can also be used to resist moments due to earthquake or other lateral loading of the structure. It is concluded that allowance for membrane action in design would be of small benefit for normal slab and beam floors and would be of greatest use when very high loads are imposed on slabs with high lateral restraint at the edges.
NOTATION
a, a'
Depths of the equivalent rectangular stress blocks at the ultimate flexural capacity of sections in regions of positive and negative moment respectively.
A
Gross area of a section.
A
s'
AI
b
S
Areas of tension steel in slab or beam sections subject to positive and negative moment respectively. Breadth of a rectangular section and of the web of a T- or L-section. Breadth of flange of a T- or L-section.
c' c
Force in conerete per unit length of a hogging moment yield line.
d, d'
Distances from the top of a section to the centroids of the tension steel, for positive and negative moments respectively. Depth of the neutral axis of a seetion, measured from the compression face.
D
Overall depth of a beam or slab.
e
Strain - subscripts used are defined by Figures 7.7 and E.1 . Young~s
modulus.
Yield stresS of steel reinforcement. Cylinder strength of concrete in compression. Tensile fracture stress of concrete. Enhancement of the load capacity of a reinforced concrete slab due to membrane aetion, i.e., F the ratio of the ultimate load with membrane action to that calculated by Johansen's yield line theory.
Fmax
Maximum attainable enhancement factor for a reinforced concrete section.
F*
Average enhancement of the moment capacities of slab sections along yield lines necessary to produce a load enhancement of F.
G
Shear modulus of concrete considered as an elastic, homogeneous material. Ratios of hogging to sagging yield moments due to steel in the short and long directions of a slab.
J
Particular values of is and i L , Subscript denoting the value of a quantity at the Johansen ultimate load. Constants defining the stress block for concret~ ) in compression as proposed by Hognestad et al.l 0 Clear span plan dimensions of a rectangular slab in the x and y directions.
m, m'
Yield moment capacities, per unit length, along sagging and hogging moment yield lines respectively ~ taken as the moment of internal steel and concrete forces about the mid-depth of the slab.
M, M'
Moments at beam sections at mid-span and support, taken as the moment of internal actions taken about the mid-depth of the beam. Values of m, M at the Johansen ultimate load. Value of M due to earthquake loading only. Maximum torsional moment in a beam.
s
Spacing of bars in slabs, or stirrups in beams.
Tx' T'x
Tensions induced in an x-direction beam due to compressive membrane action in the panels, i.e., the difference between the steel tension and concrete forces at a beam section in the span and at a support respectively~xcept as used in Appendix A).
T1 ,· .T 6 ,
TB,T:8,T:8
Also llsed to denote tension in beams.
Tm
The maximum torque in an edge beam supporting the square slab of Figure 4.4.
u
Cube strength of concrete in compression.
V
Volume of steel in a slab.
VI
Shear to be taken by stirrups.
U
W
or wM
Ultimate load per unit area of slab with membrane action.
wJ
Johansen ultimate load per unit area of slab.
W
The sum of in-plane loads acting along each half of each edge of a surround.
x, y
Rectangular co-ordinates in a horizontal plane. Vertical deflection at points F, S ... Lateral deflection of slab surround. Ratios of the effective outward movement of the edge of a surround to the half span of the slab. Parameters defining the length of top steel as in Figure 2.1. (a) In Chapter 2: Ratio of the sagging yield moment, M , in the x-direction to that in the y-directi~n, My' (b) In Chapters 5 and 7 and in Appendix A: Poisson'S ratio. Capacity reduction fae-tor as used in ACI 318-63.
Notes:
(i)
other symbols are defined in the text, generally by Equations or Figures and apply only to one Chapter.
eii) The notation used in Chapter 3 is defined in Figures 3.1 and 3.3 and within the text of the Chapter. Main symbols used·are: K
A measure of the flexibility of the restraining springs at the circumference. Moments, per unit length, in the radial and circumferential directions respectively,
Moment capacities per unit length of sagging and hogging moment yield lines respectively when no membrane forces exist at the section. '1, '1J' '1 M
Intensities of the uniform load - in general, at the Johansen load~ and at the e~Qanced load.
QJ' QM
Total loads on the circular slab,
Qr
Shear force, per unit length, at a radius, r.
To
Force in the tension steel after yield.
T e , Tr
Net tensions, per unit length, in the circumferential and radial directions.
w
Deflection of a point on the slab.
Wo
Deflection at the centre of the slab.
CONTENTS
Page
CHAPTER 1
INTRODUCTION AND SCOPE OF WORK
1.1
Introduction . .
1
1.2
Object and Scope of Work Performed
7
CHAPTER 2
MINT MUM STEEL REQUIREMENTS IN RECTANGULAR SLABS SUPPORTED ALONG ALL FOUR EDGES
CHAPTER 3
11
A STUDY OF THE EFFECT OF MEMBRANE FORCES . ON A REINFORCED CONCRETE SECTION AND ON A CIRCULAR SLAB WITH PARTIAL LATERAL RESTRAINT AT THE EDGES
3.1
Enllancement of the Moment Capacity of a Reinforced Concrete Section
3.2
24
The Effect of Membrane Action on a
.
Clamped Circular Slab Supported and Laterally Restrained at its Circumference
CHAPTER 4
31
THE EFFECT OF PANEL MEMBRANE AOTION ON THE DESIGN OF SUPPORTING BEAMS
4.1
Summary
4.2
Determination of Beam Moments .
. . . 56 . . 56
Page CHAPTER 4.3
4.4
CHAPTER 5
The Effect of Panel lVlembrane Acti on on Torsion in Supporting Beams
70
Discussion and Conclusion
77
STIFFNESS OF SURROUNDS FOR SQUARE SI,ABS
5·1
Introduction and Summary .
5.2
Method of Analysis and Cases Considel~ed
79
. . .
80
5.3
Displacements of the Loaded Edges
83
5 .L~
Stresse s in the Surround
86
5.5 Deep Beam Approximation 5.6
CHAPTER 6
86
92
Conclusions DESIGN AND CONi3TRUCTION OF A MODEL SI.AB AND BEAM FLOOR
. . · ·
·
6.1
Introduction
6.2
General Design Basis and Specifications
95
..·
99
6.3
Design of Floor Panels
102
6.4
Design of Beams
109
6·5
Construction
6.6
Material Properties and Final Slab Dimensions
CHAPTER 7
· .
· 111
· ·.· ...
. · · . . .
· 117
INSTRDllliNTATION AND TEST PROGRANl1'llE
7·1
Instrumentation
7·2
Test Programme
7·3
Reciuction and Processing of Raw Data
. .. · ·
· 122 · 131
· · . . . .
'.
··.
. .
· 134
Page CHAPTER 8
TESTS ON THE PERFORMANOE OF THE 1ffiTHOD USED TO OALOULATE SECTION AOTIONS
8.1
Summary
148
8.2
Tests on Special Control Specimens
148
8.3
The Effect of Variation in Strain Readings
CHAPTER 9
159
BEHAVIOUR OF THE NINE-PANEL MODEL FLOOR DURING THE TEST PROGRAMME
... . . ..
9.1
Summary
9.2
Test by Test Description of Floor Behaviour
. . .
. .
163
. .
9.2.1
Tests 101 , 102, 105 and 106
168
9.2.2
Tests 103 and 108
170
9.2.3
Tests 104 and 109
171
9.2.4
Tests '107, 110 and 111
172
9.2·5
Test to Failure of Floor as A Whole
9.2.6
181
Test to Failure of Outer P~nels
9.3
168
. . . . . .
190
Examination of Aspects of Floor Behaviour .
196
9.3.1
Deflections
196
9.3.2
Strains
203
9.3.3
Cracking.
209
9.3.4
Reac.tions
213
Page
CHAPTER 10
9.3.5
Moments. . . .
9.3.6
Membrane Action Effects
DISCUSSION OF TEST
.
216 232
RESULTS
1 0 .1 .Summary
253
10.2 Discussion of Test Results . .
253
10.3 Conclusions
262
CHAPTER 11
A
C01~ARISON
OF THE REINFORCING
STEEl1 REQUIREMENTS OF THE MODEL FLOOR, DESIGNED WITH AND WITHOUT ALIJOViTANCE FOR MEMBRANE ACTION 11.1 Introduction and Summary . .
266
11.2 General Basis of Comparison
26'7
11.3 Comparison of Steel Volumes
269
11.4 Discussion.
2'71
11.5 Conclu.sions
278
CHAPTER 12
GENERAL CONCLUSIONS
12.1 Conclusions from Work Performed 12.2 Suggestions for Further Research .
APPENDIX A
· 280 . . 284
DESIGN CALCULATIONS
A.1 Parkus Equations for the Ultimate Loads of Panels
· 288
A.2 Design of Panels
A.3 Design of Beams APPENDIX B
291 . . . . . · 299
MATERIAL PROPERTIES AND SLAB DIMENSIONS
Page APPENDIX B.1 Cone-rete Properties
305
B.2 Steel Properties
306
B.3 Slab Dimensions .
308
APPENDIX C
DETAILS OF LOAD INCREMENTS FOR THE TEST ON THE NINE-PANEL FLOOR
APPENDIX D
312
REDUCED DATA FROM SLAB TEST
D.1 Deflections
314
D.2 Reactions
321
D.3 Strains. . . . .
. ....
APPENDIX E
COMPUTER PROGRArmillE DESCRIPTION
APPENDIX F
RESULTS OF TESTS ON CONCRETE SLAB STRIPS .
0
•
REFERENCES
•
•
•
•
•
•
324 346
350
355
-
000 -
THE LIBRARY
UNIVERSITY OF CANTERBURY CHRISTCHURCH, N.Z.
INTRODUCTION
1.1
CHAPTER
1
AND
SCOPE
OF
WORK
INTRODUCTION In the calculation of the ultimate loads of two-way
reinforced concrete slabs, the yield line theory due to Johansen(2) has been widely adopted.
This theory does not
include the effect of forces in the plane of the slab and under-estimates the ultimate loads of slabs when in-plane compressive forces are present because the compression enhances the ultimate moment of resistance of the section. In the common case of a lightly reinforced slab the large shift of the position of the neutral axis which occurs with cracking; causes a tendency for the edges of the slab to move outward as the slab deflects further.
If the
edges are restrained against outward movement, compressive forces are induced in the plane of the slab.
The result-
ing enhancement of the load carrying capacity of the slab may be thought of as due to the enhancement of the moment capacities of the yield sections, or to an arching or doming effect in the slab as a whole. - Ockleston(3,4,5) has reported on tests on interior panels of a full scale slab and beam- floor for which the
2
ratios of experimental ultimate load to predicted Johansen load (= enhancement factor) were greater than 2.5.
The
fact that Ockleston showed that this large increase could be accounted for by the development of in-plane compressiOli has stimulated considerable research, both experimental and theoretical, into the phenomenon of membrane action in reinforced concrete slabs. powell(6) tested small scale rectangular slabs (36" x 20.57" x 1.286") with equal percentages of steel, top and bottom,in both directions.
Experimental results revealed
enhancement factors between 1.61 (for 1.53% reinforcement) to 8.25 (for .25% reinforcement). Wood(7) tested three square p1;l.nels (68" x 68 11 x 2.25) cast monolithically within a stiff reinforced concrete surround and obtained enhancement factors of 4.38 (for .25% reinforcement top and bottom) and 10.
-< Cd)
(c)
(f)
(e)
I FIGURE
2 _1
(g)
(i)
RECTANGULAR SLAB AND POSSIBLE FAILURE MECHANISMS
(h)
-
13 steel required for the slab was calculated and the conditions giving minimum steel volume were determined. a result, optimum values of i1 and i2 were found;
As
values
of ~,
>--2 were related to i 1 , i 2 ; and fie' the most economical coefficient of orthotropy was determined as dependent upon Ly/Lx' i1 and i 2 . In many cases, especially when
~e
took high values,
minimum reinforcement conditions were seen to govern.
2.2
VOLUME OF STEEL IN SLAB For a lightly reinforced section, the moments per unit
width are given by:
.... (2 .1 )
similar expressions resulting for the hogging yield moments. For very lightly reinforced sections the value of 'a' is small relative to dx or dy and for the purposes of studying minimum steelrequireme~ts it may be assumed that d = d .
x
Y
It is therefore reasonable to assume that mx and my vary directly with the area of steel, As' in exactly the same manner. . Therefore, in general, As = km where k is a constant incorporating dx and f y ' The volume of steel, V, is then given by
v
==
k L~ymy(1+;U) + 2k L~ymy(i1 A 1 #+ i2 A 2 ) ••• (2.2) (bottom steel) (top steel)
-
14
2.3
COLLAPSE MODES OF SLAB Figures 2.1(c) and (d) show the two most probable
collapse modes. In (c) full hogging moments are developed around the edges of the slab and full sagging yield moments developed along the sagging yield lines as shown. In (d) the portion of the panel without top reinforcement fails as a simply supported slab of reduced span lengths.
Collapse patterns (e), (f), (g), (h), (i), (j)
were not considered at this stage, but it will be shown that the conditions imposed by modes (c) and (d) require little or no modification when the other patterns are considered. The collapse load of mode (c) may be shown to be (7)
and since mode (d) is a special case of (c) for which i1 i2 = 0, Lx
=
(1-2~)Lx' Ly 2.4- YYly
=
= (1-2A2 )Ly :
P. 2. .... (2.4)
The minimum length of top steel may be obtained by equating Wc
=
wdo
The resulting relationship betweeni 1 , i 2 '.\.1
15
.... (2.5) Comparison of similar terms on the left and right sides of Equation 2.5 gives:
(1 -
and
2 \1 )"(1-Z'\2.)
-
(1- 2\2 2 (1 - 2..\1
-
.... (2.6) 1 + i1 1 + i 2.
.... (2.7)
Equating values for (1-2~f given by each of 2.6 and 2.7 gives:
=
(1_2~)2 (1+i 2 ) (1 +i1 )
.... (2.8)
which reduces to: \
1-21\2
_1_
=
/1 +i2\
•
D
•
•
(2.9)
and similarly 1-2'\1
~
=)1 +i11
.... (2.10)
Thus A1 and "'2 may be determined if i1 and i2 are known.
--
16
2.4
MOST ECONOMICAL COEFFICIENT OF
ORTHOTROPY,~
e
The value of)A-- which give s the least volume of steel in the slab may be determined by setting
~~=
O.
From Equation 2.2:
v
= my
and
k LxLymy (1+"u+2 i/'1fi-+2iZA2) =
w
L~ (~)2 t 2. 2..4-
.... (2.2a)
(1- 2. A'\ ) 4-
,42 (1 - 2 \'2)
from Equation 2.4.
2
- 1
in which t =
.... (2.11)
For differentiation with respect to;U-, values of i 1 , i 2 , Lx' Ly will be constant and therefore substitution for my in Equation 2.2a gives:
V= K[~~(1+2i2.A2.) + ~(1+2i1Al~
.. . =
0 for minimum V.
Now from Equation 2.11:
2lt
~
=
~(h)2(1- 2>-.,)2-
- 2.lt.-t 1)
L)(.
Substitution for fA' and
t -1-1
(1 - 2.>-'2.)2
dt.ln ~ 'dv =
~
0 gives
17 which is the condition for minimum volume. Substituion for t+1 from Equation 2.11, squaring, and collecting terms leads to the result:
..•. (2.13) which corresponds to the result stated by Wood(7).
The
validity of Equations 2.3 to 2.13 is limited to the range of i1 and i2 values for which the collapse mode is of the form shown in Figure 2.1(c).
In particular, the central
sagging yield line must be parallel to Ly . For the symmetrical rectangular slab under consideration, this condition is fulfilled if
2.5
fL
). (Lx )2.(~) \ Ly "~+-I1
.
MOST ECONOMICAL VALUES OF i1 and i2 The values of i1 and i2 which give the least volume of
steel could be determined by differentiation of Equation 2.2 but this is difficult and tedious and a simple computer programme was written to investigate the effects of i1 and i2 on the steel volume. For each Ly/Lx ratio steel volumes were calculated for a range of i1 and i 2 • For each combination of i1 and i 2 ", f£e was calculated before the volume of steel was computed. In all cases the combination of i1 = i2 = 2.0 gave minimum volume.
For a square slab, the differentiation of
-
18
the volume expression with respect to i (= i1 in fact, give i
=
=
i 2 ) does,
2.0 for minimum steel.
For L/Lx% 1.0 greatest economy is thus achieved if i1
2.6
==
i2 == 2.0, provided"ue can be attained.
EFFECT OF OTHER COLLAPSE MODES (i) Modes (c),, (d), (e), and (f) By considering modes (e) and (f) as special cases of
mode (c) it may be shown that, provided
A1
and >-2. are cal-
culated from Equations 2.9 and 2.10, modes (c), (d), (e), (f) have identical collapse loads for any given value
of~.
(ii) Modes (g), (h), (i), (j) Since these bear the same relation to each other as (c), (d), (e), (f), the collapse loads of patterns (g), (h), (i), (j) are identical.
It is thus necessary to find
the regions of Ly/Lx' i1 and i2 for which the latter modes have a lower collapse load than the former.
To achieve
this the loads of each set were computed for a range of Ly/Lx' i 1 and i2 (A1 andA 2 were calculated using Equations 2.9 and 2.10; the coefficient of or'thotropy,;U-, was always set at
~
as given by the particular values of
Ly/Lx' i1 and i 2 ). Under these conditions, modes (g), (h), (i), (j) governed only when L/Lx' i1 and i2 were such that mode (c) was not valid initially for the calculation of
JUe'
By considering the case when the collapse pattern
19 consists of diagonal yield lines, it may be shown that the initial collapse mode is valid if the value of
f0e
cal-
culated therefrom (Equation 2.13) satisfies
.... (2013a) Only in exceptional cases will this condition not be satisfied since for economy i1 i.e.
~,
~ i2 and
/Ae
-
~ 3(~:y 2
increases with Ly/Lx while its minimum
allowable value decreases. The analysis of modes (c) and (d) will therefore provide sufficient check unless Equation 2.13a is not satisfied.
2.7
PRACTICAL
IMPLICA~IONS
AND LIMITATIONS
For a slab of chosen Ly /Lx , three parameters were investigated for a range of i1 and i2 values in order to assess the practical usefulness of Equation 2.13. parameters investigated were: coefficient of orthotropYi required in the slab;
The
(i) the most economical
(ii) the volume of steel
and (iii) the range of load and i
values for which minimum reinforcement requirements do not govern. (i) Variation of Most Economical Coefficient of Orthotropy The variation of
~e
with i 1 and i2 was determined for
-
20 a given Ly /L x ratio by using Equation 2.13. Values of i1 and i2 between 0 and 3.0 were used. For each combination, A1 and A2 were determined by Equations 2.9 and 2.10 and fie calculated. Values of Ae were plotted on an i1 vs i2 graph and contours of equal
~
drawn in the regions
where the value of i2 did not invalidate the assumed failure mechanism. Figure 2.2 shows these graphs for Ly/Lx
1.0 and 2.0.
(ii) Variation of Steel Volume In this investigation the value of
~e
and'used to find the values of mx and my.
was calculated The value of my
was fouhd from Equation 2.4 and the percentage of steel, 2 p, calculated using the equation Pd fy = m, based on the assumption that the depth of the rectangular stress block, a, is zero. From 2.4 =
..•. (2.4b) in which
=
VVR is a convenient non-dimensional
measure of the load which,.for the purposes of this investigation,was kept high so that minimum reinforcement can. ditions were not encountered in the range of values of i and Ly/Lx investigated.
From the values of percentage
steel volume calculated for each combination of i1 and i 2 , contours of equal volume were plotted for valid regions on
FIGURE 2.2
21
VARIATION OF fie
(a) Ly/~=1.0
(b) Ly/Lx = 2.0 3r-----~------~------~~-
f4a contours
I FIGURE
2.3
VARIATION OF STEEL VOLUME
(a) Ly!4<=1.0
I
(b) Ly/L)(,.1.50
2~~~--H-~---1~uvl=~m
= ~Xl00 contours of equal volume
2
i2
3
~"t-----t----1.10
0
2
i2
3
O~~~~~~~~~==~==~~~==~----~-2.4 MINIMUM REINFORCEMENT BOUNDARIES I l+\--\---+
(0)
Ly /Lx = 1.0
lood contours
41-+----+ (b) Ly / Lx" 1.50
22 the i1 vs i2 graphs.
Figure 2.3 shows these c.ontours for
Ly /L x = 1.0 and 1.5. (iii) Minimum Reinforcement Restriction Imposed
"by
Codes of Practice For each value of Ly/Lx' a range of load parameters was used, for eac.h of which the contour on the i1 vs i2 diagram was found which marked the boundary beyond which minimum reinforcement restrictions would govern. The steel percentage was calculated on the basis described in (ii) above.
In all cases )&e was calculated
before my and p. Figure 2.4 shows contours of equal W for Ly/Lx = 1.0 R and 1.5. The minimum steel percentage allowed in these figures was .15 per cent as in the British Code of Practice CP114.
2.8
DISCUSSION Figures 2.2, 2.3 and 2.4 show clearly the effect of
the various parameters. The value of
~
e is not affected greatly by change in i1 and i2 values particularly in relation to the effect of
The large increase in
Ae
with
increase in Ly/Lx reduces the steel requirement in the long direction and hence far greater loads are necessary in rectangUlar slabs to avoid minimum reinforcement conditions
23 when the value of
A
is used.
To illustrate the inter-
pretation of WR values, consider a slab with (Lx/d) = 30, 2 f; = 4000 Ib/in . The ultimate load, w, is then 106.9WR psi which for WR = .025 is 2.67 psi or 374 psf. Comparison of these values with Figure 2.4 shows just how often minimum steel will govern. When minimum reinforcement conditions do not apply, the volume of steel may be seen to vary considerably with i1 and i2 for a given Ly/Lx'
For the very minimum con-
ditions of i 1 = i2 = 2.0, increase in Ly/Lx requires more steel per unit volume of slab. The advantages of using a value of i1 in the region of 2.0 are clear from inspection of Figure 2.3 and, for slabs with Ly/Lx> 1 .0, a decrease in i2 from the position,i 1 = i2 = 2.0 l brings smaller penalties than a decrease in i 1 " Penalties for decreasing i2 from this point are relatively less for a slab with Ly/Lx> 1.0, but it is clear that use of i1> i2 should be preferred. It may be concluded that although minimum reinforcement conditions will govern in many cases, values of fix/my
= I&e and
i1~ i2~
200 will give greatest economy where
these conditions can be achieved.
Furthermore, it is
apparent that variation from this optimum does not increase the volume of steel greatly but an increase in i2 may result in appreciable change in minimum reinforcement condi tions for slabs with Ly /L x /' 1 .0.
24
CHAPTER
A STUDY REINFORCED WITH
3.1
OF
THE
EFFECT
CONCRETE PARTIAL
3
OF
SECTION
LATERAL
MEMBRANE AND
ON
RESTRAINT
FORCES
ON
A CIRCULAR AT
THE
A SLAB
EDGES
ENHANCEMENT OF THE MOMENT CAPACITY OF A REINFORCED
CONCRETE SECTION 3.1.1 Introduction
The enhancement of the load carrying capacity of reinforced concrete floors by compressive membrane action is due to the enhancement of the moment capacity measured about the mid-depth of the sections along the yield lines. For any singly reinforced section there is a limit to the factor by which the application of compression enhances this moment capac.ity.
The amount of reinforcement in the
section has the greatest effect on this maximum factor and in the following sections, the effect of reinforcement and other variables on the maximum attainable enhancement factor is examined.
3.1.2 Yield Locus for a Singly Reinforced Section By examining the conditions at ultimate, on a singly reinforced section, a failure locus may be obtained
25 relating the moment and axial compression. Consider the section as in Figure 3.1(a) and having a tensile stress, fy' in the steel and a rectangular stress block £or the concrete, as shown in Figure 3.1(0). With the-notation as in Figure 3.1, noting that the moment of forces is taken about the mid-depth of the section: b~1
~
I '85td I
·1\
S\..'I"
-r-
- ---
~~
\
r.
\
P,s;: pbd
•
•
0
(a) Section
\
es
\
(;'6 y )
-'--
(c)
(b) Strains FIGURE 3.1.
Stresses
(e)
Actions
SECTION NOTATION.
p
.85f'.a - pdf
M =
.85 f'.a.(D/2-a/2)+pdf c y (d-D/2)
c
Cd) Forces
Y
. ... (3.1) .... (3.2)
and if the moment 'about the mid-depth for P = 0 is denoted l\IT
'0'
Elimination of 'a' from Equations 3.1 and 3.2 gives the yield locus as .... (3.3)
26
in which g = (.5D/d-2t)/(1-t)
.... (3.3a)
h = t/(1-t)
.... (3.3b)
t
= pfy/(1.7
f~)
.... (3.3c) dM
For M/Mo to be a maximum, dP
0, which gives
(P/To) = 2g/h
.... (3.4)
where Fmax is the maximum attainable enhancement factor, and is a function of t and Did only. 3.1.3 Relationship Between t, Did and Fmax The relationships between Fmax ' Did, p, and may be shown on one graph.
fy/f~
Oonsidering the relationship between Fmax ' t and Equation 3.5 gives
Did,
the solution for which is
t. =-
[2Fmq~-(2-~)] - J[2FT'ro.<-(2-~~2_Fvvw«~)21 4
F"n<;;K
••
0
(
3 .6 )
Therefore t may be plotted against Fmax for a given Did. From Equation 3.3(c) it is clear that it' may also be plotted against
fy/f~
to give a straight line of slope =
27
(0
~
.02r-~r-HIT~~~~~~~~r-+-~~---r-
.15
-0
C
o
u f11, (T) \I)
.O'rTrffl~~~~~74~~~~--+--------r-
0'"
W
l.4J dD
~-"""'1.2
=9.0
o
~---l-1.0
~------~----~~------~------~------o 5 10 15 20
fy/f~ and FIGURE 3.2
~ax
MAXIMUM ENHANCEMENT FACTOR
28 p/1 .7. The plots of fy/fc' and Fmax (horizontal axis) against t (vertical axis) are shown in Figure 3.2. The maximum attainable enhancement factor fo:r:. a given fy/f~, D/d and p is determined by following the appropriate vertical line corresponding to f y /f'c until the desired straight line for p is met at an ordinate t 1 . The point" wi th ordinate t1 on the appropriate curve of constant D/d has an abcissa equal to the maximum attainable enhancement factor. process is shown on the figure for and p
= .003, to give Fmax
3.1.4
=
fy/f~ =
This
9.0, D/d
=
1.20
5.3.
Condition That Steel Will Yield at Maximum
Enhancement In the above analysis it was assumed that the steel strain, e s ' exceeded the yield strain, e y ' at all times. The situation giving least steel strain is that when maximum moment enhancement is reached.
This is when the depth
of the rectangular stress block, a = .5D since any increase of 'a' beyond .5D would reduce the moment about the middepth.
Villen a
.5D, d n = D/1.7 and consideration of similar triangles on the strain diagram (Figure 3.1(b)) =
yields .... (3.7)
If the modulus of elasticity of the steel, E , s
= 30 x
-
29 106 psi, then e y = fy/30 x 106 and for yield to take place 8 > ey . This requires S .... (3.8)
For e u = .0033, the following conditions result if yield is to occur:
Did = 1 .0, fy
~
'70,000 psi
Did = 1.1, fy
~
54,500 psf
Did
=
1.25, fy
~
36,000 psi
Thus for most mild steels in sections with values of
Did
~1.2
the condition of a = .5D does not invalidate the
assumption that the tension steel is yielding. 3.1.5
Discussion and Conclusion
The variation of Fmax with f y If'c' Did and p is summed up in Figure 3.2. Quantitative data as to the reduction of Fmax with increase in p, the increase in Fmax with increase in Did, and the reduction of Fmax with increase in may be obtained therefrom.
fy/f~
This diagram serves also as a
compact qualitative description of the factors influencing the enhancement of the moment capacity of the section. is clear from Figure 3.2 that to obtain the highest enhancement factor,any design should aim at: (i)
A low value of f y-c' If'. which would be best
It
30 achieved by increasing the concrete strength since it is the concrete which is providing the enhancement. (ii)
A high value of Did gives a large enhancement
factor.
Howeverya high value of Did would cause a
reduction in stiffness and in the absolute value of the unenhanced moment, Mo'
Other factors such as
crack widths dictate that minimum cover to the tension steel is preferable and the Did value would thus be fixed within close limits. (iii) A low reinforcement content. is the most critical.
This requirement
The low reinforcement content
of reinforced concrete slabs leads to the enhancement of their load carrying capacities due to self-induced compression on the sections when the outer edges are restrained from moving laterally outward.
This great-
er enhancement for lower slab reinforcement is not due entirely to the fact that greater enhancement of the moment capacity of a section is available.
The
tendency for the edges to spread outwards as the slab deflects is greater for lightly reinforced slabs since the neutral axis is nearer the compression face of the concrete.
The benefit gained from this effect is not
apparent in the figure. Finally, it must be remembered that for slabs,the attainment of the maximum available enhancement will be impossible in practice if the net
compression is provided
31 by passive restraint against lateral movement,since the lateral movement and vertical deflection will impose the condition that a .
j::2.0
:s
I ........
«.. ....
2.0
-..
'( CD
«..
1.0
"
J:
du/Os=O.O 0 1.0
1.5
2.0 enhancement factor
3.0
(c) effect of membrane action on maximum torque [ FGURE 4.4
TORSION IN EDGE BEAMS
I
00
74 torque is imJ L/2(1 + A) as would normally be the case. However, once membrane action enhancement takes place the position of the slab centroid with respect to the shear centre of the beam becomes important.
This effect is
taken into account by the term B and the magnitude of this effect may be conveniently determined OY,calculating the ratio, H == (1 + A + B): (1 + A), ,for various F values and beamslab junctions.
The results of these calculations for a
typical beam are given below.
Calculation of theR,atio H Example beam. b
L=
.05,
Assuming F*
==
.1 ,
F.
.
Therefore 1 + A == 1.225,1 + A + B Load Enhancement Factor F 1 .0 1.25 1 .50 2.0 2·5 ~"
== 2.0,
i
==
Values of H du/Ds==O 1 .00 059 .32 -.02 -.23 -1 005
du
(F -1) (1 - D) 1.225 - 2.5 -F---.--£
:=
~1+A+B2/~1+A)
d u /Ds==1.0 1.00 1.00 1.00 1.00 1.00 1.00
These values are plotted on Figure 4.4(c).
du /Ds=2.0 1.00 1 .41 1 .68 2.02 2.23 3.05
75 The effect of change in du/Ds is understandably large and changes sign when the mid-depth of the slab coincides with the mid-depth of the beam.
The reduction in maximum
torsion is remarkable when the top of the slab is flush with the top of the beam. 4.3.3
Conclusion
Design of a slab such as in Figure 4.4 to include the effects of membrane action would require the beams to be designed for a laterally outward load in addition to the normal vertical load.
Placement of the slab flush with
the beam at the top would offset this additional requirement by reducing the torsion induced in the beams.
In the
more general case in which lateral restraint is provided by surrounding panels and the supporting beams the effect on the beams would not be as great.
The surrounding panels
would take most of the lateral force and the nett lateral force on the beam would be smaller.
Less effect on lateral
bending and torsion would result. Nevertheless, the high sensitivity of the maximum torque,even at low enhancement factors, indicates that consideration of this behaviour is important in many cases. 4.3.4
Suppression of Hogging Yield Moment Development
Along Exterior Edges of Panels in Which Membrane Action is Present In designing edge panels to develop full hogging
76 moments along the exterior edges, only the torsional strength of the edge beam need be sufficient for this to occur ultimately.
In the case of panels which may develop
membrane compressive forces which enhance the load capacity of the panel, the full development of the exterior edge hogging yield moments may not be required.
In the test to
destruction on a model nine-panel reinforced concrete slab and beam floor described in Chapter 9, the steel strains along the exterior edges of the edge panels were well below yield values at the predicted ultimate load and a reduction in the edge beam torsion was evident.
The presence of
compressive membrane forces normal to the edge would account for the reduction in torsion, and, because membrane action is not purely an ultimate phenomenon, the value of the restraining moment could have been enhanced above that which the level of steel strains would normally imply. However, the capacity of the edge beams to resist lateral force was not high and an alternative mechanism was sought.
Recently~39) it has b@'e'n report~d that the torsional stiffness of a reinforced concrete beam reduces remarkably when cracking occurs.
It is clear then that large twisting
deformations of the beam would be required before the full yield moment could be developed and the slab element would have to rotate even further to create the differential movement necessary for the development of the yield moment.
77 No attempt was made to analyse this case but the observed behaviour during the test suggested that membrane action in other regions of the edge panels provided assistance in carrying the load before sufficient slab deformation could occur to develop the hogging yield moments along the exterior edges.
Torsional moments in the edge
beams computed on the basis of the full development of these hogging yield moments could thus considerably overestimate the true values.
4.4
DISCUSSION AND CONCLUSION It is clear that membrane action in panels will affect
the flexural and torsional steel requirements of beams. The method of determining beam tensions is a simplification but values of beam section actions resulting provide adequate strength and a realistic distribution of moment between mid-span and support sections. The equations derived to determine the flexural reinforcement at critical sections of a T-beam subject to moment and axial tension require some qualification in that at sections of the beam at which the moment is zero, a sufficient amount of longitudinal steel must be placed to take the tension.
Furthermore, at the critical sections
it is necessary to check that the neutral axis lies within the section. Because of the likely adverse effect of beam deform-
78 ations on the development of panel membrane action it would be wise to ensure the beam collapse mechanisms do not occur before the panel mechanism. The effect of compressive membrane action on the torsion induced in the supporting beams is clearly considerable, and worthy of consideration in design.
For the
case in which membrane forces acting normal to the edge beam reduce the torsion, any advantage so gained could be offset by any extra provision required for biaxial bending of the beams. However, in cases where membrane action may exist in other regions of the edge panels it is likely that the torsion for which the edge beams would normally be designed will not be attained.
This latter effect, or
even the combination of the two effects discussed could provide an instance in which the neglect of membrane action leads to the overdesign of edge beams for torsion.
79
5
CHAPTER
STIFFNESS
5.1
OF
SURROUNDS
FOR
_§9JI1\._R_E_-,S_L_A_B-.;.,,s
INTRODUCTION AND SUMMARY The degree to which compressive membrane action
enhances the load carrying capacity of a reinforced concrete slab depends principally on the lateral stiffness of the elements providing restraint against outward movement of the slab edges. For interior panels of a multi-panel slab and beam floor, this restraint is provided by the panels surrounding the one in question.
Thus when the interior panel
exhibits compressive membrane action, the surrounding panels are subject to in-plane forces.
The surrounding
elements may therefore be considered as a flat slab with in-plane loads applied normal to the edges of a central hole which corresponds in size to the panel exhibiting compressive membrane action. In order to obtain some measure of the variation of surround stiffness with the size of the outer panels, slabs of analysed.
elasti~
isotropic, homogeneous materials were
The study was restricted to the consideration
of a squ~re slab with a square central hole.
80 A library computer programme employing the finite element method for the solution of plane stress problems was used to calculate the deflections of the loaded edges and the stresses within the surround. Because the rigorous plane stress analysis required large computational effort an alternative method of computing the edge deflections was sought.
Consideration of
each edge of the surround as a deep beam proved very satisfactory in this respect.
The rapid computation of
edge displacements would permit extension of theories such as that due to park(11) to include interaction of the edges of surround and slab.
5.2
METHOD OF ANALYSIS AND CASES CONSIDERED The dimensions and properties of the slab considered
in this study are shown in Figure 5.1(a).
Due to symmetry
it was necessary to analyse only the portion ABCDEF with boundary conditions as in Figure 5.1(b).
For each sur-
round shape, three separate load distributions were applied to the edges, BC and CD, each of the same total load.
The
shapes of these distributions (see Figure 5.1(c)) were chosen as representative of the possible distributions of membrane forces along the edges of a square interior slab. Analysis was carried out using an existing computer programme for solving plane stress problems by the finite element method based on a quadratic strain triangle(21).
~______~A______~~
horJ'l(_~.,,'leOl·"""
*-tie
WckMU. t ~lII l"CItIo~"J.ly ...15
E
(i) uniform
~
d i ..-~__
'----_ _-----' ~ I
w--l
c
c
B
(iii) cubic
(ii) pOl"!lbolic
B
(c) Load dlllltributionlil
FIGURE 5.1 SLAB SURROUND ANALYSED
II')
~
~ II')
~
" II')
~ II')
q
....1:1 ~
1..125
~
.125., .125,
I ~IGURE 5.2
~.125 f
~INITE
.125
~ .~ ~
.or
5
ELEMENT MESH
6t
1
I
c
82 Each of the portions ABCF and CDEF was divided into 160 triangul ar elements as in Figure 5 .2 .
Loads were applied
to nodal pOints in the y direction along Be and in the x directi on along CD . The study was limited to cases in which ax
=
a y and
= by' In the first seri es , t he slab was of uniform x thickness throughout and f our cases were considered with
b
ax/bx tak ing t he values:
. 5 ,1. 0 , 1 . 5 , 3 . 0 .
To investigate the effect of supporting beams, the case of ax/bx Be and CD.
5.2.
= 1.5
was analysed with a thick band a l ong
The dimensions of this band are sh own in Figure.
The symmetrical shape was r equired bec ause a two-
dimensional s tress syst em was being analysed, and the tapering thickness ac r os s the second row of elements was necessary t o a~oid stre ss di s continuity at the element b oundari e 5
•
. For each surround under each load the following quantiti es were determined at each n odal point . (i)
The normal stresses and strains in t he x and y direction .
(ii)
The she a r stre ss a nd s train in t he x 'or y ,
direction. ( iii ) (iv) (v)
The maximum and mi nimum principal stresses. The maximum shear stress. Displacements in the x a nd y directi on.
For reasons outlined under 5 .1, displacements of the
83 loaded edges were of particular interest in the context of membrane action and it is these that receive greatest attention in the following sections.
5.3
DISPLACEMENTS OF THE LOADED EDGES Although stress concentration at the re-entrant corner,
B, cast doubt on the accuracy of computed stresses near this point, the use of small elements in this region ensured that its effect on stresses at other points was very small, and the effect on deflections even less. In Figure
5.3
the displacements in the y direction
of the edge BC of the surrounds are shown.
The quantity,
n = ,\tE, expresses this movement in non-dimensional form. t,
~ = deflection of the edge, t
=
thickness of slab, E
modulus of elasticity of slab material, W
=
= total load
applied normal to the edge BC ( = one half of total load applied to one edge of the interior hole). Hence the deflection, ~, of the edge may be obtained from
~ =nW/tE Features to note in Figure (a)
.... (5.1)
5.3
are
The small difference between load cases (i) and (ii).
(b)
The large difference between load case (iii) and load case (i).
84 (c)
The ratio of maximum deflection (at B) to on at C is greater for low values of
defle
surround. width. (d)
ons along the edge are remarkably
Defle
constant for load case (iii), especially for ax/b x =1 00 and ·1 (e)
.5.
The maximum deflection falls off rapidly as ax/bx is decreased from
3.0, but the change
x Ib x is decreased from 1.0 to 0.5 is very small.
in maximum deflection when a
Figure
5.4
shows that load case (iii) has less
sensitivity to change in ax/bx than load case (i) and gives a clear indication that decrease in a
x Ib x lower than
ttle increase in surround stiffness.
1 00 brings
The
effect of increasing the surrourld width is further lessened when the results of Chapter 3 are recalled, viz., an inGTeaSe in surrom:d ate increase
(f)
ffness does not produce a proportion-
the enhancement factor.
The effe
of
uding the thicker edge beam on
the maximum deflection is plotted in Figure
5.4.
In both load case (i) and load case (iii) its inclusion is equivalent to increasing the surround width.
However, this
'.
effective increase in width was less than could have been achieved by using the same amount of extra material to increase the ,surround width directly.
This situation
load i load ii load iii
20
--- --- -----.......
6
........
5
deep beam values
3:
;)
...
...
...
...
.-
._._._._._._.L-.--..:-._
+'
"'-ReadingS not tatcen beyond this point
o o
12500
37000
25000 MICROSTRAlN
50000
-
121 6.6.2 (i)
Final Slab Dimensions Level of Top Surfaces:
Precise level readings
on the top surface taken to :t.005" varied from -.20 to
+.15" above the mean level.
The standard deviation of
the 169 readings was .073",
The planeness of the top
surface was better than these figures indicate since readings revealed a small overall slope. (ii) Panel Thickness:
The overall average of nine
readings per panel was 1.976" with range of a standard deviation of .048".
2:..11 11 and
At the end of the test,
thickness measurements were taken at the edges of the centre panel (E), and a corner panel (J).
centre~edge
panel (H) and a
The average of all these readings was
1.904". (iii) Beam Thickness:
Beam depths were measured at
the ends and quarter points of each span before the test. Results were:
6"
beams:
Average depth::::: 5.965", s.d.
7.1.2 II
b earn",: !=(
Average depth
(iv) test.
Cover to Steel:
7.490 11
,
s.d.
= .036
tl
.030 11
A check was made after the
Both panel and beam steel cover were generally
wi thin 1/32" of the expected value. Fuller details of the measurements of the slab are given in Appendix B.
122
CHAPTER
INSTRUMENTATION
7.1
AND
Z
TEST
PROGRAMME
INSTRUMENTATION 7.1.1
Reaction Measurements
Figure 7.1 shows details of a support B2.
The 10-
ton capacity Philips PR 9226 electrical resistance load cell is shown sitting between a two-way roller system and a
1i"
mild steel bearing pad.
Adjustment of the
nuts at the column head enabled the whole assembly to be levelled.
All reaction points were of similar form.
The roller supports for the outer ring of support points were
i"
ball bearings between hardened, ground plates.
Two of the
inner
supports (B3 and C2) had one-way-
rollers while support C3 was fixed against horizontal movement ensuring that, although the floor as a whole could not move, all reactions were vertical. Each load cell was wired to a 16-way, four-channel switch connected to a Budd Strain Indicator, and readings on each cell recorded manually.
Values of reactions
were calculated using the load-strain calibration curves obtained for each cell as a result of tests performed before and after the testing of the floor.
FIGURE 7. 1
DETAIL OF SUPPORT
FIGURE 7.4
TEST SET-UP
B2
124 7.1.2
Strain Measurement
Strain gauge positions are shown in Figures 7.2 and 7.3.
All gauges were wired to a 140-channel strain data
logger in a two-arm circuit.
The data logger (right
foreground of Figure 7.4 ) was accurate to + -5/LS .and could read automatically each gauge in turn.
A digi t8.1 vol t-
meter incorporated in the logger provided output, in microstrain units, on a typewriter and tape
punch~
The automatic switching facility required that each active gauge have its own dummy. The active concrete gauges were PhiJips PH 9810 C/11 (600 ohms,' flat grid, 1 inch gauge length,
i"
grid width)
glued directly to the concrete with Philips cement kit PH 9244/04.
Concrete dummy gauges were of the same type
and glued in the same manner to the three concrete blocks to be seen in Figure 7.4. Active steel gauges were BLH SR-4 A-12 paper backed gauges (flat grid, 1 inch gauge lengths, 120 ohms, gauge width, 3/32").
The reinforcing steel was exposed by the
removal of the cork blocks and considerable care was necessary in obtaining uniform adhesion onto the diameter bars.
i"
Grooved rubber pads were used to form the
gauge around the bars.
Nitrocellulose adhesive from the
Duco Cement kit was used throughout.
Dummies for these
gauges were temperature-compensated, 120 ohm gauges mounted on Aluminium.
Two unstressed SR-4 gauges mounted
125 in the same way as the active gauges and having dummies on the aluminium, were used to assess the effects of temperature on the strain readings. Following initial placement,
all gauges were
checked for continuity and resistance to earth, the necessary replacements being made until all gauges were satisfactorily mounted.
Gauges were then waterproofed
with wax. The 140 channels of the data logger were split into blocks of 20, the first channel of each block being wired to a Philips PR 9249A dummy strain gauge to check drift. Gauges 7-32 were placed so as to afford measurement of membrane force in the centre panel in a region of low moment.
Gauges 33-46 provided this facility in a centre-
edge panel, 47-50 in a corner panel. Gauges 78, 79, 76, 80, 82 were placed to give an indication of T-beam flange widths.
Hair cracks normal
to the line of the gauges in mid-span necessitated their placement slightly off centre. Gauges 112-120 were placed for measurement of moment and membrane action force in a region of high moment. Gauges 94, 95, 105-108 served a similar purpose for a centre-edge panel. All other gauges were placed to give an indication of stress levels at critical points and in some cases, means of calculating moments and forces at a section.
126
FIGURE 7.2 [ STRAIN
GAUGE
POSITIONS ON
1- Denotes strain gauge C - Denotes
BOTTOM
position
concrete gauge on upper surface
STEEL
I ' I -fff1='-'---rI
I t-i--t---~ ~I-I-I-
l
m~
--ru--~,
\
-+----+--t -----r-
G:
-TI,·.--+,[' _.-+-_
, II '
tm. .*-+.+
'ij 8~ I
, }j-' ~ I'
I',tt'-1 -1-
I
,I
J~7
--\-1--
-;
0
I
.
::
I
I
I
i r
I
i
I
i I
I
:
11
A I
JI
I
'j I
tv=,,'l"
H
1
~v"
~
--1----I
i-:. !---LI.'
--1------.--J ;
.!
I
i
I
I'
_-I-
i
Ii
I
I
I 192 I f:zl-I-I---r--r--r--r--t--!-l I I h2¥ilI 1217f---I-+ I--f-FI-I-f---i--I--l t- h,Ir-f--- I- r-I--f-I~~-·---- t- r- I- h I~ c_ --------'--t---h 11 1128(; I I I I -LI i i 1--1-t---. I I Ii I I I I I i i +-'-- I i i ' I 118 ~I~C-_i I J
--+U---t+-
u
I
r i - i - I - I - ' - - -,I-I-t-h r
I--:..:-ri t-
11----+.--1 . , -h.--+---'-+--1 II'
~ ___ ~
1-1-'1
\-f-~jr:-:n--j--
.-~-+---L-i I ---L---~I _.'
I-f--I-·-
I
:
I
I
+
II
I
~--+-
t---
127
-+.~
·t~l~4.r';'
I
1001 I
E
I
I
B
I
II 117C
i
I
;
I
i I i
I
1 116
110pC I I 1D5 r I "5 1(8(: I I "4 1D7 1 I th3 i 1~,ei n2
:,
IO~
,:7'1
I
I
i
I
I I
11
I
I
I
I
C 134111f.lr-:r-+,---+4.1-r+'OJ'hse -----~-+- --~>_~eJ f'~+- + -r--,c~ r'-"-fgo t-t+- --r- --tJ<:~:~~"'--------Ig:-I~ t91 j : " - ~I :3 i: : -t -- --, I I
1~~
I- I-
91
I
~
1
!1
I
:
-4
-
t - · -+-t--++ -+-_Ll -1' L--1 I I ~ I I 8B __ _ ----'---~ 1 ___ t J ~r~ - t --~- - TJ -
-r
1 _ _ _ _ '---_
n
--+-
~i ~_~
i
----r-
--
--
I I L I f~ rT,-t-t
i__ ~ . W--~TI , 1861 I I , -+-=1tTTfli3tc~f-;1:::,3j6--t--+
'-{fl -- --
-+
-'-----j--t---i-h
!
,
-~
I
-h I
;
I i ' 1I i 113 I •
F
. 98 l i e ' I I I ~ I I T T 1---.-1-J I l, I I I '-~ ~ c--t--I!,: I
I
I
~
+-+t I- t- +-t-+ t-I--l
i I : I L,-----f-rllit--f-J
h'
FIGURE 7.3 STRAIN
:
GAUGE POSITIONS ON I -
Denot~
gauge position
C--
Denot~
concrete gauge on
TOP
un~rside
STEEl
128
7.1.3
Deflection Measurement
Dial indicators mounted on a 'Dexion' frame attached near the top of the supporting columns were used to measure deflection at critical points. One at the centre of each panel and one at the centre of each beam span provided vertical deflection data. Gauges to measure horizontal movement at supports B3, B2, 02 were placed to measure movement in the direction of the rollers at the support points (North-South at B3, East-West at 02 and both North-South and EastWest at B2).
7.1.4 ·Load Application and
Measuremen~
Water-filled bags placed between a reaction platform (erected over the slab and tied down to the laboratory floor) and the top surface of the slab provided means of load application. Nine bags (one over each panel) were made with a
3 11 high wall and covered the whole top surface of the slab when placed and filled with water. Pressure was applied by forcing water into the bags. The four corner bags were
inter~connected,
there being no
provision to have one corner panel at a higher load than the other three. panel bags. Figure
The same was true of the
centre~edge
The centre panel bag was a separate system.
7.5
is a diagrammatic representation of the
129 hydraulic loading system,
Apart from the main feed hose
which was 1" diameter, all hoses were
1\:-"
diameter plas-
tic tubing. The main feed hose came from a constant head device which could be adjusted to any level to suit the load requirements, providing an effective means of maintaining the load at the set level. Due to scaling down in the model, there was a difference of
75 psf between the self weights of proto-
type and model.
Another constant head device fixed at
the appropriate height above the level of the slab was used to apply this difference so that the self weight of the model plus the applied "dead 10ad l1 was equal to the prototype dead load.
This load was the starting point
for all tests. For pattern loads where two different load levels were required, panels not loaded with live load were switched to the IIdead-load-onlytl constant head device leaving the variable device for setting of the live load on the others. When the lower load of a pattern was greater than the prototype dead load, a mercury manometer was used to set the load and the dead load device was not used. The mercury manometer served also as a means of checking the reading on the calibrated variable head device.
--<>
Mains
Voriable Head
O-Valve
IIMen:ur~ Manome
l -_I
Constant Head
N-+-
J
6
7
8
10
I ..... .... ..... C~ 'OJ
.... ....,
Comer
Pcnei BOIs
>. . .
..... C
""
"I
Centre-edge f'bneI Bags
) ...... ..... ....
2
I
.---1-.
0
.... 1
3
I 4
Centre Panel Bag
5
-"'"'a'"
,...-
....
I FIGURE 7.5 LOADING SYSTEM I
l
I
-
."" 124.25", 27.25".1 31.25-
I FIGURE
I 31.25" I
27. 25"
124.2~1
7.6 MOMENT LINES ACROSS SLAB
I
....W
o
131 7.2
TEST PROGRAMME 7.2.1 The
Dead Load Reactions
75 psf difference between the prototype and
model self weights was applied to the model throughout the programme.
The application of this load provided a
more stable arrangement in reducing the difficulty of setting and maintaining the dead load reactions at the required level.
These reactions were set several times
before the testing programme was started, for as long as the tendency for the corners to lift
re~ained.
When
tolerable stability had been achieved, Tests 101 and 102 were performed. In the design, moment redistribution was kept to a minimum and for this reason the dead load reaction
for
each support was taken equal to the reaction at ultimate load, scaled down linearly. An initial setting of reactions was made for self weight only, before a more accurate setting was performed for prototype dead load.
Successive trials were made
until the required value at each point was obtained. The corner reactions tended to reduce due to uplift and reactions were reset after
~est
102 because this
tendency was then less and small differential movement of reaction points had caused some redistribution. this no further reaction adjustment was made.
After
132 7.2.2
Load Tests Performed
The overall test programme, carried out between 6th and 22nd May 1968 is summarised in Table 7.1 . Table 7.1.
Summary of tests performed.
Maximum Panel Loads No. of Increments (~) Test Load ~ Stage Centre Centre Corner !I£ Down - - - -Edge Nos. 101
1-10
102
13-23
103 104
25-38
225 225 225
51-63 1A-13A
75 225
76-94
375
105 106
225
225
225
225 225
75 225 225
75 225
375
375
7 7 7
3
7
6 6 10
7 12
5 6
95-109
375
375
375
7
13
108 114-132
400
75
400
10
9
107
109 133-151
200
375
200
10
12
110 152-167
450
450
450
11
10
11/1 168-189
375
375
375
9
112 189-220
775
775
775
16
113 221-227
850
850
6
114 228-239
600
966
1170
8
Remarks
12
(Live load (removed from (outer panels (with centre (panel load at (375 (Corner and (centre panel (loads adj(usted to give (upward corner (reactions (Outer panels (held at 225 (while C,P, (loaded to 375 (and back. All (then loaded to (375 (66 hours at 375 (then loaded to (775 (To failure of CePe (To failure of (centre=edge (then corner (panels
133 Figures in psf are applied loads including the
75
psf difference between model and prototype self weights. Three hundred and seventy-five psf is dead load plus full live load.
Seven hundred and seventy-five psf is twice
dead load plus twice full live load. Full details of all load increments are given in Appendix
c.
7.2.3
Procedure at Each Load Increment
The load was set using the hydraulic system described in 7.1.4, a period of a few minutes being allowed for the system to settle.
A check between mercury mano-
meter readings and 'Tariable head device setting was used to ensure that a static state had been achieved. Dial indicator readings were then taken, the load cell readings taken once and one cycle (140 channels) of strain readings performed.
The whole floor was then
checked for cracks, new cracks being marked with the corresponding load stage number.
Load cell readings
were taken again and if considerable cracking or reaction distribution had taken place since the start of the increment a further cycle of strain readings was taken. On the completion of reading the load was set for the next increment and during the time taken for the load of the next increment to settle graphs of load versus deflection and load versus strain were drawn for some critical points.
7.3
REDUCTION AND PROCESSING OF RAW DATA
7.3.1
Deflections
Readings were taken in ten thousandths of an inch and punched into cards.
The start of Test 104 was used
as datum in the reduction of all readings.
7.3.2
Reactions
Bridge readings taken for each reaction point were punched into cards and the reactions at each point computed on the assumption of a linear relation between load and bridge reading.
Calibration of each load cell
provided the constant relating the two quantities. As a check, the sum of the reactions was compared with the total applied load plus self weight, in which the total applied load was the sum of the products of the nominal bag pressure and full panel areas.
In all
cases the sum of the reactions was the smaller quantity since the bags could not be made to apply pressure over the whole area, due to curvature of the bag walls. effective loaded area was surface area.
The
II
The
94 per cent of the total top
c l ear span" area of slab panels was
81 per cent of the total top surface area so that the load applied represented some loading arrangement in between the total area and the clear span area of the panels.
Placement of the bags was such that the unloaded
area was directly above the beams and therefore each slab panel was subject to the full measured bag pressure over
its total clear span area.
No reduction of this value
was therefore necessary to obtain the pressure sustained by the panels. Figure
7.6
shows the lines along which moments were
calculated from the reaction values and applied loads, the latter being scaled down by the ratio of effective loaded area to total top surface area. The moments so calculated were used in checking the results of moment computations from strain readings and in assessing moment redistribution.
7.3.3
strain Readings
Readings of each gauge in microstrain were punched onto paper tape and processed by computer.
The raw
strain readings were reduced in the following manner. (i)
Datum correction A particular load stage was chosen as datum,
and for each gauge and the reading at the datum stage was subtracted from all other readings. (ii) Drift correction The first channel of each block of 20 gauges was a standard strain gauge of high stability.
The
variation of reading in these gauges was used to assess the electrical drift of the Strain Data Logger. ation was not great
(se~
Vari-
Appendix D listing of gattges 1,
21,41,61,81,101,121).
The readings of gauge 81 were
taken as representative and the
datum~corrected
reading
of this channel at any load stage was subtracted from the readings of all other channels at that stage. (iii) Temperature correction Dummy gauges for the active concrete gauges were of the same type and mounted on similar concrete blocks. Thus variations of length due to temperature were assumed to be compensatory and the concrete strain gauge readings assumed to require no correction for temperature. The active steel gauges had dummies which were temperature compensated and mounted on aluminium.
Temper-
ature could therefore be expected to affect the readings of the steel gauges.
To compensate, two steel gauges,
(Nos. 139 and 140), mounted in the same fashion as the active gauges were used.
These were of the same type and
were mounted on steel reinforcement embedded in a block of concrete.
The blocks remained unstressed by external
forces and had identical dummies to the active gauges. The
datum~corrected
reading of channel 140 was subtracted
from all steel gauge readings at each load stage to correct for temperature. (iv) §Eecial drift correction At the beginning of each test, up to LS 151 (see Table 70'1), the corrected reading of each channel was compared with the corrected reading of that channel at the end of the previous test.
If any differencE occurred,
the readings of the gauge in the test to follow were
137 corrected by this difference. (v)
Zero correction
Initial balancing of the gauges was performed when the total load on the slab was 100 psf.
Allowance for
this initial load was made by computing the difference in readings of each channel at LS1 (75 psf applied) and LS5 (175 psf applied) and adjusting all readings of that channel by this amount. At sections at which measurement of normal force and moment were to be made one gauge was mounted on the main steel and one mounted on the opposite face of the concrete.
This permitted the determination of the strain
profile, assumed linear, across the section.
This linear
strain profile as given by the corrected strain readings was used, in conjunction with section properties, to determine the actions on the section.
Computer sub-
routines were written to compute the steel and concrete forces resulting. was assumed to be
The stress-strain curve for the steel tri~linear
and the stress-strain
relationship for the concrete was assumed to be of the form proposed by Hognestad et al.(20).
The derivation of
the subroutines is described more fully in Appendix E. 7.3.4
Computation of Section Actions from Strain
Readings 7.3.4.10
General basis
The subroutines, CONACT and STEEL, described in
138 Appendix E were written to calculate the concrete and steel action in a section whose strain profile was linear. In the computation of section actions from the test readings, a linear profile was defined by a measured concrete strain and a measured steel strain. The steps in the computation were as follows: (i)
Reduction of strain readings
This was done by the method outlined in Section
7.3.3.
(ii) Computation of strains for equivalent strain profile (a)
~uivalent
steel strain: (e ) s
When two steel strain readings were taken at the section at the same level, the average value was taken. In cases where only one reading was taken, this was assumed to be the strain in the section at the level of the centroid of the steel. (b)
Equivalent concrete strain:
(e ) c
When only one gauge was used it was placed parallel to the steel bar and the concrete gauge reading was taken as the section strain at the face of the concrete. When two concrete gauges were used, one was at right angles and the other parallel to the reinforcement. Poisson effect was considered in reducing e to the relation:
c
according
where e 1 = concrete strain measured parallel to reinforcement e
2
= concrete strain measured perpendicular to the reinforcement
;tv = Poisson's ratio For the two sections for which three concrete gauges were used these were in 120 0 I1rosette" form and two-dimensional strain analysis was used to obtain the principal strains which were used to obtain the equivalent strain component parallel to the reinforcement. The strain profile was then defined, and was as shown in Figure
7.7.
1/
•
A 65_, e~
Section
Strain Profile
FIGURE 7.7.
STRAIN PROFILE
0
strain~
Values of equivalent top and bottom concrete and
e~
were computed directly and stored.
and e' were likewise storedo s strains
ec~
e~,
e s and
e~
Values of
ec 8
S
For each load stage, the
corresponding to the strain
profile defined by the values computed for e
c
and e s were
calculated and stored. (iii)
Computation of concrete actions
The values of e
c
and
e~
were used in the subroutine
CONACT as top and bottom concrete strains and the concrete forces determined. (iv)
Computation of steel actions
Arrays of top and bottom steel strains had been stored.
The loading and unloading performed during the
test necessitated the examination of the strain history to determine the plastic portion of the indicated strain. The method used to determine this is shown in Figure
e
5
e
---I r;/
5 til
depends on whether e 6 - e eyo p5 Each array of top and bottom steel strains was searched in the manner outlined and only the elastic portion of the strain retained for input into subroutine STEEL for computation of steel forces. (v)
Calculation of section moments and forces
Output from the subroutines CONACT and STEEL were in non-dimensional form, giving the steel and concrete forces and moments, acting at and about the non-dimensional values,
M/f~bD2
and
mid~depth.
T/f~bD,
plied by the appropriate values of f ' bD c
2
These
were multi-
and fibD c
respectively. (vi)
Cracked and uncracked sections
In tension the concrete stress-strain curve was assumed to be linearly elastic with a modulus of elasticity as given by the secant from 0 to 1000 psi on the compressive stress-strain curve.
142 For each load stage the concrete was assumed first to be uncracked in which case concrete tensile stress was assumed to be proportional to concrete tensile strain, no matter how large the strain.
Actions were then computed
and the section assumed to be cracked.
In this case
concrete was assumed to have no tensile strength and the actions were again computed. (vii)
Factoring of concrete strains
The reduced and corrected value of concrete strain parallel to the reinforcement, e ' was factored by 1.0, c 1.20 and 1050 for each load stage for panel sections only. In regions of steep strain gradient, the gauge length of
1" would lead to an average strain value, when in fact the maximum strain was required. Concrete gauges on the undersides of the panels were thought to suffer most from this effect but this factoring made little difference to the computed actions along panel edge sections and only in uncracked sections away from the edge where strain gradients were probably insufficient to warrant this factoring, was any appreciable difference evident.
Special measures had to be taken to obtain more
realistic values of panel edge section actions as described below. (viii)
Effect of
T~
and L-beam flange width
For all beam sections the procedure described above was used to determine the actions on the rectangular
portion of the section only.
The effect of flange width
was determined by assuming the flange to be of plain concrete and that the strains in the flange sections were the same as those in the rectangular portion at the same level. Thus from the strain profile of Figure 7.7 the concrete strains at the top and bottom of the flange were calculated and used as input in the subroutine CONACT. each load
stage~
For
total flange widths of 100,2.0 and 3.0
times the web width were used in computing section actions. 7.3.4.2
Modified method for calculation of panel
edge section actions The modified method to be described was necessary because the steel strain and concrete strain measured near a panel edge section did not apply to the same crosssection.
This is illustrated in Figure 7.9(a).
The steel
strain measured corresponded to the cracked section at BB but the concrete strain to the uncracked section at AA. Further~
at the end of the test the zone of crushing at Y
was no wider than
itl~
and as the small values suggested,
the concrete strain measured was not that existing at y. The key to the modified method is given by the forces on the section at BB shown in Figure 7.9(b). required were the action at
mid~depth,
The values
mE and C , resultE
ing from the steel tension Ts and combined steel and concrete compression, c . c
Even for large variations of c , c
B X
I
144
Steel gauge
W ~--------~;i~--------------------~ Slab
y Concrete gauge
(a) Section showing
Beam
difference in gauge positions
IFIGURE 7.9
ACTIONS AT EDGE OF PANEL
I
o q (\/
(a) Concrete strip element
I FIGURE
7.10
I
(b) Computed strain profiles
I
Tension"
•'It •
o
C\l
I
(V)
0"":
f
/
/'
----- -.. '" .."..-. ........... . ~.', -;;::
/
~
.... .
.",,' ... . ./
' ....
.. '
--- GG _._. LL ........ AA (gauge position) --DD
-----
.-
145
,
the level of its line of action will not alter significantly and may reasonably be assumed constant.
The value
of mE' however, is dependent very largely on the magnitude of C = CD' the shift in the line of action of Cc being E of the second order. Thus, if moments are taken about the assumed line of
act~on
of cc' the magnitude of CE need not be known for an accurate assessment of mD to be made. o
It is reasonable to assume that the moment and force at Section AA will be equal to those at Section BB.
If
moments are taken about the level D at Section AA, these should sum to mDo On the basis that moments about D, at the level of the bottom steel, are equal at AA and BB it may be seen that
(mD)
AA
= (mD) =T 1 BB s a
Knowledge of the bottom concrete strain at AA was used to obtain mE and C
E in two ways as follows.
(a)
Assuming full bond transfer of steel force
between BB and AA. If full bond transfer occurs between BB and AA concrete and steel strains at the level of the top steel are equal at Section AA. steel force at BB.
mD was calculated from the known
The section at AA was uncracked and
the strain profile was found which satisfied the condition that (mD) = (mD) and having the bottom concrete strains AA BB
146
equal to that given by the reduced and corrected value of concrete gauge reading at AA.
This was done by increasing
the top strain from zero in steps of .000005 unttl (mD) = AA (mD) . The actions, mE and CE for this strain profile BB were then determined. (b)
Assuming no bond transfer between BB and AA
This assumption meant that at AA, the top steel force
was equal to that at BB.
A strain profile in the
concrete, having the strain at the bottom surface equal to that given by the gauge at AA, may be found such that (m ) = (mD) • D AA BB Since the contribution of the top steel force to mD
is the same at AA as at BB, the required strain profile would result in the'moments of the concrete forces at BB, about the level
D~
being zero.
For D at a level of .94
of the total depth the linear strain profile which gives zero moment of concrete force s about D has e ~ ~ - 2. e c (Figure 7.7) for a material linearly
elastic in both
tension and compression. To check the linearity of strain profile, the strip of concrete WXYZ, assDlTIled to be of elastic material, was analysed for a load at Y and the relationship between top ahd rJottom concrete strain at AA was found. Figure 7.10(a) shows the unit width concrete strip analysed, and the load assumed to be acting upon it. Analysis of this element was done using a library finite
element c;omputer programme, with elements as drawn. It was fOUIld that the strain in the concrete at the top surface was approximately equal to
-.5
times the
concrete strain on the bottom surface when the strain profile was linear.
At AA the strain profile was not
precisely linear but very nearly so as may be seen in Figure 7.10(b) which shows the computed strain profiles, In this case, therefore, the concrete strain profile was calculated directly from e
c
:=
~
e' where e was c c
known, and the section actions computed. The difference between methods (a) and (b) above was not large and in the analysis of results, values of method (a) were used.
CHAPTER
TESTS
ON USED
8.1
THE TO
8
PERFORMANCE CALCULATE
OF
SECTION
THE
METHOD
ACTIONS
SUMMARY In this chapter two types of test on the method used
to compute the moment and normal force on a section are described.
The first was on a series of three specially
cast slab strips with identical strain gauges to those on the model floor.
Known actions were applied to the
gauged section and these were compared with the values computed from the strain readings. The second test was on the sensitivity of the actions on a section to change in strain reading in order that the likely effect of electrical drift and other unwanted components of the strain reading could be assessed.
8.2
TESTS ON SPECIAL CONTROL SPECIMENS 8.2.1
Introduction
In order to check the suitability of the method used to compute axial force and moment in slab sections, a series of three slab strips of the same mortar mix used for the model floor was testedo
The strips had the same
14-9 depth and bottom reinforcement as the model panels.
Strain
gauges on the steel and concrete were of the same type and mounted in the same way.
A range of moments and axial
forces was applied to the gauged section.
Gauge readings
were processed by the method described in Section 7.3 and the computed and applied axial forces and moments compared. 8.2.2
Strip Dimensions and Test Set-up
Each strip was 36 11 x 8i" X 1.98" with two lead bath annealed bars as reinforcement.
%11
apart~
symmetrically placed in plan
at the bottom.
i"
diameter
The bars were with 3/16" cover
Each strip was loaded at the third points
of the 33" span. Vertical load was applied with a screw jack through a proving ring and spreader beam. At each end of the strip a steel end block was attached, covering both the ends and the end portions of the underside in order to transfer both the vertical reaction and the applied axial compression.
,
Axial compression was applied by tightening each of the two tie rods.
Two diametrically opposite strain gauges
on each rod placed parallel to the longitudinal axis provided the means of force
measurement~
each rod being
thoroughly checked and calibrated before and after the tests. Force in the rods was transferred to the slab strip through beams across the ends.
These beams consisted of
150 two steel flats with a gap for the rods.
Force from these
onto the end blocks was transmitted through two half-round mild steel pieces for which the end blocks were shaped. This arrangement ensured that the tie rods remained horiz ontal throughout the te st.
Figure 8. '1 shows the te st
set up at the end of a test and Figure 8.2 shows strips 1 and 2 after testing. 8.2.3
Instrumentation
The mid-span section of each strip was strain gauged with a gauge on each steel reinforcement bar and a gauge on the top concrete surface above each bar. Each tie rod had two electrical resistance strain gauges cemented to it which were used to measure axial force.
Dial gauges were used to measure the vertical
displacement at
mid~span
the roller support.
and the horizontal movement at
Proving ring readings provided a
measure of the applied vertical load. 8.2.4
Tests Performed
Each load increment represented a combination of moment and axial force at the mid-span section.
Incre-
ments of proving ring force were 50 lbs giving moment increments of
275
lb~in.
Axial force increments were
approximately 1000 lb or 118 lb/in width. Table 8.1 gives a summary of tests performed on the three strips.
FIGURE 8. 1
TEST SET-UP FOR STRIPS
STRIi-' N" 1
FIGURE 8.2
STRIPS 1 AND 2 AFTER TESTING
-152 Table 8.1 .
Strip Test Summary. Axial Force
Strip Load Range No. lb.
Ran~
Strip Load No. Range lb.
lb.
1
0-800 3500-4000
3*
0~700
5000
0-5000
2*
0-700
3000
3*
0-600
4000
5000
2*
0-550
2000
3*
0~500
3000
100-450 0-5000
2*
0-400
1000
3*
0-400
2000
0
3*
0-400
1000
3*
0-350
0
0
1*
0-450
1*
0-400
1000
2*
0-200
2
0-400 0-5000
2*
0-700 2000-4000
3
0-550
2
*
Strip Load Axial --No. Range Force lb. Range lb.
2
0-450 0-5000
1*
1*
Axial Force Range lb.
400-950
5000
0
3* 350-4-50 0-2000
Denotes cracked section.
8.2.5
Behaviour During Tests
All strips developed a single crack near mid-span which led to high steel strains.
The cracks in strips 1
and 2 did not form directly beneath the centre of the concrete gauges on the top surface and these readings were low as a result. exactly at
For strip 3 a groove was made in the underside
mid~span.
This ensured that the cracking took
place at mid-span and that the region of highest concrete strain was near the middle of the gauges. The very small percentage of reinforcement made the ultimate moment less than the cracking moment and cracking was accompanied by large increases in steel strainjespecially when the applied axial force was low.
153 When the vertical load was increased for a set value of axial force, the outward spread of the ends caused an increase in axial force, but only in cases where variation became large was any adjustment made. 8.2.6
Results
Only the comparison of calculated and applied values of moment and normal force is presented in this Section. The full results are given in Appendix F. (a)
Determination of Applied Actions:
Moment at the mid-span section about its mid-depth was computed from the three components: moments~
(i) Dead load
including allowance for the weight of the proving
ring and spreader beam;
(ii) Moment induced by the verti-
cal applied load and (iii) Moments induced by the eccentricity of horizontal force applied at the ends of the strip. (b)
Determination of Section Actions from Strain
Readings: The method of Section
7.3 was used.
Section strain
values were taken as the average of the two taken on the steel and on the concrete.
Both the normal method and the
second modified method (Section analysis.
For strip
7.3.4) were used in this
3, the normal method was used since
the crack formed exactly at
mid~span
but for strips 1 and
2, cracking was not exactly at mid-span and a situation
similar to that described in Section 7.3.4.2 arose whereby concrete strain readings greatly underestimated the maximum concrete strain.
The second modified method, in which no
bond transfer was assumed, was used. (c)
Comparison of Applied and Calculated Section
Actions: Figures 8.3 and 8.4 show graphically the results of In Figure 803(a)~ the ratio of calculated
this comparison.
moment to applied moment is plotted against applied moment for the uncracked sections. Results from each strip are recorded, and for strips 1 and 2, values of the ratio for applied axial compression, N app
=
0 and N app
=
5000 lb are given for each level of
Mapp$ Figure 8.3(b) is a similar plot for the ratio of calculated and applied normal forces, Ncalc/Napp for an uncracked section. plotted for Mapp
For each value of Napp ' the ratio is
= 330 lb-in and Mapp
=
2550 lb-in.
Figure 8.4 shows similar plots for a cracked se
on.
The range of normal force and moment was not as great as that applied before cracking and less points are shown. Because the cracks were close to the gauge"the results of strips 1 and 2 after cracking were calculated with a linear strain profile approximating that given for Section GG of Figure 7.10.
For this the strain at the level of the
bottom surface was taken as -1.8 times the top surface
o DenotJ
2
• Strip 1 • Strip 2 A Strip3
Napp = 5 kip
0.. 0..
2
(a) Forces
o
Nap? = 0
ex
z
"c:t "0 zo
1
~
M app =330Ib-inj O-Mapp= 26001b-in
® ______________~______________~~_______ ®
®
(0) Moments
OL-______
~
______
~
o
______
~
2
II
______
~
3
________
~
_____
O~------~------~------~
o
5
4
______ ________ ~
2
~
5
4
Mepp (kip-in) FIGURE B_3 ACTION COMPARISON BEFORE CRACKING .... Mapp
2
=330 Ib-in; 0
FIGURE B.4 ACTION COMPARISON
Mapp = 2550 ib-in
2
AFTER CRACKING
(b) Moments
0.. 0..
o
2:
€I
'u-
.·-Napp=Oj O-N app =4kip
~
"8 1 I------....:.:...--~:,..-_rr----=----------------2: ® ® c!>® ®
®
®
(b) Forces
OL---____
o
~
______
~
2
______
~
3
______
~
________
4
~_
5
o~
o
______
~
______
~
2
______
~
______
3 Mapp(kip-in)
~
________
4
~
5
156 strain reading. Further evidence of good correlation of results for strips 1 and 2 is given in Figure
8.5.
This shows plots
of moment, both calculated and applied, versus proving ring force.
The results cover tests on strip 2 in which
the proving ring force was increased with the applied compression set at 2000 lb and later 3000 lb, and a similar test for strip 1 for Napp = 1000 lb. 8.2.7
Discussion
Results from all strips before cracking showed agreement within ~ 20 per cent for most cases, many of which corresponded to wi thin 10 per cent.
Before cracking, strain
was easier to measure in that the effect of finite gauge length was not as great as after cracking.
This led to
more accurate calculation of section moments and forces but the low values of strain at this stage made the effects of electrical drift and other strain reading errors relatively greater. The position of the crack in relation to the concrete gauges was obviously significant
0
For strips /1 and 2 where
the cracks did not form at the gauge points, it was possible to obtain good correlation of applied and calculated values of section actions.
These were still dependent on
the position of the crack since it was shown that the strain profile across a section near the crack was not linear.
By taking account of this factor 1 satisfactory
4
FIGURE 8.5 MOMENTS FOR STRIPS 1 AND 2
FIGURE 8.6 GAUGES 118,119.120 MOM ENT AND FORCE
60
VARIATION
Cracked section
--Moment --- -Compression
3
~-------+------~~~~~--~---------
400
a - applied
c -computed - - - - Strip 1 Napp.. l000
Strip 2 Napp " 3000 _'_'_'Strip 2 Nopp .,2000
o
200
400
600
800
Load (psf)
o ~------~-----------------------o 200 400 600
FIGURE 6.7 GAUGES 73,77 MOMENT AND FORCE
Proving ring force (Ib)
VARIATION
<40
10
---M
----T
o
o~------~------~------~--------~-200 400 600 800 Load (pst)
o
'158
correlation of results was achieved, especially for strip 2 where the crack corresponded almost exac.tly to the end of the concrete gauges. The results of strip 3 did not compare as favourably. The measurement of concrete strain directly above the crack was not simple because of its rapid imriatiort along the gauge length. This method of calculation of panel section action could be expec,ted to give results wi thin approximately 30 per cent of actual values. In assllillingthe applicability of this conclusion to
results of the model floor, the following points must be borne in mind: (i)
For the uncracked se
ons of the model the variation
of strain reading due to electrical drift and other time effects may have had a signifJ.canteffect. (LL) For sec
were see
ons at the panel edge the concr'ete gauges 8
011
considerable d:L
anae from the cracked,
and therefore the assumption of a linear
strain profile for the concrete afforded a closer approximation to actual behav:Lour than a similar assumption used for these strip tests. (iiDEach strip was tested over a period of several hours. Strain reading errors introduced 'by time-dependent effects
the slab model
st may therefore have
been greater than in the strip tests.
159
8.3
THE EFFECT OF VARIATION IN STRAIN READINGS Of importance in the interpretation of results is the
sensitivity of the values calculated to change in strain reading. Considerable variation in strain readings between the end of one test and the beginning of the next test on the following day was detected and although this was accounted for in reducing readings, it pointed to the possibility of appreciable discrepancies between strain readings and the true strain due to stress alone.
In order to examine the
effects of strain reading variation on the section actions, four typical sections were chosen and the raw readings of steel and concrete gauges varied from the actual values. The resulting moments and forces were compared. The four sections taken were as follows: Description of Section
Gauge Numbers 118,
119~
120
Centre panel edge, modified method used on cracked section.
8, 10
Centre panel span, normal method used. Section uncracked almost throughout test.
73, 77
Interior beam, centre span, at mid-span, normal method used.
126, 129
Interior beam, centre span, at support, normal method used.
Four runs were performed for each section using different values of strain deviation,
D.
In each run the value,
6 ,
was subtracted from datum-corrected readings of concrete
strain gauges and added to datum-corrected readings of steel strain gauges.
The four values of Dused were:
-20,
0, +20, +40 microstrains. Results are compared graphically in Figures 8.6 to 8.9 inclusive.
These show the increase of section actions
with increasing load after load stage 168. At 775 psf the ratio of moment or normal force for
6=
+40 to the moment or normal force at
~= ~20 had the
following values: Ratio Mmax1M. mln
Section
Ratio Nmax/Nmin
118, 119, 120
1 .23
1 .40
8, '10
1 .40
1 .60
73, 77
1 .05
1 .04
126, 129
1 .08
'I .0"7
The curves shown represent a variation of 60 microstrain in both concrete and steel strain.
The fact that
steel strain was increased and concrete strain decreased could be expected to produce a greater effect on the moments than on the axial forces. Variation in the panel sections was greater than in the beams.
Gauges
/1-18~
119 and 120 were at a centre panel
edge section where the modified method of computation was used.
The 60 microstrain variation in this case is a large
proportion of the measured concrete strain.
The same is
true of the section at gauges 8, 10 which did not crack
161
FIGURE 8.B GAUGES 126,129 MOMENT AND FORCE VARIATION
100
75
15
--Moment
.=-~-=-.=J Tension 50
10
-0_.
o
O _ _ _ _.....I...._ _ _.....-l._ _ _ _.........._ _ _
o
200
400
600
~---
800
Load (pst)
FIGURE 8.9 GAUGES 8,10 MOMENT AND FORCE VARIATION
A =-20ps
----C ---M
o~------~--------~------~~------~----200 o 600 400 BOO Load (pst)
162
until late in the test. cent and
Lf,O
The variation of moment (23 per
per cent) and
normal force (40 per- eent and
60 per cent) for the 60 mic.rostrain variation indicates the appreciable sensitivity of the computed actions to strain reading variat;ions.
For this same strain variation? the
beam actions show markedly less change.
The variation of
le ss than /10 per cent in the se actions due to the 60 microstrain variation is clear evidence of their insensitivity to such change. The magnitude of likely strain variation in the slab test is difficult to determine exactly but variations in the temperature and zero gauges during the test suggested that 30 microstrain would be an upper limit.
Most of the
actions computed from strain readings during the test would therefore vary by less than one half of the above values.
163
CHAPTER
BEHAVIOUR
OF
THE
DURING
9.1
THE
9
NINE - P ANEL TEST
MODEL
FLOOR
PROGRAMME
SUMMARY This chapter describes the behaviour of the floor
during the test programme in terms of the strains, deflections, loads and reactions measured at each load stage, and the examination, during and after the test, of physical effects such as cracking.
!
The floor had been
designed for an ultimate load of 800 psf including membrane action.
This load was twice the Johansen ultimate load of
the centre panel,
1.35 times the Johansen load of the
centre-edge panels and equal to the Johansen load of the corner panels.
Pattern loads were applied in early te'st
runs but beyond 400 psf all panels were loaded equally until the centre panel failed at almost 850 psf, a failure brought about by the transition of the panel from a state of predominantly compressive membrane action to one of predominantly tensile membrane action. were intact at this stage.
The outer panels
Following the failure of the
centre panel the load fell to 540 psf.
Load on all
p~nels
was then increased to 710 psf at which time the centre
164 panel loading bag was in danger of bursting through the full depth cracks which had formed at the centre of the panel.
Load on the centre panel was then reduced to 600
psf and the load on all outer panels increased to 960 psf when the centre-edge panels failed in a combined panel and beam mechanism.
The centre-edge panels were held at
960 psf while the loading on the corner panels was increased.
The end spans of the interior beams developed
plastic hinges at 1170 psf* and the test was stopped at this load with panel failure mechanisms in the corner panels incompletely developed. Symmetry of behaviour was excellent throughout the test programme until after the centre panel failure. During the loading to failure of the outer panels the plastic hinges in the end spans of the interior beams did not develop simultaneously and the symmetry was upset noticeably. Reactions were not affected greatly by moment redistribution and values remained close to those expected.
Summation of reaction values indicated that
the loading bags applied load over only 90 - 95 per cent of the total top floor area due to rounding of the bag edges. * Figures quoted indicate the intensity of load applied to the panels of the model floor. This includes the 75 psf difference between prototype and model self weights but does not include the self weight of the, model (= 25 psf).
165 Compressive membrane action enhancement was exhibited by all panels.
Measured compressions at the edge of the
centre panel were of the order used in design and beam tensions in the centre spans of the interior beams were accordingly large.
Tension in the exterior beam centre
spans indicated the presence of compressive membrane forces in the long direction of these panels as expected. measured along the interior long
edge~
Forces
parallel to the
short side, were large enough to suggest considerable membrane action in this direction.
Beam tensions in the
centre spans of both interior and exterior beams were large and may have been larger if all panels had failed simultaneously.
Values were higher at the support than at mid-
span. Initial cracking of the undersides of the outer panels produced a marked effect on the centre panel.
The loss of
lateral restraint caused an increase in deflection and strain values in the centre panel. hours of sustained loading at
Stability during 66
375 psf was good but instab-
ility was evident at 550 psf when the undersides of the corner panels cracked for the first time and cracking of the centre-edge panels extended. The effect of membrane action on the torsion in the edge beams was evident in the slow increase in strain in the panel reinforcement at the sections adjoining the beams.
Torsional deformation in the beams was not large
166
until after the centre panel had failed and the end spans of the interior beams had developed large cracks prior to the full development of plastic hinges there. Moments along lines traversing the whole floor were calculated from reactions and applied loads.
The rise with
load of the 'free' moment so calculated was linear and agreed well with the expected value throughout the test. , Initially, moments along mid~span sections were relatively high in comparison with the support values but as load increased, the rate of increase of mid-span moment fell and a corresponding rise in the rate of increase of support moment took place. Moments at beam sections computed from the strain readings showed a similar trend.
The line moments, cal-
culated from the sum of beam section moments, showed satisfactory agreement with those calculated from the reactions and applied loads.
The effect on line moment
calculations of the net beam tension and net slab compression was large since each force was taken to act at a different depth below the top surface of the slab.
The
two forces thus formed a couple which had to be taken into account. The rate of increase of compression at the edges of the centre panel was similar to that of the increase in supporting beam tension, being small at low load levels and increasing with load.
167 Although strain readings afforded some measure of panel and beam section forces and moments which in many cases compared satisfactorily with values determined by other means, the accuracy of the results was not sufficient to distinguish any difference in centre panel behaviour with varying load pattern.
Only the variation
of steel strain at the middle of the centre panel showed signs of the surround being slightly stiffer laterally when the outer panels were not loaded fully. Deflections at DL + LL* were small for all components with the centre panel showing greatest deflection to span ratio.
This ratio first exceeded L/360 at 450 psf when
considerable surround stiffness loss occurred with the initial cracking of the underside of the centre-edge panels.
This loss and the extended cracking of the
centre panel caused appreciable unrecoverable deflections. strain levels were also low at DL + LL.
At this
load, after the floor had been loaded to a maximum of 450 psf, strains at the panel edges ranged up to of yield values.
two~thirds
Steel strains in the beams were approx-
imately one half yield values.
The application of 450 psf
had yielded the steel at the middle of the centre panel. At the stage of failure of the centre panel most of the beam steel had yielded or was near to yield, and steel "'The abbreviation DL denote s the prototype dead load II II LL II " II li ve II
=
=
100 psf 300 psf
168 at the edge of the panel was well beyond yield. edge steel along the exterior
beams~
Panel
however, showed
very small values of strain. Panel deflections at this stage were large in both centre-edge and centre panels though rapid increase had not commenced until 600 psf.
The deflections of the
centre spans of all beams were le ss than 1/360 of the span and other beams deflected little in excess of this. The applied load of almost 850 psf at the stage of failure of the centre panel was 10 per cent in excess of the design ultimate load. Details of slab behaviour follow in the next two sections.
In Section 9.2 the behaviour of the floor during
each of the 14 tests performed is described.
Although
chronological order of te sting is :not strictly adhered to
j
this section is intended to give a cliar impression of the floor behaviour as the test progressed and to indicate the effect of different load patterns. A more detailed analysis of particular aspects of the floor behaviour is given in Section with all-over load.
903 which deals mainly
Deflections, strain readings, cracking,
reactions, moments and membrane ac.tion effects are dealt with in turn. 9.2
TEST BY TEST DESCRIPTION OF FLOOR BEHAVIOUR 9.2.1
Tests 101
~
102, 105 and 106
In these tests the slab was loaded over its whole
169 surface to a maximum of dead load plus full live load (375 psf applied). 9.2.1.1
General
Tests 101 and 102 (to DL +
~LL)
were preliminary
tests only, serving to test the loading system, data recording devices and the symmetry of response.
Embedment
of the load cell ball bearings into the mild steel bearing plates caused differential settlement of the reaction points and redistribution of reactions.
This, and the
tendency for the corners to lift made resetting of the reactions necessary after Tests 101 and 102.
No further
adjustment was made. Test 105 was a repeat of Tests 101 and 102, performed after the pattern load tests (103 and 104) had been performed. In Test 106 the load was increased to DL + LL before being released in stages. 9.2.1.2
Deflections
Load-deflection relations were linear for all vertical deflection gauge points.
In Tests 101 and 102
full recovery was not achieved due to the embedment of the load cell bearings into the mild steel bearing plates. Deflection levels in Tests 101,102 and 105 were very low. The maximum deflections occurred in Test 106 when DL + LL was applied.
Values of the largest deflection
occurring within each of the element type groups are given below (Table 9.1).
"Q:J:Eur~ 6.2 ( 0)
Reference Mark
MaximUIll Deflection ~
Interior Beam
Beam NS3
.0285
/1: 2/190
or Beam
Beam NS1
.0154
1 : 1+050
~Panel
Panel C
.0210
1:2120
Centre-edge Panel
Panel H
.0403
1: 11 00
Centre Panel
Panel G
.0616
"I :/lcY12
Elem~i,~lle
Exte Corner
Deflection: §li3?-_~__IE~ )
-----------~-----~--.-
9,281.3
Membrane Action Effects
There was no cracking of the underside of the centre panel during these tests but the cracking at the edges could be expected to cause the development of compressive membrane forces normal to the edge.
fJ:lension in the beams
was
high hut compressiV'e membrane forces were
iable.
Figure 9.1 shows the increase of this edge
compres
apprec~
on during Test 106.
9.2.2
Te sts 1 03 and "108 General
In both these tests the load on the centre-edge panels was held at
75 psf while the centre and corner
panels were loaded to 225 psf (DL +
.5T~)
in Test 103 and
to 400 psf (DL + 1.08LL) in Test 108. 9.2.2.2
Deflections
With one exception the deflections of beams and panels were smaller than those occurring at LS85 (see Table 9.1).
The cracking of the underside of the centre
panel which took place during Test 107 produced nonrecoverable deflections and the maximum deflection to span ratio rose to 1:678. 9.2.3
Tests 104 and 109
In Test 104 the centre-edge panels only were loaded to 225 psf, while the load on the other panels was maintained at 75 psf. In Test 109 the load was taken up to 375 psf on the centre~edge
panels.
The load on the other panels was kept
at 75 psf until, at a load of 250 psf on the centre-edge panels, when the corner reactions were about to become zero, the load on the other panels was increased to 200 psf to ensure that the corner points of the floor did not lift off their supports.
When 375 psf was reached on the
centre-edge panels the load on the other panels was reduced from 200 to 150 psf at which stage the corner reactions were again nearly zero.
-
This latter condition represented
the most severe loading applied during Test 109.
Test 104
brought no further cracking but in Test 109 cracks appeared near the centre of the middle spans of the exterior beams
172 NS4 and EW1.
9.2.4
Deflections during both tests were not large. Tests 107, 110 and 111
9.2.4.1
General
The loading sequence in these tests was designed specifically to examine the behaviour of the centre panel in the presence of reduced load on the surrounding panels, the effect of reduced edge moment restraint and modified surround stiffness being of particular interest.
Time
dependent effects were examined during two periods in the course of these tests. The whole floor was loaded to a maximum of 375 psf in Test
107.
This was followed by reduction of the load
on all but the centre panel to 75 psf, the original condition of 75 psf allover then being achieved by reduction of the centre panel load in stages. All panels were loaded equally in Te applied load was first increased to 450
psf~
110.
The
reduced then
to 375 and held for 22 hours and then reduced to 75 psf. In Test 11"1 the load on all panels was taken up to
225 psf, the centre panel load then being raised to 375 psf and reduced again to 225 psf.
All panels were then
loaded to 375 psf and held at this load for 66 hours while time=dependent effects were examined.
90204.2
Deflections
Cracking produced by the higher load levels eaused
173 larger deflections than in previous tests.
Much of the
deformation in the centre panel was not recovered on release of load.
The effect of cracking is clear from the
examination of the maximum element deflections for each test (see Table 9,2) but the different load conditions must be taken into account in making any comparison. Table 9.2. Maximum Deflections of Elements for Tests '107, 11 0 and 1 /j 1 . Element Type
Reference Mark (Firure 602 a))
Test No.
Interior Beam
NS3 EW3 EW3
Exterior Beam
NS1 NS'l NS1
Corner Panel
Centre~edge
J J J
Panel
Centre Panel
*
**
***
F F D G G G
Span: Deflection
Maximum Deflec,tion "Cinches)
~atio)
107* 110* * 11'1***
00305 00550 .0470
2040 1140 1330
107
O'1c:.9 ,'0 ,0399 .0309
3700 1570 2020
002'19 00305 .0197
2030 1460 2260
00380 .0732 .0546
1~170
.0871 66 02 1 37
720 289 29 4
c..l
"110 11/!
/107 '\10 111
10'1 1'10 1 'l/t
107 110
0
111
375 psf on centre panel; 75 psf on all others. 450 psf on all panels,@ 375 psf on eentre panel; 225 psf on all others. The effe
of the application of
4~~50
psf (rye; psf I /
excess of the design serviee load) on the centre panel
610 810
174
deflections was very marked.
The maximum deflection of the
centre panel before design service load was exceeded was only 1/600 of the span. 9.2.4.3
Crackigg
Cracking in the centre spans of beams and in both the centre and centre-edge panels took place during these testso First cracking of the underside of the centre panel occurred when the load on the outer panels was reduced to 275 psf and left for one hour (LS100).
Three cracks
radiated from the centre and extended well towards the edges of the panel (see Figure 9.2 (a)). reduction of outer panel
load~
one of these cracks took place. the centre
panel~
On further
considerable extension of The maximum crack width in
measured when the load on the outer
panels was 75 psf, was 0002110 The maximum load of 450 psi (DL + 1 025LL)J applied in Test 110,produced further cracking in the centre panel, in the centre spans of all beams and c,aused the initial cracking of the underside of the
centre~edge
panels.
At 400 psf and 425 psf (LS156 and 157) small extensions in the centre panel cracks were noticed and the maximum crack width was .003".
Some new cracks appeared
in the centre spans of interior beams. At 45.0 psf (LS"158) eracks formed in all elements except the corner panels and their supporting spans. the middle of each
centre~edge
In
panel a crack ran the full
( /
a)
Centre Panel - Test 107 - 375
sf
(c)
Centre panel
Test
o-
450 psf
\
Cd)
Interior beam
FIGURE 9.2
centre span - 550 psf
CRACKING DURING TESTS 07
(b)
Centre-edge panel - Test
o-
450 p
10 A.ND
176 width,parallel to the short sides,and in one panel two smaller cracks formed in the L-beam flange, either side of mid~span
(Figure 9.2(b».
Cracking at the supports of the centre spans of the NS interior beams was notieed for the first time and several new cracks appeared at the middle of these spans. Each centre-edge panel crack caused increased cracking in the centre panel and since an appreciable time elapsed before all four centre-edge panels had cracked, a similar time passed before cracking of the centre panel ceased. Comparison of Figures 9.2(a) and 9.2(c) reveals the extent of cracking in the centre panel produced by this load increment.
Figure 903 shows the crack pattern for the whole
slab at this stage
o·
New cracks appeared in Ere some beams supporting the centre panel steeply inclined cracks (marked
0
0
Further, smal
b
In beam EW3 two
"177" in Figure 902)
appeared at the thir1 points of the these did not
age /177
load
om
span. the
One
am.
r cracks of a similar nature appeared
after the load of 375 psf on all panels had been maintained for nine hours.
After 21 hours 1Tery few new cracks had
formed and after 29 hours the extension of existing cracks was negligible.
ulr__________-LnL'______~;~~~cl~i____~L~till~LI__________~iI
EW1
o... --.'
EW2
<
-.fJ.'.
r-------li------
I
Ii
I
II
I
II II
I
2
5
: II I: I I II II I L ________ L ______ 3.. ___ ....:_J L _______ J
J
,--------l---J1:s------M ,------, I
I1
I
I I
4
6 .5
2
II
I
I
I
I
I'll
iI
11[7f5
I
II
I
450 PSF APPLIED LOAD :l'::..:
I
I I
Maximum crack widths shown in .001" units for load at 375 pst
1 .1
I II
L _______ J L _________
:-------1
I
I I
I II II
I
II
CRAD< PATTERN AS AT
::1<; ~ I~ I
I~
I
I I
FIGURE 9.3
I I
I I I
L______ --l
i-----~--~-:
:-------l
II
I I
I
II
I
I
I
J
I
I I I II IL ________ -.JIIL ______
I:
2 _____
'"
I
JI
I
II
I L _______ J
z EW3
EY>'4
b Ij
a·
t.
[] Lr
yr;'L(I~ E 1.5\.5
L~ Y
3ll
Ul
.U
@
8
!I
w
z ~
--
-....J -....J
Maximum :Load Table 9.3 shows the values of strains and computed moments and normal forces for four critical load stages. IJoad stage 98 preceded the first cracking of the under·· side of the centre panel and much of the beam cracking. strains are low as a result.
The effect of the application
of 450 psf at LS158 was to increase steel strains sharply, particularly in the centre panel.
At both panel and beam
sections, sharp increases were evident in the values of moments and mambrane forces. Loads applied in Tests 111 produced relatively little change, 9,2.4.5
Membrane Action Effects
Small changes in beam tensions and panel membrane forces resulted from the removal of load from the outer panels in Test 107. The difference in behaviour between the load configuration in Test 111 and that in which load was applied equally to all panels was not large enough to be detected. The most vivid expression of membrane action came in Test 110 with the application of 450 psf when the loss of surround stiffness resulting from the cracking of the centre-edge panels caused a marked change in the -behaviour of the centre panel.
The cracks in the centre-edge panels
(Figure 9.2(b)) ran along radial lines from the centre of
TABLE
GAUGE
ELEHENT
9.3 - STRAINS, HOi"ENT·S AND NORMAL
POSITION GAUGE NO'S
MAXIMUM CCNC. STRAIN
,?ORC;;:S [,1'
Lon:: STA'}r;::; ,,3, 15S, 17G end if,')
MAXn11lM S7EEL STRAIN
MONE1·;T
YORC;;; C:i;cs -cr.::
LSN Ext.
Beam
C.Span
Ext.
Beam
C.Span
Ext.
Beam
E.Span
133,134 Mid-Span 53,55 Mid-Span 59,62 Support
Int.
Beam
C.Span
Int.
Beam
C.Span
Int
Beam
E.Span
126,129 73,77 Mi:1-Span 64,67
Centre Panel
Centre
Bottom
Centre Panel
Edge
Centre Panel
Edge
Centre Panel
9" from Edge
Support
Hid-Span
Edge
Panel H Centre
Edge
Panel H ::;dge
-In t. Long
Edge
Panel HEdge
Ext.Long
Edge
Panel H Edge
Short
Bottom
Corner Panel
Edge
Int.
Corner Panel
Edge
Ext.
6 114,115 118,119 20,22
-45
-246 -76 -63
176 -122 -20 -31
-225 -45 -51
-97 -62 -44
-163 -92 -63
-75 -31 -38
-124 -50 -59
-54 -117 40
-52 -161 62
-14 -115 61
-18 -128
-108
-182
-107
98 -143
-70
98
409
371
596
50 l !
109
164
423 549 101
73
279 306 338 -38
1650 1441 1431 -91
1575 1305 1261 -'18
1638 1422 -1390 -110
39 -133
172
8
-3 719 16
195
390
-42 681 -29 328
-13 776 -13 404
246 14
870 12
780 -21
919
All Strains Corrected for Drift (Gauge 81) and 100 PSF Initial Load. Steel Strains Corrected for Temperature (Gauge 140). Refer to Figures 7.2 and 7.3 For Gauge Positions.
92 297 508 50
106
91 86
Moments and Forces Calculated From Strains Shown.
176 231 486 21
339 102
5
94,95 97 102
158 .)03 534 53
-1
-3.6 -13.0 6.4 7·5 4.6 +3.2 -13.6
12.4 7.4
-23.6 12.6
-17.,.,
-
-c"./
1.7
-.9 -2.6
-21.7
-.5
1.:
-141 -300 -171
-188
-302
-'17 -257
-137
-239
1.6 -~,. "I
.l. j
•
fJ
-i .. ~
-.? .6
11; -106
-74
73 -109 -207
-121
-232
-1:'1
5·5
-114 -206 -74
15S
0
-13.1
-76 -172
117 -6~
-213
-158
-180 the floor.
This, and the presence of more cracks towards
the outside edge indicated that these panels were acting as deep beams and ties against the compressive membrane forces in the centre panel.
Each of these long cracks was
seen to produce immediate and sharp deterioration of centre panel behaviour. 9.2.4.6
Effects of Sustained Loading
At two stages during these tests, design service load was maintained on all panels for an appreciable time.
The
application of 375 psf for 22 hours at LS161 revealed negligible time effects, but the 66 hours at this load (LS189) produced detectable changes. Figure 9.4 shows defleetion in the centre panel, steel strain at its edge and steel strain at the mid-span section of an interior beam plotted against time for LS189. The effect at zero time has been set to zero for each curve and steel strains have been corrected for temperature variation.
The total increases over the 66 hours are small and
a general trend towards stability with time is apparent. The values of the centre panel parameters continued to rise at the end of 66 hours but at a decreasing rate. Changes in compressive membrane forces in the centre panel and changes in beam tensions were slight.
181 9.2.5
Test to Failure of Floor as a Whole
9.2.5.1
General
This test was performed in two parts:
the slab was
first loaded to the predicted ultimate load of
775 psf
applied after which the load was reduced to 400 psf.
The
load was then taken up again until failure of the centre panel at almost 850 psf.
Failure of the centre panel was
deemed to be the failure of the floor as a whole.
At this
stage the deflection of the centre panel increased markedly as tensile membrane action took over from compressive membrane action as the principal load carrying mechanism.
Centre-
edge panels were showing only moderate signs of distress at this stage and the corner panels even less. 9.2.5.2
Deflections
Maximum levels of deflection for the floor elements at the predicted ultimate load are shown in Table 9.4. Most elements showed a sharp increase in deflection at 550 psf when the first cracking of the underside of the corner panels occurred, and a general loss of stiffness was evident after this stage. The deflection of the centre panel increased steadily with load and was approximately equal to the depth of the slab when failure occurred and tensile membrane behaviour became predominant.
The centre spans of all interior beams
and the EW exterior beams tended to stiffen in the latter stages of the test,immediately prior to the failure of the
Compression - lb/in ~
1001
, / , ...............
~
o
7r
0
I .
!I
.
I I
I
I ! I ::::.....6---'"
!l __
75 I
,/,
7
ll\)~a;OO J
"
IOU!
C
11 .....
tE~ (Il-
::J
....
:::
o o o
50 I
iI
~/
cll)
o~
:;Sc
~.-
;;::: E
~ U) 25 I
I
oH
I
o
l
_. "
I
12
r
,
24 48 36 Hours since load first attained
60
~
[flGURE-9.4 TIME EFFECT =1..:5189] -J>
CD
I\)
centre panel. Table 9.4.
Deflections at LS220 (775 psf). Reference Mark (F:igure 6.2(a))
Maximum Deflection (inches)
Interior Beam
NS2
.143
438
Exterior Beam
NS4
.123
510
Corner Panel
A
·523
85
F
.649
68
Element T2:I2e
Centre~edge
Panel
Deflection: ~pan ratio)
Centre Panel
E
Interior Beam) East Span )
EW2
0430
104
Exterior Beam) East Span )
EW'1
0135
330
9.2.5.3
1 .32
47
Cracking
No fresh cracks appeared until a load of 500 psf was applied when a small number of cracks appeared in the beams.
At 525 psf one further crack in the centre span of
beam NS4 appeared. The application of 550 psf (LS200, 200A, 200B) produced cracks in almost all elements with dramatic effect. Further cracks appeared in the centre panel (Figure 9.5) and some
centre~edge
panels.
Cracks appeared in all
beam spans and first cracking of the undersurface of the corner panels occurred (Figure 9.7). Again each new craek in any of the panels surrounding
184
the centre panel
brought further cracking in the under-
surface of the centre panel.
The cracks in the corner
panels were limited to one per panel running along the diagonal passing through the middle point of the centre panel giving the floor panel crack pattern an even more radial nature and further indicating the effect of compressive membrane action in the centre panel.
The low reinforce-
ment content of the panels meant that the cracking and ultimate loads of the corner panels were almost equal, and cracks formed rapidly and extended almost the whole distance from one corner to the other.
In some cases the
crack formation caused dull thuds. Cracking in the beams at this stage was also significant.
In the exterior beams cracks appeared over some
interior supports and at the middle of some centre spans. In the end spans 1 cracks appeared very near the corner r (lower right of Figure
9.7)
indicating the considerable
bending moment induced by the twisting moment in the adjacent exterior beam at right angles.
Similarly induced
cracking took place in the end spans of interior beams (Figure
9.6
left middle;
Figure
9.7).
Cracks also
appeared over the interior supports of these beams. Although the crack widths in the panels and end spans of the beams shown in Figures
9.9, 9.10, 9.11 and 9.12
(taken at the end of the test programme) are very much greater than they were at failure of the centre panel,
FIGURE 9.5
CENTRE PANEL
FIGURE 9. 7
FIGURE 9.8
FIGURE 9.6
CENTRE-EDGE PANEL
FIGURES 9.5 to 9.9
FIGURE 9.9
CRACKING AT LS 200 (550 PSF)
CORNER PANEL
EXTERIOR BEAM, END SPAN
INTERIOR BEAi'1, CENTRE SP Ai\)"
186 the figures do show that much of the new cracking after LS200 (550 psf) was confined to the outer panels and end spans of beams (numbers lower than 228 indicate the extent of cracks formed before centre panel failure).
The centre
spans of interior beams showed few further cracks but the centre panel cracking extended considerably on the application of loads greater than 550 psf. Load increase beyond 550 psf brought more cracks in the centre panel, radiating from the centre
j
extending
further towards the edges of the panel as the tensile membrane region extended.
These cracks became wider near
the centre of the panel, becoming full depth before loading of the panel was stopped.
The state of cracking in the
centre panel after loading of it had been stopped may be seen in Figures 9.11 and 9012.
The zones of crushing on
the top along the diagonals show the effects of the large 11
circumferential 11 compression.
Full depth cracks extended
further towards the edges of the panel in regions away from the diagonals. 9.205.4
Moment, Strain, and Normal Force Levels at
Maximum Load The wide load range of this test
c,aused great changes
in the levels of the above quantities and only a brief description of the changes occurring is given here.
A
detailed description follows in later sections (9.3.2, 9.3.5, 9.3.6).
187 Table 9.5 shows the values of
strains~-i
moments and
normal forces for LS220 (775 psf) affording comparison with Table 9.3. Almost all steel strains tabulated are either past yield or very close to it and the values of these at midspan and support of the centre spans of both interior and exterior beams are nearly equal indicating little postyield moment redistribution.
Formation of cracks in the
end spans did not coincide with the steel gauge position and some erratic strain values resulted. In the centre panel, the strain in the steel at the centre increased to beyond the limit of the data logger. Steel strains at the edge were well in excess of yield as was the case for most points along the edges that were continuous over the beam supporting them.
Values of steel
strain at the panel edges supported by the exterior beams were remarkably low, especially in the corner panels. Values of beam moments compared favourably with design moments at the supports but were lower than design values at mid-span.
Tensions at the supports of the centre spans
of both interior and exterior beams were larger than those at mid-span.
The average magnitude compared well with
design values but exterior beams carried a greater portion of the total than was expected. 9.2.5.5
Evidence of Membrane Action
The application of 550 psf at load stage 200 provided
188 Table 9.5.
Strains, Moments, and Normal Forces at Load Stage 220. Gauge Position
Gauge Max. Max. Moment Force Numbers COlle. st. steel K" or K or Strain
;US Ext.
)kS
Ib"/"
Ib/"
'1256 1584
30.2 13.1
10.6
1978
-304
1447 1270
-92.0
Beam~
C. Span Support Co Span Mid-span E. Span Mid~span Into Beam: C. Span Support C. Span Mid-span E. Span Mid-span Centre Panel: Centre Bottom Edge Edge
9" from edge
133,134, 53,55 59,62
19 -11
126,129
-572 -209 -*
73,77 64,67
6
114,115 118 ,11 9 20,22
Edge Panel H: Centre Bottom 5 Edge Int. Long 94,95 Edge Ext Long 97 Short "102 Edge 0
Corner Panel: Edge Int. Edge Ext.
-212
91 86
*
7.3 1.7
15.3 9.6 3.6
43.5 9·3
-187 -295 48
*6979 8158* 2428* -71
-352 -509 -1'13
=197
=222
2306 2'136
-405
-195
~149
-308
'193 4137
2723 103
All strains corrected for drift (gauge 81) and '100 psf initial load. Steel strains corrected for temperature (gauge '140) Moments and forees calculated from strains shown. * Values off scale or gauge broken. 0
189 clear evidence of the reliance of the centre panel on compressive membrane action.
Later in the test, as failure
of the centre panel approached, the level of beam tension and slab compression rose steadily but finally fell away rapidly with the
push~through
of the centre panel.
At LS200 cracking in the outer panels was radial in nature indicating the effect of membrane action in the centre panel.
Again the effect of outer panel cracking was
observed - the thuds produced 'by the cracking of the corner panels were coincident with sharp increases in centre
pane~
:.;.
deflection. The
three~quarters
of an hour taken to settle at LS200
was a measure of the instability of the centre panel and a static situation prevailed only when all outer panels had ceased to crack. steeply inclined shear cracks in the interior beam centre span were indicative of the presence of considerable tension. 9.2.5.6
Failure Mec,hanism
The centre panel IIfailed li with the progression of the tensile membrane region towards the slab edges and panel edge compression decreased with consequent loss of beam tension. fened.
Centre spans of interior beams accordingly stifTowards the outside edges of the centre panel a
wide region of slab at high (tlcirumferential tl )
compression
developed to support the tensile membrane area at the centre.
~!,
As the panel was pushed on into a more complete tensile membrane stage this region became narrower until crushing occurred... The centre-edge panels at this stage were in a fairly advanced stage of forming a composite panel and beam mechanism as can be seen in Figures 9.10, 9.11, 9.12.
(The
predominance of the beam mechanism evident in this illustration developed as a result of later loading.)
Corner
panelf3 had cracked across both diagonals and were forming
a panel mechanism, but again, later loading brought about the predominance of the beam failure mechanism. Beams showed little sign of distress at this point. Centre spans of interior beams became increasingly stiff with the reduction in tension c.arried and the cracks in their end spans, whic:h were later to develop into wide cracks at pl.astic hinges, were at;]'ll narrow (see Figure
902.0:,
Test
t;c;.
E'a:Llure of Outer 1)8.ne18
-~------~-
In this test the centre panel load was kept constant at 600 psi" while the outer panel load. was increased. until the centre-edge panels "failed" at 966 psf.
Corner panel
load was then increased with centre panel load still 600 psf and centre-edge panel load 950 psr.
Failure of the
corner panels occurred at 1170 psf. Only a general description of the floor behaviour and failure modes is given.
Many steel gauges had gone off the
Exterior beam
Interior beam EW2
FIGURE 9
Al'1S AT E
OF TEST
FIGURE 9. 11
LOADED SURFACE AT END OF TEST
FIGURE 9. 12
UNLOADED SURFACE AT END OF TEST
194
data logger scale and cracks had rendered many concrete gauges useless. Figures 9.10, 9.11, and 9.12 show the final state of the floor. The formation of plastic hinges in the positive moment regions of the end spans of the interior beams brought about a pronounced folding mechanism in the
centre~edge
panels and affected the Icorner panels similarly at a later stage. load
The centre edge panels showed no sudden drop in capacity but at 966 psf on all outer panels, the rate
of deflection under constant load was so great that the load on these panels was not increased further.
Full depth
cracking at the middle of the centre-edge panels had only just developed at this stage, the principal cause of the loss of load capacity being the full development of the combined beam and slab mechanism, evident when concrete crushing occurred at the interior supports and on the top surface above the wide cracks in the end spans. The load on the corner panels was taken up until these panels could sustain no further load.
Again, beam mechanisms
were respnnsible for this inability to sustain further load.
Concrete crushing at the supports of the interior
beams continued and large rotation of plastic hinges in the end spans caused considerable twisting of the exterior beams and the development of the combined torsional and flexural hinges near the corners resulted (see Figures 9.10, 9.11
195 and 9.12). The test was stopped at 1170 psf when the deformation rate was excessive. An interesting feature of this test was the lack of development of yield moments along the slab edges supported by the exterior beams. The following steel strains indifate the degree to which yield moments were developed along these edges. Readings were taken at LS239. Position Corner panel: Interior edge
Exterior edge
Centre e~panels: Interior edge (short)
Exterior edge
Gauge No.
Microstrain
83 91 84 90 85
'1870 44-40 2130 2000 2300
86 88 87 89
240 2200 220 970
98 99 100 102 96 97
2970 3100 2040 8000+ 1860 2220
At the end of the test the principal crack in each centre-edge panel was that along an arc between cracks in
196 the end spans of the interior beams supporting the panel (see Figure 9.12).
This crack was full depth for the middle
24" but T-beam flange effects caused closure at the top of this crack in the region of the beams (see Figure 9.11). Cracks along the exterior edges of the corner panels were measured at .002" at the end of the test programme. Development of flexural hinges in the end spans of interior beams allowed large torsional rotation of the exterior beams and torsional resistance was provided only by the end spans of the exterior beams.
Each such span
showed the effect of this with the development of a torsional and flexural hinge, to a greater degree in some beams than others.
Figure 9.10
shows the most fully
developed of these at the end of the test programme.
9.3
EXAMINATION OF ASPECTS OF FLOOR BEHAVIOUR DURING TESTING
9.3.1 Figures
Deflections
9.13 and 9.14 show load-deflection plots for
the full range of load applied over the whole top surface. Each curve is typical of its group and the values plotted include residual deflections.
The difference between NS
and EW exterior beams is apparent when the curves for the east and south exterior beams are compared. The curves for the centre, corner, and centre-edge panels all show the effects of cracking at 550 psf but at
197 450 psf, only the centre-edge and centre panel deflections show a marked increase. Loss of stiffness of all panels after 550 psf is clearly seen and the similarity of shape of the corner and centre-edge curves after this stage show the effects of supporting beam deflection.
The centre panel
load~
deflection curve indicates the push-through failure that occurred at a deflection of nearly 2 11
,
Because 850 psf
was not fully attained the path of the load-deflection curve for the centre panel was not accurately determined. load~deflection
The
plot for the tensile membrane stage is
close to a straight line through the origin. The curious shape of the curves for the centre spans of interior and exterior beams is due to the effect of tension in these spans.
The loss of tension in the beams
towards the end of the test is evident in the steepening of these curves, especially in the interior beams in which deflection decreased with increase in load near the end of the test.
However, this was due in part to the formation
of plastic hinges in the end spans of these beams. The varying scale used in Figures comparison of stiffnesses difficult.
9.13 and 9.14 makes Figures 9.15 and 9.16
show load-deflection plots for Tests 1 06 and 110 in which deflection at the start of Test 110 has been set equal to that at the end of Test 106 and the constant horizontal scale makes direct comparison of relative stiffnesses
--.
9
f---
- - f--
8
.-
f----
/'
7
/'
6
~
3
-- r - - -
-l--
-
,/ {(
!
2
/
o o
VI
1-/!f
/
~
-
~~
.----
I
8
V
I
f
7
/
/'
6
DEFLECTION
-
--
t
21
24
27
30
12
24
36
48
00 .01 in. units
DEFLECTION
(al
. 9
9
8
8
7
7
e -
6
5
5
~4
8. 4
/ ' -----
~
),
S
I J
...J2
U
V
f I I
II
~3
--.----~
J
f-
-
96
'108
120
( b) /
i--
--
/
L--
j/ III I J
PANE
-l---j----
84
72
F
- - t-----
-.--~
t------ I---
o o DEFLECTION
17
.01 in. units (c)
--
j' fj
PANE
_.
, o
15 .01 in. units
V ~ -
I
r
I
12
l.-----
-
1
...
-
I f
BEAM NS4 CentrE span
9
l..--- hr
If
J " 6
--
i
rh
f
3
1
9 I---
I FIGURE
9.131
34 51 DEFLECTION
68 85 102 .01 in. units (d)
119
136
153
170
900
+-
900
800
--
f
7
-1--I
,
o
(§
~Jt--.HII-+--+---+----'f----";c;F-.~"",,,,---1--+
---1--+-"--
~300~~~--~--~--+----~~~~-~---1~-~---+---
O~
o
7
14 21 DEFLECTION -
28 35 .01 in units
49
42
63
56
__
~
__
3
70
~
____
~
__- L_ _
8
7
--r( /
/
1~
r-~
I
21
9 8
...... ......
"
" "
7
V
----V
1------
-
18
24
27
30
(b)
I
V
_ _ _ _~_ _- L_ _ _ _~_~_ _
6 9 12 15 DEFLECTION - .01 in. units
(al
9
~
6
(
V
~
i (r--
~
?
I----
-
j/
1/
V
I
l'
,I 2
CE TRE
I
I
~NEL
IIbrth 'SiIlIilI1
I
I
U-
BEAM 1'>151 I I
2 I
o o
o 12
30
24
42
48
54
60
o
5
DEFLECTION - .1 in units (c
l
I FIGURE
9.141
10 15 20 25 DEFLECTION - .01 in. units
(d)
30
35
40
45
50
200 possible. Table 9.6 gives the deflection readings for all gauge points.
In some cases, pattern loading caused upward
deflections (negative values) and the sharp increases in deflection at LS158 and LS200 are noticeable, especially in panels and centre spans of beams. It is of interest to compare the deflections in the table with ACI code requirements for deflections.
ACI-318-
63 Clause 209 specifie s '1/360 of the span as the maximum allowable for floors carrying plaster ceilings, 1/180 of the clear span otherwise.
The most stringent requirement
for long term loading is the allowance of an additional deflection of twice the short-term deflection. The first values which exceed the L/360 requirement are marked. with an asteri sk in Table 9.6.
When the first
values of deflections at design service load are factored by 3.0 to allow for long term deflections, the centre panel just exceeds the L/360 requirement. However~
the dependence of the centre panel on compres-
sive membrane action makes the assessment of long term deflections a special case in which the deflection is unusually sensitive to outward creep of surrounding elements which provide the necessary lateral restraint.
At
service load, however, the magnitude of the membrane forces may not be high and it is reasonable to conclude that the deflections of the floor at service load would not be
T,1,BLE 9.6 LOAD STAGE
1·:AX LeAD
DEFLSCTIONS AT SELECTED LOA;) STf..GES FOR ALL Gt.U:;E [rIFTS (.or01 n:CH lI!HTS)
32
57
225
225
375
3
PAT'rERN
14,0
104
7A
225
155
177
375
3'75
.?13
375
400
5
2
25
107
25
15
64
46
99
97
20
o
68
43
110
375
375 3
500
217
600
?20
22"
775
825
5
GAUGE NO.
-16
23
39 48
-19
3 4
37 41
-8 -6
33
5
42
-12
6
41
7
42
-9 -18
8
34
-17
29
21
10
27
64 57
1c
107
21
9
59
76
15
116
26
8
71
30
69
20
110
25
10
51
20
24
55
21
101
21
6
53
71
56
30
79
30
102
36
25
77
98
29
59
2
94
-6
-21
39
49
65 22
77 90
512
268
52
69
90
13 4
57
P6
109
345
80
98
212
72
91
1'+1
90
110
46
70
195 129
212
9('0
1435'
1000
1560'
5'7
877
1346*
2136
410
1160
1900·
436
591
881
425
950
1630*
391
559
1029
40
136
61
101
112
111
124
175
123
155
213
719
1591"
2397
;;3
144
47
104
119
115
143
190
130
176
231
475
11P:?
23(';2*
2943
4,:c7
6447
11
17
21
,9
112
37
87
103
97
106
147
103
147
191
492
1(,42'
12
20
21
40
121
55
111
108
116
161
150
221
256
478
801
1811"
2731
13
22
26
41
111
43
93 90
99
93
71
63
54
131
163
391
821
1701"
2661 2645
14
24
24
45
120
32
84
944
1795'
122
37
87
155 154
495
43
115 122
198
25
93 95
152
25
95 101
106
15
106
151
194
379
757
1462'
2232
16
11
21
32
111
31
86
87
80
103
141
93
141
191
318
691
1571'
2461
17 18
32 40
40
75
262
230
220
250
240
310
550
470
510
650
1342
1452
85
285
266
273
235
257
315
409
465
445
1135
1186
1245
19
25
30
60
261
227
210
240
231
290
540
445
490
555 612
963 861
1234
45
902
1090
1147
1220
20
40
35
80
253
210
205
225
225
274
435
365
405
555
901
1235
1425
1555
21
-35 -30
70
40
110
15
-37
141
162
121
320
240
300
371
580
562
425
73 65
50
140
-18
165
34c
260
)45
1228
1540
-22
223
184
367
293
176
435 464
1()~7
110
170 164
200
40 6-,.
30 34
685
590
440
154-
-:09
-1
189
~--
T79
--399
309
384
479
969
1139
1204
22
24
-33 -21
23
81,
25
100
-5
85
205
40
250
115
85
215
293
400
886
1975'
4500
6820
100
-5
210
48
268
120
89
210
193
296
400
1230
2405'
4342
8130
27 28
100
5
90 82
305 291
197
26
42
75
190
282
195
282
383
1242'
2770
5230
7650
L/360 :::: 1240 uni ts for short spans
-1
79
31
239 229
101
92
185 184
94
69
189
279
179
269
372
785
2074'
4279
6579
L/360 :::: 1730 uni ts for long "'~pans.
29
20
122
122
365
.105
90
400
417
430
732
920
2050'
3855
6490
9230
-10
133
122
335
121
96
402
415
330
672
535 440
700
30
648
840
1924'
3390
6330
8882
31
5
145
138
345
97
75
367
390
385
727
546
705
924
1802'
3385
5428
7537
32
75
135
135
403
114
95
378
385
391
725
543
694
907
1227'
3054
5655
7775
33
211
-6
214
616
524
976
216C*
2137
?139
2746
921
..
First value in excess of L/360
FIGURE 9.15
DEFLECTIONS-T~~TS 106J11~ I
~~--+----t--+---t---t----r-----t---r----t BEAM
I
EW2 WEST SPAN
l~'~~~~--~-----+----~r----r---i-r~~iI~--t-----t I
"tI
~
~~~--~~~-----+~Lh~~~-r-r~-r--Jti-~B~~M-t-----t EW3
CENTRE SPAN
FIGURE 9·16 DEFLECTIONS - TESTS 106 • 110
I
CENTRE
a300~--~~~-4---.H+T----r----~--~4-r---T-----T----I
./ oL---~-----J----~----~----~----~----~----~----
I~
203 excessive in the long term. 9.3.2
strains
9.3.2.1
General
Readings in microstrain at selected gauge positions are presented in Table 9.7.
Load~strain
curves for the
latter part of the test programme appear in Figures 9.17 to 9.21.
Appendix C contains readings for all channels
at all load stages. 9.3.2.2
strain Levels
Table 9.7 shows the generally low level of strain at 375 psf applied at LS85, no underside panel cracking having taken place· at this stage, but the shr;inkage eraeks present before testing
commeneed~
show up in the higher strain
values at the panel edge seetions.
Craeking of the under-
side of the eentre panel at L.8100 (as the outer panel load ~as
reduced) brought a sharp increase in strain in the
een'tre panel bottom steel (eogo gauge 6) with little effect on the strains at the edge of the panel. Strain values were still comparatively low at LS155, gauge 5 recording the highest level at less than half the yield strain.
The effect on the strain levels of the
application of 450 psf is seen most clearly in Figures 9.18 to 9.21 inelusiveo
Centre~edge
and eentre panel strains
showed a partieularly large rise with the occurrence of underside eraeking in these
panels~
yield being reaehed in
TABLE 9.7 STRAINS '_T SELECTED POINTS GAUGE No.
YIELD STRAIN
57 52 63 73
1340 1340 1340 1340
124 136
LSN
104
375/3
155 375/1
158 450/1
175 350/5
189 375/1
193 475/1
200 550/1
213 600/1
218 725/1
220 775/1
226 825/1
-27 97 279
-19 117 97 265
33 107 132 331
47 548 161 596
7 478 96 497
38 529 136 549
77 641 193 680
83 996 985 794
134 1145 1636 982
1566 1362 2098 1271
2135 1478 4713 1270
2159 1685 6629 1297
87 87
110 104
135 175
168 216
373 439
330 379
377 452
443 543
496 618
804 846
1206 1206
1358 1462
1630 1916
9 260
-39 126
-64 165
-52 306
-54 390
-35 633
-83 542
-71 659
-27 813
-27 890
20 1133
599 1786
840 1922
1046 2000
18
_12
-17 242
383
2 870
-35 779
-19 919
13 1079
11 1196
21
233
2 216
_6
128
1513
35 2623
91 2723
2655
221 177
246 625
278 683
237 494
361 671
1272 1650
1063 1540
1223 1638
1528 1909
1764 2264
1921
1945
1976
1600
32 225/2
57 225/3
7A 225/1
85 375/1
375/5
123A 400/2
49 11 73 125
-9 95 65 101
23 71 83 160
45 112 125 323
-5 18 26 299
1430 1430
44 43
49 54
63 65
111 135
96 98
1690 1690
_6 55
9 88
-3 97
88 91
1690 1690
13 89
6 83
7 118
111 6
1690 1690
113 68
85 29
139 56
LOAD
142B
84
128
All values corrected for temperature and drift. Refer to figures 7.2 and 7.3 for gauge positions. I\.)
o
.t:.
205
__ - 7
600
... ----
r-
J
J
600
I
)
/
/
-
I
~ II
C\
..::I
200
{ ,
, ---...".-
...--,
//)
I
,
I
/
/
IFIGURE
9.17 LOAD V. STRAIN-GAUGES 128.130 ]
{'
o
2000
1000 MICROSTRAIN
FIGURE 9.18 LOAD v. STRAIN GAUGES 51,53,70,71
0
0
500
1500
1000 MICROSTRAIN
--600 /' I
/
600
124)-'/
II! '!-
I
·. ./'26
~
~400
s::
FIGURE 9.19 LOAD v. STRAIN - GAUGES 124,126,133.136
200
MICRQSTRAIN
0 0
~
---128 --130
I
o
~
V
L ,-
I;~"
0400
-
-
.. ..... - ... -'
500
1000
1500
2000
206
\
800 -
V
~
600
~
~---
~
" v~--~~ .-
~
't;; 400
.....-:.
0-
/
I
r
,.- - --
1121
/
,/
/
o
>-
(
.3
/
a
t-~'116
/
"0
/
I
I
I
200
I
I
FIGURE 9.20 STEEL STRAIN AT EDGE OF CENTRE PANEL
If o
I 1000
I
I
I
2000 Microst.rain
:] w >-
400~~--+-~------~--------+-~
200 H--IJr-----_+_
FI GURE Q.21 STEEL STRAI NAT EDGE OF
CENTRE-EDGE PANEL
o
1000
2000 Microstrain
207 the bottom steel at the middle of the centre panel. Levels of strain continued to rise with load but not always at the expected rate, a feature particularly noticeable at the edges of the centre panel. at the edges of the panels (see Figure
The slower increase
9.20)
coupled with
the steady increase in beam steel strain indicate quite clearly the effect of membrane action. A slow rate of steel strain development in the edges of the centre-edge and corner panels was also evident
0
In
the centre-edge panel, steel strain development along the short edges lagged that along the interior long edge but yield was reac.hed before floor.
5
psf was applied to the whole
At this stage, however 9 gauge 96, on the exterior
long edge showed only
990fLS
and it was only during the
test to failure of this panel that the steel along this edge yielded. Along the outer edges strains were small through to failure
the corner panels steel and even in the later test
these panels these strains never became
large. The reason for this low value of strain
at the edges
was clearly a result of the smaller edge restraint afforded by the edge
beams~
retarding the development of moment.
Although the edge "be,ams were sufficiently strong to carry the torsion induced
the full yield moment, the rotation
required to achieve this was too great and compres
208 membrane action developed in the panels to compensate for the slow development of the full yield value.
In the
corner panels the beam mechanism formed completely before the panel mechanism was fully developed and so the yield value was never reached at the edge. Figure 9.19 shows the strains at the supports of the centre spans of both interior and exterior beams while Figure 9.18 shows the steel strains at mid-span of both beams.
The similarity between exterior and interior beams
is good,and after cracking,the separate curves of load versus strain have nearly identical shape for both mid-span and support. Beam steel strains at vicinity of yield.
775
psf are seen to be in the
This has two important implications.
The closeness of both mid-span and support values to yield indicates that little moment redistribution was required and secondly, the fact that yield of this steel was accomplished is indicative of large beam tensions of the order expected. Steel strains in the interior beams show a tendency to reduce with increasing load beyond about
750
psf, a clear
indication that the maximum tension had been r.eached and was reducing.
This effect was more marked at mid-span than at
the support where steel was required to take the moment due to the load on the end spans. Generally the concrete strains were of little value as
209 a means of determining the maximum strains in the concrete. This was particularly noticeable for the panel edge sections where the region of high concrete strain was confined to about one eighth of an inch width at the beam-slab junction. The gauge was not therefore in the correct position to measure the maximum strain.
Even if the gauges had been
correctly positioned the small area over which high strains took place would have led to reduced readings since the gauge reading would be an average over the 1 inch gauge length.
These factors did not render the readings useless
for the purposes of computing membrane forces as is discussed in Section 7.3.4. Beam concrete strains were less sensitive to this effect and Figure 9.17 shows the strains as measured by gauges 130 and 128 followed through increasing load.
The
curve for gauge 130 has a continuous form and the two curves are almost identical up to 550 psf, the small increase at 450 psf showing up in both curves.
At 550 psf when the
corner panel cracked on the underside for the first time and the centre-edge panels cracked further, the curves part, the strain in gauge 128 dropping significantly in magnitude. This drop, with the slower increase that followed it give a clear indication of the presence of tension in the beams. 9.3.3 9.3.3.1
Cracking General
Development of crack patterns in the elements of the
210 floor during testing has been described fully in the preceding sections on test by test behaviour of the floor (see Sections
9.2.4.3, 9.2.5.3, 9.2.6).
This section is devoted
to the examination of the serviceability of the floor at design service load with respect to crack widths.
9.3.3.2
Crack Width Serviceability
Only at Load Stage
161 were widths measured in detail.
These are shown in Figure
9.3 (p. 185).
Although the
measurements were taken at an applied load of
375 psf (DL +
LL), the maximum load sustained up to that stage was psf.
450
This overload had little effect on the beams and
centre-edge panels, but the cracking in the centre panel that took place at
450 psf was considerable and crack
widths were substantially larger than the first DL + LL values. The values shown in Figure
9.3 are maximum values and
as such may not be compared directly with the quoted limits of ACI
318~63
Clause
1508 which gives .015" as the maximum
mean crack width for interior members,
0010 11 for exterior
members. In this comparison, the effect
~
scale and of the
relation between maximum mean crack widths and maximum crack widths was accounted for by adjusting the code ues.
Average crack widths are generally taken as
val~
two~thirds
of the maximum values and the ACI code values were therefore increased by
50 per cent for comparison with maximum
r 211 prototype crack widths. Allowance for scale was made in two ways:
(i) on the
assumption that crack width varies as the square root of the scale factor and (ii) on the assumption that crack width varies directly with the scale factor. The maximum allowable model crack widths resulting from the above adjustments are summarised in Table 9.8. Table 9.8.
EX:Qosure Condition
Maximum Allowable Crack Widths For the Model Floor. Maximum Allowable Crack Widths ~inches) 1 2 4 3A 3
1
Scale
4A
Max. Observed Crack Width }!'igure 9.:2
Interior
.015 .022 .005
.013 .011
.028
.015
Exterior
.010 .015 .004
.010 .007 .018
.015
The numbers 1=4 in the table refer to the difference adjustments, as follows: 1.
Normal ACI 318-63 Code values for maximum allowable mean crack width.
2.
Values in 1. increased by 50 per cent for comparison with absolute maximum crack widths on prototype.
3.
Values of 2. adjusted for scale variation directly proportional to the scale factor.
4.
r-
Values of 2. adjusted for scale variation proportional to the square root of the scale.
"
212 Columns 3A and 4A require further explanation. crack widths in Figure
9.3
The
were measured at the design
service load but only after the overload to 450 psf had produced a marked effect on the centre panel cracking. Values of columns
3 and 4 were factored by the ratio of
steel strain in the bottom steel of the centre panel for design service overload.
load~
before and after the application of
This was found to be approximately 1:2.5.
Since crack width is proportional to steel stress, the relationship between the values of columns 3A or 4A and those of Figure
9.3
may be assumed to be the same as the
relationship between centre panel crack widths before overload~
and the actual maximum allowable crack widths for the
model. Comparison of values of Table
9.3
9.8
with those of Figure
reveals that for an assumed variation proportional to
the square root of the scale factor, all beams and outer panels satisfy the more stringent limit of .007",
Centre
panel crack widths do not satisfy either exposure condition but when adjustment is made according to the ratio of steel stresses, all centre panel crack widths are seen to be less than the more stringent exterior exposure condition for maximum allowable crack widths. It was concluded that if crack widths were assumed to vary as the square root of the scale factor, the
service~
ability of the model floor with respect to crack widths was
213 more than adequate for loads not in excess of the design service load.
Two qualifications must accompany this con-
clusion, viz: (i)
Exposure to exterior conditions would result in
marginal serviceability if crack widths varied directly as the scale factor. (ii) The dependence of the centre panel on compressive membrane action to improve its load carrying capacity and serviceability. 9.3.4 9.3.4.1
Reactions General
Measured reactions came close to expected values throughout most of the test programme.
The prime use for
these was in the calculation of moments across full width sections of the floor (see next section).
In this section,
the variation of reaction with load is discussed briefly and a comparison with expected values is made. 9.3.402
Variation of Reaction with Applied Load
Figure 9.22 shows plots of reaction value against load for support points C3,
A2~
and D1 respectively.
The scale
for reaction value was chosen to make direct comparison of the three figures possible.
The reaction ratio plotted is
the value of the reaction measured,divided by the expected value of reaction at the point for a load of 775 psf applied over the effective loaded area of the floor surface.
The
true origin for the graph is at -33 psf (= dead weight/
1.0
.8
tv Support C3 o
.6
~
(b) Support D1
- - EXPECTED
- - - - - LS1l4-B2 _·_._.-LS 133-151 - _ . - - LS 166-227
~
j
~ .6
(c) Support A2
i ~
o~Ar-----~----~-7~~----+-----+
~l
FIGURE 9-22 REACTION vs LOAD
r o
100
200
400 LOAD-PSF
600
215 effective loaded area) and the straight line joining the origin with the point (775~ 1.0) provides a useful reference. For equal load on all
panels~
the corner support
reaction, D1, was uniformly higher than expected up to
550 psf, thus when the initial discrepancy was allowed for, the variation with load almost exactly corresponded to that of the expected reference line.
After 550 psf the rate of
increase of this reaction became greater at the expense of the other reactions.
The degree to which this affected the
other reaction was exaggerated by the scaling effect used to plot the reaction variation. A detailed study of the variation of the reaci:;:ions provided no reliable information as to the distribution of loading along the beams.
9.3.403
Method of Calculation of Line Moments from
Reactions and Applied Loads The
moments~
M1
@.
0oM10 , about lines 1 .... 10 of Figure
'7.5 (p. 130) were calculated using the measured values of reactions and the known values of the loading pressures on the panels.
Nominal values of applied load could not be
used because the sum of all 16 reactions was always less than the sum of the products (nominal applied load x corresponding total available
area)~
indicating that the
bags did not exert pressure over the full area 9 rather 92~96
per cent of it.
It was most likely that the unloaded
area was in the region of the beams (at the edges of the
216
bags) but in calculating the line moments, load was assumed to act over the full area with reduced intensity.
Calcul-
ation of the line moments was therefore both accurate and straightforward. Moments General Two sources were available for the determination of moments in the slab.
Strain readings were used (see Section
7.3.4.3) to compute the moment and normal force at sections of the slab and beams.
The other source was the reaction
values and applied load, which provided a means of determining the moment along a full width seetion of the floor. Moment computation from the strain readings gave the values at particular seetions whereas those computed from the reactions afforded only the total moment along the section line.
The results of the latter method were in-
herently more accurate than those of the former.
Comparisons
between the two methods were made by summing the section moments across the floor and plotting the load-moment curves for both methods.
9.3.5.2
Basis of Calculation of Line Moments from
the Sum of Section Moments Consider the element of floor cut out by lines 2 and of Figure
7.5.
Eaeh line cuts through four beams and
exposes three lengths of panel edge.
The total moment
3
217
acting along each of these
as given by the strain
lines~
readings, was calculated by summing the individual beam and slab moments across the line. Let Figure 9.23 represent one of the beams and enough slab section on either side of it to make the sum of the slab compression equal to the tension in the beam, with the following notation: Mb
= moment in the beam at the support about the mid-depth, as calculated from the strain readings.
Tb
= tension in the beam at the support acting at mid-depth as calculated from the strairi readings.
0 sum '
=
sum of slab compression over the length of slab . O~um
considered such that m'sum
= Tb.
= sum of slab moments acting about the mid-depth of the slab, summed over the length of slab as defined by O§Uffi
=
Tb
o
Mb , Tb , 0sum' msum are similar values at mid~spano The moment of the exposed actions in the beam, Mtotal is therefore given by 0
•••
(9.1)
and so across the full line 2, the moment is the sum of the computed beam moments plus the sum of the computed slab moments across the full width minus the sum of the products
218
-b
w'1
~
on\
y-
F"
-r-
o If)
1_-
..-
0\
M'
-
I
,
Mid-span Line 3
Support Line 2
FIGURE 9.23 COMPUTED ACTIONS ON
A TYPICAL
SLAB AND BEAM ELEMENT
w per unit length
a
t
c
b
r
wa 2 - F-
a -a
I ...
a/2
-l" a/2. I
b/2 .. ~
b/2
[FIGURE 9.24 FREE MOMENTS
B
•v
219 of Tb(D-D )/2 for each beam, bearing in mind that D may s vary from beam to beam. At mid-span the total moment, Mtotal' of the actions exposed in the beam of 9.23 is ... 0(9.2) and a similar summation of these quantities, beam by beam yields the total moment along a line such as line 3. For checking purposes, a most useful quantity is the "free" moment, which for any symmetrically loaded span is the average of the two support moments plus the moment at mid-span.
This moment should equal the moment induced at
mid-span by the same load on a simply supported span of the same length.
The case of a uniformly distributed load is
illustrated in Figure 9.24. Referring to this figure and denoting the total moment at the opposite support (not shown) as
Mtotal~
Free moment ~ Mf = t(Mtotal + Mil total) + Mtotal == .1- (M' 2
9.3.5.3
11 ) b + Mil) b +.1-2 (M'sum + Msum
Comparison of Line Moments
Each line cut through four beams,not all of which
220 were strain-gauged sufficiently to determine moments at the sections cut.
The sum of moments in the beams cut by any
line was computed by assuming complete symmetry of floor, behaviour, eogo,when only two beams Cone exterior and one interior) were suitably gauged, the sum of the two known moments was doubled.
Unknown tension couples were similarly
treated.
m~um
The slab moments, msum ' were not known at all and for only one edge of the centre panel was suitably gauged.
Thus assessment of the contribution of slab moment was not at all accurate and in most cases the difference between the full moment as calculated from the reactions and the sum of beam moments only was examined to ensure that it was of reasonable magnitude. Figures 9.25 to 9.28 show moment-load curves for lines 2, 3, and 4 calculated from reactions and applied loads and from strain readings.
All curves are for increasing load
from LS168 upward to LS227. described Figure
~nd
Each figure is more fully
discussed below.
9.25~
All curves in this figure are calculated from the reactions and applied load by the method described above C9.3.4).
Line 1, line 2 and line 3 moments are shown
individually, together with the free moment in the end span Ci line 2 + line 1) and that in the centre span CiCline 2 + line 4) + line 3).
The latter may be seen to compare
221 2 favourably with the w1 /8 values. Line 3 moment increased linearly from the outset but curled over to reach a maximum value at approximately 800 psf.
Line 2 accordingly showed the reverse tendency,
increasing more sharply after 650 psf. The larger value of mid-span moment initially suggested a relatively large EI value in this region, probably due to the contribution of flanges in the T- and L-beams, and the subsequent reduction in the rate of increase of moment was probably due to the decreasing role played by the flanges, and to the increased cracking at mid-span. Figure
9.26~
Comparison of line 2 moments is made in this figure, the curve for moments calculated from the reactions and applied loads being the basis for comparison (curve 3). Along line 2, only two beams were gauged to give values of beam tension and moment (gauges exterior beam;
126~129
133~134
in the interior beam).
in the
Curve 1 is
the sum of moments only in these two beams, calculated from the strain readings and doubled to allow for the other two beams.
For curve 2, curve 1 values were reduced by the
total value of the tension couples as given by Equation
9.1.
The difference between curve 3 and curve 2 represents
the sum of slab moments along line 2. No slab edge moments were measured along line 2 but it is reasonable to assume that the centre panel edge moments will be approximately equal to those along the centre panel
222 edge at right angles to line 2. 114,115;
116,117,
Gauges 118,1'19,120;
showed nearly equal values of slab edge
moment at LS227 and it is reasonable to take this value as acting along the whole length of the centre panel edge. Further, since no moment values were obtained for the edges of the centre-edge panel, the values of moment per unit length of edge obtained for the edge of the centre panel were taken as representative of the centre-edge panel values.
The total length of edge over which this moment
could act was thus 62.5 + 2(4405)
=
151.5 inches.
At load stage 227 the slab edge moment per inch given by gauges 118,119 etc., (650 Ib/in) required multiplication by 130 inches to make values of curve 2 + slab moments equal to curve 3.
This same factor was applied to the slab
moment at the other load stages,resulting in curve 4 which compares favourably with curve 2. The factor of 130 inches implies a high value of moment (500 Ib/in) along the short edge of the centre~edge panels.
Some estimate of the feasibility of this value may
be gained by comparison of the expected normal forces along these edges (340 Ib/in. in the centre panel; the centre-edge panels (see Figure 6.3)).
270 lb/in. in The centre panel
edge forces given at these sections from which the moments were taken are all of greater magnitude than expected at LS227.
If it is assumed that the expected and actual mem-
brane force values compare as well in the centre-edge
3
FIGURE 9.25 MOMENTS CALCULATED FROM LOADING ___ __ .~
400
• 5(UNE 2
2
:i
3
4 5 6 7
$
~_~~
~~~~;;;bI;-----1I-,r~-L--+----
LINE 4) ~ LINE ;3
I, .~~".,..,,- . .
.",..1.
.92W(62.5"~/8
:2
LINE 3 LINE 2
.92W(44.5,,)2/a .5L1NE 2 .. LINE 1 LINE 1
/
....- ....-
....- ....-
.-' 100
200
300
400
7
600
500
TOTA L LOAD - PSF
800
700
"W
250
IFIGURE
9·26 MOMENTS ALONG LINE
21
200 1- BEAM MOMENTS ONLY "
2- "
MINUS
~<. 5 T( D· 1:\»
3- AS CALCULATED FROM LOADING
4- CURVE 2 .. 130" x SLAB MOMENT
100
oL-____-L______L-____- L______
~
o
100
200
300
____
~
____
500 400 TOTAL LOAD - PSF
~~
600
____
~
____
700
~
____
800
~
221+ panels,an enhanced moment of 500 lb/in along the short edges is reasonable. Figure 9.27: This figure shows line moment values for the middle sections of the floor in both directions (lines 3 and 8). The lower two curves are
plo~s
of the sum of beam moments
only, in each case this sum comprising the two separate interior beam values and twice the one exterior beam value obtained.
Both these curves fell well short of the curves
calculated from the reactions (upper two curves). The two middle curves compare far more favourably with the top pair, since for these the necessary addition of tension couples has been made, according to Equation 9.2. The difference remaining represents msum and is well within the capacity of the slab portion. Figure 9.28: The lowest curve (1) in this figure is the graph of, the sum of beam moments only along line
3~
again being
twice the exterior beam value + the two separate interior beam values.
No tension couple adjustment was made for
this or for curve 2 which is a plot of line 2 moments. Curve 3 shows the sum of these. Equation
9.3 that Mtotal
=
If it is assumed in
Mil T - TI total' B - B
curve represents the total free
momen~
•
less the portion of
+ ~(m U + mil )) . sum sum When 130" times the centre panel edge moment per unit
moment taken by the slab sections (m
sum
FIGURE MOMENTS ALONG LINES 3 AND 8 A - AS CALCULATED FROM LOADING B - SEAM MOMENTS .~lr(D·D;J/:2) C _" • II ONLY --UNE :3 ----LINE
---- ...."
a
---- ----- --OL-____
o
~
______
~
____
~
______
~
______
~
100
____
~
______
~
____
~
______
700
FIGURE 9.26 MOM ENTS ALONG LINES 1,2 AND 3
EXPECTED FREE MOMENT
I-LINE .'3 (BEAM MOMENTS ONLY) :2 -LINE :2 ( " ' ,,) 3 - LINE :1/ • LINE 3 (BEAM MOMENTS ONLY) <4 - " (ADJUSTED FOR ,5(D-~T AND l3O"x SLAB EDGE MOMENTS)
.100
200
3QO
500
TOTAl LOAD - F'SI"
700
800
~
226 width is added to this curve, values are significantly in excess of the theoretical free moment (solid line).
How-
ever, when each beam moment was adjusted separately for the Tb (D-D s )/2 terms and the same slab edge moment term added, curve 4 resulted. 9.3.5.4
Variation of Section Moments with Load
Values of section moments as calculated from the strain readings are shown for several cri tic,al, sections in Table 9.9, for increasing load in pattern 1 after LS168 and for critical load stages before LS168.
Some of the former
values are plotted in Figures 9.29 to 9.31 inclusive. Figure 9.29 shows the beam moments plotted against load.
For the exterior beam centre spans, the support mom-
ent rose far more sharply than the mid-span values.
In the
interior beam centre spans, values at support and mid-span showed marked equality up to 550 psf after which a sharp rise in support moment occurred and the rate of increase in mid-span moment fell.
The values of design moment are
marked in the figure and may be seen to compare favourably at the interior beam centre span and the exterior beam support.
Values at the exterior beam mid-span were less
than the design moment, while those at the interior beam support were greater. In Figure 9.30, the moments at sections along the edge of the centre panel are plotted.
The curves for gauge posi-
tions 118,119,120 and 114,115 (near the middle of the edge) are
T!lBLE
9.9
BEAll STEEL GAUGE LSN
71
73
70
-
... J\.ssuminr; complete bond transfer.
HO!v:E~;TS AT SELECTED LCJ..D ST1\GES (CRt,eKED SECTIC'NS BF=1.0 FOB BEtlMS)
HnlENTS
72"
-
KIF-IN
... ·Concrete eauee not at mid-span. SUB tCll'lNTS lb. in/in.
127
126
132
136
116"
114"
112"
107"
105"
LOAD
32
225/2/75
9.1
12.9
11.6
12.1
-124
-116
-82
-49
-67
-82
10.3
11.0
10.0
-5.0
-3.8 -6.4
1.6
7.4
-3·5 -4.2
+1.3
225/3/75
-6.9 -6.9
-5.0
57
0.4
6.1
-82
-49
-25
-99
9.6
15.4
13.0
13.0
-9.2
-7.1
-4.9
-6.7
"-6
4.3
-129
-82
-40
-97 -100
-103
225/1
-79 -131
-114
-128
7A
-86
79
225/1
8.8
15.2
12.7
13.2
-9.2
-7.0
-5.0
-6.3
3.9
3.9
-141
-124
-83
-37
-103
-115
-130
84
350/1
15.4
10.8
6.3
-190
-179
-127
-51
-13.2
-11.4
7.2
7.0
-20'5
-193
-135
-58
-131 -142
-165
12.2
-7.9 -8.9
6.5
16.9
-15.5 -16.9
-10.1
375/1
14.3 16.3
-12.5
85
17.2 18.4
-178 -188
96A
225/1
12.3
8.7
13.1
12.1
-11.6
-7.8
-6.4
-7.6
3.2
375/1
16.8
12.4
17.8
17.4
-17.9
-13 .. 6
-9.4
-11.6
6.2
3.7 6.4
-71 -114
-21
98
-87 -128
104
375/5/75
13.8
-3.7
-4.5
-130
11.3
15.5 14.1
-5.4
400/2/75
14.9 12.2
-9.4
123A
9.9 8.2
-113 -169 -114
-11.9
-6.9
-4.7
-4.0
-107
-134
142B
375/3/150
-13.4. -11.2
-6.3
-10.'5
-151
-176
9.7
13.2 11.8
-6.4
225/1
7.3 6.2
11.3
153A
10.1 8.6
--4.0
-4.8
-135
155
375/1
13.9
10.4
18.9
17.5
-18.9
-11.7
158
450/1
23.0
18.6
23.8
168 171A
75
9.8
8.5
23.5 10.6
225
15.1
12.1
15.5
183A
225
11.0
185
275
13·7 15.4
14.9 16.6
15.2 14.2
-30.0 16.0 21.4
187
325
189H
375
17·1 18.9 22.7
20.2 22.6
12.0
10.3
-171
-128
-109
-1.0
-206 -194
-192 -172
-86
-43 -26
-5.3
-213
-184
-81
-23
-175
-132
-36
19
-7.2
3.9 -1.6
-62 -114
.4
-162
-123
-30
-12.5
2.3
-208
-166
-190
-9.5
-15.0
-141
-94
-167
-229
-302
8.4 16.2
10.3
7.9 10.6
3.4 1.0
-176 -241
-57
-23.9
3.7 6.4
-83 -112
-113
-8.6
19 18
-77 -128
-170
3.5 4.6
-293 -114
-71
-80
-7
-53
-48
-145
-47
-73
-82
-119 -188
-109
21.5
16.4
10.8
10.7
1.9 1.6
4.3
-187 -184
-143
-44
-72
-170
23.4
17.6
11.5
11.7
1.9
4.7
-212
-165
-63
-83
-79 -102
-129
15.7 17.4
-148
-195
25.3
12.4
12.8
2.3
5.3
-237
-192
-78
-90
-118
13.1 13.5
13.9 14.7
5.3 6.4
-270
-213 -217
-132 -146
20.6
32.9
25.2
14.1
16.0
6.9
-293
-234
-97 -97 -104
-93 -103
24.5
-113
-170
-217
26.8
22.8
35.2 38.5 40.3 37.3
27.4
15.3 16.4
2.6 3.3 3.8 4.0 4.6
-270
19.4
27.6 30.6
-170 -188 -196
-213
19.0
19.5 21.7 23.4
7.5 8.2
-310 -341
-256 -285
-124 -145
-142
-182
-242
-246
-200
-265
5·0 5.6
8.4 8.5
-370 -370
-305 -305
-159 -165
-286 -424
-213 -208
-277
-257 -283 -302 -326 -323
6.6 7.0
9.3
-3~4
-340
-160
-221
-322
9."
-405
-362
-195
-472 -494
-236
-335
-313 -297
18.4
191
425
24.3
13·7 15.2 17.3 18.6
193
475
26.6
20.5 23.6
30.1
24.9 27·2
31.9 35.7
25.9 27.4 30.1
2~.0
30.5 40.3
32.8 36.0
-239 -232
3~.1
32.0 39.2 41.4
20.0
17.3 19.4 20.0 23.3 24.4
40.6
44.0
20.5
25.9
38.4
45.2
50.6
16.8
28.5
6.9
-369
-205
-519
-270
-385
-241
51.6
60.5
20.8 26.9
-494
-586 -726
-375
-443 -428
-374
63.2
47.6
75.0 91.9
-263 -281
43.5
59.9 74.8
-447 -492
-358
46.5
7.4 7.8
-451
57.0
29.7 30.0
9.5 10.9 12.9
-388
44.5
-515
-494
-352
-869
-356
-434
-405
47.5 47.2
So.3
96.1
37.5 38.4
13.0
66.6
29.5 31.4
9.4
45 •.7 47.5
11.5
-512
-506
-418
-923
-420
33.3
40.5
11.6
-537
-554
-511
-1024
-370 -401
-476
99.2
13 •.2 12.8
-514
-445
50.3 40.4
~1.3
46.9
33.7 34.0
42.2
12.0
13.0
-560
-1175
-418
-459
43.8
12.2
13.1
-650
-555 -659
-575
39.1
98.6 108.8
102.5
67.0
-1953
-1866
-542
-541
-474 -509
199
525
200
550
20b
575
29.4 38.2 32.7 35.6
213
600
3< .1
31.5
214
625
4c.3
35.4
42.0
216
675
44.3
38.9
49.5
218
725
47.6
42.1
220
775
48.7
224
775
51.0
225
800
52.4
226
825
54.8
227
850
50.8
68.2
30.2
106.5
17.2 18.5
-302
-316
J
,
/
/
/
100
i
I
i
a- GAUGES 1IS,l1Q,120
( I
b-
)
..
..
c-..
I I
d-
w
..
116,117 112,113 114,115
J
, II
~
Q. 52
.
a-INT. BEAM b -" • .. c - EXT. BEAM d - ..
I-
Z iii
:::r: 0 :::r:
/
Y I
I
-
GAUGES 105, 106
94,95 107,108
G ..
SUPPORT(12e,12Q) I MlD-SPAN(71.75) " SUPPORT(13a,138 / b MD-SAt.N (53,55)
2
O~----~------~----~------~---
o
200
400 600 APPLIED LOAD - PSI'"
800
FIGURE 9·29 MOMENTS AT BEAM SECTIONS
o~~--~------~----~------~-
o
200
400 600 BOO APPUED LOAD - PSI'"
FIGURE 9-30 MOMENTS AT EDGE OF CENTRE PANEL
o~----~------~----~--
o
200 400 600 Af'PU ED LOAD - PSI'"
____ __
FIGURE 9_31 rvlOMENTS AT EDGE OF CENTRE -EDGE PANEL
~
229
similar but towards the corners the initial rate of increase of moment was lower.
Membrane compression
became enormous near the corners towards the end of the test, a fact which explains the steep rise in moments in this region. Moment values at sections along the interior long edge of the centre-edge panel showed good grouping as Figure 9.31 shows.
The effect of the crack along the
centre of each of these panels was evident in the discontinuity in the curve for gauges 94,95 which were located on the line of this crack. 400 lb lt /
It
The moment values of
at 775 psf indicate considerable compressive
membrane action in this direction.
9.3.5.5
Discussion
Line moments calculated from the measured reactions and applied loads were expected to be reliable and accurate and were shown to be so.
Values of line moments computed
from the strain readings showed similar trends to the reaction-load line moments and the values compared well. The effect of beam tension and slab compression on the line moments was shown to be large and the conclusion that the strain moments compared well with the reaction-load moments was based on values of strain moments in beam sections only, corrected for the couple induced by the beam tension and slab compression. The assumption that concrete in a cracked section
230 carried no tension in regions of tensile strain once the extreme fibre strain exceeded the nominal cracking strain could have been the cause of the general tendency for values computed from the strains to rise slowly until relatively high loads were reached.
Cracks did not form
exactly at the gauge positions and bond transfer from steel to concrete between the crack and the gauge position would have reduced the tension in the steel, while at the same time inducing some tension in the concrete.
The loss of
steel tension was accounted for inherently in the strain reading but the "no tension" assumption ignored the corresponding tension in the concrete and therefore the section moment and tension would have been underestimated in the most likely case of this tension having its centre of action near the tension steel.
At higher loads,
further cracking would result in reduction of the force transferred by bond and -better values from the computation would result. Residual strains at LS168 have clearly led to high initial values, though not in all cases is this fully applicable.
Cracking of the floor in previous tests caused
redistribution of moments which remained effective when load was removed 1 causing significant alteration in the initial conditions of subsequent tests. The effect of variation of flange width on the computed moments at the
mid~span
sections was small
0
For a
section at the mid-span of an interior beam section, the maximum change in moment due to an increase in flange width from 1.0 to 3.0 times the web width was 14 per cent.
All
values plotted are for a flange width equal to the web width.
9.3.5.6
Conclusion
Line moments calculated from reaction values and applied loads showed good agreement with expected values. Lack of slab moment data made direct comparison of line moments difficult.
Nevertheless,! favourable results were
obtained from comparisons of line moments calculated from the two·· independent methods. Beam section moments followed the same trend as the line moments and compared well,overall,with the design values. Mid-span moments for both lines and sections tended to be large initially showing a slower rate of increase at higher load while support moments showed the reverse tendency. Slab moments were well in excess of Johansen values due to enhancement by compressive membrane action.
Vari-
ation of these moments with load was approximately linear for sections near the middle of the edges but values nearer the corners showed slow development initially followed by a sharp rise at higher load values. All moment values calculated from strain readings
232 contained inherent inaccuracies of sufficient size to make it difficult to draw firm conclusions from their variation with load, but good results were obtained in the comparisons of these values with the moments calculated from the reactions and applied loads.
The slow rise in moment in
some beam sections was attributable to the transfer of steel force to concrete, a factor not accounted for in the method of moment computation from the strain readings.
9.3.6
Membrane Action Effects
9.3.6.1
General
The effects of membrane action were the subject of particular study in this experiment.
Analysis of strain
readings, deflections, beam and slab sec,tion moments and axial forces provided quantitative data on the membrane action in the floor while effects observed both during and after the test provided qualitative information on the action of membrane forces. Values of forces computed for the slab and beam sections compared well with those expected but no useful information was obtained from these as to the effect of varying load forces.
patte~n
on the distribution of membrane
However, study of the centre panel deflection and
the strain in the bottom steel at the middle of the centre panel did reveal some differences. Effects of sustained loading were negligible, except for the centre panel where deflection and strains increased
233 by detectable amounts and redistribution of compressive forces took place. In Section 9.3.6.2 the effects of membrane action on each of the floor elements is examined in
detail~
Section
9.3.6.3 deals with the comparison of membrane forces, and some aspects of particular interest are discussed in Section 9.3.6.4.
Table 9.10 shows values of net
force on critical
sections at selected load stages. 9.3.6.2
Effect of Membrane Action on Floor Elements
Compressive membrane action caused enhancement of the load carrying capacities of all three panel types.
At what
was deemed to be failure of each panel type, the following values of the ratio of actual load to Johansen load were obtained~
Centre panel:
•
i· } ~~
-~
2018 cf 200 required in design.
Centre-edge panel:
1·55 cf '1 .35
II
Ii
Ii
Corner panel:
1 046 cf 1000
Ii
Ii
II
Although values for the
centre~edge
and corner panels
were affected by large beam deflections, it is clear that for these panels, the enhancement due to membrane action was significantly in excess of that required. Compressive forces in the panels gave rise to considerable tensions in the
beams~
thereby reducing their
flexural capacity. The fact that the panel types had unequal ultimate loads eased the burden of tension carried by the
beams~
T/,ELE
9.10
-
SECTIm: FORCES ;·.T SELECT:::n T,OAr. S'!'t.3ES
SLr\B STEEl. GAUGE _ _ _
71
73
70
72
2.7 2.6
32
225/2/75
1.6
4.0
57
225/3/75
1.1
2.6
127
126
132
136
54
53
2.6
-.2
1.1
-1.1
-.7
-.2
.8
-1.1
-.5 -1.6
-1.1
2.2
-1.9
-106
-11?
-68
-30
-"2
-62,
-Po
-62
-17
-34
-102
-90
-118
-122
-68
-106
-71
-33 -23
-93
-106
-102
-109
-125
-.f,
-181
-1'54
-115
-36
-128
-141
-181
-175 -112
-166
-110
-27
-131
-159
-121
-1.5
-.7 -.3 _1.4
-90
-3'7
-83
-103
-121
-1.3
-.9
-.9
-172
-163
-76
-4 -12
-118
-153
-181
-1.9
-3.2
-2.9
-169
11
-64
-106
-125
20
-137
-1.2 -1.8
.5 .1
-?3
2.6
84
350/1
2.7
1.5
3.5
1.5
-2.4
-.2
-2.4
-1.P
85
375/1
3.0
1.8
;,.6
2.0
-2.5
-.3
-2.0
-1.4
-.G
96A
225/1
2.3
1.6
3.0
-2.4
-.4
375/1
3.0
1.9
3.8
-2.8
-·5
-2.7 -0.6
-1.4
98
1.9 2.4
375/5/75
2.6
1.7
3.5
2.5
-1.0
-4.4
? .1 2.8
2.0
-3.7 -1,.9
-.:.'.2
1.h
-G.4
-3.0
-7·2 -8.1
2.6
1.5 2.2
-6.9
-2.9 -1.4
-8.8
5.0
3.8
2.9
3.0
-1.9 .6 .2
3.7 4.0
-1.3
3.7
-2.3 -2.3 -2.4
-2. ~
-0,.5
-4.7
-170
-132 -142
-3.5
-3.0
-141
-1no
-53 -38 10
-4.2
-4.9
-4.1
-7(;
25
58
-6.8
-2.h
-3.6
-3.0
-132 -124
-87
33
-2.4
-1.0
.8
1.4
-118
-51
-1.2
.8
1.0
1.7
+52
1.2
1.9
+9 -78
73 +125
.0
1.0
1.7
1.0
1.7
-25 -50
10
-.1
-.3 _.4 + .6
1.0
-66 _60 -79 -88
-19 -26
-8
117 117
123A
400/2/75
1.9
1.5
375/3/150
1.6
.9
153A
225/1
1.3
.u
155
375/1
2.2
1.2
158
450/1
4.0
168
75
4.4 2.4
3.0
171A
225
3.4
3.7
4.1
3.5
183A
225
3.0 3.2
3.9 4.1
-.4
3.3
-.7
187
275 325
3.3 3.4
3.2
185
3.5
3.7
4.4
3.5
-.9
189H
375
3.8 4.6
3.9 4.8
4.6 5·1
3.7 5.1
-1.0 -.1
5.0 5,0
-.7 -.7
94
-128
3.0
2.9
142B
105
-.6
4.8 4.8
2.9
1.0
-6.0
107
-1.0
1.2
225/1
o
112
-1.4
225/1
104
- } b/in.
114
-59 -122.
.1
79
7A
FC'R~ES
116
-40
-84
-11)1'
-128
-175
-71
-103
-140
84
-100
-144
-187
67
-120
-131
-232
119
75 71
3 -41
-31
-33 -99
122
74
-32
-34
-99
130
70
-42
-62
-115
115
79 70 69
-61
-69
-140
-73 -86
-94 -94
-158 -140
-95 -111
-156 -175
191
h25
4.8
5.0
5.0
-.6
1.5
-30
123
475
5.2
5.3
5.3 5.6
-.3
193
5.3 5.6
1.7 1.6 2.3 2.4
-.2
5.2
-.6
.4
1.7
2.6
-113
-36
122
69 40
-09 -105
199
525
5.5
.2
5.S
2.0
-66
-100
-120
-175
.4 5.4 6.6
7.8 7.8
,t,
-110
-159
105 87
-97
1.2
2.3 2.6
-135
G.o
6.2 6.9
-37 -(8 -(9
118
6.1 6.1
3.0 3. 3 3.3
-116
5.7
-.5 .3 .3
.5
550
8.9 6.1 6.7
6.1
200
-222
-11
-130 -A9
-1208 -152
0.9
o .u
1.~
3.2
-1~9
-113
'/4
-20
-U""j
-122
7.2
10.5
-195
-123
59
-290
-0
-98
-100
11.3
-171
28
-308
-22
-162
-21
_""Z.oP:
-1LJ1
-22?
-71
-569
-1(-;2
-234
-159
-195
208 213
575 600
6.9
7.2
7.8
7.9
214
625
8.0
8.9
216 218
675 725
220 225
775 775 800
226
825
227
850
224
1.1
1.6
7.3 7.4
'/.)
R.6
8.0
9.4
7.5 0.4
9.9
_?4h
10.0
7.9
9. 2 9.7
12.2
-290
- ~14
7.7 7.4
9.6
7.2 6.0
9.9
13.0
15.5
-303
-320
-149
-722
-161
-254
9.4
14.7
14.6
11.9
-320
-234
-812
-177
-293
-210
7.2
9.0
5.5
10.7
F
1?7
1C·7 ,).8
-41G
-255
-933
-213
-352
-228
6.7 4.9
8.5
4.4
-272
-?52
7.~
r
10.4
11.1
13.G
p.4
12.6
9.?
10.6
).8
1n.7
12.4
f. 'i
,
-
0.1
6.4 6.')
7.3 7.3
-334
-82
7.2
- 390
-422
-10')9
-::~34
7.1
_")('7
-2017
_1c n ')
-378
-,7,
-308
235 particularly the centre spans of the interior beams, because the degree of development of compressive forces in each panel type was different, i.e., by the time the centre-edge and corner panels were in greatest need of compressive membrane action enhancement, the centre panel had failed and formed a tensile net which tended to reduce the tension in the beam spans supporting it. The interaction of elements was important "in assessing the effects of membrane action in the floor, a fact revealed in the following element by element analysis of the effects of membrane action on the floor behaviour. (i)
Centre Panel
The development of membrane forces in this panel was slow at low load levels but increased very rapidly after the Johansen load was exceededo
As the collapse load was
approached, evidence of a significant reduction in membrane compression near the centre of the panel was detected. Edge forces continued to rise during this stage and strains which indicated the level of "circumferential" compression near the edges increased sharply. The variation of edge force with increasing load beyond LS168 is shown in Figure
9.32, from which it may be
seen that the delay in rise of compressive force is more marked near the corners of the panel. Values at
775 psf compare reasonably with the pre-
dicted average force of 340 Ib/in.
The rate of increase
236 in this edge force showed no sign of reduction while the load was increasing, but the level of compression did drop with the push-through failure of the centre panel. Figures 9.11 and 9.12 (pp.
192~3)
illustrate clearly
the transfer of compressive force from the middle region to the edge.
Zones of crushing on the diagonals reach
only half way along, decreasing almost linearly in width. An increase in crack width is then evident.
These photo-
graphs were taken after this panel was loaded well into the tensile membrane range but they only exaggerate an effect which was evident at initial failure.
'.
Strains parallel to the edge provided clear evidence of the increasing compression and subsequent decrease. Readings of gauges placed in this direction were examined and the readings of the two rosette stations were resolved into components parallel to and perpendicular to,the edge. The variation of some of these with load is shown in Figure 9033.
This figure shows the variation for channel 78 and
the components perpendicular to gauge 14 and gauge 25, the latter showing larger values by virt;ue of its closeness to the diagonal, where compression was higher. For comparison, values for gauges 82 and 76 are plotted.
Gauge 76 represents the strain in the top of the beam,
directly above the web, and values are very much less than those for gauge 78, indicating that the strain in gauge 78
237
500 I
FIGURE 9.32 FORCES AT EDGE OF CENTRE PANEL
,, I
400
300
--Gauges 118,119,120
.£
:a
---"
-. - "
I,
1#
116.117 114,115
c:
o
'(ij
200
tI)
QJ t-
o. E o u 100
I
I
)
I /
-..,.,- -' ( I
1 60,0
800
" , ./ /
-.
_,_,_o~
-
. /'---._.-.-"
)
I
.J' ........... \
\ \
,
1000
14*
_...l_.;rr-
8
--
.--___-:r.
.........
.... \I)
0.. "-'
1:1 0
.2 600 1:1 .~
Q.
0..
<{
FlGURE 9.33 CONCRETE STRAINS IN THE CENTRE PANEL) PARALLEL TO THE BEAMS Gauge nos. shown beside curves
*-
Component parallel to gauge
o~------~------~--------~------~--------~------~--------~------~--------~o 100 200 300 400 500 600 800 1000 1200 Compressive strain
~S)
239 was due principally to centre panel action and not to compression in the T-beam flange.
Gauge 82 shows a little
of this effect of "circumferential" compression. the strain in gauge
78
Whereas
became tensile as the centre panel
failed, gauge 82 strain remained compressive. Study of Figure 9.11 reveals that the spread of the tensile membrane is greater away from the diagonals, revealing an important difference between circular and square slabs in this respect. for strain in gauge
78
Comparison of the curves
and the strain perpendicular to
gauge 25 illustrates that this effect was present at the time of initial failure, and again this effect is merely exaggerated in the photograph. The effect of surround stiffness on the centre panel behaviour was particularly marked at three stages during the test programme as described earliero The effect of loss of surround stiffness on steel strain at the middle of the centre panel is shown in Figure
9.34.
Causes of the large increases are shown on
the figure. At 550 psf readings of strains were taken several times and the variation of strain with time during this unstable period was quite erratic;
i.e., there was no
suggestion that the cause of the increase in deflection was due to creep deformation of the surround. This graph shows clearly the effect of having a lower
FIGURE 9.34 LOAD v. STRAIN - GAUGE 6 (CENTRE OF CENTRE PANEL)
First cracKing in corner panels. Extensive cracking elsewhere.
600
Stable conditions reached after 75 minutes. Cracking of rectangular panels
40
45
5565
75
/
Time at which reading taken - minutes since load first attained
/
/
5
load on outer
/ / / /
j
400 I.L.
tri a.:
z
panel
300
Q
« 0
...J
200
100
o
3
6
9
12
15
18
21
24
27
STRAIN
30
33
36
39
42
100 microstrain units
45
48
51
54
57
60
63
66
69
72
2~
load on the outer panels.
In spite of reduced moment
restraint at the edge of the centre panel, the increased surround stiffness resulting from the lower load on the outer panels caused a slower rise in strain with load. The effect of the application of 375 psf for 66 hours is shown to be small and a recovery to the original path on application of further load is evident.
The increasing
stability with time during this test was illustrated previously (see Section 9.2.4.6). The membrane forces perpendicular to the diagonals appeared to vary approximately linearly.
Some assessment
of these forces was made by measuring the depth of crushing along the diagonals at the end of the test.
Results of
this investigation are given in Figure 9.35.
Measurements
were taken along each of the four diagonals.
The average
force per Unit width was determined and plotted against the distance from the centre.
Variation is almost linear
and very large values are reached near the edge of the panel. (ii)
Centre-edge Panels
These panels sustained loads well in excess of their Johansen loads due to enhancement by compressive membrane action but evidence of this was not as clear as in the centre panel. Compressive forces were expected parallel to the long side and although no provision for measurement of these
r 242
forces was made, the presence of tension in the centre spans of the exterior beams indicated that panel
compres~
sion in this direction was considerable.
9.36 shows
Figure
the values of measured compression along the interior long edge of one of these panels, plotted against load.
The
rate of rise was steadier than for similar sections in the centre panel but there was still the evidence of a steeper rise for loads greater than 600 psf. Values are considerably less than those in the centre panel but indicate substantial compressive membrane action in this direction, although much of this force was a reaction to the centre panel forces.
At the end of the
test programme the panel mechanism was considerably developed as Figures 9.11 and 9.12 show.
These Figures
show clearly the zones of concrete crushing in the end spans of the interior beams and a study of the panel crack joining two of these zones revealed the presence of mambrane action of a different nature.
Near the beams, this crack
was present on the underside of the panel only, while the concrete at the upper surface above this crack had crushed. At its middle, the crack penetrated the full slab depth so that the panel sections along the crack were subject to high compression near the beams and tension in the middle. (iii)
C
r Panels
The assumption that membrane action would not enhance the load capacity of these panels was clearly conservative.
243
FIGURE 9.35 AVERAGE COMPRESSION NORMAL TO THE DIAGONALS OF THE CENTRE PANEL AT THE END OF THE TEST
""'2000 c
~
Force :: .85f~
c
X
(average depth of crushing)
,2 lI)
~ 1000 E o u
Ob-------L---____
o
400
6
~~~
__
~
____
~~
12 18 Distance from centre
______
24
30
{in}
FIGURE 9.36 CENTRE-EDGE PANEL FORCES AT THE INTERIOR LONG EDGE
300 Gauges 107, 108
§
,.,
/I
"
"
105,106 94.95
200~------~------~--------+-~--~Hr-----
'iii lI)
tl
t-
o..
E o U
100~------~r-----~~~~~~~r----r-------
O~~~--~------~--------~------~-------
o
200
400
600 Applied load (psf)
800
~
____
~k_
36
244 The lower span to depth ratio partially compensated for the more flexible surround of these panels.
The T-beam
flange effect described for the centre-edge panels was also evident in these panels and the panel mechanism was not as fully developed even at the end of the test programme, suggesting that more membrane action enhancement may have been available if the beams had not failed. The low steel strains at the exterior edges of the panels indicate the degree of development of the full yield moment along these edges.
Extra membrane compres-
sion would account for this, the moment required being taken by the moment of concrete compression force about the mid-depth rather than by steel force moment.
High
concrete compression along these edges affects the torque induced in the exterior beams for the reasons discussed in Section 4.3. (iv)
Interior Beams
In the absence of any net-
axial tension, the centre
spans of these beams were capable of carrying a far greater load than 800 psf but in fact steel strain readings reached yield values at this load,indicating the presence of large tensions in the spans. Analysis of section strains showed larger values at the support than at mid-span as may be seen from Table 9.10 and Figure 9.37(a).
The rate of rise in beam tensions
followed a similar pattern to .that followed by the panel
245 edge compressions, tensions being small at low load levels and rising steeply in regions of higher load. During the test,the tension gave evidence of its presence with the formation of steeply inclined shear cracks (see Figure 9.10).
Values of tension measured
in the end spans were not reliable and the behaviour of these spans during the test showed no conclusive signs of the tension implied by the presence of compressive forces along the interior long edge of the centre-edge panels. After the centre panel pushed through to form a well developed tensile membrane, the effect of reduction of induced tension was clear.
Deflections and mid-span steel
strains reduced considerably, but this was not wholly due to the effects of reduced tension because at this
time~
the end spans were showing signs of development of plastic hinges in the positive moment regions and the resulting increase in support moment tended to produce the same effect as the reduced tensions.
As loading progressed this
effect became more marked and the tensions continued to reduce. (v)" Exterior Beams Visually detectable effects of membrane action on these beams were few and confined to the
cent~e
spans.
(
IA.
The forces measured at the sections showed a similar trend to the interior beams in their variation with load, as may be seen in Figure 9.37(b).
Values were appreciably less
/
14 l-
I
FIGURE 9.37(0) TENSION IN INTERIOR BEAMS
12 I- 1- Gauges 2 " If I' 3 ,. 4 "
"
i~
I
.I 1\\
/r \
126,129} Support 127,128 section 73,77 ] Midspa-n 71,75 section
i/
I/ i i / /.", /
31:
'
111
c
.Q
c:
~
6
~
11
I-
/J ,-
--'"". ..... ..... /r;../
If
;
.....
"
53,55JMidsp an 54.56 section
10
"-
"\
~2
,.-3 \
\\
'.
'[8
\
\
\
i \
;g
3
c 0 'iii c
,
~ 6
/
\ 4
-~.,...j-::I J'
1- Gauges 132,133,134} Support 136,137, 138 section
4
\
""/
4
12
2 3
4/\
if
1/1
'y--1
/ f2
10
].8 l-
FIGURE 9.37(b) TENSION IN EXTERIOR BEAMS
/
",.
",.'
.
".. ",.'
/-
2
2
0 0
---
----
1_ _ _ _ /
1-
----- -'- ___ J
-,
200~ "--""
/'
-
",.
..... '"
0
600
Applied load (pst)
800
800 (pst)
1000
I\)
~ (J)
247 than interior beam values and again the mid-span tensions were less than those measured at the support.
In spite of
this, even the lowest mid-span value was larger than the design tension. The presence of membrane forces in the centre-edge and corner panels reduced the torque which would normally have been induced in the exterior beams and slowed the development of steel strains at these edge sections.
It
was only after the formation of plastic hinges in the end spans of the interior beams that the torsion in the end spans of the exterior beams showed up clearly.
The
reduction in torque to be carried would cause the beams to have greater flexural strength but the presence of axial tension would offset this advantage. 9.306.3
Comparison of Measured Membrane Forces
Membrane forces were compared in two ways:
Panel
and beam forces ,were compared with the design values, and the beam tensions compared with the sum of slab compressions along a line traversing the full width of the floor. (i)
Comparison of Membrane Forces with Those Expected
In designing the beams, the mean membrane forces were taken as 340 Ib/in for the centre panel; long direction of the centre-edge panels;
270 Ib/in in the 'zero in the
short direction of the centre-edge panels and in the corner panels. Figure 9.32 shows that, at 775 psf, centre panel forces
248
were close to 340 lb/ino
No measurement of forces was
made in the long direction of the centre-edge panels but Figure 9.36 shows clearly that the membrane action force in the short directions was underestimated and had a value of approximately 200 lb/in at
775 psf, set up principally
as a reaction against the compressive forces in the centre panel. Expected and actual beam tensions did not show particularly good agreement at
775 psf as may be seen from
Table 9.11: Table 9.11.
Beam Tensions at Predicted Ultimate Load (775 psf).
Interior Beam Centre Span Exterior Beam Centre Span
Expected Value K
17.6
5.2K
Measured
QijIid~s12an)
8.7K 6.6K
Measured (SuJ2J2ort) K 1409 11 .4K
26.3
Total
22.8
(ii)
Comparison of Tensions with Compressions
Line 2 was the only line along which sufficient data existed to allow such a comparison to be made.
Figure 9.38
shows the variation of the sum of support section tensions in the exterior and interior beam sections cut by line 20 As in the summing of moments along this line, the figure of 130" was used to factor the panel edge forces per unit
width.
The force at each of the panel edge sections wa,;
weighted according to the length of centre panel edge it covered (half the distance to the next gauged section on either side). Between 550 psf and 775 psf, the values compared well but after 775 psf, panel compression continued to rise while the tension in the beams fell.
In this comparison
it was assumed that the centre-edge panel forces were of a similar order to that expected, which leaves room for considerable variation.
However, unreliable strain read-
ings in the panel edge sections seem to be the only possible source of this divergence of values.
Values of
compression at the panel edge did fall sharply as the pushthrough failure -took place and continued to fall as the tensile membrane was forced to spread from the centre. The curves show good correlation of rates of increase for the greater part of the load range. 9.3.6.4
Conclusions and Discussion
In spite of some inconsistent results, the measured tensions and compression8 and the variation of strain readings with load served to illustrate how enhancement of panel strength by membrane aetion was achieved and the effect it had on the supporting structure. The mechanIsm by which panel compression was resiDtod by beam tensions was clearly active.
The presence of com-
pressive membrane action in the corner panels and the short
250
70
I-
,I
FIGURE 9.38 COMPARISON OF BEAM AND SLAB FORCES
I
J
ALONG LINE 2
60
,
I-
I I
- - - Twice the sum of beam tension as given by gauges 126,129 and 133,134
,I
1\
501130" x (weighted average of compressions at the edge of the centre pane!.)-
I
40 - • At positions of steel gauges
I
J
112 / 114,116 and 118.
I I I I
201-
~I , //, I
I
I
10~-------+--------4------4~~-------+--------+-
o
lODe
direc
on of the centre-edge panels
produc~d
a more com-
plex distri ()lition of membrane forces than would have resulted from the simple mechanism assumed in the design of the floor. Beams would have shown greater distress if all panels had reached failure simultaneously since all panels would then have exhibited maximum compressive membrane action. Craeking, especially
cracking, of the panels
surrounding the centre panel had a marked effect on the centre panel behaviour:
The centre panel deflection
increased with the loss of surround stiffness. As centre panel deflection increased? membrane com-pressbre forces normal to the diagonals increased near the corners but decreased towards the (';entre of the' panel. The def
etion at failure of
almost equal to the panel in Park's equations. Ii
depth~
The
le to do with this high
fle
centre panel was as compared with .5D used ons
the beams had
The more flexible
surround clearly allowed greater deflection, while the development
high compres
on near t;he edges of the slab
provided sufficient enhancement; for the panel to sustain the required load in spite of a small central tensile membrane region.
The low reinforcement content meant that
a relatively low compressive force was needed to produce the requ:lred el1D.ancement and the high span to depth ratio and low surround stiffness combined to affect the geometry
252 of the mechanism in a way that allowed the development of the tensile membrane before concrete crushing occurred. These factors also led to the higher stability in the centre panel, though at the expense of load enhancement. The instability brought by a stiffer surround was seen at LS200 when the deflection of the centre panel increased in jumps and it was clear that had the surround not cracked and become less stiff the ultimate load of the centre panel
..
would have been greater than 850 psf. The
l~rge
difference between measured tensions at the \
...,
mid-span and support sections of the beams could be due, in part, to the effect of horizontal shear along the junction of beam and panel. Further difference could have resulted in the different accuracies of the "no-tension" assumption in the computation of moments due to the different flexural bond requirements at these points.
253
CHAPTER
DISCUSSION
OF
10
TEST
RESULTS
1 0 .1 SUMMARY The effect of membrane action on the behaviour of the model floor was the subject of particular interest. The preceding three chapters have dealt with the design, construction, instrumentation and behaviour under load, of the nine-panel model floor.
In this chapter the principal
findings of the test are discussed.
Particular reference
is made to the adequacy of the ,design method.
General
conclusions as to the behaviour of the floor and the suitability of the design method are drawn.
10.2 DISCUSSION OF TEST;' RESULTS 10.2.1
Beam Behaviour
Without question, the behaviour of all spans of all beams was satisfactory at service load.
Flexural steel
in the end spans was known to be excessive and it was the behaviour of the centre spans of both interior and exterior beams which was of chief interest.
That all the additional
longitudinal steel placed in these spans was required to resist:, the induced tensions was evidenced by the high
254 tensions measured and the high level of steel strains recorded simultaneously at mid-span and at the support, immediately prior to the failure of· the centre panel. The behaviour of these centre spans showed that the assumption that the concrete would take no shear was conservative for the interior beams at least.
This was
probaply true for the exterior beams but the incomplete development of yield moments along the edges of panels adjoining these spans reduced the torsional load on the exterior beams and the situation was not as clear. The proportion of the total tension taken by the exterior beams at design.
775 psf was larger than calculated in
Design values, computed assuming the two corner
and one centre-edge panel to be a continuous beam of K
constant flexural rigidity, were 17.6
and 5.2
K for the
interior and exterior beam respectivelY5 a ratio of 3.4 to 1.
Had the deep surrounding beam,formed by the outer
panels, been assumed rigid and the tensions distributed according to the concrete section areas of the beams, K values of 12.8 and 10K would have resulted giving a ratio of 1.28 to 1.
Ratios of measured values were 1.32 at mid-
span and 1.30 at the support.
Both methods of calculation
are simplifications but the measured values suggest-that the latter method is a closer approximation to the actual relative distribution of tension.
255 10.2.2
Lateral Restraint of the Edges of the Centre
Panel Outward movements of the elements surrounding the centre panel had a marked effect on the centre panel behaviour.
The outer panels clearly contributed to the
lateral surround stiffness by flexural and shear action. The fact that cracking of the undersides of the outer panels produced large increases in centre panel deflections, strains and cracking, was a vivid illustration of this. The reduction in lateral stiffness produced by this cracking was not measured but the sharp change in centre panel behaviour after cracking suggested that the uncracked surround would have had sufficient lateral stiffness to enhance the load well beyond that finally attained. 10.2.3
Centre Panel Behaviour
The behaviour of this panel at service load was very satisfactory but an illustration of the possible effects of surround stiffness loss which could occur with time was obtained when the load was raised 75 psf above the total service load of 400 psf.
Cracking of the panels surround-
ing the centre panel caused large increases in strains, deflections and cracking in the centre panel.
The pos-
sibility that this could have resulted from sustained service loading can not be overlooked, but the centre panel did show adequate stability during the application of service load for 66 hours.
256
The increase in membrane compression normal to the edges was sharper at higher loads, most probably as a result of the high cracking load and the greater tendency for the edges to spread outward after underside cracking. With the loss of lateral stiffness of the surround, edge compression reduced and the deflection increased, but the sensitivity of the panel to further loss of surround stiffness was reduced.
There was an increased tendency for
the panel to form a tensile net at the centre, supported by an outer region of high compression.
With very stiff sur-
rounds, much greater enhancement factors than 2.0 may be achieved for lightly reinforced panels and the failure is far more unstable.
The failure of the centre panel in
this case was not sudden, due to the gradual spread of the tensile membrane region and large deflection which took place before very high compressive forces normal to the diagonals crushed the concrete in these regions and brought about failure.
It was because this region of high com-
pression did not extend the full length of the diagonal and because the deflection at the middle was already high that "failure" was comparatively gentle. The ability of the centre panel to sustain more than twice its Johansen load in this practical situation was encouraging, especially in view of the near equality of experimental and predicted ultimate loads.
2~
10.2.4
Centre-edge and Corner Panel Behaviour
These panels sustained well in excess of 800 psf due to enhancement by compressive membrane action.
Values of
measured membrane forces suggested that membrane action enhancement was active in both directions.
The low level
of hogging moment along the exterior long edge indicated that enhancement due to membrane action was larger than expected.
Two possible causes account for the low moments
along this line. Membrane forces perpendicular to this edge would have reduced the torsion in the exterior beam and at the same time enhanced the moment at the edge beyond that indicated by the low level of steel strains. If no compressive membrane forces had existed perpendicular to the beam at the edge of the
slab~
the edge mom-
ents would have been very low and the load carried by membrane action in the long direction would have had to be greater.
Hogging moments would not develop because of
the reduced torsional stiffness of the exterior beams after cracking. Which of these situations applied was not clear. Large membrane forces were measured perpendicular to the interior long edge and for equilibrium it appears that forces perpendicular to the exterior long edge must be of similar magnitude
0
However, it is unlikely that the
exterior beams alone could withstand a lateral force of
258
200 Ib/in without deflecting sideways. It seems more likely. that only small compressive membrane forces acted normal to the exterior long edge and that the large forces in this direction at the interior long edge were distributed to the beams in the manner shown in Figure 10.1, leaving very little compressive membrane force reaction against the laterally flexible edge beam.
Compressive membrane force in the centre panel acting on centreedge panel. ~----
FIGURE 10.1.
Forc.e transferred to interior beams.
MEMBRA..NE FORCE DISTRIBUTION IN SHORT DIRECTION OF CENTREEDGE PANEL.
Compressive membrane action accounts for the high load capacity of the corner
panels~
although in this case
membrane action was not allowed for in either direction. The low span to depth ratio and the relatively higher lateral stiffness of the edge beams (due to their shorter spans and to the effect of the extra steel placed in sections where cutoff could not be achieved) favoured the
259 development of compressive membrane action.
Again, forces
perpendicular to the exterior edges of the slab could not develop to any large degree and the reaction due to the centre-edge panel membrane action could enhance the load in the manner shown in Figure 10.2.
Reaction on corner panel edge due to centre-edge panel membrane action in long direction. Compression across corners enhances the moment capacity of sections in this +egion.
FIGURE 10.2. 10.2.5
CORNER PANEL MEMBRANE FORCES.
General Behaviour of Floor
The method of design for the centre panel and beams proved adequate in that the expected loads were sustained and the extra steel placed in the beams was required.
The
interaction of elements was seen to be of critical importance when membrane action is to be relied upon to enhance the load carrying capacity of slab panels.
Had the
enhanced capacities of the outer panels all been 850 the centre panel may not have carried this load.
psf~
Increased
deflection of the outer panels would have reduced the
260 lateral stiffness of the surround restraining the centre panel edges.
Such a situation could have been achieved by
increasing the size of the outer panels but this would have meant designing the corner panels with an enhancement factor substantially greater than 1.0.
The effects of an
increase in outer panel size would be beneficial to the centre panel initially but after the outer panels cracked and their deflection increased,the advantage of greater breadth of surround would be lost. approximately the same load
Had all panels shown
capacity~
beam tension would
have been higher as a result of the simultaneous ac.tion of high membrane forces. Extrapolation of the results of this test is therefore difficult in view of the unknown effect that transverse loading of the outer panels has on their lateral stiffness.
However~
the results indicated that it would
be possible to assess the contribution of membrane action provided due allowance was made for the sensitivity of the panels to loss of surround stiffness. 10.2.6
Measurement of Moments and Forces
at Slab
and Beam Sections The methods used to measure forces and moments at sections by means of strain readings proved satisfactory. The performance of the method in this test pointed to several ways in which the method could be improved. Measurement of concrete strain must be made over a
length sufficient to average the effects of aggregate size. In lightly reinforced slab sections the length over which very high concrete strains occur is much smaller than the desirable length for gauges.
Furthermore, in any section,
measurement of concrete strain at the point where crushing will occur, must be subject to doubt when high strains are reached because of the steep strain gradient likely to occur along the length of the gauge.
It appears more
reliable to place the concrete gauge in a position of low strain gradient and relatively low maximum concrete strain, In the centre panel, the compression perpendicular to the diagonals reached a very high value and increased with distance from the centre.
Measurements of this compression
during the test would have been valuable, especially when the initial failure of the centre panel was imminent. In tests carried out over a period of days, electrical drift and time effects in the concrete are likely to introduce errors into strain readings.
Reduced sensitivity
to such errors may be achieved by placing gauges in regions of relatively high strain. In
cases~
such as in the centre panel of this floor,
when membrane forces will be only moderately high, tests of short duration would probably yield more reliable results. 10.2.7
Technical Aspects
The methods used for recording and measuring reactions,
262
strains and deflections were entirely satisfactory. The use of water in the loading bags presented problems in the manufacture of absolutely water-tight bags but this did not outweigh the advantage of safety arising from the use of water instead of air in the pressure bags.
The
simplicity of setting and maintaining the load with waterfilled bags proved a great advantage.
The flexibility of
the reaction frame did reduce considerably the sensitivity of the loading system to rapid fall-off in load and although this meant that the falling branch of the loaddeflection curve of the centre panel could not be followed exactly, the use of water permitted satisfactory control of load during failure.
10.3
CONCLUSIONS On the basis of the behaviour of the model floor
under load and the above discussion,the following conclusions as to the behaviour of the floor and design method were drawn. (i)
The design method for the panels proved satis-
factory but left no margin for deterioration of behaviour of the centre panel under the action of service load for an indefinite period. (ii) Compressive membrane action enhanced the load carrying capacities of all panels.
The corner and centre-
edge panels carried well in excess of the required 800 psf.
263 In the centre
panel~
compressive membrane action more than
doubled the load capacity and enabled it to perform satisfactorily at a total service load of 400 psf. (iii)
Membrane forces measured at the edge of the
centre panel were of the same order as those predicted by the theory due to Park. (iv)
The deflection of the centre panel at failure
was approximately equal to the slab depth and occurred after tensile membrane forces had developed at the centre. (v)
Only moderate lateral restraint was provided by
the panels surrounding the centre panel, resulting in large deflection at failure and the formation of a partially self-equilibrating system of a central tensile region supported by an outer region of compression. (vi)
Cracking of the panels surrounding the centre
panel caused a significant loss of lateral stiffness. (vii)
Compressive membrane forces in the floor
panels were carried almost entirely by tension in the beams.
Outer panels provided stiffness against outward
bowing of the surround but carried little or no tension after they had cracked. (viii)
The tension induced in the beams was of the
same order as designed for. (ix)
It is conservative to neglect shear taken by
the concrete in beams carrying axial tension but some account must be taken of the reduced shear capacity due
264 to the effect of axial tension. (x)
Membrane action in the outer panels suppressed
the formation of hogging yield moments along those edges supported by exterior beams:
the panel deflection required
to develop sufficient membrane action was less than that required to develop full hogging yield moments against the torsionally flexible edge beams. (xi)
Stability of the centre panel under 66 hours
of sustained service load was more than satisfactory but extrapolation of this result to predict behaviour under loading sustained indefinitely is difficult in view of the sensitivity of the centre panel behaviour to very small increases in outward movement of the panels surrounding it. (xii)
Had the outer panels been larger, their
increased deflection and cracking at any given load would have reduced the surround stiffness even further 9 and the simultaneous demand of all panels for high membrane action enhancement would have increased the tension induced in the beams. (xiii)
Measurement of compressive membrane forces
perpendicular to the diagonals would have provided interesting information as to the extent of the tensile membrane at the time of initial failure.
The large values that
these attained would have made their measurement easier. (xiv)
The interaction of the different elements of
265
the floor was particularly noticeable in this case
~
an
example of the value of testing structural systems rather than separate elements.
266
CHAPTER
A
COMPARISON OF
THE
OF MODEL
THE
REINFORCING
FLOOR,
WITHOUT ALLOWANCE
11
FOR
STEEL
DESIGNED MEMBRANE
WITH
REQUIREMENTS AND
ACTION
1 '1 ."1 INTRODUCTION AND SUMMARY When compared with normal design procedure, the method of design used for the model floor resulted in, a saving of panel reinforcement but an increased amount of steel in the beams.
In view of the satisfactory behaviour of the model
floor it is of interest to compare the actual amounts of steel in\lolved and to determine whether a net gain results.
loss or
Such an analysis was performed using
straightforward procedures to calculate the steel volumes required.
The analysis showed that for the model floor,
approximately
7
per cent extra steel was required for
membrane action design. Howeyer, in cases where the beam steel for earthquake moments CEQ + DL + Seismic LL) exceeded that required for full service load moments (DL + LL) plus tension induced by membrane action, a saving of total steel could be made by allowing for membrane action in the design of the panels. It was concluded that when earthquake moments go\rerned the
267 strength of the beams, design of the panels for a service load of DL + j-LL (without allowance for membrane acti on) could be considered.
11.2 GENERAL BASIS OF COMPARISON In calculating the volume of longitudinal steel in the beams, the area of steel at any section of the beam was found by linear interpolation between the critical points (see Figure 11.1).
No allowance was made for
anchorage length or for standard bar sizes. One quarter of the positive steel was continued to the support unless a greater amount was required for torsion.
One third of the negative steel was extended a
distance of one tenth of the span past the point of inflexion.
The additional steel required for beam tension
was calculated from the difference between steel areas at the critical sections with and without tension.
The extra
volume is represented by the shaded areas on Figure 11.1. The volume of shear and torsion steel stirrups was calculated according to the actual area of the shear force and torsion diagrams.
No extra torsion steel was required
for membrane action design but extra shear steel was required because,when the beams were required to carry tension, the shear force carried by the concrete was assumed to be zero. Panel steel was calculated from the lengths actually
268
"-3"
:£
0 - -_ _ _:::..1-_
1'-0"
~I
Interior beam
4'-0"
2'- 9"
(\I
C
.~
(\J (\J
~ q
..
~
('t)
I~ "it
If')
r
l'-6 N
I FIGURE
a)
0
Exterior beam
11.1 LONGITUDINAL STEEL IN BEAMS
--:
269 used.
For design with no membrane action the volume of
steel in the slab was taken as the volume allocated for the membrane action design multiplied by the enhancement factor appropriate to the panel.
11.3 COMPARISON OF STEEL VOLUMES 11.3.1 (a)
Panel Steel
Bottom steel as placed: 35 lengths of
i"
diameter each way~ 13079 ft.
long Volume =142.2 in3 (b)
(51? off each end for
Top steel as placed: anchorage) 80 lengths of 80
11
II
iii
diamet;er each way x
11
II
Ii
/j
9.5"
x 11 .0"
Ii
Vollrne = 5708 in 3 Distribution between the panels was the same for top and bottom steel as follows: Centre panel
.165 times total area
4 Rectangular panels
.485
4 Corner panels
.350
Ii
11
II
"
1.000
Without membrane action, panel steel volumes were increased according to the enhancement factors,
vizo~
1.00 for the
corner panels, 1035 for the rectangular panels and 1097 for the centre panel.
270 Table 11.1 summarises the results of the panel steel comparison. Table 11.1.
Panel Steel Volume Comparison. Panel(s) Centre
Rectangular (4)
Corner (4)
All
Membrane Action Design: Top Bottom Total
9.5 23.5 33.0
28. -1 6900 97·1
18.7 4-6.3 65·0
38.0 93.0 131 .0
9.2 22.8 32.0
909 24.0 ~7, 9 5..1
57.8 142.2 200.0
Conventional Yield Line Theory' Design: Top Bottom Total
20.2 49.7 69.9
76.9 189.0 265.9
fference: Top Bottom Total
All
volu~es
11.3.2
0
in cubic inches. Overall Comparison
The volumes of longitudinal beam steel were calculated from the are as shown in Figure 1-1.1. The total
vol~lIDe
of shear steel in the beams was
assumed to be proportional to the total area of the shear force diagram, less the area represented by the shear taken by the concrete.' Similarly the total volume of torsion stirrups was assumed to be proportional to the net the torsion diagram.
area of
271 The results of calculations of beam and slab steel volumes are given in Table 11.2. Table 11.2.
Steel Volume Comparison - All Elements.
Element
Steel Volume Cin3) Membrane Y.L.T. DifferAction Design ence
Per Cent Saving
DeSIgn
SLAB Centre Panel 4 Recto Panels 4 Corner Panels All Panels
32.0
49.3
33.0 97·1 69·9 200.0
65·0 131 .0 69.9 265.9
33.9
0.0 65.9
25·9 0.0 24.8
Long. Steel Shear Steel Torsion Steel
1Tl.8
24.8 56.0
153.9 16.8 56.0
-17.9 -800 0.0
-11 .3 -47·7 0.0
EXT. BEAM TOTAL
252.6
226.7
-25.9
-11
Long. Steel Shear Steel Torsion Steel
213.6 40.6 0.0
146.6 20.4 0.0
-67.0 -20.2 000
~4508
INT. BEAM TOTAL
245·2
167.0
~87.2
-52.2
BEAM TOTAL
506.8
39307
.1
-28.8
FLOOR TOTAL
706.8
659.6
EXTERIOR BEAMS
.3
INT. BEAMS
~113
-9901 000
47.2
1-1 .4 DISCUSSION The figures of Table 11.2 show that for the membrane action design the volume of the additional steel required in the beams exceeded the volume of steel saved in the
272 difference was 7.2 per cent of
panels and that the net
the total steel volume which would have been required for normal yield line theory design.
Before conclusions can
be drawn from these figures several aspects require discussion. (a)
Method of Steel Volume Calculation
The method of longitudinal steel volume calculation represents a compromise between the exact following of bending moment,
tension,and torsion variation,and the
restraints imposed by practical considerations.
The
method used for shear and torsional stirrup requirements, in following exactly the variation of shear force and torsion, took no account of minimum codes of practice.
ste~l
requirements of
There was therefore a tendency to
underestimate the total stirrup steel volume and to overestimate the additional steel required when tension was present. (b)
",
Adequacy of Design Assumptions
The volume of steel required for membrane action design was computed on the basis of the steel used in the model floor.
High steel strains in both interior and
exterior beams confirmed that the longitudinal steel placed was not excessive.
However the beams did not show
signs of failure at any stage and a small percentage of this steel ,could have been UllneCessary. said of the stirrups.
The same may be
No strain measurements were taken
273 on this steel and the degree of excess was difficult to determine. It is reasonable to conclude that a small amount of the steel placed in the beams was unnecessary and that the net
loss would be no greater than the 7.2 per cent quoted
in Table 11.2. (c)
Dual Use of Beam Steel Required for Earthquake
A.ctions The model floor was typical of a floor in a multistorey reinforced concrete frame building but the beams were designed for vertical loads only.
In
earthquake~
prone areas, beams supporting a typical floor of a multistorey frame building would be required to resist considerable earthquake moments.
The steel placed to
counter these moments could well be more than 'is required to resist the moments and tensions due to vertical loads alone.
Two principal reasons exist for this.
Firstly,
the earthquake moments in the beams may act in either direction requiring additional steel at the bottom of the support sections.
The second factor is the allowable
reduction in live load when earthquake forces are considered. NZSS 1900 Chapter 8(36) allows design for the combination of earthquake forces and the vertical loads to be for a load of Dead load + (DL +
'1- LL
~
live load + earthquake
+ EQ)
'"'174-. c.
for buildings with relatively low live loads and DL +
%LL
+ EQ
for high live loads. Thus, when earthquake beam moments are relatively high, it may be possible to design some interior floor panels by ordinary yield line theory for a load of DL + ~
LL or DL +
% LL
having ensured that when vertical load
only is considered: (i)
Membrane action in the panels will provide
sufficient assistance to carry the balance of live load. (ii) The beam steel required for the full live load condition does not exceed that required for DL + part LL
Condition (i) will be satisfied if the surrounding panels provide sufficient lateral stiffness, and if the beam steel is sufficient to carry the tension induced by the membrane action.
Satisfaction of condition (ii) will
depend upon the relative magnitudes of earthquake and full live load moments. For the steel areas involved in the model slab, the ratio (MEQ/MDL+LL) was determined for which, at the sup~ port section of the beams: (Steel for DL + part LL + EQ)
=
(Steel for DL + LL)
The method by which comparison was made is described below. Let A
=
total area of steel at section for moment and
275 tension at DL + LL. Am == that part of A I'equired for moment only. At
=
A - Am
=
F
=
ratio of panel ultimate load to Johansen load.
v
=
the proportion of live load considered to act
extra steel required
fo~
tension
concurrently with horizontal earthquake loads. The moment acting on the section is directly proportional to the load and the following assumptions were made in determining the steel areas: (a) Am was directly proportional to moment.
This is
true to a first approximation since the distance between the lines of action of the steel and concrete forces in the section is relatively insensitive to change in moment. In fact, an increase in moment will cause a reduction in this distance and the actual amount of steel will be slightly more than assumed here. (b) At was directly proportional to tension (c.f. Figure 4.3 (b)). (c) Tension was directly proportional to the amount by which the applied vertical loading exceeded the Johansen load.
This implies that compressive membrane aetion in the
panels commences when the Johansen load is reached, which, although open to question is reasonable in this context.
(d) The tension induced in the centre spans of beams K at the ultimate load was 5.2 for the exterior beams and K K 1706 for the interior beams. It was assumed that 5.2 of
.,
.,
276 the 17.6K was due to the compression in the centre-edge panel. Calculation of MEQ/MDL+LL(=Z) for the Interior Beam for
v =
1-
At the support: =
1.25 Am'
2 A = .54 in
In this case DL
ultimate load
=
800 psf.
2 11 Am = 02 -r l'n :, =
At
=
0
301'n2
100 psf, LL = 300 psf.,
Johansen load of centre-edge
panel = 800 71.35 = 594 psf. panel = 800 7 2.0 = 400 psf.
Johansen load of centre DL +
~LL
= 300 psf which
requires design for an ultimate load of 300 x 2.0 = 600 psf. By ass-umption (c) there is no membrane action in the centre~edge
panel at this load and from assumption (d)
the tension in the interior beam will be reduced by 5.2K . Membrane action in the centre panel will be reduced to (600-400)/(800-400) =
i
of its full load value.
The
contribution of the centre pa.nel to the interior beam tension at full load is, by assumption (d), = 1706K - 502K 12.4K . Therefore the tension must be reduced by half of K this = 6.2.
=
Thus the tension in the beam at 600 psf is
a.ssumed to be T
=
KKK K 17.6 - 5.2 - 6.2 = 6.2 . ~
Using assumption (b) it is found that the area of steel required for this tension is (6.2/17. 6 )A t = o35At = •44Am , The steel required for moment will be (600/800)
Am
=
o75Am
277 and the steel required for earthquake moment only is equal to Am (MEQ/MDL+LL)
=
Am' Z
The total steel required for DL + j-LL + EQ will thus be Am (.44 + .75 + Z) whereas for DL + LL the total required is Am (1+1.25)
=
2.25Am,
Hence for earthquake conditions to govern: 1 .19 + Z
>
2.25 or Z /' 1 006.
Conditions for this case and the others were: DL + j-LL
DL + j-LL
Interior Beam:
Z
> 1 .06
Exterior
Beam~
Z
> .72
Interior Beam:
Z
> 1 .75
Exterior Beam:
Z
>- .97
It is important to note that whereas the DL + j-LL condition required extra tension
steel~
the DL + j-LL condition did
not, However~
any disadvantage in the former ease is offset
by the presence of earthquake steel required for the reversal of loading which was not included in the above analysis.
Since the values of Z shown above are frequently
exceeded in practice, it would be possible in many cases to design the panels by normal yield line theory to sustain substantially less than the full live load. Such a design procedure would require careful
278 consideration of the conditions of lateral restraint at the edges of the panels which would make the method less attractive.
But the above analysis indicates that existing
floors satisfying the necessary conditions for membrane action would have a considerable reserve of strength.
11.5
CONCLUSIONS The preceding analysis has shown that membrane action
design requires more steel to be placed in the supporting beams than could normally be saved in the panels. In situations in which beam steel is required for other loads sueh as earthquake loading, a net
saving of
steel could be achieved. Floors in which this saving could be made would have to: (i)
Be required to sustain live loads high enough for minimum steel requirements not to govern the determination of panel steel.
(ii)
Be part of relatively tall frame buildings in which earthquake moments are high.
(iii)
Contain panels whose edges meet a high degree of restraint against lateral movement.
An important corollary to the conclusions above refers to panels of multi-panel slab and beam floors in which the beams have been designed to resist earthquake moments, viz., many of these panels? even when designed by yield
279 line theory, will be capable of carrying loads which are very much greater than those for which they have been designed.
Furthermore, this will apply to cases in which
adjacent panels of the floor are loaded simultaneously, provided the supporting columns are not overloaded.
280
CHAPTER
GENERAL
12
CONCLUSIONS
12.1 CONCLUSIONS FROM WORK PERFORMED Conclusions have been drawn at the end of each section, some of which are included in the following general conclusions. (a)
Concrete slabs reinforced with the minimum of steel
required by Codes of Practice can sustain high loads without assistance from compressive membrane action.
The
benefits of enhancement of load due to membrane action will therefore be of greatest significance for slabs which are required to carry high loads. (b)
For
a rectangular, orthbtropically reinforced· slab
with equal hogging moments along opposite edges, a ratio of hogging to sagging moment in each direction equal to 2. 0 gives the least volume of slab steel.
Negati '.re moment
steel should extend into the slab for a fraction, the span from each edge such that 2A
1
=
1 - 71 + i'
~
, of
, where
i is the ratio of hogging yield moment to sagging yield moment in that direction.
This length of top steel results
in identical collapse loads for all four symmetric.al yield line patterns for the panel.
281
(c)
The assumption of rigid-plastic materials in the
analysis of a clamped circular slab with its edges restrained elastically against outward movement is not accurate when the edge restraint is small.
For very
stiff surrounds the assumption is sufficiently accurate to compare well with experimental results. (d)
The absence of top steel at the edges of laterally
restrained slabs has little effect on the ultimate load. The complete omission of top steel may not be wise but its length could be reduced in slabs subject to compressive membrane forces. (e)
Assessment of the effective surround movement should
include the effects of slab shortening, creep and shrinkage.
As the flexibility of the surround increases, it
becomes increasingly important to account for vertical slab deformations occurring prior to the full development of yield lines. (f)
When compressive membrane action is exhibited in two
adjacent panels of a slab and beam floor system, the common supporting beam must be designed to accommodate the tension induced.
Design of the critical sections of such a beam
may be performed using the ultimate strength method and limit analysis.
It is recommended that in these cases
moment redistribution should be kept to a minimum to guard against the adverse effects of beam deformation on the development of compressive membrane action in the panels.
282 (g)
Extra longitudinal beam steel is required in beams
which are designed for tension as well as flexure. ever~
How-
the extra steel required is less than would be
required for a pure tie of the same length as the beam. (h)
Membrane forces in slab panels can have an appreciable
effect on the torsional moments induced in the beams supporting them. (i)
The outward deformations of the sides of a square
surround of elastic material subject to in-plane loads can be closely approximated to those of an equivalent deep beam.
Such a simplification would greatly assist in the
development of a theory for membrane action which takes into account the interaction between membrane forces and surround movement.
(j)
The theory due to Park proved satisfactory in
designing a nine-panel model floor.
High margins of safety
were required when the outward movements of the surrounding panels were calculated on the basis of an elastic,uncracked surround. (k)
Failure of the centre panel of the model floor took
place at a higher deflection than the O.5D used in Parkis theory and although membrane forces at the edge were of the same order as predicted by the theory, the tensile membrane stage had commenced before failure occurred. (1)
The extra steel added to the beams of the model floor
to take the tension induced was no more than sufficient,
283 indicating that the beams must be designed to resist the tension induced and that the magnitude of the tensions was satisfactorily predicted and designed for in this case. (m)
Strain gauge measurements on the steel and concrete
afforded a successful means of measuring compressive membrane forces in the panels and the tensile forces in the beam sections. (n)
The serviceability of the model floor designed by
membrane action theory met code requirements as to deflections and crack widths at service load.
The stability of
the central panel under sustained service loading was encouraging, especially in view of the high span to depth ratio of 32.
More knowledge of the effect of long term
loading on restrained slabs in practical situations is required before confident predictions of the long term behaviour of such slabs can be made.
(0)
Consideration of membrane action in the design of the
nine·-panel model floor re sul ted in a considerable saving of slab steel but the extra beam steel required for tension exceeded that saved in the panels. stances, however, a net
In favourable circum-
saving of steel could be achieved
by using the beam steel provided for earthquake moments to carry the tension induced. (p)
When design for earthquake allows a reduction in live
load, the steel required for earthquake moments in the beams can be used to carry the
tens~on
induced by panel
284 membrane action.
The panels could be designed for reduced
live load by yield line theory provided the capacity of membrane action to take the balance is ensured.
12.2 SUGGESTIONS FOR FURTHER RESEARCH An additional margin for safety exists in panels, designed by yield line
theor~
but having boundary con-
ditions conducive to the development of membrane action. This additional capacity may be utilised by permitting floors to be loaded in excess of the design live load in favourable circumstances. However, the design of such panels to allow for membrane action is another matter, requiring a reliable and accurate means of assessing the enhancement that membrane action will produce.
Although the ultimate load of the
central panel of the ''Irl:tia-panel model floor described in ..' Chapter 9 was accurately predicted by an existing theory for membrane action, more research would be required before a reliable design method could be derived.
Existing
theories and methods of analysis which have been developed principally for slabs with high lateral restraint allow quite accurate prediction of the behaviour of such slabs. But in floor slabs where only moderate lateral restraint exists, these methods cannot be regarded as reliable. In the case of floor panels in buildings, extremely high design loads would be required before the benefits of
285
membrane action could be fully exploited.
In floors
where the enhancement by membrane acti.on could be used, it appears that the overall economic advantages would not be great.
The development of a reliable design procedure
for slab and beam floors would require: (i)
The development of a theory which accounts for
the interaction of membrane forces along the boundary of the slab and the outward movement of the restraining medium. This in itself would not be sufficient because the increasing deflection at ultimate load with decrease in surround stiffness would have to be recognised.
In particular,
future theories should recognise the tendency for a tensile membrane region to form at the middle of the slab before the ultimate load is reached. Extension of Park's theory(11) using a more refined strip approximation, possibly using the results of GUrfinkel(16), could provide a solution, but the assumption that the membrane force is constant along each strip would require close examination. (i~)
Further investigation of the effects of tension
on the behaviour of beams, particularly as to the flexural and shear
reinforc~ment
requirements.
In cases where the
beams provide much of the lateral restraint, knowledge of the effect of tension on the axial stretch would be valuable. (iii)
Experimental studies of the effects of long
286
term loading of slabs with surrounds of reinforced concrete. These would do much to remove the uncertainty inherent in the sensitivity of membrane action to loss in lateral stiffness. This work would probably not be warranted in the case of floors where lateral restraint at the edges is usually low and the design loads insufficiently high.
For
struc~
tures such as pressure vessels and blast resistant structures, where the surround stiffness is high, the rigidplastic theories incorporating an allowance for edge movement (e.g.,thetheory due to Park) will give
,+,c
satisfactory results, but research on (i) and (ii) above could provide useful improvements for this situation. The study of the effects of panel membrane action on other parts of the structure is important whether or not the enhancement of the panel is allowed for in design. Further research, particularly experimental, on the effects of membrane action on the torsion induced in the supporting beams could lead to less stringent design requirements for torsion in edge beams in some cases. This points to the need for further studies of whole structural systems.
Tests on separate structural elements
have the advantage that the actions applied to the element may be accurately determined because forces due to the interaction of elements may be eliminated.
Membrane,',!(:
action is a very good example of a case in which these
f 287
interaction effects are beneficial to a degree which is worth considering in design, even if very high safety margins must:be imposed. In studies of whole structural systems it is not sufficient to rely on the equality of steel tension and concrete compression in a beam or slab section and greater importance must therefore be placed on the role of the concrete strain gauges.
Further research into the measure-
ment of actions on a reinforced concrete section would be valuable in providing a reliable means of interpreting the experimental data obtained.
288
APPENDIX A.1
A
DESIGN
CALCULATIONS
PARK'S EQUATIONS FOR THE ULTIMATE LOADS OF PANELS For the design of the centre and centre-edge panels
of the nine-paneL: model floor, the equations derived by Park(11) were used to estimate the contribution of compressive membrane action to their load carrying capacities. The equations were derived using the approximate yield .-"
line
pattern(~df
Figure A.1 (a).
The slab was envisaged as a
series of strips in each direction and the sum of axial, creep and shrinkage strains, G, and the outward boundary movement, t, was the same for all strips in the same direction.
Conditions of geometrical compatibility and
equilibrium of horizontal forces were used to obtain the actions at the critical sections of each: strip (see Figures A.1(b), (c), (d)). Analysis by virtual work for a slab with all edges restrained and with an empirical value of O.5D for the central deflection at the ultimate load gave
w~~ (3\:; ~ 1) ~ ~'R3uD2{~(183~ .281"2) + (-479 -.490R2) + 6~ ( L)Se. r2. !::z. (3R2. - 1) + h -1l 11:::> D/ L Lx j
::3
-'c (T~ -Tx - Cf + ~x[-e C~- cj,,) +~} + T~ {e( d~x -~) - ~}
- 2
-I-
u. [
TY(d1Y- ~)
x +-
+
+
CsxY-
4-
r
&( Lx) 1b J5
2
(7 k2. - 3) - R2.8 kLx.1>7 (Lx 14 IE." 2. + lx
Ly
Lx
+(T; - Ty
-
C~y + Cs y ) z. ]
C;x { ~~dix) + ~}
-
C$Ye~~ d~y)
Ty (d 1'y -
:;)
+
4-
~Gly2l} J Lx
.
1x {e(d1X -
~) ~~]
C~y( 2; -d~y) •••• (A.1)
4
(b) O.5Lx
Hogging moment
yield
lin~
yield
line
-
MECfolANISM OF A STRIP.
O.5Lx
- - Sagging moment
-
COLLAPSE
"'-"'-"'-"'-Fully
fixed
pl ...
O.5£(l~2p)l ...
t
edge
ASSUMED YIELD LINE PATTERN
~lrfll]irl~~1 " " , I-
c~
-- - ----" ,
__ ;
~--~::::::::::::i:::::--1-
~_=t _________ -~_ ~
D"",,-,,,-,,,,,,,,,-C x direction or -
Yield
STRIPS (a)
strips
(c)
sections OF
EOUIVALENT
UNiFORMLY LOADED WITH ALL EDGES
unit width
I 0
0
0
0] 0
0
SLAB
SECTIONS
FIG. 2
TWO -WAY SLAB FULLY FIXED
I~l(-!
:j f"WI
CONDITIONS
O.5d
n,D
~eu~ral_
ox's
f~
Strain Distribution
Elevation
AT
fsc
M
0
Cross - Section (d)
INTERNAL ACTIONS AT YIELD END PORTION OF STRIP.
A
SECTION
ON
A
VIELD
Cc
--f>
T
\ centroid
, Inhrnal Actions
Stre s s Distribution
LINE WITH
FIGURE A.1 STRIP APPROXIMATION DUE TO PARK
_ Cs ~~)
A rectangular section was taken'in this case Since the torsion was induced by yield moments which developed at the ,junction of the beam and slab. For the middle span of the exterior beams the ratio of nominal torsional stress to nominal shear stress (ACI 318-63 Clause 1(101) was approximately .7:
~3.
This ratio
was used_ in ,:listributing the shear and torsion taken by uncracked concrete sections. Maximum allowa'ble nominal shear stress in concrete from ACI 31c1~63 Clause 1701 = 2jl1
jif ""
135 psi (.0"".945)
Shear taken by concrete = 850 lb. Foree 1n two legs of stirrup = 1000 lb. V.~,s
for verti,cal stirrups
:0
6000 lb.in (ACI Equa-
t Jl,' on -1 r{7 _il"T ') U 'V u.s f" or
45 0 S-lrrups t' = 8500 Ib .. in (ACI Equation
17.6)
sMt for vertleal torsion stirrups = 12,700 lb.in 2 (Australian Code Equation (25) with yield stress
r
1'- 8" 1.70
t06
Moment (lb-inx10 4 )
1'-6"
Moment (Ib-in x 104 )
1'- 5"
I-
4'- O·
I I
.34
~
EXTRA STIRRUPS 1.25
4-0
11
.62
2'_9"
Torsion
Ob-inx 104 ) IFiGURE A.3 EXTERIOR BEAM ACTIONS
I
IFIGURE AA INTERIOR BEAM ACTIONS
I
302 used in lieu of permissible stress) Maximum torque in end spans
10630 lb-in
==
Maximum torque in centre spans lViinimmn stirrup spacings:
==
12500 lb-in
end spans:
s,
mU1.
-1.2 in
centre spans: s mln 1.0 in Maximum longitudinal steel required for torsion (Equation 26 of Australian Code), ASh == .172 in2 for 0
end f:rparw
~ Ash
2 .205 in
==
~,'
/ d- )\
Allocation of Flexural Steel
(i)
In the centre span with tension:
In Equations 4.14 and 4.15: D
==
6.0 11
,
f! '" LJ200 c
At the support:
fy
~osi,
==
o ,--
1-
3 • 5" , b/b f
==
.4,
LI-2000 psi,
T/f,;'bd:= .064, M/f~,bd2
Equation 4.14 gave pi
::::
==
.089
1.28 per cent,, AIs
, 2
==
.25
:=
" 2 010 :t.n
l.n
Longitudinal torsional steel required at top 2 at
i"
2 at
dia.
, 2
-lit elia.
ll1
r)
M/f'bd c
L -
.044, T/f,!bel v
==
.064
Equation 4.15 gave p = .85 per cent, Ac
o
2
.16 In
.:0
" 2 In
3 at 1l-1i dia. (ii)
In the end spans (no tension): At the support M == 3.29 x104 1b t1 , from ACI 318~63 Equation "16~1, As == .15 in2 Torsional steel 2 at
iii dia.
, 2
In
==
r, .-'
.22 inC-,
303
But the 2 -
-a:"
dia. bars on the other side of the
support could not be curtailed and As supplied was .32 in
2
At the position of maximum positive moment 2 4 M = 1.55 x 10 Ib-in, As = .07 in 2 at
-a:"
dia.
A.3.2 Interior Beams (a)
Design Actions
Figure A.4 shows the design actions which were obtained as for the exterior beam except that the moment applied at the end due to edge beam torsion was the sum of the yield moments from the adjacent half span on each side of the beam. (b)
Size of Cross Section
This was set at 7.5" deep and 3.5" wide. ACI Clause 906(b) imposed the condition that the maximum flange width should not exceed was therefore taken as 66+4
(c)
(d)
=
i
of the span which
16.5 in.
Allocation of Shear Steel Shear taken by concrete
= 2460 lb
Force in two legs of stirrup
=
1000 Ib
VI s for vertical stirrups u·
=
7500 Ib-in
VI s for 45 0 stirrups u·
=
10600 Ib-in
Allocation of Flexural Steel (i) b
=
In the centre span with tension: 3.5", D
At the support:
=
7.5", b/d = .2 f 2 M/f'bd = .140, T/f~bd c
=
.171
304 Equation 4.14 gave p':=2.33 per cent, A' s
.57 in
2
:= ·54 in2
4 a t : II di a . + 2 at -2, II di a . ')
At mid-span:
]jf'bd c
c
:=
.034, T/f'bd
.171
c
,-,
Bquation 4.15 Gave p := 1.30 per cent, A "" . 32 in'::::
x
it
2 at (ii)
II
di a. + 2 at
-l-
II
eli a .
In the end span: 4 M := 7.00 x 10 lb-in, A
i"
2 at
s
dia.
::IovJever, on the other side 4 at vided and only 2 at 2 := .32 in .
i"
i" +
2 at
ill
were pro-
were cut off so that As provided
At point of maximum positive moment: 4 3.03 x 10 lb-in, As
M 2 at A.3.3
1 " 4"
dia. + 1 at e;1 " dia.
.11 in .11 in
2 2
Graphical Allocation of Shear and Torsion Steel
The quanti tie s V~. s and
r\~t'
s for each stirrup repre sent
an area on the shear force and torsion diagrams res-pectivly.
Stirrup spacings were determined by dividing the
areas into elements of area V's u or Mt.s.
For shear the
total area covered was that representing the shear not taken by the concrete.
For torsion the gross torsional
moment was used after the total torque applied exceeded the torsional capacity of the uncracked concrete section. Maximum spacings as governed by ACI 318-63 Clause 1706 (b) were 3.0 in for the exterior beams and 3.7 in for the interior beams. Final steel placement is shown in Figure 6.1 (-p. 96).
SLAB
DIM:ENEUONS
I-'HOPEW2 TES
Is
'Mix
Certified Coners
;31
, , l' EtT!:.L 0
6
("
'12
of 'J
-
-,
'; j
72
~506
r
(
'\
d)
Cylinder strength
Age When Tested (12 a;ys)
----
21+ 28 163 181 198
2 2 4 6 1
Cube Strength
Age
Number Te ste-d
Number
-~-
163 181 198 (e)
B.10,3
Modulus of Rupture
163 18'1 198
4
5 3
Ave.f' ]2s1) c,
-G~
')77S 3850 4350 4350 '- I
~
L~420
Ave, u
---.~-=--=--.,.
53'10 pt1i 4890 II 4700 If
L~
3
710 psi 0 11
2
(n~o
It
Modulus of Elasticity
'rable B.1 gives a summary of readings taken on test cimens to determine the modulus of elasticity of the concrete used,
Ree/dings of shrinkage in the unreinf::lI.'ced specimens are
B.2
sed in Table B.2.
STEEL PROPERTIES The results of tensile tests on the reinforcing bars
used are givBn in Table B.3.
A Baty extensometer was used
to measure extension over a two inch length. were tested without being machined.
All bars
SUMMARY OF MODULUS OF ELASTICITY RESULTS ON MORTAR MIX SPECIMENS
SUMMARY OF SHRINKAGE AND TEMPERATURE MOVEMENT READINGS TAKEN ON SAMPLES OF UNREIN FORCED
~ STRESS
CONCRETE SPECIMENS
No. 4
(psi)
(OS)
0000
0000
STRESS
(psi)
0000
SHRINKAGE IN MICROSTRAIN TEMP.
145
20/12/67
68
1300
328 514
21/12/67 21/12/67
66 68
0900
122
100
93
1200
129
108
103
735 1015
22/12/67
66
144
128
193
163
1308
69 68
0900 1700
171
22/12/67 23/12/67
1100
211
184
151 168
0000
0000
155 314
353
124
102
500
707
242
216
1000
1061
366
1500
1414 1768
498 640
350 502 624
2120
778
2475 2830
1172
774 952 1146
3180
1398
1374 1544
4125
1720
4250 4375
2230
956
DATE
0000
2000
2500 3000
532 688 900 1124
TIME
1700
23/12/67
69
1800
219
189
172
1436
1990
24/12/67
63
1300
243
196
1644
2332
26/12/67
1100
1789
2664
27/12/67
63 68
300 346
258 296
2010
3102
29/12/67 1/01/68
66
1200
68
0200
395 464
331 356
3500
3530 3710
1944
1670 1852
3890
2222
2082
3/01/68
69
1100
583
446
175 234 269 300 338 406
2312
4/01/68 6/01/68
72 64
1600 1600
642 670
479 488
449 452
8/01/68
66
0900
745
543
498
66
0900
505
70
0900
754 803
551
Prisms were 18" x 6" x 6" with 8" demec gauge readings.
9/01/68 10/01/68
The above readings are the average of two taken, in each case I from opposite sides of the specimen.
11/01/68 12/01/68
69 70
0900
597 586
539 528
13/01/68
74
1100
609 644
15/01/68 17/01/68
69
1400 1500
18/01/68 19/01/68
73 67 64
1400
907 993 972 945
551 586 602
23/01/68
70
0900
987
29/01/68 3/02/68
65 65
0900
7/02/68 21/02/68
67
0900
925 974 1068
3360
4070 4250
2742
Cylinders
Sample 1:
3750 4000
12" x 6" diD.. with 4" demec gauge readings.
Average of 3 readings
1.94" thick strip_
Sample 2:
Average of 6 readings
two blocks 18" x 6" x 3.5".
Sample 3:
Average of 6 readings
two block.s 18" x 7.5 x 3.5".
All readings in the above table have been corrected for temperature
variations and thus represent the unrestrained shrinkage of the .specimens. A value of 8 micros trains per degree Fahrenhei t was taken in reducing
13/03/68 1/04/68
the readings to an equivalent reading
27/05/68
at 68°F.
67
1500
0900
1700
1300
790 804 881
651 691 693 698 741 739 793 855 900
658 644 640 681 679 729 794 836
1500
1055
0900
1043 1034
935 938
869
0900 0930
1205
1102
1020
875
w
o'-l
308 Table B.
.
Tensile 'I'e st s on Reinforcement.
Yield Modulus Steel Nominal Yield No. Where --Diameter Force Tested Stress of Used - - - - (lbj CJ2sij Elasticity: .211 '4890 A 44,400 31 .0 x 106 Beams 8 8 Only 1 II 40,100 29.9 x 106 Beams A "1920 -4 9 Only 1 II 616 A 8 50,300 29.8x10 6 Slab 8 Beams 1 11 502* B 39,000 29.6 x 106 All 5 E'Stirrups
~;
s.d.
~-
1.65 3.3 3.16 2.4
* .2% proof stress. Type A: yield
British ateau.
Type B:
New Zealand soft drawn wire.
B .3
steel~
lead bath annealed to give extended
SIJAB DIMENSIONS B.3.'1 Pan~
Leyel of ,Top Surface of Floor and Beam and DE?,Pth§
Readings of level on the top surface before the test are shown in Figure B.1.
Panel and beam depth measure-
mentis taken before the test are shown in Figure B.2. Average thicknesses taken from the nine readings per panel were:
\ 12" !!l
.p
I I ," >"
ll6.5
O
12"
12" \
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12" \16.5"
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12"
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FIGURE B.l LEVELS ON TOP SURFACE
~
r.,
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10
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:-
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;f?
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t
'(?t:P 4}
1o:P"
~
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kP
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~ 1# 125 1005 75 150 75 68 54 75 55 69 1020 7"> 175 56 75 200 75 70 1050 57 75 225 75 70 1105 75 71 ' 1125 58 75 200 75 59 75 175 71 1138 1152 75 60 75 150 71 75 75 125 72 1208 61 75 75 1222 63 75 71
APPENDIX C DETAILS OF LOAD INCREMENTS FOR THE TEST ON THE NINE PANEL FLOOR Column
Load Stage Number.
Column 2-4
Panel loads in lb/ft. 2
Column 5
Temperature.
lA
Column 6
Time at which load first attained.
Column 7
Time at which load changed for next stage.
2A 3A 4A 5A 6A 711 8A 9A lOA 11A 12A 13A 76 17 77A 78 19 80 81 82 LOAD STAGE NO 83 84 85 86 87 8a 89 90 91 9111 92' 9211 93 94
75 125 150 175 200 225 250 275 300 CEIIIT PNL LOAD 325 350 375 350 325 300 275 250 225 200 175 150 125 75
95 95A 96 9611 97 9711 978 98 99 lCO 101 102 102A 103 103A 104 105 106 107 108 109
75 125 175 225 275 325 350 375 375 375 375 375 375 375 375 375 325 275 225 150 75
7811
LOAD CENT RECT CRNR - TE~P STAGE PNL PNL PNL DEG F NO LOAD LOAD LOAD 75 75 75 11 1 11 2 100 100 lCO 72 3 125 125 125 4 150 150 15C 11 10 5 175 175 175 70 6 200 200 200 225 7 225 225 200 200 2eC 8 9 150 150 15C 75 75 69 10 75
13 14 15 16 17 18 19 20 21 22 23
75 100 125 150 175 200 225 200 175 125 75
75 100 125 150 175 200 225 200 175 125 75
75 1CO 125 150 115 20C 225 200 175 125 75
65 67
26 27 28. 29 30 31 32 33 34 35 36 37 38
75 100 125 150 175 200 225 2eO 175 150 125 100 75
75 75 75 75 75 75 75 75 75 75 75 75 75
75 100 125 150 175 20e 225 2Ce 175 150 125 loe 75
70 71 71 71 71 72 72 72 72 72 72 72 72
68 69
68 68 68 69 70
TIME ON 1340 1400 1425 1450 1515 1540 1605 1640 1655 1708 1005 1025 1041 1057 1114 1129 1143 1155 1208 1222 1345 1400 1415 1430 1445 1509 1530 1545 1602 1616 1632 1645 1700
TI"E OFF 1400 1425 1450 1515 1535 1603 1640 1655, 17C6
TESHOt 6/5/68
lCCO 1020 lC37 1052 1109 1125 1140 1153 1205 1219
TESTl02 715/68
1355 1410 1430 1443 15CO 1525 1540 1557 1613 1626 1642 1656
TESTl03 715/68
75 100 125 150 175 200 225 200 175 150 125 100 75
75 100 125 150 175 200 225 2'0.0 175 150 125 100 75
75 100 125 15C 175 20C 225 200 175 150 125 100 75
72 72
72 72
72 72
72
72 73 72 71 72 71
75 75 68 125 125 68 150 15-0 70 70 175 175 200 20C 10 225 225 70 250 25C 10 275 275 70 300 300 72 RECT CRNR TE"P PNL PNL DEG LOAD LOAD F 325 325 72 350 350 72 375 375 72 350 350 72 ,325 325 72 300 30C 72 275 275 72 250 25C 72 225 225 12 200 200 72 175 175 72 150 150 72 125 125 72 75 75 72 75 125 175 225 275 325 350 375 325 275 225 175 150 125 leo 75 75 75 75 75 75
75 125 175 225 275 325 350 375 325 275 225 175 15C 125 10C 75 75 75 15 75 75
67 67 68 68 69 69 70 70
11 71 71 72
13 73 73
73 73
72 71
12
1400 1412 1425 1440 1455 1515 1526 1547 1559 1612 1624 1640 0910 0928 0943 1005 1024 1040 H05 1133 TI~E
ON 1155 1215 1350 1415 1432 1443 1458 1520 1530 1545 1555 1608 1625 1640 0950 1005 1040 1055 1112 1128 1140 1212 1225 1355 1410 1435 1455 1525 1540 16CO 1612 1620 1635 1650
TIllE
OFF
0918
0942
lCCO 1015 1045 11CO 1120 1134 1147 12C5 1215
TESTl04 8/5/68
1355 1408 1422 1433 1450 15C2 1522 1544 1554 1607 1620 1630
TESTl05 8/5/68
Cge6 0921 0932 0958 1011 1035
TESTl06 9/5/68
nco
1125 1147 Ufo'E OFF 1207 1235 14C8 1432 1443 1458 1518 1530 1545 1555 16C8 1623 1635 0945 lCOO 1022 1050 n05 1122 1135 12C8 1222 1350 1405 1430 1450 1520 1535 1550 1608 1617 1630 1645
TESTl06 9/5/68
TESTl07 10/5/68
W
--l>
I\)
LOAD CENT RECl CRNR TE{IIP PNl PNL DEG STAGE PNL F NO LOAD LOAD LOAD 66 114 75 75 75 67 75 125 115 125 116 175 75 68 175 68 75 225 117 225 118 250 75 68 250 69 75 275 119 275 300 75 30C 120 75 70 121 325 325 70 75 122 350 35C H 123 375 75 375 72 123A 400 75 40C 75 375 72 1236 375 75 74 124 350 35C 74 75 325 125 325 126 300 75 74 3CO 275 74 127 275 75 75 128 250 15 25C 129 225 75 225 75 75 130 175 75 175 15 125 15 131 125 15 75 75 75 132
-
133 134 135 136 137 137A 138 139 140 141 142 142A 1426 143 144 145 146 llo1 147A 148 149 150 151
15 75 75 75 75 200 200 200200 200 200 175 150 150 150 150 150 150 15 75 15 75 15
75 125 175 225 250 250 275 300 325 350 375 375 375 350 325 300 215 250 250 225 175 125 75
75 75 75 200 20C 200 20C 20C 2ce 175 150 150 15C 15C 150 15C 75 75 75 15 75
152 152A 153 153A 154 15loA 15lo6 15loC 155 156 157 158 159 160 161 162 164 165 166 161
75 125 175 225 215 300 325 350 315 400 425 lo50 425 loOO 315 350 325 275 225 175 125
75 125 175 225 275 300 325 350 315 400 425 450 425 400 375 350 325 215 225 175 125
75 125 175 225 275 300 325 350 315 400 lo25 45C 425 400 315 353 325 275 225 175 125
168 169 170 111 172 173
15 125 175 225 250 275
75 125 175 225 225 225
75 125 175 225 225 225
163
15 H
72 72 73 73 73 73 74 74 74 74 75 15 15 15 74 75 15 15 75 16 75 14 14 74 14 74 74 74 13 73 73 73 73 72
72 12 71 11
11
12 12 71 71
71 7.1 70 71
TI~E
ON
ana
0940 0953 1010 1027 1103 1120 1130 1150 1210 1225 1356 1415 1430 1445 1457 1521 1535 1550 1610 e9lo0 0955 1015 1028 1050 1105 1120 1140 1157 1212 1410 1425 1440 1455 1523 1538 1550 16CO 1612 1622 1636 1650
TI ~E CFF e'll5 0935 0948 1004 1022 1057 1112 1125 1145 12C7 1220 1352 14C9 1425
14lo0 1454 1504 1530 15lo5 1600
0930 0953 1010 1023 10lo0 11CO 1118 1135 1152 12C7 1410 1420 1435 1450 1505 1533 15lo7 1556 1606 1618 1630 IM5
0915 0920 0932 0935 0945 0950 10C3 1005 1020 1040 1105 1110 1120 1125 1140 1145 12eo 1205 1355 1400 1430 1432 1545 1550 1600 1605 1610 1615 /14CO 1405 1418 1423 1437 1440 1455 15CO 1520 1530 1540 1545 1555 1600 1615 1620 1640 1645 1650 1655 10915 e920 C9" 093A 0941
LOAD CENT REel CRNR TEfJP 1I1"E TII'E STAGE PNL PNL PNl DEG CFF ON F NO LOAD LOAD LOAD lCC8 0950 72 225 TESTl 08 174 300 225 1013 1030 72 225 13/5/68 175 325 225 1045 nco 176 350 225 225 72 1105 1125 225 177 375 225 1130 1138 12 225 178 350 225 1143 1150 72 225 179 '325 225 1155 12C3 13 180 300 225 225 1208 1215 14 225 225 181 275 14 121'5 1220 182 250 225 225 1225 1350 14 225 225 183 225 75 1355 14C5 184 250 250 250 1418 1410 75 275 185 275 275 75 1423 1430 186 300 300 300 1445 1435 75 325 325 325 181 75 1450 15CO 188 350 350 35C 20/5/68 68 1515 lOCO 189 315 375 375 !ESTl12 61 1005 1025 190 400 4CO 400 20/5/68 1045 1104 191 425 425 425 61 1110 1128 192 450 450 45C 67 1135 1155 193 415 475 lo75 68 12eo 1230 194 500 500 '500 69 1235 14CO 30C TESH09 195 300 300 1418 69 1408 14/5/68 196 400 400 4CC 69 1425 1lo32 450 450 197 450 69 1440 1453 198 500 500 5CC 69 1458 1535 199 525 525 525 1540 1705 200 550 550 55C 69 1710 1720 201 400 400 40C 21/5/bB 1725 /l015 61 202 200 200 2ce 66 1020 1C30 300 300 30C 203 1105 1045 66 204 400 4CO 400 66 1110 1122 500 5CO 500 205 11lo0 H21 66 206 525 525 525 61 1145 12C2 201 550 550 550 1223 1210 67 515 575 575 208 67 1230 1345 209 375 375 315 67 1350 14CO 210 500 5CO 500 llo 10 67 1405 550 550 550 211 67 1415 1425 515 575 575 212 68 1430 1512 600 600 600 213 68 1520 1540 625 214 625 625 69 1545 1600 650 650 215 650 1610 1633 675 675 216 ~~6 1640 1703 1CO 700 211 1710 1735 725 125 218 725 1750 1740 15C ,750 150 219 TESTl10 1830 1755 220 775 115 115 15/5/68 TESTl13 62 1830 /C940 221 400 400 40C 2215/68 62 0945 0955 600 600 222 600 1COO 1020 750 63 223 750 750 1025 10lo5 224 775 775 775 1050 1108 225 800 8CO 800 65 1115 1135 226 825 825 825 1140 67 850 850 850 227 67 1200 1340 556 556 556 228 67 1345 1355 229 543 543 543 1351 1420 600 60C 230 600 68 1425 1445 231 660 660 660 68 1450 1515 232 710 710 HC 16/5/68 TEST1l4 68 1520 1535 233 600 825 825 68 1540 1550 850 85C 234 600 68 1550 16C5 875 815 235 600 69 1610 1620 900 900 236 600 68 1625 1635 231 600 925 925 68 1640 17CO 600 950 95C 238 239 600 966 966 600 950 1110 TESTlll 16/5/68 850 PSF NOT ATTAINED DUE TC FAILURE CF T~E CENTRE PANEL. 1 715/68 fAILURE OF RECTANGULAR PANELS. FAILURE OF CORNER PANELS.
,
'"
** *"'*
**'" **'"
!.AiI
314
APPENDIX
REDUCED
D.1
DATA
FROM
D
SLAB
TEST
DEFLECTIONS Reduced readings of all deflection gauges with Load
Stage
51
as datum are tabulated below.
The first column
contains the Load Stage Numbers, columns 2, 3 and 4 show the nominal panel loads in psf and the subsequent columns contain the deflection data in .0001 inch units.
The
numbers at the head of these columns refer to the dial gauge positions as given in Figure D.1.
4-
5
b
N
"
2.7
14-
t
2.4
....31
13
,,3 0
:1 a7
13
2.0
15
7
,28
8
FIGURE D.1.
I'"
12-
'<.~
""tor it36
" 3)
18
t7
~34
.32.
9
21
.
II
II
2.9
£2
10
.2.':.
1
DIAL GAUGE POSITIONS.
315
"i T t.G~ (\Hf<
1.0 7". 2.0 ICO. "l, .n 125. '.0 150. " .0 17<]. f..O 2eo. 7. (] 22S. 7.1 225. 7.2 225. Q.I]
q.
r:
1 r:.O 13. () Ui.r; 1'J .0
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17,0 IR.f)
10. ()
20. () 21. I) 22.1)
n.n
ch .0
2eo.
15(").
':II.n
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')P .0 5" .r; hf';. f)
h 1.r} 63. r] 1,4 2.4 :3 .4 4.4 5.4
t,.4
7.4 P .4 9.4 10.4 l1.4 12.'-1 1 ~. '-I 76.0 77. ()
77,1
78.n
7P.l 7Q.O
CPlj'<
7, •
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lee.
1 {":; ,
15(;.
11 ').
2en.
22"1, 22'> • 225. 221) • 225. 225. 20U. 200. 150. 1"(',. 1'-) • 75. 7'? • 75. Ion. ICC. 12'1. 12'"1. 15C. 15e. 1 7S. 17S.
,'"
7'. 75. 75.
". 75.
75. 75. 75. 75.
?'. 2eo. ?'. 22:>. 7" • 2Cf) • 75. 17,). 7'1, 1 c:;c, 7'). 12"; • 7" • lCr, • 7'1. 7'1. 7") , tcc, 7", 7'. 12'1. 1 'lC 7" •
".
0
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,
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75.
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7, .
125. 150. 17') • 2Cr}. 22'5.
RC TR
CP'f\)~
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25e, 27'5. 3CC. 32'5. 35C. J • In.o 75. 75. 13lj. n 75. 12"). 75. 75. 115. 1 3'i.0 75. 7"; • 11f-.O 75. 221) •
-164 -1'10
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75. 7" •
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75.
QO. () 2')C. 'll.n 275. P2.r} 100. H.3 .0 325. '14.f) 350.
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STAGE CNT!.l.
8'1.0
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-17('; -21,6 -?6C -254 -?';C
- ')')<)
20C. 22 S
75. 1 ')0. 7". 12'. 75. 7". 75. 75. 100. ICC. 12'5. 125. 150. lSf':. 175. 175. ,C(';. zoe; • 225, 2Z'; • 2CO. 2GG. 175. 17'1. 15C. ISO. 125. 12"'. 100. tor:: • 7'1. 75. 7'j. 75. 12'S. 12'5. 1 'JC. 150,. 175. 17'). 200. 2orl. Z2"1, ?2"i •
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63 60 58
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00 09 95 107 lC 3
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94
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1 -3 -4 -6 -4 -4 -3 -3 ?
4 7 11 17 2l 26 13 26 22 17 15 7 I.
4 10 26 27 11 35
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19 15 15 16 16 17 21 17 42 57 70 70 76 e5 91 101 lOR 116 111 112 106 97 94 87 RO 7?
50 41, 2R
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10 13 19 1 1 21 13 3 3 9
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6 10 11 20 20 20 24 21 21 l' 11 11 6 1
15 21 29 31
10 23 26 26
39 4" 51 60 62 62
12 38 42 51 56 51 -77g 55 55 52 51 46 41 40 36 31 26 21 11 1 21
22
28 30 29 25 20 17 11 8
11
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69 65 SA
5A 52 50 47 45 40 31 29 25 15 25 35 40 50 59 61 '6 60 51 51 45 3A 35 30 25 20 20 20 20 21 22 21 32
'.5
60 65 72 ao 87 95
105 110 105 105 105 99
9? BA
8, 75 r.1 50 37 30 25 20 15
,
32
39 42 51 52 57 52 46 46 41 36 31 28 21 21 21 l' 16 16 l' 10 19 33 52 60 66
71 71
86
9' 101 98 98 95 90 86 81 72 69 54 43 31 21 14 11 6
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THE LOAD STAGE
USED AS ZERO FOR THIS RUN IS
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346
APPEiiDIX E.1
COMl)UTER PPROGRAMME
E
DESCRIPTION
PROGRAII]VIE TO CALCULATE ACTIONS ON A REINFORCED celT CRETE SECTION This programme is described in Sections 7.3.3 and 7.3.4. The values of steel strain and concrete strain measured
at any section defined a linear strain profile from which the strains at the top and bottom surface could be calculated for input into subroutines CONACT and srl'EEL which were used to compute the concrete and steel actions within the section.
E.2
SUBROUTINES CONACT AND STEEL The following assumptions were made in deriving the
subroutines: (i)
The strain profile was linear across the section.
(ii)
When the neutral axis lay within the section, all concrete subject to tensile strain was assumed to carry no tensile stress once the maximum tensile strain in the concrete exceeded the fracture strain.
(iii)
When strain across the section was tensile at all points in the section, all concrete subject to a strain greater than the fracture strain was assumed to carry no tension.
(iv)
The stress-strain relationship for steel was tri-· linear as defined by Figure E.1(a).
347 (v)
The stress-strain curve for concrete in tension was linear up to a tensile fracture strain of e t at which the fracture stress was c
f'/~
c
(see
Figure E .'1 (b)) . (vi)
The stress-strain curve for concrete in compression
~as that proposed by Hogenstad et al.,(2)
and shown in Figure E.1(c). (vii) (viii)
The section was, in general, that of aT-beam. Moments were considered as acting about the middepth of the section where the net axial force was considered to act.
E.2.1
Subroutine STEEL
Section properties were known and the strain at the levels of top and bottom steel were calculated from the two known values of strain at the top and bottom surface. Stresses could therefore be found from the assumed stressstrain relationship. E.2.2
Subroutine CCNACT
Knowledge of the top and bottom surface strains enabled the determination of concrete actions by use of the assumed stress-strain relationships for concrete in tension and compression. Subroutine CONACT was written as four separate cases depending On the sign of top and bottom strains: CASEA
Both top and bottom strains tensile
CASEB
Top strain compressive, bottom strain tensile
Elastic (
348 Strain hardening
Yield plateau
(a) Steel
O L-~----------------------~~----------------~----e ey ef Strain
Fracture
(b) Concrete in tension
~.~ ect
Q
~--------~----------------e e'ct Strain
FIGURE E.1 ASSUMED STRESS -STRAIN RELA TIONSHIPS
(c) Concrete in compression
o Strain
349
CASEC
Both top and bottom strains compressive
CASED
Top
stra~n
tensile, bottom strain comres-
sive. For each case integration of the stress-strain curve over the appropriate ranee of strains was performed analytically and the results used in the subroutines for the relevant case. In computation the flange overhangs of "r-- or L-beams were ignored initially and the actioris on the rectangular portion of the section were found. To calculate the flange contributions, the strain at the level of the bottom of the flange was used in place of the strain at the bottom of the beam.
The flange sec-
tion was then considered as a rectangular section of different depth and breadth.
The subroutines were then
used to calculate the moment and force in the flange as related to its mid-depth which were transformed to equivalent actions at the mid-depth of the beam. Values of the parameters defining the stress-strain curves used in computation were as follows (notation referring to Figure E.1): f'
c
4350 psf, f y /f'c
=::
9.78 (beam steel),
11 .82
(slab steel) ey e ct
.0014 (beam steel)
=::
==
.0028,
.~
e sh == .0100, e f
1 .85, eo
== ==
.00169 (slab steel) .0028, e
==
.0101 , C
==
O.
u
==
.004
350 APPENDIX
F
CONCRETE
F.1
SLAB
STRIPS
DESCRIPTION OF TABULATED RESULTS ~~he
results oJ the test;s on three slab strip specimens
are tabulated be low.
Irhe quant:itie s I i sted are:
Load Stage Number
= Vertical deflection at mid·-span in ,0001 11 units DHZ
Horizontal movement of the free end in .0001
CST
11
uni tf;.
= The average of the two concrete strains in microstrain.
The values listed include a
correction of --20 microstrain to account for initial loading. The average of the two steel strains in mierostrain,
The values listed have
included a correction of +15 microstrain for initial load. rrotal load, in pounds, applied to the strip
PRF
through the proving ring. The sum of the forc.es, in pounds, measured in the tie rodE; w:oed to apply the axial compression to the section. NCAI,C
;:
Axial cornpreesioll cnicnlaL;cd from strain readj ngs.
MAPLJ)
Total moment about
mid~depth
applied to
the mid-span section. MCALC tions
Moment calculated from strain readings. om the strain readings were computed using the
subroutine s CONACT and 'sIJ.'EE.L.
The value s of the para--
ml3ters defining the stress-strain eurves are given in the table and correspond to those given in Figure E.1. variable, SR, is defined by SR
==
£' If!
Y
c
==
FY/FCDASH
trhe
S TR P
Sl A8
ECT: EY: 5R
:
JIj L Y
NC.l
80fiE-03
EO
:
O.250~-C2
Ee
0.600f-02
0.169E-02
E~
:
O.2~OE-Ol
FF
0.100f-Ol
Q ..
11.40QFCDA5fl
:
l'J68
Q
C.OOCO C.C
44CC.0
l SN
DVT
GHl
CST
'S3 'S4 55 56 57 58 59 6C 61 62
225 235 256 251 250 241 240 240 4129 3940 3680
4 -5 -4 3 10 19 2'; 32 395 396 382 389 399 371 270 -7 -57'; -555 -538 -408 -562 -279 -262 -358 -485 -529 -552 -562 -560 -558 -552 -549 -545 -525 -429 -331 -172 -621 -608 -598 -500 -382 -355
- 2 54 -281 -29'l -277 -2'S6 -232 -211 -112 -105 -79 -';3 -26 -72 -124 -168 -176 -19S -245 -300 -361 -19P -,6 -30 -69 -10'; -137 -163 -1 R1 - C59 -226 -243 -261 -204 -17'; -143 -106 -63 -196 -264 -237 -213 -181 -135 -95 -307 -282 -251 -217 -170 -351 -321 -299 -255 -202 -380 -384 -341 -290 - 51 -100 -117 -134 -149
C3
64 65
6C
lSN
DVT
Lllll
CST
SST
PRF
1
0
0 -2 -11 -p -25 -29 -25 -20 -12 -8 0 5 2
-20 -44 -61 -83 -108 -131 -150 -129 -107 -gz -58 -39 -54 -76 -99 -122 -144 -167 -182 -162 -141 -119 -97 -7<; -8H -IQ9 -133 -154 -17P -202 -219 -198 -176 -154 -132 -111 -126 -149 -111 -194 -216 -241
IS 0 -24 -40 -52
O. O. O. O. O.
-5'; -4C -23 -4
50. 50. 50. 5C. 50. 50. lCO. lCO. 100. lCO. lCO. 100. 15 O. 150. 150. 150. 150. 150. 2CO. 200. 2CO. 200. 200. 2eQ" 250. 250. 25C. 250. 250. 250. 300. 300. 3CO. 3CO. 3CO. 300. 350. 350 .. 350. 350. 350. 150.
2 3 4
5 6 7
8
9 10 11 12 13
1<1
15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
,9 4C 41 42 43 44 45 46 47 48 49 50 5l 52
a
0 4 10
15 43 38 32 29 26 20
4l
50 51 58 62 10 92
90 85 80 76 11 91
96 101 109
113 120 142 139 134 130 126 120 145 156 160 165 171 179
2CO 194 191 J. 88 185 180 202 209 215 220
0
-5
-8 -ll
-19 -19 -15 -9 0 9 14 20 11 9 2 -2 -15 -15 -12 -6 2 11
20 25 20 15 12 ';
-5 -4 1 5 14
22 30 30 26 20 12
-25e
-235 -216 -194 -172
-lSI
-166 -186 -210 -232
-7~
1"1
78 40 25 9
-7 -22 -40 -26 -p
6 24 41 <;7 71
53 39 22 'i
-9 5 20 36 S4
71 89 105 100 83 106 49 33 45 61 78 96 120 138 154 136 120 103
O.
1-t00 ..
4CO. 4CO. 4CO.
lO
~CALC
~APlC
~CAlC
0 1007 2019 2995 3993 500'; 4987 3999 29')9 1915 899 34 32 1016 2031 3035 4013 Sal! 5000 39')9 3015 2003 1012 35
101. 1164. 2363. 3434. 4464. 5691. 5587. 4591. 3501. 2258. 1124. 199. 215. 1235. 2291. 3369. 4371. 5540. 5464. 4410. 3452. 2340. 1301. 285. 214. 1248. 2253. 3294. 4405. 5440. 5431. 4438. 3422. 2338. Ut8. 290. 222.
323 323 323 324 326 330 619 613 6C7 6e3 6CO 598 873 878 883 890
333. 45C. 370. 422. 541. 552. 937. 888. 849. 801. 716. 646. 905. 1005. 1099. 1173. 1241. 1265. 1566. 1558. 1490. 1440. 1358. 1266. 1519. 1586. 1712. 1768. 1846. 1938. 2241. 2187. 2114. 2054. 1911. 1905.
~AP
37
1032 2029 3017 4022 5027 5016 4034 3032 2035 1025 32 35 10 16 2037 3035 4023 5025 5012 4030 3037 2036 lOll 45 5'i 1036 2062 3046
921.
1928. 2996. 4021. 5107. 5169. 4144. 3182. 2120. 916. -74.
-167. 817. 1841. 2869.
A97
908 1193 1183 1173 1164 llS5 1148 1423 1432 1443 1455 14e8 1483 1769 1754 1738 1724 1710 1698 1973 1988 2005 20 2 3 2041 2062 2348 2326 2306 2286 2U6 2248 2524 25<;4 2567 2590
2191.
2387. 2472. 2554. 2626. 2724. 2996. 2914. 2817 • 281'i. 2791. 2721. 3013. 30t5. 3174. 3246.
67 68 69 70 71 12 73 74 75 76 77
78 19 80 81 82 83 84 85 86 87 88 89 90 91
n
,
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 HI H2 113 114 ll5 ll6 117 118 119
120 121 122 123
124 125 126 127 128 129 130 IiI
33';8
2812 2640 2C30 700 419 510 649 1065 430 1678 1679 1290 695 -482 421 391 423 452 485 515 468 532 10 18 1641 2385 355 478 5ll 951 1679 2429 3125 638 950 1653 2755 3355 IC30 1470 2995 3512 4605
1419 3602 4271 5038 4346 3959 4138 4320 4522 4732 4971 5197 5379 5655 6065 6400 6808 730') 8255 8912 9579 10366 11140 12,)OR 13370
-198
-658 -588 -450 -320 -248 -610 -521 -342 -351 -242 -667 -660 -615 -639 -590 -661 -660 -651 -646 -635 -622 -615 -b09 -607 -626 -627 -572 -540 -588 -';54 -755 -754 -754 -797 -P90
13? 16g22 -F?!
-Ho -184 - 202 -217 -234 -242 -244 -247 -248 -241 -234
-224 -201 -181 -149 -14 '3 :l~!
SST
P RF
'APl8
NCAle
~APlC
~CAlC
°7 72
400. 4CO. 450. 45C. 450. 450. 45C. 450. 3CO. 200. lCO. O. O. O. O.
4041 5042 'S047 4041 3062 2038 1046 52 55 51, 56 ,8 1038 2030 3023 4029 <;035 5023 5061 5086 4762 42 -42 957 1976 2941 3917 4969 4959 4951 4947 4945 3961 2983 1930 969 -25 4910 4968 3948 2964 1953 nl -3 4969 3929 2944 1934 953 4961 401B 2932 1973 949 4936 3971 2941 1953 4 1000 1042 1095 ll45 11 94 1239 ln9 1305
18510
2613 26t,) 2127 2899 2874 2247 2823 2799 1995 1'<44 893 3'i2 614
3327. 3454.
R7
103 120 137 156 teC 2150 1840 1458 1044 '559 178 -3'32 -722 -813
-739 -569 -41 -785 346 830 325 -297 -561> -733 -797 -776 -754 -728 -698 -690 -525 109 716 1208 -811 -709 -598 2 672 U84 1678 -524 -69 581 1253 1723 -gz 371 1274 1712 2239 159 1237 1776 2238 977 622 898 106'1
1279 1482 1709 1921 20n 2278 2312 2304 2293 2384 2141 20C6 1761 157C 1199
14.9 7
1462
P!~
O.
O. 150. 300. 450. O. O. O.
O.
O. O. C.
O.
50. lCO. ISO. 2CO. lCO. lCO. 100. lCO. O.
O.
200. 2CO. 2CO. 2CO. 200. 200. 3CO. 300. 300. 3CO. 3CO. 4CO. 4CO. 400. 4CO. 4CO. 450. 45C. 450. 450. O. O. 50. 100. 150. 2CO. 250. 3CO. 350 ..
400. 397. 381. 370. 355. 302. 3D!. 287. 281. ZQ6D
-;]9. ,47. 35':.
131.,0
1415 1460 1490 1496 1418 1301 1130 P81 590 12, 38
- - 3?
4946"
4934. 3954. 2954. 1902. B62. -241. -151.
-1900
-514. -491. 398. 1143. 1956. 2320. 2574. 3012. 3420. 3615. 2584. 122. -293. 476. 1297 • 1816. 2198. 2503. 6838. 2837. 2984. 3137. 2574. 2166. 1384. 569. -229. 2583. 3175. 2831. 2158. 1353. 520. -244. 3454. 2882. 2108.
1280;(>
814. 3558. 2936. 2067. 1658. 819. 3648. 2900. 2165. 1678. -184. 516. 576. 594~
592. 614. 1027. 1051. 1070. 1133. 1213. 1238. 1275. 1270. 1222. 1233. 1313. 897 ..
823. 44().,
398. __ 48~.
858
936 605 533
1404 2301
3339
527 330 "315 446
460 181 490 517 807 1096 1387
1677 1058 1031 1069 10!? 311 499 1660 1624 1704 1750
16"6 1 f~ 2 2
2290 2346 2459 2505
2292 ): 13
3033
1 3215
3{~O
2960 3498
4228 4054 3781 324 718 1029
1346
1665
1988 2313 2i:37 n49
328, "3364
3354
37610
3687. 3630. 3535. 3486 ..
3490. 1771. 1391.
12 /i4 ..
8CO. 919. 1047. 1081. 917. 1004. 1410. 1901. 2632. 1042. 476. 708. 716. 632 ..
717. 815. 969.
4183. 1264. 1402. 15" 9. 11"1. 1022. 1149. 122C. 1177 •
1013 • 1564. 1433. 1603.
1753. H06.
1689. 1916. 20590 2210. 2349 .. 2010. 2533. 2588. 2950. 2623. 2491. 2878. 3524. 3484. 312.;.
953. H21.
13e4.
1632. 1864.
2107. 2055. 2308. 2518. 2124. 2780. 2790 e
2981 2783 zeC7 2340
2806. 2825. nE 9. 2645. 2432. 2<14. 2170. 1984.
?~3Q
. P~25"
3369
3370 31:5 31~5
2283
1925.
JULY
NC .3
ST~P
SLAB
ECT=
0."00E-03
I:C
C.2I)OF:-C2
EU
0.650E-02
0
c.sceo
tY=
0.169E-02
t~
0.2COEc-01
EF
0.10 rlE-01
C
0.0
SR =
11.400rCOASH =
4400.0
lSN
DVT
DHI
C 51
SST
PRF
~APlC
1 2 3 4 5 6 7 8
0 20 38 55 75 95 114 131 ISO 170 18? 209 57D 371 51 71 94 115
0 3 4
-20 -35 -49 -04 -79 -94 -IIC -123 -140 -157 -173 -190 -CIO -331 -165 -186 -20; -214 -247 -270 -298 -325 -3P -4010 -464 -530 -616 -716 -811 -2C7 -184 -204 -222 -242 -265 -294
15 Zg 41 54
O. 50. 1CO.
0 6 8 6
9
10 11 12 13 14 15 16 17 18 19 2C 21 22 23 24 25 26 27 28 29 30 31 32 , 3 34 35 36 37 38 39 40 41 42 4~
44 It 5 46
47 4" 49
SC 'il
e2
138
158 179 2CO 225 2'51 285 321 368 419 470 70 60 81 101 124 145 170 199 226 265 3G8 35'1 40'1 470 61 55 7h
lCO 121 150 1 PI
nr 2P~
.,
8 9 10 10 10 II 13 16 130 121 8'3
a5
83 83 83 83 83 83 76 75 78 80 81 e5 90 50 75 79 80 81 81 83 86 89 90 9'5 100 105 114 71
79 00 80 81 H4
P'J
'13
1 1:r
-~37 -~8~
- 4 52 -531 -t23 -712 -n4 -191 -166 -IP6
-?C7
~9
PC 95 109 123 137 156 lAC 1751 lC41 -129 -113 -~4
-79 -1,0 -3'1 -14 11 54 102 lI'. q
248 350 469 5~9
-126 -99 -78 -60 -/-tC
-16 Ih 65 12') 211 J09
432 'i57 719
-qg -7S -'>'5 -31
-?3~
-273 - ~ ?')
-L.l":
_cr'
)
I ..
e
l?~ ~ (t
1
572
l,)C.
2CO. 250. 3CO. 150. 4CO. 4'5C. 5CC. '350. 2C2.
O.
O. 50. ICO. 150. 2C0. 250. 'CO. 35C. 4CC. 45C. 5CC. 55C. 6CC. 650. lCD.
O.
O. 50. lCO. 150. 2ee. 250. 300. 3?C. 4CO. 45C. '>CC. 'i5C.
bCC. O.
O.
n.
O.
C.
n. r.
O.
r·
~
9 9
A
10 11
13
C Al C
"APlC
"CALC
101. 113. 98. 109. 69. 128. 113. 42. 53. 87. -54. -294. 2591. 1257. 8319. 8330. 8198. 8177 • 8118. 8C26.
323 598 873 1148 1423 1698 1913 2248 2523 2798 3013 B48 1437 324 348 1'33
333. 600. 856. 1122 • 1403. 165;:. 1941. 21<;3. 2489. 277e • 3101. 3471. 4105. 2275. 284. 660. 1048. 1397. 1829. 227< • 2784. 3197 • 3653. 4103. 462e. 5171. 584 I. 6584. 7235. 745. 797. 1214. 1586. 1995. 2460. 2927 • 3436. 3901. 4469. 5095. 5777 • t4C3.
~
16 66 40 5020 5018 5016 5016 5014 5012 794~. 5007 5006 7884. 5C08 7882. 8031. 5012 5018 8320. R677 • 5C28 9264. 5043 5065 ~992 • 5096 10550. 92'18. 4992 4003 7822. 7658. 4003 4C04 7522. 4008 7379. 4011 720'1. 4009 7062. 7074. 4C 14 7193. 4022 749<) 0 4C34 40 S4 8014. 8595. 4021 410'1 n42. 4151 9645. 7985. 4CCtl h638. 3CC7 6475. 300'l 30C5 6244. 3rl0 59"4. 3')1, 'ie7'> • 3024 595". ~C47 h 313. 3r: 7 -J
r
h )4f-:. •
no
1205 1492 1777 20£02 n48 2 c-; 5 2q23 ,216 3509 38ce 4110 4412 3~7
347 630 913 1197 14et 171:6 20<;2 2339 2629 2922 3219 351t 3p18 347 3~9
(,20
gr::3
1~4
410R 7~2
043
~'34
70£::5.
868. 873 • 1293. 17'54. 2313. 2~IC.
33;3.
40~7.
4762.
lSN 53 54 55 56 57 58 59 60 ft
62 t3 64
65 H
c7 68 69 70 71 72
73
74 75 76 77 78 79 PC ~1
B2
P3
84
~5 ~6
87 88 89 '10 91 <)2 93 94 95 9f;
97 98 99 ICC 101 102 103 104 105 lOt lC7
DVT
OHI
348 408 470 56 59 82 116 11:5 231 3C2 381 440 5C5 68 105 183 259 335 406 479 548 610 692 170 340 431 506 589 702 840 1005 12CO 1408 1772 2CCR 2516 3COP 3502 4012 4505 5COO 5'550 6010 6500 6750 7010 7?75 7505 7755 A014 8120 "260
109 116 121 BC 8R 89 90
R4~9
9001 9490
lOB 11']004
109 10'529 llC 11530 I I I ll4S'J 112 174 hO 113 f'
n
105 115 126 132 141 89 100 112 123 135 145 158 163 16A 170 116 1'39 171 182 192 202
221
239 255 2H 312 332 370 404 435 462 490 512 521 ';29 530 529 529 529 529 529 52g
529
')28
52R 024 ')24 '522 522 'i2l
SST
PRr
-611 533 -702 6~4 -796 R6~ -168 -73 -153 -32 -184 4 -233 73 - 318 2C2 -419 374 ')65 -520 -622 771 -695 933 -780 111~ -161 -24 -209 191 4D~ -306 (11) - 391 -478 8n -560 lCIC -645 1208 -727 IB9 -804 157C -903 17'06 -266 267 t;'jr; -343 -444 12('3 -526 1424 -60'1 16b1 -69, 1943 -773 221h -p 59 2'523 -95R 2'336 49Cf) -1029 'i3'ir. -1159 7'](;C -1262 -15CC -6U3 3(21'; -1755 -2C17 30ze -2297 l02e -2575 3020 -2863 3020 ,02e -3150 -33B9 3PC -31:26 3 C2C -3749 3020 -3859 3020 -3962 lC20 -4C4P 307r; -4139 1U20 -422P 3C2Q -427'> 3n2~ 3()?f) -4323 -4,95 10'1) ";)r:21"' -4~34
4CO. 450. 5CC.
CST
-4f:'1"J
-47" Q -4t:;2? -'i[7':
'in -sr: 11
S:?C -SOli 0 2t3(]
1r")f'
3C7( 1C;~C
3C ?r~
1 czr ~nzr
3 (" (
O. O.
50. 100. 150. 2CO. 25C. 3CO. 350. 4CO. O.
O.
50. ICO. 15C. 2CO. 250. 1CO. 350. 4CO. O.
O.
5C. ICC. 150. 2CO. 25C. 3CO. 350. 368. 390. 386. 434. 462. 476. 4se. 5C9. 50<;. 495. 4A4. 4,,1. 4106. 463. 453. 436. 473. 413. 4 0 6. 408. 19~.
J73.
147.
-1
~
1.
HI. ,cr. lC l. 4Ch. 0,.
PlC
~CAlC
"APlC
3120 3156 3203 3011 1993 1998 2014 2035 2077 2125 2179 2216 2266 2006
7525. 7941. 8345. 6607. 4945. 4636. 4494. 4773. 5249. 5710. 6160. 6426. 6766. 4877. 2668. 3024. 3377. 3781. 4145. 4536. 4907. 5272. 5861. 3185. 1600. 2096. 2445. 2753. 3133. 3502. 3869. 4339. 2836. 3374. 2585.
2631 2926 3223 339 334 614 896 1181 1470 1762 2056 2345 2637
~A
993
1047 1099 1157 1204 1250 1294 1337 1397 IOc3 44 46 49 52 51 48 45 53 78 181 279
'325*******
779 1026 1257 1449 1625 1791 1894 1979 2017 2046 2066 2079 20'10 2089 2089 20P 2065 2016 1941 IP 32 It 7 il 1313 374
14275. 16819. 19404. 2l7n. 2397C.
25922. 27393. 28722 • 293"8. 29921. l0420. 30823. 31234. 31627. HAZ8. 32C31. 32327. 32P79. 33343. l3P?3. 342 8S. ,47 9 2. ·1404b.
17 310 P ')C.
-2115
1232.
"CALC
548e. 6079. 6662. 919. 1211. 1842. 2423. 3127. 3861. 4541. 5198. 5640 • 6154. 33e 1382. 1979. 333 617 2629. 901 3189. 1186 3766. 1471 4298. 1757 4844. 2043 5362. 5846. 2329 6433. 2619 341 245C. 2420. 324 599 3112. 3653. e75 ll51 4181. 1426 4536. 1702 4847. 1917 5159. 2254 5547. 2359 4506. 2497 4958. 2504 431>7. 2842**· .. ••• 3099 8725. 3302 10234. 3513 11639. 3777 12793. 3912 13713. 4042 14399. 4120 14830. 4255 15153. 4245 15286. 43C5 153P7. 4315 15467. 4283 1:525. 4267 15576. 4270 1'5619. 4254 15638. 4281 15655. 4265 15677. 4191 157C7. 4C74 15721. '39PA 15724. 3799 15715. 34q6 15692 • 2491 15bRS. 25ee 1:425. 323 -96";.
W Ul W
L SN
OVT
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 [7
0 0
[P
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 H 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 59 59 60 6[ 62
0
64 65 66 67 68 69 70 71 72
73
74 75 76 77 78 19
eo
81 82 83 R4
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86
e7
98 89 00 91
92
93 9<, 95 g6 'J7 08
5 8 10 19 40 60 7B 95 115 132 150 170 29 15 35
52 70 89 105 122 140 160 20 13 15
52 70 89 lC5 122 140 160 19 9
28 48 63 90 100 118 [35 152 10 0 20 40 60 76 95 112
no
lH 6
a
19 38 56 72 90 110 [28 145 0 22 98 170 [90 210 230 249 266 288 305 325 348 370 393 450 499 612 88 90 7b
55 08
11n 138 160 188 21b 250 28' 321 355 390 413
OHZ 0
-5 -11 -IR -21 -25 -22 -21 -20 -18 -15 -12 -10 -9 -?5 -22 -20 -19 -15 -12 -10 -8 -4 -[
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3 8 10 12 16 20 -1
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2 5 10 11 15 11 20 23 1 -24
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3 9 10 13 19 20 21 25 30 32 4[ 50 70 -12 10 1 -?8 -22 -20 -19 -14 -10 -5 C R
14 20 25 30
C 5T
SST
PRF
-2C
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-127 -144
15 2 -14 -33 -4 R -b4 -51
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-206 -222 -23P -253 -135 -112 -12f. -140 -15') -170 -186 -201 -216 -230 -112 -92
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-151 -165 -lAO -195 -209 -93 -67 -93 -99 -113 -126 -141 -156 -170 -186 -69 -46 -61 -74 -90 -103 -118
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-68 -84 -98 -113 -126 -141 -25 -134 -194
-2';6 -273 -287 -303 -320 -!35 -'3'32 -368 -;R4 -404 -424 -451 -'S10 -539 -578 -184 -75 - 64 -139
=m -208 -221'1
-24'1 -269
-202 - 311,
- 340
-362 - 3"16 -4 C51').
3CO. 35C. 400. O. C. 50. 1 co. 150. 2CO. 250. 1CO ..
350. 'ICO. O. O. 50. 100. 150. 200. 250. 3eo. 3'50 .. 400.
O.
O. 50. 100. 150. 2eo .. 250. 3CO. 350. 4CC. O. O. 50. 1 CO. 150. 2CO. 250. 3CO. 350. 4CO. O.
O. 50. 100. 150. 2CO. 250.
3eo. 350. 400. O. O.
2eo.
400 .. 450. 5eo. 550. 600 .. e50. 70C. 750.
eco ..
850. 9CO. 950. 950. 950. Slb7.
O. O.
o.
O. 1CO. 1 '50 ..
76
2eo.
331::
350. 4CO. 450"
103 144 104 260 4Cl 460
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250.
3CO ..
sen
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,,-CALC
,.. APl C
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REFERENCES
1.
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2.
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3.
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4.
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6.
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7.
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8.
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9.
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10.
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11.
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13.
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1964 . . 1LI_.
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-
000 -