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A plate model for composite slab analysis
Thin-Walled Structures 10 (1990) 299-328
A Plate Model for Composite Slab Analysis
H. D. Wright Department of Civil and Structural Engineering, University of Wales College of Cardiff. Newport Road, Cardiff CF2 IXH, UK (Received 19 May 1989; accepted 28 September 1989)
ABSTRACT Composite floor decks constructed with profiled steel sheeting acting as formwork and reinforcement to a concrete slab are increasing in popularity. In this paper an elastic folded plate method of analysis has been applied to model their behaviour. The analytical model idealises both the steel sheeting and the concrete as a collection of thin-walled plates and separates the bending and shear action between horizontal and vertical elements. The effects of concrete in tension and slip between the concrete and steel have been included. The method has been used to analyse the results of 32full-scale slab tests and is found to give good results.
INTRODUCTION Composite slabs formed using profiled steel sheeting as a permanent formwork and tensile reinforcement to a concrete slab have now become a c o m m o n form of construction for floor decks in steel-framed buildings. A full description of the system has been presented in an earlier paper. I The design is carried out in two stages. The first ensures that the profiled steel sheeting will span between supporting steel work and can carry the weight of wet concrete and construction loads (construction stage loading). Following the construction stage loading, the concrete hardens and the finished slab is required to carry service loading. The system takes 299 Thin-Walled Structures 0263-8231/90/$03.50© 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain
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H. D. Wright
load in a composite manner with mid-span bending action being resisted by the concrete in compression and the profiled steel sheet in tension. The shear transfer between steel and concrete is provided by chemical and mechanical bond at the interface. The mechanical bond is provided by embossments rolled into the webs of the profile. A folded plate method of analysis has been used to model the behaviour of the profiled steel sheeting during the construction stage.: This paper describes an adaptation of this method which has allowed the behaviour of the composite slab to be modelled successfully. A detailed description of the structural behaviour of composite slabs will be presented first, followed by the analytical procedures adopted. Comparisons between the theory and test results will then be made. Finally, alternative uses for the analytical method will be suggested. The structural behaviour of the composite slab is influenced to a great extent by the concrete which forms the major part of the slab. In typical slabs the concrete is about 100-150 mm thick and the slab spans up to 3 m (see Fig. 1). The action of the concrete during loading is dependent upon many factors such as strength, density, aggregate type, workability during casting and even stress history. It is therefore very difficult to predict with accuracy the exact way in which the composite slab will behave. From tests carried out by the authors it has been possible to establish the general behaviour of a typical slab as it is loaded to failure. This will now be described. For very low loads the composite slab acts as an elastic beam with a single neutral axis generally in the concrete portion near the top of the section. Figure 2(a) shows the elastic stress block. This behaviour continues until the concrete in tension reaches its yield or cracking stress.
A
./ //
bght mesh reinforcement to control shrinkage
Fig. 1. The compositeslab system.
embossments
301
Plate modelfor composite slab analysis
k_eL:Concrete
stress
P'-I
Steel stress
(a)
u.d.l
:
L
,_•_. __NA.
x
i
/ F Z./
_
mid span stress
(b)
L÷
N.A.
cracked concrete mid span stress
(c)
(il Shear resisted by concrete in compcession. (ii) Shear resisted b y concrete aggregate i~erlock. (iii)Shoor resisted by steer.
Fig. 2. Composite slab failure mechanism.
This stress is generally in the range o f 0 . 5 - 3 . 0 N / m m 2 so it can be concluded that the fully elastic behaviour occurs only for relatively l o w load levels. The second stage o f behaviour occurs with the onset o f cracking. The cracks may initially be micro cracks but even so once these occur the concrete loses its tension capacity and this zone can be assumed to take no part in resisting moment. As a result, the neutral axis moves upwards,
302
H. D. Wright
(d)
Le÷+
/ mid span strains
(e)
sti~
(f)
Leeee~
Fig. 2. --contd
as shown in Fig. 2(b). It should be noted that as the tensile stress varies from maximum at mid-span to zero at the support the depth of the cracked concrete zone will also vary. Therefore the depth to neutral axis will also vary and the slab will no longer act as a prismatic beam. This gives rise to a certain degree of non-linear behaviour although in practice this is very small and is normally ignored in most methods of analysis. One further aspect of this stage of the behaviour is the transfer of shear in the cracked concrete and between the concrete and profiled steel sheeting. Indeed, the remaining stages of composite slab behaviour are often dominated by the loss of shear resistance. As the concrete in the tension zone cracks, its capacity to carry shear diminishes and the total shear on the section is redistributed to the uncracked concrete in compression above the neutral axis and to the profiled steel sheeting. As a result, the shear capacity of the section is reduced and shear deformation occurs. This is seen in load tests when the orientation of cracks propagating at about quarter-span of the beam change from vertical to become diagonal. This is shown in Fig. 2(c), where the inset diagram shows the diagonal tension mechanism.
Plate model for composite slab analysis
303
Taylor3 identified the proportion of shear resisted by each part of a conventional reinforced concrete beam as (a) 20--40% by the concrete in the compression zone, (b) 33-50% by aggregate interlock in the cracked concrete, (c) 15-25% by the dowel action of the reinforcing bars. Assuming that the composite slab is similar, it can be deduced that the concrete in the tension zone and the profiled steel sheet take a substantial part of the shear load. Although shear is carried by the aggregate interlock along the length of the shear cracks, the complementary longitudinal shear must be carried by the concrete-steel interface. The chemical bond formed by the cement paste with the steel is surprisingly strong and for low loads is adequate to carry this shear. However, as the load increases this bond begins to fail and more reliance is placed on mechanical interlock between the concrete and steel. The interlock resistance between the concrete and steel is greatly enhanced by the formation of mechanical keys in the webs of the profiled steel sheeting. These embossments, shown in Fig. 3(b). have derived from earlier attempts to provide shear transfer such as that shown in Fig. 3(a). The critical nature of the geometry of these embossments has been established in earlier papers 4. 5 and will be discussed below. As the load on the slab increases the embossments are called upon to transfer more and more shear. This causes some slip between the two materials as the concrete tends to rise up and over the embossments, as shown in Fig. 2(e). The amount of slip depends to a great extent on the embossment geometry, especially its depth, but up to 2 mm of movement can be resisted before collapse occurs. Crisinel et al. 6 identified two types of collapse: the first, a brittle collapse, occurs when the embossments can carry little load, which leads to a slab failure soon after the chemical bond between steel and concrete breaks down. The second, a ductile collapse, occurs when the embossments continue to carry shear load after the chemical bond has been broken. Figure 2(0 shows the typical final collapse mechanism for both failure types. The load to mid-span deflection relationship for the composite slab as it is loaded through each of these stages is shown in Fig. 4. It can be seen from this figure that the stiffness of the slab depends upon the stage of behaviour reached. However, as the behaviour of the slab is relatively unpredictable once the embossments are taking their full capacity and involves considerable plastic deformation the work here will be restricted to the description and analysis of slab stiffness up to the end of stage d.
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H. D Wright
(a)
R~nforcement attached to steel by welding
ssments
(b)
Fig. 3. Methods of providing shear transfer.
LOAD End slip
r
Mid span deflectlorL
OISPL.A~I~NT
Fig. 4. Typical load deflection and load slip relationship.
Plate model for composite slab analysis
305
Bending stiffness As described in the previous section, the stiffness of a composite slab is dependent upon the extent of cracking in the tension zone of the concrete. Although this cracking may occur at relatively low levels of tensile stress the exact extent of cracking is often difficult to detect. In addition, although the depth of cracked section may vary along the beam, the effect of this is small in comparison with overall behaviour. Consequently, a reasonable assumption is that the composite slab will behave as a prismatic beam and that the concrete in tension carries no direct bending stress. This would appear to be at least a safe assumption although, as concrete can take some tension, it is possibly conservative. The British code of practice for the design of composite slabs, BS5950: Part 4,7 recognises this apparent conservatism and allows the analysis to take an average stiffness of the cracked and uncracked section. This assumes that the concrete will be carrying some tension. BS5950 goes further in suggesting that the stiffness of the slab in either its cracked or uncracked form can be calculated using simple beam analysis with a transformed section or modular ratio method. The basis for this method assumes that the concrete cross-sectional area can be transformed into an equivalent steel area and that the resulting section may be treated as an elastic prismatic element. Figure 5 shows the steps in the analysis for an uncracked composite beam and Fig. 6 shows the corresponding steps for a cracked section. The depth to the neutral axis depends upon the height of tension cracks in the cracked section analysis and this can only be found once the stress block has been determined. Consequently, either an iterative approach or the formation of a quadratic equation is needed. Figure 6 shows the steps in analysis leading to quadratic eqn (1): dna2 (bm/2) + d n a a s t - ,4stdnast = 0
(1)
Fortunately, there is normally only one real root to this equation. The tedious calculations required for both cracked and uncracked section analysis have been programmed for computer solution by the author and this program has been used later in a comparison with the folded plate method of analysis.
Shear stiffness The simple beam analysis allowed in the code is based on the assumption that shear deformation is small enough to be ignored and
H. D. Wright
306
strona
conc stress
steel stress
N.A
1)
UNCRACKED
Ac
= Area concrete
Act
m
=mA c
=
modular
ratio
Ast = Area steel Ic
= Second morn. area concrete
Ict = m . I c
Ist = Second morn area steer. dna c = Depth to N A . concrete
(from top of slab)
dnast = Depth to N.A. steet (from top of slab) dna
= {Act ~ dna c ° A s t x dnast) /
rst
= Absotute value of
dna- dnast
rct
= Absolute value of
dna-dnac
Is
= Ist •
(Astxrst 2 )
•Ict
*
(Act
* Ast )
( A c t = rct 2 )
Fig. 5. Uncracked section analysis.
the deflection can be calculated using the single term d = x (I~L3/E1)
(2)
where x is the coefficient describing the loading condition, e.g. 5/384 for uniformly distributed loads. This assumption is very nearly correct for most common structural materials and proportions as shear deformation is normally very small in comparison with that caused by bending. However, in the case of the composite slab the cracked concrete and the slip between the concrete and steel may give rise to relatively large shear deformations which have a much greater effect on the overall deflection of the slab. Figure 7(b) shows the exaggerated deformation pattern under pure shear, assuming that the concrete has not cracked and acts in an elastic manner. It can be seen that the shear stress gives rise to both vertical and longitudinal strain. Once the concrete begins to crack the longitudinal strain can be resisted only by the shear bond transfer through the friction afforded by the embossments as shown in Fig. 7(c). This transfer is often
Plate mode/for composite slab analysis
307
b r
/
¢0n¢ stress
strain
NA. .
f 2)
7
steel str. .
Jl
_
Z-I
CRACKED
A c = Area concrete = b= dna
Act = rn.Ac
rn= m o d u l a r
ratio
Ast = Area stee| [c
= Second morn. area concrete
!ct = rn.I c = m.b.dna3 / 12
Ist = Second morn area steel dna c = Depthto N A concrete = d n a / 2 dnast = Depth to N A steel (from bop of slab1 dna
=(Act K clnac .
Ast=clnast) / (Act • Ast )
Clna 2 ( b.rn/21 -Ast dna - Ast clnast= 0 Is
calculatecl as before. Fig. 6. Cracked section analysis.
relatively flexible and allows slip deformation, which has a tendency to open the cracks still further. As the cracks widen the capacity for vertical shear resistance through aggregate interlock is reduced and more of the shear is redistributed from the concrete to profiled steel sheet interface. The amount of shear deformation is therefore dependent upon the extent of cracking~ the width of the cracks propagated and the stiffness of the longitudinal shear transfer between the concrete and profiled steel sheeting.
The effect of ponding So far in this description of the composite slab it has been assumed that the overall cross-section is uniform along the complete length of the slab. This is not always true, as the slabs are normally cast simply supported and considerable deflection often occurs because of the self weight of the wet concrete. The effect of ponding on the bending stiffness of the composite slab may be demonstrated using a simple plane frame stiffness approach to obtain the deflection of three beams under the same load.
H. D. Wright
308
B.Md~,
(a)
(b)
t .... (c)
Fig. 7. Assumed deformation pattern. Let us consider the three beams shown in Fig. 8. The first simulates the shape of a beam that has ponded under construction loading, the second assumes that the effective thickness is the nominal thickness at the support and the third assumes that the effective thickness is that at midspan ofthe beam. The results ofapplying identical uniform loads to each of these beams are also shown in Fig. 8. It can be seen that the error in using the support thickness amounts to 64%, whereas the error in using the mid-span thickness is only 6%. This high error will, of course, be reduced for profiled steel sheets that are stiff enough to resist large ponded deformations during construction. Consequently, it would appear sensible to use the ponded depth in calculating the section modulus for bending stiffness determination. This comparison does not hold for shear deformation as the maximum shear load occurs at the support, the thinnest section of the ponded slab. However, as shear deformation is generally less than
Plate model for composite slab analysis
309
~9
(a)
(b)
l
~
1
B
(c) 0"5.
i
2
~ DEF ram6'-
Fig. 8. ldealisationof ponding. bending deformation the error in assuming the ponded thickness for shear as well as bending deformation is acceptable. Ultimate load determination
This review of the determination of the stiffness of composite slabs is restricted to code requirements and the author's observations on these. This is due in part to the limited emphasis placed on stiffness in comparison with ultimate load determination.
H. D. Wright
310
The prediction of ultimate load capacity has been based on the US code for reinforced concrete design 8 which, in common with the British code BS8110fl uses limit state methods of design. Both codes deal primarily with reinforced concrete, which, in general construction, is a relatively stiff medium. Serviceability checks are usually restricted to satisfying maximum span to depth ratios. For simply supported slabs the maximum span to depth ratio is 20, although this may be modified if the percentage of tension reinforcement is high. Composite floor slabs occasionally exceed this ratio and would therefore require special attention according to BS8110. The special attention includes not only a deflection check assuming a fully cracked section but also control of crack widths so that the buried, unprotected reinforcing bars do not rust. This latter requirement does not apply to composite slabs, and, as suggested earlier, a fully cracked section analysis may be conservative if shear is assumed to be negligible. The British code specific to composite slabs, BS5950: Part 4." is also a limit state code and, as described earlier, has specific requirements on the deflection limit state. By far the largest part of the code deals with the ultimate limit state calculations for load capacity. A detailed study of the code procedure and of several other methods ofanalysis has been made.4 5 In the references it is shown that the procedures for design are based on empirically derived equations which rely on expensive test information for each type of profiled steel sheet used. Consequently, many researchers have turned their attention to either refining the code method or alternatively trying to derive new, more theoretically based, solutions. Seliem and Schuster ~°and Prasannan and LuttrelP ) used widely differing methods to reduce the reliance on test information. Although a definitive theoretically acceptable solution to the ultimate load capacity of composite slabs has not yet been obtained, the author believes that a greater understanding of the flexural behaviour will be of considerable benefit.
THE FOLDED PLATE ADAPTATION
General description Goldberg and Leve first presented the folded plate method in 1957 ~2 when they considered the specific case of loads applied to the edges of the system of plates. Evans u extended this to include the action of uniform loading on the plates, and also described the method in some detail. The method uses elastic plate flexure and plane stress theory in
Plate model for composite slab analysis
311
conjunction with a matrix stiffness approach to the solution of the equations of equilibrium and compatibility. It would, at first, seem less than ideal to attempt to model the solid body geometry of the composite slab using this thin plate model. However, it will be shown in this section that the concrete can be modelled as relatively thin horizontal plates jointed to the steel sheet by vertical plates carrying only shear.
Solid body modelling using plate elements The folded plate method of analysis is ideally suited to structures formed from several thin plates joined along their edges. Two assumptions made in the analysis would appear to conflict with its potential use to model solid bodies. The first is the requirement that component plates may be represented by uniform thin plates exhibiting similar properties. The second is the requirement that component plates be joined at a single node point. These two requirements and the difficulties associated with each are described below. (a) The basis for the derivation of the stiffness matrix for each plate in the folded plate structure lies in the assumption that the behaviour of the plate accords to simple elastic plate flexure and twodimensional membrane theories. Both of these rely on the assumption that forces and moments can be assumed to act along the mid-plane of the plate. These assumptions are found to be valid where the span and breadth to thickness ratio of the plates are high and the radius of curvature of the deformed plate is large in comparison with the thickness. In deriving these theories the normal and shearing stresses through the depth of the plate have been ignored. Although the concrete elements shown in Fig. 9 have low breadth to thickness ratios their length to thickness ratio is very high even for short spans. Consequently, the analysis is reasonably accurate when considering longitudinal forces. In addition, the basic theory is restricted to plates of uniform thickness. In modelling a composite slab the sloping webs of the profiled steel sheeting can give rise to plates representing concrete of variable thickness as shown in Fig. 9(a). It is possible to develop the method to cater for plates of variable thickness but this additional complication has not been carded out here. Instead, a stepped plate model has been adopted where necessary, as shown in Fig. 9(b). (b) It will be shown later that with little loss of accuracy the folded plate method can be used to model relatively thick plates.
312
H. D. Wright
Plate re~esenting.
a) .,~ stepped plates of / / uniformthk:knes.
bl
~
duplicatedarea
¢) Fig. 9. Folded plate idealisation.
However, this brings to light the second difficulty associated with solid body modelling. In Fig. 9(a) the profiled steel sheeting and the concrete are shown as two sets of thick plates acting at the centreline of the actual plate. Each joint or node between plates is shown as a single point. With thick steel plates little error occurs with this assumption as the duplication of plate area, as shown in Fig. 9(c), is very small. However, the concrete plate and the steel plates cannot be connected together without changing the effective plane of the plate. A change of this nature would entail major rewriting of the theory and computer program and has not been attempted. As an alternative, a more flexible approach involving the use of'dummy' elements has been developed. The final configuration of the model used is described in the next section.
Plate modelfor composite slab analysis
313
The composite slab model As outlined earlier, the main bending action in a composite slab is registered by a couple formed from compression in the concrete and tension in the steel. Shear resistance, although an essential part of the slab action, has less effect on the overall stiffness than the resistance of this couple to moment. Consequently, a system of plates can be envisaged that separate the bending action and shear action of the system. Figure lO shows this system of plates for a single pitch of a composite floor slab. It can be noted that the main horizontal plates will take the bending action and the vertical plates the shear. In this simulation it is assumed that most of the concrete in the tension zone has cracked and the remaining concrete carries mainly compression. The actual thickness of the concrete plate carrying compression is dependent upon the depth to the neutral axis. Initially, this has been calculated using simple beam theory. Consequently, the concrete in bending compression can be modelled as a thin plate lying at the midheight of the uncracked concrete.
concreto
I| ossumed concrete crocked
~"
'@mmY'sn~. / /,~elements
sIN| p|otes
;,6 e
230
J
Fig. tO. The final folded plate model.
1-
314
H. D. Wright
The 'dummy" elements connecting this plate to the profiled steel sheeting transfer shear between the concrete and the steel. To do this, they have to possess sufficient shear stiffness to be equivalent to the concrete they represent. If these plates were given the actual dimensions and material properties of the concrete they would also have significant bending stiffness which, as the concrete has cracked, is not correct. The 'dummy" plates must, therefore, combine high shear stiffness with low bending stiffness. In an isotropic elastic material the relationship between bending stiffness and shear stiffness is related by the laws of elasticity. In the two-dimensional plate theory used in the basic folded plate formulation this relationship is G = E / 2 ( 1 + v)
(3)
Any change in the Young's modulus of the dummy plate will cause a proportional change in the shear modulus. Unfortunately, the change in Young's modulus causes an equal change in bending stiffness which is to be avoided. A change in Poisson's ratio will also affect the shear modulus, but in this case it will have little effect on the bending stiffness. In the extreme case, the Poisson's ratio may be made negative. The shear modulus of concrete assuming that it is an elastic material is approximately 10 000 N/mm:. A plate of concrete b mm wide would have a shear stiffness of 10 000 b N/mm. The two dummy shear plates, shown in Fig. 10, which have the same shear stiffness, can be evaluated by equating this stiffness using eqn (3): 10000b = E × 2 t / 2 ( l + v ) Putting v -- - 0.999 and t = 0.1, the equivalent dummy plate Young's modulus becomes lO0/b. This is shown in Fig. 11. This principle has been applied to the composite slab model shown in Fig. 10(a). The bending stiffness of the dummy shear plates amounts to 0.08% of the bending stiffness of the complete section as derived using simple beam theory and a cracked section. The folded plate analysis gave a mid-span deflection of 11.08 mm for an imposed load of 20 kN/m 2. The simple beam theory gave a deflection of 11.25 mm for the same load. This shows that the two theories give approximately equal deflection. The additional deflection (1.5%) found in the folded plate method is most probably due to shear deformation of the concrete. This is not included in the simple beam analysis. One further observation that must be made is in the deformation pattern of the dummy shear plates. Most of the plate is in tension as well as shear and consequently the negative Poisson's ratio has the effect of expanding the element. Consequently, the breadth of the dummy plate
Plate modelfor composite slab analysis
315
\ X
N i",,
6 r. l O 0 0 0 b
t
X
N
t
G~ E I Z t
.V. -- -t)999 Fig. ! !. Equivalent d u m m y shear elements.
increases, giving rise to a slight discrepancy between the concrete plate deformation and the profiled steel sheet deformation. This is very, small, less than 0.03 mm, and is ignored. With the variation between the folded plate analysis and the simple beam method amounting to only 1.5% it is possible to develop the folded plate method further to allow for more accurate analysis of the shear bond behaviour. The cracked concrete section
In the last section the thickness of the concrete plate was predetermined using the simple beam analysis before substituting this value in the folded plate analysis. There is, however, no requirement to predetermine the concrete plate thickness as it can be established using the iterative approach.
316
H. D. Wright
The neutral axis of a composite slab normally falls within the area of concrete above the top flange ofthe profiled steel sheeting. If an arbitrary thickness of concrete in compression is chosen as the depth of concrete from the surface of the slab to the top flange of the sheeting, an initial folded plate analysis can be carried out. The results of this analysis will give a stress distribution within the concrete plate similar to that shown in Fig. 12(a). The stresses may be evaluated from the longitudinal forces and bending moments which are obtained from the analysis. If the thickness of the concrete plate is reduced and the length of the d u m m y shear plate extended, a second analysis will result in the stress distribution shown in Fig. 12(b). By changing the thickness of the concrete plate and the length of the d u m m y shear plate to suit, eventually a stress distribution will result which gives no concrete in tension. This procedure has been incorporated into a specific folded plate program, and the flow chart for this procedure is shown in Fig. 13. With (a)
\
/
(b) •
I
I
~1
~ - ~
steel
stress.
concrete compressive stress. concrete tensile stress. Fig. 12. Cracked concrete modelling using folded plate analysis.
Plate model for composite slab analysis
317
INPUT INITIAL GEOMETRY
ANALYSE
COMPUTE 1,410SPANSTRESS
A.MEND GEOMETRY
GREATERTHAN . /
I PRINT RESULTS Fig. 13. Computer flow chart for modelling concrete cracking.
this iterative method the concrete can either be assumed to be unable to take tension or can be assumed to be capable of carrying a limited value of tensile stress. This can be achieved by adjusting the 'checking limit" as shown in the flow chart. To test this method the slab described in Fig. l0 was analysed. The convergence of the method to an acceptable level is complete within three iterations and the final results were only 1.48% different from those achieved using the simple beam analysis. Test observations of composite slab stiffness The alterations to the basic folded plate analysis described in the previous section have been made as a result of observed behaviour of slabs under test. In the development of the analysis mainly qualitative
318
H. D. Wright
considerations have so far been incorporated. In this section quantitative comparisons are made between the method developed and test information. Four series of tests have been made on composite slabs and this experience has been used in developing the method. The test series. reported in Refs 14-17, were all carded out to evaluate the ultimate performance of particular profiled steel sheet slabs according to BS5950 Part 4.
Comparisons between analysis and test results Using the test data the slabs were analysed using the folded plate method. The ponded depth of each slab was evaluated using the folded plate method developed in Ref. 2 and this depth was then used as the slab thickness. The Young's modulus of concrete was determined from the cube strength and density quoted in the test reports. The results of the analysis are shown in Fig. 14 as comparisons with the test results. The equivalent second moment values used on both axes relate to the initial slope of the graph produced by plotting load against mid-span deflection. It was assumed that this equivalent value could be determined from the simple beam relationship I = 5wL4/384Ed
(4)
A perfect correlation between test and analysis would result in all the results lying on a 45 ° line. It can be seen that, in general, the analysis overestimates the stiffness of the slabs. This discrepancy is most probably due to slip between the concrete and steel as described above.
Shear deformation and slip The load to deflection response, which is the subject of this investigation, is a serviceability check in the British standard and consequently this information is produced only in passing in these tests. In fact. the stiffness values shown in Table 1 are produced after each slab has been loaded and unloaded some 10 000 times. This cyclic loading has the effect of reducing the chemical bond between the concrete and steel so that interface slip will be likely to occur throughout the load range. Even so, the end slip, measured in the tests, appears not to start until the slab is heavily loaded. Figure 15 shows a typical load slip response for one of the tests and it can be seen that end slip occurs in a large jump before increasing more
Plate modelfor composite slab analysis
319
/.0(:30
Equwalent I cm6/m. crocked section futly composite f p. method. . 300C
4,
'~
4,
2000,
I000
Equivalent I
test L
1000
2000
3000
l.O00
Fig. 14. Comparison between fully composite folded plate model and test results.
steadily to failure. It should be noted that the stiffness of the slab is less than even the fully cracked section analysis would suggest, indicating that slip is occurring throughout the load range. An explanation for the lack of end slip early in the loading range may lie in the fact that friction between the concrete and steel sheet occurs in the support region due to the reaction. This may be sufficient to prevent movement up to this load. Incorporating end slip into the folded plate model Shear deformation, whether elastic in the concrete or due to slip, has been shown to have a similar effect in Fig. 7. The total deformation could therefore be thought o f as either elastic deformation or slip.
H.D. Wright
320
Load kN/m 2
30
20
deflect ion slip. crackcKI section
10
onalysis.
cl~flection (ram) end slip =101mml v
5
10
15
20
25
30
Fig. 15. Load deflection and load slip curve for Test 2. Series l.
Elastic deformation can be modelled analytically in the folded plate analysis by controlling the shear stiffness of the dummy shear plates. A low value of shear stiffness will give rise to large shear deformation equal to that caused by slip. This is shown diagrammatically in Fig. 16. The shear stiffness of the dummy elements in the slabs in test series one was adjusted so that the deflection produced matched the test results. The resulting reduction in dummy plate shear stiffness was remarkably consistent, as shown in Table 1. Only slab 1 showed any inconsistency. This was the first slab tested in this series and in fact was cycled in excess of 30 000 times. Consequently, the behaviour of this slab was assumed to be affected by this procedure and the results were ignored in calculating the average dummy shear stiffness for the test series. It was therefore thought possible that the deflection of the slabs in the other series of tests could be predicted more accurately by using dummy shear stiffnesses evaluated for each test and the average values used.
Plate modelfor composite slab analysis
321
concrete shear
I
+
slid
I m
equivalent
total
i
Fig. 16. The summation of concrete shear and slip.
Figure 17 shows the resulting predicted stiffness of each test speciment in comparison with the test values. The improvement is marked when the results are compared with Fig. 14 (which shows the same results without modification by the reduced dummy shear stiffness). There are, however, some slabs that fall below the 45 ° line, which indicates that they are more stiff than the analysis would suggest. These slabs were, in fact, some ofthe longest thinnest spans and were also those slabs for which it appeared that the bond between the concrete and steel did not completely fail during cycling. This manifested itself in a more progressive ductile failure.
The practical application of the folded plate method It has been shown in the last section that the application of a folded plate approach using a dummy shear stiffness obtained from test results significantly improves the prediction of slab stiffnesses. It must be remembered, however, that the method has been developed from test data and that the tests used do not entirely represent the situation that the
H.D. Wright
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TABLE I D u m m y Shear Stiffness for Slabs Tested
Test series
Slab
Dummy shear stiffness assuming full interaction (N/mm)
Dummy shear stiffness to match test result (N/mm)
i 2 3 4 5 6 7 8 9 I0
7 840 15 525 9 928 14 270 17 348 14 691 I I 925 12 580 13 376 16 594
220 140 230 350 350 220 220 150 230
I 2 3 4 5 6 7 8 9 10
9 075 6 078 8 546 6 177 8 789 5 873 8 442 7 224 7 563 9461
90 500 1200 500
1 2 3 4 5 6
8 429 8372 8 372 8 273 8 273 8 273
a 350 550 450 305 285
388
i 2 3 4 5 6
23131 23131 23131 23131 23131 23131
° ° a 236 198 150
195
~,,,~ ,,tin 800
Average for series (N/mm)
234
680
a
°Slabs were found to be as stiffas, or stiffer than, fully composite cracked section analysis a n d were ignored when calculating average d u m m y shear stiffness.
Plate model for composite slab analysis
323
t.O00
Equwalent I crn6/m. crocked section f.p. method incorporating slip 3000
2000
1000
.//
+* *
Equivalent I test 1000
2000
3000
&O00
Fig. 17. Comparison between partially composite folded plate model and test results.
slab would occupy in actual construction. It is unlikely that the slab would be fully simply supported as in the test and it is also unlikely that the cyclic load regime of the tests would occur in practice. However, it does provide a lower bound solution and may indicate a first slip load. Let us consider the end distortion of one of the slabs as obtained from the analysis, as shown in Fig. 18. It is possible to evaluate the effective end slip from the relative positions of the nodes as shown in this figure. Consequently, an effective end slip can be found for each load. The shear load and slip occurring in the slab may be affected by the geometry of the embossments. This is possibly true as long as the slip is small. Figure 19 shows the assumed action of the concrete shearing against an embossment. Initially, the increase in load causes the concrete to rise up over the embossment. The shear load is proportional to the applied load and consequently the rate of slip is proportional to the applied load. This will continue until the concrete has risen to the full
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H. D. Wright
)P q9 q8 G7 0:6 0~5.~ 0:3 0.~ 03
0.1 02 0:3 0~.0:5 ~.6 q7 if0_ ram. 1
J ~22/
/ /
jo
equivallnt end slip
Exaggerated Qnd displacements slab 2
Fig. 18. Equivalent end slip evaluation.
height of the embossment, when a large a m o u n t of slip will occur. This sudden slip will most probably cause failure. The relationship between slip a n d shear load is complicated by the fact that the embossments are placed on the web of the profiled sheeting. The concrete needs to displace the steel sheet sideways as well as lifting it vertically. Consequently, it would be difficult to attempt to evaluate a slip modulus from the geometry of the sheet, although it may be possible to obtain a value from small-scale tests. Several researchers have c a r d e d out experimental studies on small-scale tests. Hassaine, 18 Jolly and Zubair, 19 Crisinel a n d Daniels 2° a n d Brekelmans 21 have all c a r d e d out pilot studies. Unfortunately, most of the information relates to the ultimate load c a r d e d and none is specific to the profiles tested by the author. If both the slip modulus and the ultimate slip load can be determined from model tests then these may be readily incorporated in the folded plate method. By restricting the end slip to a small value likely to be the strain limit for concrete adhering to steel, we may also be able to predict a first slip load. This will be slightly conservative unless the frictional resistance of the support is taken into account. The use of small-scale model tests to evaluate parameters that may be used in a theoretically meaningful analysis is of significant economic
Plate modelfor composite slab analysis
325
/
\ /
L~od
E Fig. 19. Assumed shearing action of concrete against an embossment.
benefit to producers of profiled steel sheet. The current empirical test regime is expensive, time-consuming and is criticised for its lack of a theoretical basis. It is hoped that further work on model testing will allow the method described here to be developed to expand the ideas proposed in this section. Conclusions
The application ofthe folded plate method of analysis to composite slabs formed with profiled steel sheeting has been outlined. The work concentrates on the development of the basic method to overcome the problems associated with solid body geometry, the lack of tensile capacity in the concrete and the effects of slip between the concrete and steel. As a result of this work, the folded plate method has been found to be a suitable tool for the analysis of solid beams. This has been achieved by separating the bending action and shear action present in the beam. Horizontal plates take the major longitudinal forces associated with pure bending. The shear action is taken by vertical 'dummy" shear plates. The 'dummy' shear plates allow a more accurate analysis of shear deformation and allow a closer prediction of lateral deformation of the steel sheeting.
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By using an interative process the action of concrete in bending has been successfully modelled. This involves the use of a tension cut-off principle although limited tensile stiffness could be incorporated easily. In composite slab behaviour the presence of slip between the two materials causes non-linear behaviour. This behaviour has been modelled by incorporating the slip as a part of the shear distortion in the slab. By introducing a low shear modulus to the dummy elements it has been found possible to model closely the behaviour of 32 composite slab tests.
Alternative applications Several fundamental structural actions have been analysed using the folded plate method. These actions are not unique to the composite floor system described here. Other structural systems involve geometric nonlinear behaviour, and include concrete and other elements which are only partially and often discretely connected together. One of the most widely used structural systems is reinforced concrete. The similarity with the composite slab makes it an ideal first example to analyse using the folded plate method. Kristek and Smerda '2 used the folded plate method to investigate not only reinforced concrete bridges but also prestressed and post-tensioned structures. In addition, they studied the effects of creep and shrinkage. Another composite structure is the sandwich panel formed from two profiled steel sheets filled with a lightweight low-density foam. The 'dummy' shear elements described here could be adapted to model this foam core, which has a very low shear stiffness. A different type of sandwich panel is the double skin composite structure proposed for immersed tube tunnels? 3This comprises two steel plate skins separated by a concrete infill. Connection between the skins and the concrete is provided by stud shear connectors. Again the use of the 'dummy' vertical shear element would allow analysis of panels with incomplete or flexible connection.
REFERENCES 1. Wright, H. D., Evans, H. R. & Harding, P. W., The use of profiled steel sheeting in floor construction, Journal of Constructional Steel Research, 7 (1987) 279-95. 2. Wright, H. D., & Evans, H. R., A folded plate method of analysis for profiled
Plate modelfor composite slab analysis
3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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steel sheeting in composite floor construction, Thin-Walled Structures 5 (1987) 21-37. Taylor, H. P. J., Shear strength of large beams, Proc. American Society of Civil Engineers, 98 (ST(II))) (1972) 2473-90. Wright, H. D., & Evans, H. R., Observations on the design and testing of composite floor slabs, Steel Construction Today. 1 (1987) 91-9. Wright, H. D., Evans, H. R. & Harding, P. W., Composite floors: comparisons of performance testing and methods of analysis, Proc. International Symposium Steel in Buildings. IABSE-ECC, Luxembourg, 1985, 219-25. Crisinel, M. J., Fidler, M. J. & Daniels. B. J., Behaviour of steel deck reinforced composite floors, Proc. IABSE Colloquium. Thin-Walled Metal Structures in Buildings, Stockholm, 9-12 June 1986, 279-90. BS5950: Part 4, Structural use of steel in building. Code of Practice for design of floors with profiled steel sheeting, British Standards Institution, London, 1982. American Concrete Institute, (A.C.I. 318-77) Building Code requirements for reinforced concrete, A.C.I. Detroit, 1977. BS8110, Structural use of concrete, Part I: Code of practice for design and construction, British Standards Institution, London, 1985. Seliem, S. S. & Schuster, R. M., Shear-bond capacity of composite slabs, Proc. 6th Specialist Conference on Cold-Formed Structures, University of Missouri-Rolla, November 1982, 511-31. Prasannan, S. & Luttrell, L. D., Flexural strength formulations for steel deck composite slabs, Report of the Department of Civil Engineering, West Virginia University, 1984. Goldberg, J. E. & Leve, H. L., Theory of prismatic folded plate structures, Publ. Int. Assn. Bridge and Struct. Engineers. 17 (1957) 59-86. Evans, H. R, The analysis of folded plate structures. PhD thesis, University of Wales, Swansea, 1967. Evans, H. R. Wright, H. D. & Harding, P. W., Full scale load tests on P.M.F. Ltd. CF46 composite flooring profile, Internal Report UCC/PMF/CF60, 1986. UCC Composite Structures Group, Assessment of P.M.F. CF60 for both construction and service loading, Internal Report UCC/PMF/CF60, 1986. UCC Composite Structures Group, Composite tests on Alphalok decking, Internal Report UCC/Alpha/Alphalok 1, 1988. UCC Composite Structures Group, Assessment of the composite performance of PMF/CF60 under service conditions, Internal Report UCC/PMF/ CF60/2, 1987. Hassaine, N-E., Shear-bond performance of profiled steel sheets when used as composite floor decks. MSc thesis, University College, Cardiff, 1987. Jolly, C. 14,. & Zubair, A. K. M., The efficiency of shear-bond interlock between profiled steel sheeting and concrete, Proc. Conference on Steel and Aluminium Structures, Cardiff, 8-10 July 1987, 127-36. Crisinel, M. J. & Daniels, B. J., Verbal presentation of small scale tests on composite slabs, ECCS TC 7.6 Working Group Meeting, Delft, November 1987.
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21. Brekelmans, J. W. P. M., Verbal presentation of small-scale tests on composite slabs, ECCS TC 7.6 Working Group Meeting, Delft, November 1987. 22. Kristek, V. & Smerda, Z., Changes of the state of stress of a prestressed concrete bridge assembled of precast segments, Prestressed Concrete inCzechoslovakia Inzeuyrska Stavebuy, May 1982. 23. Narayanan, R., Wright, H. D., Francis, R. W. & Evans, H. R.. Double-skin composite construction for submerged tube tunnels, Steel Construction Today. 1 (1987) 185-9.
A Plate Model for Composite Slab Analysis
H. D. Wright Department of Civil and Structural Engineering, University of Wales College of Cardiff. Newport Road, Cardiff CF2 IXH, UK (Received 19 May 1989; accepted 28 September 1989)
ABSTRACT Composite floor decks constructed with profiled steel sheeting acting as formwork and reinforcement to a concrete slab are increasing in popularity. In this paper an elastic folded plate method of analysis has been applied to model their behaviour. The analytical model idealises both the steel sheeting and the concrete as a collection of thin-walled plates and separates the bending and shear action between horizontal and vertical elements. The effects of concrete in tension and slip between the concrete and steel have been included. The method has been used to analyse the results of 32full-scale slab tests and is found to give good results.
INTRODUCTION Composite slabs formed using profiled steel sheeting as a permanent formwork and tensile reinforcement to a concrete slab have now become a c o m m o n form of construction for floor decks in steel-framed buildings. A full description of the system has been presented in an earlier paper. I The design is carried out in two stages. The first ensures that the profiled steel sheeting will span between supporting steel work and can carry the weight of wet concrete and construction loads (construction stage loading). Following the construction stage loading, the concrete hardens and the finished slab is required to carry service loading. The system takes 299 Thin-Walled Structures 0263-8231/90/$03.50© 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain
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load in a composite manner with mid-span bending action being resisted by the concrete in compression and the profiled steel sheet in tension. The shear transfer between steel and concrete is provided by chemical and mechanical bond at the interface. The mechanical bond is provided by embossments rolled into the webs of the profile. A folded plate method of analysis has been used to model the behaviour of the profiled steel sheeting during the construction stage.: This paper describes an adaptation of this method which has allowed the behaviour of the composite slab to be modelled successfully. A detailed description of the structural behaviour of composite slabs will be presented first, followed by the analytical procedures adopted. Comparisons between the theory and test results will then be made. Finally, alternative uses for the analytical method will be suggested. The structural behaviour of the composite slab is influenced to a great extent by the concrete which forms the major part of the slab. In typical slabs the concrete is about 100-150 mm thick and the slab spans up to 3 m (see Fig. 1). The action of the concrete during loading is dependent upon many factors such as strength, density, aggregate type, workability during casting and even stress history. It is therefore very difficult to predict with accuracy the exact way in which the composite slab will behave. From tests carried out by the authors it has been possible to establish the general behaviour of a typical slab as it is loaded to failure. This will now be described. For very low loads the composite slab acts as an elastic beam with a single neutral axis generally in the concrete portion near the top of the section. Figure 2(a) shows the elastic stress block. This behaviour continues until the concrete in tension reaches its yield or cracking stress.
A
./ //
bght mesh reinforcement to control shrinkage
Fig. 1. The compositeslab system.
embossments
301
Plate modelfor composite slab analysis
k_eL:Concrete
stress
P'-I
Steel stress
(a)
u.d.l
:
L
,_•_. __NA.
x
i
/ F Z./
_
mid span stress
(b)
L÷
N.A.
cracked concrete mid span stress
(c)
(il Shear resisted by concrete in compcession. (ii) Shear resisted b y concrete aggregate i~erlock. (iii)Shoor resisted by steer.
Fig. 2. Composite slab failure mechanism.
This stress is generally in the range o f 0 . 5 - 3 . 0 N / m m 2 so it can be concluded that the fully elastic behaviour occurs only for relatively l o w load levels. The second stage o f behaviour occurs with the onset o f cracking. The cracks may initially be micro cracks but even so once these occur the concrete loses its tension capacity and this zone can be assumed to take no part in resisting moment. As a result, the neutral axis moves upwards,
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H. D. Wright
(d)
Le÷+
/ mid span strains
(e)
sti~
(f)
Leeee~
Fig. 2. --contd
as shown in Fig. 2(b). It should be noted that as the tensile stress varies from maximum at mid-span to zero at the support the depth of the cracked concrete zone will also vary. Therefore the depth to neutral axis will also vary and the slab will no longer act as a prismatic beam. This gives rise to a certain degree of non-linear behaviour although in practice this is very small and is normally ignored in most methods of analysis. One further aspect of this stage of the behaviour is the transfer of shear in the cracked concrete and between the concrete and profiled steel sheeting. Indeed, the remaining stages of composite slab behaviour are often dominated by the loss of shear resistance. As the concrete in the tension zone cracks, its capacity to carry shear diminishes and the total shear on the section is redistributed to the uncracked concrete in compression above the neutral axis and to the profiled steel sheeting. As a result, the shear capacity of the section is reduced and shear deformation occurs. This is seen in load tests when the orientation of cracks propagating at about quarter-span of the beam change from vertical to become diagonal. This is shown in Fig. 2(c), where the inset diagram shows the diagonal tension mechanism.
Plate model for composite slab analysis
303
Taylor3 identified the proportion of shear resisted by each part of a conventional reinforced concrete beam as (a) 20--40% by the concrete in the compression zone, (b) 33-50% by aggregate interlock in the cracked concrete, (c) 15-25% by the dowel action of the reinforcing bars. Assuming that the composite slab is similar, it can be deduced that the concrete in the tension zone and the profiled steel sheet take a substantial part of the shear load. Although shear is carried by the aggregate interlock along the length of the shear cracks, the complementary longitudinal shear must be carried by the concrete-steel interface. The chemical bond formed by the cement paste with the steel is surprisingly strong and for low loads is adequate to carry this shear. However, as the load increases this bond begins to fail and more reliance is placed on mechanical interlock between the concrete and steel. The interlock resistance between the concrete and steel is greatly enhanced by the formation of mechanical keys in the webs of the profiled steel sheeting. These embossments, shown in Fig. 3(b). have derived from earlier attempts to provide shear transfer such as that shown in Fig. 3(a). The critical nature of the geometry of these embossments has been established in earlier papers 4. 5 and will be discussed below. As the load on the slab increases the embossments are called upon to transfer more and more shear. This causes some slip between the two materials as the concrete tends to rise up and over the embossments, as shown in Fig. 2(e). The amount of slip depends to a great extent on the embossment geometry, especially its depth, but up to 2 mm of movement can be resisted before collapse occurs. Crisinel et al. 6 identified two types of collapse: the first, a brittle collapse, occurs when the embossments can carry little load, which leads to a slab failure soon after the chemical bond between steel and concrete breaks down. The second, a ductile collapse, occurs when the embossments continue to carry shear load after the chemical bond has been broken. Figure 2(0 shows the typical final collapse mechanism for both failure types. The load to mid-span deflection relationship for the composite slab as it is loaded through each of these stages is shown in Fig. 4. It can be seen from this figure that the stiffness of the slab depends upon the stage of behaviour reached. However, as the behaviour of the slab is relatively unpredictable once the embossments are taking their full capacity and involves considerable plastic deformation the work here will be restricted to the description and analysis of slab stiffness up to the end of stage d.
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(a)
R~nforcement attached to steel by welding
ssments
(b)
Fig. 3. Methods of providing shear transfer.
LOAD End slip
r
Mid span deflectlorL
OISPL.A~I~NT
Fig. 4. Typical load deflection and load slip relationship.
Plate model for composite slab analysis
305
Bending stiffness As described in the previous section, the stiffness of a composite slab is dependent upon the extent of cracking in the tension zone of the concrete. Although this cracking may occur at relatively low levels of tensile stress the exact extent of cracking is often difficult to detect. In addition, although the depth of cracked section may vary along the beam, the effect of this is small in comparison with overall behaviour. Consequently, a reasonable assumption is that the composite slab will behave as a prismatic beam and that the concrete in tension carries no direct bending stress. This would appear to be at least a safe assumption although, as concrete can take some tension, it is possibly conservative. The British code of practice for the design of composite slabs, BS5950: Part 4,7 recognises this apparent conservatism and allows the analysis to take an average stiffness of the cracked and uncracked section. This assumes that the concrete will be carrying some tension. BS5950 goes further in suggesting that the stiffness of the slab in either its cracked or uncracked form can be calculated using simple beam analysis with a transformed section or modular ratio method. The basis for this method assumes that the concrete cross-sectional area can be transformed into an equivalent steel area and that the resulting section may be treated as an elastic prismatic element. Figure 5 shows the steps in the analysis for an uncracked composite beam and Fig. 6 shows the corresponding steps for a cracked section. The depth to the neutral axis depends upon the height of tension cracks in the cracked section analysis and this can only be found once the stress block has been determined. Consequently, either an iterative approach or the formation of a quadratic equation is needed. Figure 6 shows the steps in analysis leading to quadratic eqn (1): dna2 (bm/2) + d n a a s t - ,4stdnast = 0
(1)
Fortunately, there is normally only one real root to this equation. The tedious calculations required for both cracked and uncracked section analysis have been programmed for computer solution by the author and this program has been used later in a comparison with the folded plate method of analysis.
Shear stiffness The simple beam analysis allowed in the code is based on the assumption that shear deformation is small enough to be ignored and
H. D. Wright
306
strona
conc stress
steel stress
N.A
1)
UNCRACKED
Ac
= Area concrete
Act
m
=mA c
=
modular
ratio
Ast = Area steel Ic
= Second morn. area concrete
Ict = m . I c
Ist = Second morn area steer. dna c = Depth to N A . concrete
(from top of slab)
dnast = Depth to N.A. steet (from top of slab) dna
= {Act ~ dna c ° A s t x dnast) /
rst
= Absotute value of
dna- dnast
rct
= Absolute value of
dna-dnac
Is
= Ist •
(Astxrst 2 )
•Ict
*
(Act
* Ast )
( A c t = rct 2 )
Fig. 5. Uncracked section analysis.
the deflection can be calculated using the single term d = x (I~L3/E1)
(2)
where x is the coefficient describing the loading condition, e.g. 5/384 for uniformly distributed loads. This assumption is very nearly correct for most common structural materials and proportions as shear deformation is normally very small in comparison with that caused by bending. However, in the case of the composite slab the cracked concrete and the slip between the concrete and steel may give rise to relatively large shear deformations which have a much greater effect on the overall deflection of the slab. Figure 7(b) shows the exaggerated deformation pattern under pure shear, assuming that the concrete has not cracked and acts in an elastic manner. It can be seen that the shear stress gives rise to both vertical and longitudinal strain. Once the concrete begins to crack the longitudinal strain can be resisted only by the shear bond transfer through the friction afforded by the embossments as shown in Fig. 7(c). This transfer is often
Plate mode/for composite slab analysis
307
b r
/
¢0n¢ stress
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= Second morn. area concrete
!ct = rn.I c = m.b.dna3 / 12
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=(Act K clnac .
Ast=clnast) / (Act • Ast )
Clna 2 ( b.rn/21 -Ast dna - Ast clnast= 0 Is
calculatecl as before. Fig. 6. Cracked section analysis.
relatively flexible and allows slip deformation, which has a tendency to open the cracks still further. As the cracks widen the capacity for vertical shear resistance through aggregate interlock is reduced and more of the shear is redistributed from the concrete to profiled steel sheet interface. The amount of shear deformation is therefore dependent upon the extent of cracking~ the width of the cracks propagated and the stiffness of the longitudinal shear transfer between the concrete and profiled steel sheeting.
The effect of ponding So far in this description of the composite slab it has been assumed that the overall cross-section is uniform along the complete length of the slab. This is not always true, as the slabs are normally cast simply supported and considerable deflection often occurs because of the self weight of the wet concrete. The effect of ponding on the bending stiffness of the composite slab may be demonstrated using a simple plane frame stiffness approach to obtain the deflection of three beams under the same load.
H. D. Wright
308
B.Md~,
(a)
(b)
t .... (c)
Fig. 7. Assumed deformation pattern. Let us consider the three beams shown in Fig. 8. The first simulates the shape of a beam that has ponded under construction loading, the second assumes that the effective thickness is the nominal thickness at the support and the third assumes that the effective thickness is that at midspan ofthe beam. The results ofapplying identical uniform loads to each of these beams are also shown in Fig. 8. It can be seen that the error in using the support thickness amounts to 64%, whereas the error in using the mid-span thickness is only 6%. This high error will, of course, be reduced for profiled steel sheets that are stiff enough to resist large ponded deformations during construction. Consequently, it would appear sensible to use the ponded depth in calculating the section modulus for bending stiffness determination. This comparison does not hold for shear deformation as the maximum shear load occurs at the support, the thinnest section of the ponded slab. However, as shear deformation is generally less than
Plate model for composite slab analysis
309
~9
(a)
(b)
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~
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Fig. 8. ldealisationof ponding. bending deformation the error in assuming the ponded thickness for shear as well as bending deformation is acceptable. Ultimate load determination
This review of the determination of the stiffness of composite slabs is restricted to code requirements and the author's observations on these. This is due in part to the limited emphasis placed on stiffness in comparison with ultimate load determination.
H. D. Wright
310
The prediction of ultimate load capacity has been based on the US code for reinforced concrete design 8 which, in common with the British code BS8110fl uses limit state methods of design. Both codes deal primarily with reinforced concrete, which, in general construction, is a relatively stiff medium. Serviceability checks are usually restricted to satisfying maximum span to depth ratios. For simply supported slabs the maximum span to depth ratio is 20, although this may be modified if the percentage of tension reinforcement is high. Composite floor slabs occasionally exceed this ratio and would therefore require special attention according to BS8110. The special attention includes not only a deflection check assuming a fully cracked section but also control of crack widths so that the buried, unprotected reinforcing bars do not rust. This latter requirement does not apply to composite slabs, and, as suggested earlier, a fully cracked section analysis may be conservative if shear is assumed to be negligible. The British code specific to composite slabs, BS5950: Part 4." is also a limit state code and, as described earlier, has specific requirements on the deflection limit state. By far the largest part of the code deals with the ultimate limit state calculations for load capacity. A detailed study of the code procedure and of several other methods ofanalysis has been made.4 5 In the references it is shown that the procedures for design are based on empirically derived equations which rely on expensive test information for each type of profiled steel sheet used. Consequently, many researchers have turned their attention to either refining the code method or alternatively trying to derive new, more theoretically based, solutions. Seliem and Schuster ~°and Prasannan and LuttrelP ) used widely differing methods to reduce the reliance on test information. Although a definitive theoretically acceptable solution to the ultimate load capacity of composite slabs has not yet been obtained, the author believes that a greater understanding of the flexural behaviour will be of considerable benefit.
THE FOLDED PLATE ADAPTATION
General description Goldberg and Leve first presented the folded plate method in 1957 ~2 when they considered the specific case of loads applied to the edges of the system of plates. Evans u extended this to include the action of uniform loading on the plates, and also described the method in some detail. The method uses elastic plate flexure and plane stress theory in
Plate model for composite slab analysis
311
conjunction with a matrix stiffness approach to the solution of the equations of equilibrium and compatibility. It would, at first, seem less than ideal to attempt to model the solid body geometry of the composite slab using this thin plate model. However, it will be shown in this section that the concrete can be modelled as relatively thin horizontal plates jointed to the steel sheet by vertical plates carrying only shear.
Solid body modelling using plate elements The folded plate method of analysis is ideally suited to structures formed from several thin plates joined along their edges. Two assumptions made in the analysis would appear to conflict with its potential use to model solid bodies. The first is the requirement that component plates may be represented by uniform thin plates exhibiting similar properties. The second is the requirement that component plates be joined at a single node point. These two requirements and the difficulties associated with each are described below. (a) The basis for the derivation of the stiffness matrix for each plate in the folded plate structure lies in the assumption that the behaviour of the plate accords to simple elastic plate flexure and twodimensional membrane theories. Both of these rely on the assumption that forces and moments can be assumed to act along the mid-plane of the plate. These assumptions are found to be valid where the span and breadth to thickness ratio of the plates are high and the radius of curvature of the deformed plate is large in comparison with the thickness. In deriving these theories the normal and shearing stresses through the depth of the plate have been ignored. Although the concrete elements shown in Fig. 9 have low breadth to thickness ratios their length to thickness ratio is very high even for short spans. Consequently, the analysis is reasonably accurate when considering longitudinal forces. In addition, the basic theory is restricted to plates of uniform thickness. In modelling a composite slab the sloping webs of the profiled steel sheeting can give rise to plates representing concrete of variable thickness as shown in Fig. 9(a). It is possible to develop the method to cater for plates of variable thickness but this additional complication has not been carded out here. Instead, a stepped plate model has been adopted where necessary, as shown in Fig. 9(b). (b) It will be shown later that with little loss of accuracy the folded plate method can be used to model relatively thick plates.
312
H. D. Wright
Plate re~esenting.
a) .,~ stepped plates of / / uniformthk:knes.
bl
~
duplicatedarea
¢) Fig. 9. Folded plate idealisation.
However, this brings to light the second difficulty associated with solid body modelling. In Fig. 9(a) the profiled steel sheeting and the concrete are shown as two sets of thick plates acting at the centreline of the actual plate. Each joint or node between plates is shown as a single point. With thick steel plates little error occurs with this assumption as the duplication of plate area, as shown in Fig. 9(c), is very small. However, the concrete plate and the steel plates cannot be connected together without changing the effective plane of the plate. A change of this nature would entail major rewriting of the theory and computer program and has not been attempted. As an alternative, a more flexible approach involving the use of'dummy' elements has been developed. The final configuration of the model used is described in the next section.
Plate modelfor composite slab analysis
313
The composite slab model As outlined earlier, the main bending action in a composite slab is registered by a couple formed from compression in the concrete and tension in the steel. Shear resistance, although an essential part of the slab action, has less effect on the overall stiffness than the resistance of this couple to moment. Consequently, a system of plates can be envisaged that separate the bending action and shear action of the system. Figure lO shows this system of plates for a single pitch of a composite floor slab. It can be noted that the main horizontal plates will take the bending action and the vertical plates the shear. In this simulation it is assumed that most of the concrete in the tension zone has cracked and the remaining concrete carries mainly compression. The actual thickness of the concrete plate carrying compression is dependent upon the depth to the neutral axis. Initially, this has been calculated using simple beam theory. Consequently, the concrete in bending compression can be modelled as a thin plate lying at the midheight of the uncracked concrete.
concreto
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~"
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;,6 e
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Fig. tO. The final folded plate model.
1-
314
H. D. Wright
The 'dummy" elements connecting this plate to the profiled steel sheeting transfer shear between the concrete and the steel. To do this, they have to possess sufficient shear stiffness to be equivalent to the concrete they represent. If these plates were given the actual dimensions and material properties of the concrete they would also have significant bending stiffness which, as the concrete has cracked, is not correct. The 'dummy" plates must, therefore, combine high shear stiffness with low bending stiffness. In an isotropic elastic material the relationship between bending stiffness and shear stiffness is related by the laws of elasticity. In the two-dimensional plate theory used in the basic folded plate formulation this relationship is G = E / 2 ( 1 + v)
(3)
Any change in the Young's modulus of the dummy plate will cause a proportional change in the shear modulus. Unfortunately, the change in Young's modulus causes an equal change in bending stiffness which is to be avoided. A change in Poisson's ratio will also affect the shear modulus, but in this case it will have little effect on the bending stiffness. In the extreme case, the Poisson's ratio may be made negative. The shear modulus of concrete assuming that it is an elastic material is approximately 10 000 N/mm:. A plate of concrete b mm wide would have a shear stiffness of 10 000 b N/mm. The two dummy shear plates, shown in Fig. 10, which have the same shear stiffness, can be evaluated by equating this stiffness using eqn (3): 10000b = E × 2 t / 2 ( l + v ) Putting v -- - 0.999 and t = 0.1, the equivalent dummy plate Young's modulus becomes lO0/b. This is shown in Fig. 11. This principle has been applied to the composite slab model shown in Fig. 10(a). The bending stiffness of the dummy shear plates amounts to 0.08% of the bending stiffness of the complete section as derived using simple beam theory and a cracked section. The folded plate analysis gave a mid-span deflection of 11.08 mm for an imposed load of 20 kN/m 2. The simple beam theory gave a deflection of 11.25 mm for the same load. This shows that the two theories give approximately equal deflection. The additional deflection (1.5%) found in the folded plate method is most probably due to shear deformation of the concrete. This is not included in the simple beam analysis. One further observation that must be made is in the deformation pattern of the dummy shear plates. Most of the plate is in tension as well as shear and consequently the negative Poisson's ratio has the effect of expanding the element. Consequently, the breadth of the dummy plate
Plate modelfor composite slab analysis
315
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t
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t
G~ E I Z t
.V. -- -t)999 Fig. ! !. Equivalent d u m m y shear elements.
increases, giving rise to a slight discrepancy between the concrete plate deformation and the profiled steel sheet deformation. This is very, small, less than 0.03 mm, and is ignored. With the variation between the folded plate analysis and the simple beam method amounting to only 1.5% it is possible to develop the folded plate method further to allow for more accurate analysis of the shear bond behaviour. The cracked concrete section
In the last section the thickness of the concrete plate was predetermined using the simple beam analysis before substituting this value in the folded plate analysis. There is, however, no requirement to predetermine the concrete plate thickness as it can be established using the iterative approach.
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H. D. Wright
The neutral axis of a composite slab normally falls within the area of concrete above the top flange ofthe profiled steel sheeting. If an arbitrary thickness of concrete in compression is chosen as the depth of concrete from the surface of the slab to the top flange of the sheeting, an initial folded plate analysis can be carried out. The results of this analysis will give a stress distribution within the concrete plate similar to that shown in Fig. 12(a). The stresses may be evaluated from the longitudinal forces and bending moments which are obtained from the analysis. If the thickness of the concrete plate is reduced and the length of the d u m m y shear plate extended, a second analysis will result in the stress distribution shown in Fig. 12(b). By changing the thickness of the concrete plate and the length of the d u m m y shear plate to suit, eventually a stress distribution will result which gives no concrete in tension. This procedure has been incorporated into a specific folded plate program, and the flow chart for this procedure is shown in Fig. 13. With (a)
\
/
(b) •
I
I
~1
~ - ~
steel
stress.
concrete compressive stress. concrete tensile stress. Fig. 12. Cracked concrete modelling using folded plate analysis.
Plate model for composite slab analysis
317
INPUT INITIAL GEOMETRY
ANALYSE
COMPUTE 1,410SPANSTRESS
A.MEND GEOMETRY
GREATERTHAN . /
I PRINT RESULTS Fig. 13. Computer flow chart for modelling concrete cracking.
this iterative method the concrete can either be assumed to be unable to take tension or can be assumed to be capable of carrying a limited value of tensile stress. This can be achieved by adjusting the 'checking limit" as shown in the flow chart. To test this method the slab described in Fig. l0 was analysed. The convergence of the method to an acceptable level is complete within three iterations and the final results were only 1.48% different from those achieved using the simple beam analysis. Test observations of composite slab stiffness The alterations to the basic folded plate analysis described in the previous section have been made as a result of observed behaviour of slabs under test. In the development of the analysis mainly qualitative
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considerations have so far been incorporated. In this section quantitative comparisons are made between the method developed and test information. Four series of tests have been made on composite slabs and this experience has been used in developing the method. The test series. reported in Refs 14-17, were all carded out to evaluate the ultimate performance of particular profiled steel sheet slabs according to BS5950 Part 4.
Comparisons between analysis and test results Using the test data the slabs were analysed using the folded plate method. The ponded depth of each slab was evaluated using the folded plate method developed in Ref. 2 and this depth was then used as the slab thickness. The Young's modulus of concrete was determined from the cube strength and density quoted in the test reports. The results of the analysis are shown in Fig. 14 as comparisons with the test results. The equivalent second moment values used on both axes relate to the initial slope of the graph produced by plotting load against mid-span deflection. It was assumed that this equivalent value could be determined from the simple beam relationship I = 5wL4/384Ed
(4)
A perfect correlation between test and analysis would result in all the results lying on a 45 ° line. It can be seen that, in general, the analysis overestimates the stiffness of the slabs. This discrepancy is most probably due to slip between the concrete and steel as described above.
Shear deformation and slip The load to deflection response, which is the subject of this investigation, is a serviceability check in the British standard and consequently this information is produced only in passing in these tests. In fact. the stiffness values shown in Table 1 are produced after each slab has been loaded and unloaded some 10 000 times. This cyclic loading has the effect of reducing the chemical bond between the concrete and steel so that interface slip will be likely to occur throughout the load range. Even so, the end slip, measured in the tests, appears not to start until the slab is heavily loaded. Figure 15 shows a typical load slip response for one of the tests and it can be seen that end slip occurs in a large jump before increasing more
Plate modelfor composite slab analysis
319
/.0(:30
Equwalent I cm6/m. crocked section futly composite f p. method. . 300C
4,
'~
4,
2000,
I000
Equivalent I
test L
1000
2000
3000
l.O00
Fig. 14. Comparison between fully composite folded plate model and test results.
steadily to failure. It should be noted that the stiffness of the slab is less than even the fully cracked section analysis would suggest, indicating that slip is occurring throughout the load range. An explanation for the lack of end slip early in the loading range may lie in the fact that friction between the concrete and steel sheet occurs in the support region due to the reaction. This may be sufficient to prevent movement up to this load. Incorporating end slip into the folded plate model Shear deformation, whether elastic in the concrete or due to slip, has been shown to have a similar effect in Fig. 7. The total deformation could therefore be thought o f as either elastic deformation or slip.
H.D. Wright
320
Load kN/m 2
30
20
deflect ion slip. crackcKI section
10
onalysis.
cl~flection (ram) end slip =101mml v
5
10
15
20
25
30
Fig. 15. Load deflection and load slip curve for Test 2. Series l.
Elastic deformation can be modelled analytically in the folded plate analysis by controlling the shear stiffness of the dummy shear plates. A low value of shear stiffness will give rise to large shear deformation equal to that caused by slip. This is shown diagrammatically in Fig. 16. The shear stiffness of the dummy elements in the slabs in test series one was adjusted so that the deflection produced matched the test results. The resulting reduction in dummy plate shear stiffness was remarkably consistent, as shown in Table 1. Only slab 1 showed any inconsistency. This was the first slab tested in this series and in fact was cycled in excess of 30 000 times. Consequently, the behaviour of this slab was assumed to be affected by this procedure and the results were ignored in calculating the average dummy shear stiffness for the test series. It was therefore thought possible that the deflection of the slabs in the other series of tests could be predicted more accurately by using dummy shear stiffnesses evaluated for each test and the average values used.
Plate modelfor composite slab analysis
321
concrete shear
I
+
slid
I m
equivalent
total
i
Fig. 16. The summation of concrete shear and slip.
Figure 17 shows the resulting predicted stiffness of each test speciment in comparison with the test values. The improvement is marked when the results are compared with Fig. 14 (which shows the same results without modification by the reduced dummy shear stiffness). There are, however, some slabs that fall below the 45 ° line, which indicates that they are more stiff than the analysis would suggest. These slabs were, in fact, some ofthe longest thinnest spans and were also those slabs for which it appeared that the bond between the concrete and steel did not completely fail during cycling. This manifested itself in a more progressive ductile failure.
The practical application of the folded plate method It has been shown in the last section that the application of a folded plate approach using a dummy shear stiffness obtained from test results significantly improves the prediction of slab stiffnesses. It must be remembered, however, that the method has been developed from test data and that the tests used do not entirely represent the situation that the
H.D. Wright
322
TABLE I D u m m y Shear Stiffness for Slabs Tested
Test series
Slab
Dummy shear stiffness assuming full interaction (N/mm)
Dummy shear stiffness to match test result (N/mm)
i 2 3 4 5 6 7 8 9 I0
7 840 15 525 9 928 14 270 17 348 14 691 I I 925 12 580 13 376 16 594
220 140 230 350 350 220 220 150 230
I 2 3 4 5 6 7 8 9 10
9 075 6 078 8 546 6 177 8 789 5 873 8 442 7 224 7 563 9461
90 500 1200 500
1 2 3 4 5 6
8 429 8372 8 372 8 273 8 273 8 273
a 350 550 450 305 285
388
i 2 3 4 5 6
23131 23131 23131 23131 23131 23131
° ° a 236 198 150
195
~,,,~ ,,tin 800
Average for series (N/mm)
234
680
a
°Slabs were found to be as stiffas, or stiffer than, fully composite cracked section analysis a n d were ignored when calculating average d u m m y shear stiffness.
Plate model for composite slab analysis
323
t.O00
Equwalent I crn6/m. crocked section f.p. method incorporating slip 3000
2000
1000
.//
+* *
Equivalent I test 1000
2000
3000
&O00
Fig. 17. Comparison between partially composite folded plate model and test results.
slab would occupy in actual construction. It is unlikely that the slab would be fully simply supported as in the test and it is also unlikely that the cyclic load regime of the tests would occur in practice. However, it does provide a lower bound solution and may indicate a first slip load. Let us consider the end distortion of one of the slabs as obtained from the analysis, as shown in Fig. 18. It is possible to evaluate the effective end slip from the relative positions of the nodes as shown in this figure. Consequently, an effective end slip can be found for each load. The shear load and slip occurring in the slab may be affected by the geometry of the embossments. This is possibly true as long as the slip is small. Figure 19 shows the assumed action of the concrete shearing against an embossment. Initially, the increase in load causes the concrete to rise up over the embossment. The shear load is proportional to the applied load and consequently the rate of slip is proportional to the applied load. This will continue until the concrete has risen to the full
324
H. D. Wright
)P q9 q8 G7 0:6 0~5.~ 0:3 0.~ 03
0.1 02 0:3 0~.0:5 ~.6 q7 if0_ ram. 1
J ~22/
/ /
jo
equivallnt end slip
Exaggerated Qnd displacements slab 2
Fig. 18. Equivalent end slip evaluation.
height of the embossment, when a large a m o u n t of slip will occur. This sudden slip will most probably cause failure. The relationship between slip a n d shear load is complicated by the fact that the embossments are placed on the web of the profiled sheeting. The concrete needs to displace the steel sheet sideways as well as lifting it vertically. Consequently, it would be difficult to attempt to evaluate a slip modulus from the geometry of the sheet, although it may be possible to obtain a value from small-scale tests. Several researchers have c a r d e d out experimental studies on small-scale tests. Hassaine, 18 Jolly and Zubair, 19 Crisinel a n d Daniels 2° a n d Brekelmans 21 have all c a r d e d out pilot studies. Unfortunately, most of the information relates to the ultimate load c a r d e d and none is specific to the profiles tested by the author. If both the slip modulus and the ultimate slip load can be determined from model tests then these may be readily incorporated in the folded plate method. By restricting the end slip to a small value likely to be the strain limit for concrete adhering to steel, we may also be able to predict a first slip load. This will be slightly conservative unless the frictional resistance of the support is taken into account. The use of small-scale model tests to evaluate parameters that may be used in a theoretically meaningful analysis is of significant economic
Plate modelfor composite slab analysis
325
/
\ /
L~od
E Fig. 19. Assumed shearing action of concrete against an embossment.
benefit to producers of profiled steel sheet. The current empirical test regime is expensive, time-consuming and is criticised for its lack of a theoretical basis. It is hoped that further work on model testing will allow the method described here to be developed to expand the ideas proposed in this section. Conclusions
The application ofthe folded plate method of analysis to composite slabs formed with profiled steel sheeting has been outlined. The work concentrates on the development of the basic method to overcome the problems associated with solid body geometry, the lack of tensile capacity in the concrete and the effects of slip between the concrete and steel. As a result of this work, the folded plate method has been found to be a suitable tool for the analysis of solid beams. This has been achieved by separating the bending action and shear action present in the beam. Horizontal plates take the major longitudinal forces associated with pure bending. The shear action is taken by vertical 'dummy" shear plates. The 'dummy' shear plates allow a more accurate analysis of shear deformation and allow a closer prediction of lateral deformation of the steel sheeting.
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H. D. Wright
By using an interative process the action of concrete in bending has been successfully modelled. This involves the use of a tension cut-off principle although limited tensile stiffness could be incorporated easily. In composite slab behaviour the presence of slip between the two materials causes non-linear behaviour. This behaviour has been modelled by incorporating the slip as a part of the shear distortion in the slab. By introducing a low shear modulus to the dummy elements it has been found possible to model closely the behaviour of 32 composite slab tests.
Alternative applications Several fundamental structural actions have been analysed using the folded plate method. These actions are not unique to the composite floor system described here. Other structural systems involve geometric nonlinear behaviour, and include concrete and other elements which are only partially and often discretely connected together. One of the most widely used structural systems is reinforced concrete. The similarity with the composite slab makes it an ideal first example to analyse using the folded plate method. Kristek and Smerda '2 used the folded plate method to investigate not only reinforced concrete bridges but also prestressed and post-tensioned structures. In addition, they studied the effects of creep and shrinkage. Another composite structure is the sandwich panel formed from two profiled steel sheets filled with a lightweight low-density foam. The 'dummy' shear elements described here could be adapted to model this foam core, which has a very low shear stiffness. A different type of sandwich panel is the double skin composite structure proposed for immersed tube tunnels? 3This comprises two steel plate skins separated by a concrete infill. Connection between the skins and the concrete is provided by stud shear connectors. Again the use of the 'dummy' vertical shear element would allow analysis of panels with incomplete or flexible connection.
REFERENCES 1. Wright, H. D., Evans, H. R. & Harding, P. W., The use of profiled steel sheeting in floor construction, Journal of Constructional Steel Research, 7 (1987) 279-95. 2. Wright, H. D., & Evans, H. R., A folded plate method of analysis for profiled
Plate modelfor composite slab analysis
3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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steel sheeting in composite floor construction, Thin-Walled Structures 5 (1987) 21-37. Taylor, H. P. J., Shear strength of large beams, Proc. American Society of Civil Engineers, 98 (ST(II))) (1972) 2473-90. Wright, H. D., & Evans, H. R., Observations on the design and testing of composite floor slabs, Steel Construction Today. 1 (1987) 91-9. Wright, H. D., Evans, H. R. & Harding, P. W., Composite floors: comparisons of performance testing and methods of analysis, Proc. International Symposium Steel in Buildings. IABSE-ECC, Luxembourg, 1985, 219-25. Crisinel, M. J., Fidler, M. J. & Daniels. B. J., Behaviour of steel deck reinforced composite floors, Proc. IABSE Colloquium. Thin-Walled Metal Structures in Buildings, Stockholm, 9-12 June 1986, 279-90. BS5950: Part 4, Structural use of steel in building. Code of Practice for design of floors with profiled steel sheeting, British Standards Institution, London, 1982. American Concrete Institute, (A.C.I. 318-77) Building Code requirements for reinforced concrete, A.C.I. Detroit, 1977. BS8110, Structural use of concrete, Part I: Code of practice for design and construction, British Standards Institution, London, 1985. Seliem, S. S. & Schuster, R. M., Shear-bond capacity of composite slabs, Proc. 6th Specialist Conference on Cold-Formed Structures, University of Missouri-Rolla, November 1982, 511-31. Prasannan, S. & Luttrell, L. D., Flexural strength formulations for steel deck composite slabs, Report of the Department of Civil Engineering, West Virginia University, 1984. Goldberg, J. E. & Leve, H. L., Theory of prismatic folded plate structures, Publ. Int. Assn. Bridge and Struct. Engineers. 17 (1957) 59-86. Evans, H. R, The analysis of folded plate structures. PhD thesis, University of Wales, Swansea, 1967. Evans, H. R. Wright, H. D. & Harding, P. W., Full scale load tests on P.M.F. Ltd. CF46 composite flooring profile, Internal Report UCC/PMF/CF60, 1986. UCC Composite Structures Group, Assessment of P.M.F. CF60 for both construction and service loading, Internal Report UCC/PMF/CF60, 1986. UCC Composite Structures Group, Composite tests on Alphalok decking, Internal Report UCC/Alpha/Alphalok 1, 1988. UCC Composite Structures Group, Assessment of the composite performance of PMF/CF60 under service conditions, Internal Report UCC/PMF/ CF60/2, 1987. Hassaine, N-E., Shear-bond performance of profiled steel sheets when used as composite floor decks. MSc thesis, University College, Cardiff, 1987. Jolly, C. 14,. & Zubair, A. K. M., The efficiency of shear-bond interlock between profiled steel sheeting and concrete, Proc. Conference on Steel and Aluminium Structures, Cardiff, 8-10 July 1987, 127-36. Crisinel, M. J. & Daniels, B. J., Verbal presentation of small scale tests on composite slabs, ECCS TC 7.6 Working Group Meeting, Delft, November 1987.
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21. Brekelmans, J. W. P. M., Verbal presentation of small-scale tests on composite slabs, ECCS TC 7.6 Working Group Meeting, Delft, November 1987. 22. Kristek, V. & Smerda, Z., Changes of the state of stress of a prestressed concrete bridge assembled of precast segments, Prestressed Concrete inCzechoslovakia Inzeuyrska Stavebuy, May 1982. 23. Narayanan, R., Wright, H. D., Francis, R. W. & Evans, H. R.. Double-skin composite construction for submerged tube tunnels, Steel Construction Today. 1 (1987) 185-9.