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Shear analysis of reinforced concrete shells using degenerate elements
PERGAMON
Computers and Structures 68 (1998) 17±29
Shear analysis of reinforced concrete shells using degenerate elements M.A. Polak Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 Received 31 October 1995; accepted 1 January 1997
Abstract The paper presents a formulation of degenerate shell ®nite elements which take into account three-dimensional states of stress and strain throughout the element. The formulation was developed with the aim of modelling reinforced concrete shell structures subjected to high transverse shear stresses. It requires implementation of a threedimensional constitutive model for concrete and it allows to use out-of-plane reinforcement in the elements. Through the consideration of three-dimensional stress states, the elements can model the in¯uence of in-plane stresses on transverse shear deformations and the in¯uence of transverse shear stresses on the capacity of shell structures. The formulation was veri®ed against experimental results. The specimens analyzed were shell panels subjected to out-of-plane (transverse) and in-plane shear and reinforced concrete shells subjected to punching shear. # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved.
fcx, fcy
Nomenclature [B] [D] [D*] [Dp] [Dcp] [Dc'] [D'st] [Dst] [K] [Te] {d} Ei Gij R
strain±displacement matrix material stiness matrix (modi®ed to 5 5) in element's axes (x,y,z) material stiness matrix (6 6) in element's axes material stiness matrix in principal axes material stiness matrix for concrete, in principal axes material stiness matrix for concrete (6 6), in element's axes transverse reinforcement material stiness matrix in reinforcement directions transverse reinforcement material stiness matrix in element's axes element stiness matrix strain transformation matrix vector of nodal displacements of an element secant Young's moduli for concrete shear moduli for concrete radius
fc1, fc2, fc3, fci fcr f'c fc2max fsx, fsy, fsi fs fy fscr u, v, w v1, v2 t ec e0 ecr e1, e2, e3 es ex, ey, ey gxy, gxz, gyz
concrete normal stresses in x- and ydirections concrete principal stresses cracking stress for concrete compressive cylinder strength of concrete compressive strength of cracked concrete reinforcement stresses in x, y and `i' directions transverse reinforcement stress yield stress of reinforcement reinforcement stress at a crack displacements in global directions shear stresses in concrete thickness concrete strain concrete strain corresponding to peak compressive stress cracking strain for concrete principal strains reinforcement strain normal strains shear strains
0045-7949/98/$19.00 # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 0 2 7 - 3
18
rs rsx, rsz, rsi s1, s2, s3 sx, sy, sz txy, txz, tyz ysci
M.A. Polak / Computers and Structures 68 (1998) 17±29
transverse reinforcement ratio reinforcement ratios in x, z and `i' directions principal stresses normal stresses shear stresses angle between the reinforcement and the normal to the crack
inforced concrete shell structures subjected to high transverse stresses. The model proposed in this paper utilizes the formulation of the displacement-based, degenerate shell elements. The modi®cation is in the approach to constitutive modelling used in the element formulation. The rotating crack model is adopted where the transverse shear strains in¯uence the principal directions of strains and stresses. At every point of the shell element, a three-dimensional state of stress and strain is assumed allowing a model of the in-plane versus out-of-plane shear interaction.
1. Introduction Degenerate shell elements have been used extensively for the global analyses of shell structures ever since they were introduced by Ahmad et al. [1]. These elements were developed because of the diculties in ensuring C1 continuity of displacements in the development of shell ®nite elements based on thin shell theories. Consequently, a dierent type of element was derived which requires only C0 continuity. These C0 elements are able to model the behaviour of shells where transverse shear deformations are signi®cant. Such behaviour is important in thick shell structures, in shells subjected to concentrated transverse loads and in plates supported by columns. In these cases, transverse shear can be the governing failure mechanism. The degenerate shell elements which include transverse shear strain energy in their formulation, have received considerable attention in the recent developments in the ®nite element analysis of shells [2±5]. The elements are based on a simple concept of `degenerating' threedimensional elasticity to a shell `situation' and, although the problems of shear-locking and zero energy modes must be overcome in the successful analysis, the elements perform well for both thick and thin shells. The degenerate shell elements' formulations are based on a theory equivalent to the Mindlin plate bending theory. They include transverse shear strain energy in their formulation. In order to model nonlinear material behaviour (e.g. concrete), the shell elements are divided into layers. However, it is usually assumed that each concrete layer remains in the state of plane stress and the transverse shear eects are decoupled from the in-plane strains and stresses [6±8]. It is also assumed that the transverse shear stresses do not aect the in-plane behaviour of concrete and reinforcement. Such models can predict the magnitude of transverse shear strains and the in¯uence these strains will have on the response (e.g. de¯ections) of a structure, but cannot predict when the transverse shear becomes responsible for the failure mechanism. The emphasis in the presented study is on the ability of shell ®nite elements to model the behaviour of re-
2. Degenerate shell elements The degenerate shell elements are derived from the equations of three-dimensional continuum mechanics [9]. The three-dimensional stress and strain conditions are degenerated to shell behaviour by adopting assumptions representing a typical shell behaviour (Fig. 1). These assumptions are ®rst, normals to the midsurface remain straight, but not necessarily normal to the midsurface after deformation and second, stresses normal to the midsurface are negligible [Fig. 1(b)]. The ®rst assumption has two implications: ®rst, that transverse shear strains exist and are included in the strain energy functional; and second, that these transverse strains are uniformly distributed through the
Fig. 1. Degenerate shell element. (a) reference system and degrees of freedom; (b) assumptions.
M.A. Polak / Computers and Structures 68 (1998) 17±29
thickness of a shell. This is a crude approximation of the real distribution of transverse strains. In the case of homogeneous linear elastic shell, this distribution takes a parabolic shape along the thickness of an element and it can be accounted for in the element formulation (to correctly assess transverse strain energy) by using a shear correction factor equal to 5/6. In the case of nonlinear, fracturing material this distribution of strain along the thickness is more dicult to assess. One way of ensuring the correct magnitude of transverse shear strain energy is the proper formulation of the shear moduli for the material [e.g. Eq. (16)]. The ®rst assumption also results in elements with 5 degrees-of-freedom per node: three translations (u, v and w) and two rotations (yx, yy). The two rotations are not dependent on the translations. The second assumption implies that the strain energy associated with the out-of-plane, axial deformations (ez) is ignored. It is approximately valid in most real situations and in fact most transverse shear theories for beams, slabs and shells make use of this assumption [10, 11]. It implies that there are ®ve nonzero stresses: fsg fsx ; sy ; txy ; txz ; tyz g
1
and ®ve independent strains: feg fex ; ey ; gxy ; gxz ; gyz g:
3. The three-dimensional formulation of the degenerate elements The presented formulation follows the assumptions and derivations of the degenerate ements. The ®ve independent strains [Eq. (1)] culated in the local coordinate system of an (Fig. 1) and they are equal to:
In the presented formulation it is assumed that strain ez is present and can be found from the condition sz=0. The degenerate shell elements use lagrangian or serendipity shape functions (depending on the number of nodes per element) for both translational and rotational degrees-of-freedom. They adopt isoparametric formulation for modelling geometry. This enables modelling of any shape of a shell structure. The stiness matrix is evaluated using gaussian integration. Reduced and selective integration can be used for thin shells where shear locking can become a problem [12]. In the nonlinear formulations, integration through the element's thickness is facilitated using a layered model where each layer has one integration point at the depth of its midsurface. The major advantages of the elements are that they are independent of any complex shell theory, their formulation is inherently simple, and yet they account for transverse shear deformations.
general shell elare calelement
2
3 @u 6 @x 7 6 7 6 @v 7 7 2 3 6 6 ex @y 7 6 7 6 ey 7 6 7 6 7 6 @u @v 7 7 6 7; g feg 6 xy 6 7 6 7 4 gxz 5 6 @y @x 7 6 7 @u @w 6 7 gyz 6 7 6 @z @x 7 6 7 4 @v @w 5 @z @y
3
where u, v and w are displacement components in the local system. Using the appropriate formulations [1], these local strains are calculated from the derivatives of the global displacements. The strain±displacement matrix [B] is formed which relates: feg Bfdg;
2
19
4
where {d} is a vector of element's nodal displacements in the global coordinate system and {ee} is a vector of strain as de®ned by Eq. (3). The stiness matrix of a degenerate shell element is found using the matrix [B] and the material stiness matrix [D], from the wellknown equation: Z K
v
BT DBdv:
5
The description that follows concentrates on the formulation of the material stiness matrix [D] in Eq. (5). The formulation is applicable for any nonlinear material and it takes into account the in¯uence of the transverse shear deformations on the orientation of the material's principal axes. The assumption of the normal stress as equal to zero (sz=0), permits the calculation of the normal strain (ez). It depends on the other ®ve strains [Eq. (1)] and for linear elastic material it is equal to:
20
M.A. Polak / Computers and Structures 68 (1998) 17±29
ez
ÿv ex ey 1 ÿ v
6
In the general case of an anisotropic material and the fully populated material stiness matrix [D*] [see Eq. (11)], the out-of-plane strain can be calculated as:
ez
2
D*11 6 D*21 6 * 6D 31 D* 6 6 D* 6 41 4 D* 51 D*61
D*12 D*22 D*32 D*42 D*52 D*62
D*13 D*23 D*33 D*43 D*53 D*63
D*14 D*24 D*34 D*44 D*54 D*64
D*15 D*25 D*35 D*45 D*55 D*65
3 D*16 * 7 D26 7 D*36 7 7: D*46 7 7 D*56 5 D*66
11
Matrix [D*] relates: fsx ; sy ; sz ; txy ; txz ; tyz gT D* fex ; ey ; ez ; gxy ; gxz ; gyz gT ;
ÿD*31 ex D*32 ey D*34 gxy D*35 gxz D*36 gyz : D*33
12 7
The material principal directions are assumed to coincide with the directions of principal strains. These are calculated from the six strains in the element's coordinate system (Fig. 1):
Before including matrix [D*] in the element stiness matrix equation [Eq. (3)] it must be modi®ed to enforce the zero normal stress condition. Modi®cation of the [D*] requires removal of row 3 and column 3 and the modi®cation of the material stiness coecients:
feg fex ; ey ; ez ; gxy ; gxz ; gyz g:
Dij D*ij ÿ
8
D*i3 D*3j D*33
:
13
Knowing the magnitude of the principal strains, (e1, e2, e3), the principal stresses, (s1, s2, s3), and the material stiness constants can be found using the constitutive model adopted in the formulation. Any rational three-dimensional constitutive model can be used in the element's formulation. The material is assumed to be orthotropic. The stress±strain relationship in the principal directions takes the form:
The new matrix [D] is a 5 5 matrix and it relates strains [Eq. (2)] to stresses [Eq. (1)]
fs1 ; s2 ; s3 ; t12 ; t13 ; t23 gT Dp fe1 ; e2 ; e3 ; g12 ; g13 ; g23 gT ; 9
The formulation presented in this section is a nonlinear procedure for the analysis of reinforced concrete.
fsx ; sy ; txy ; txz ; tyz gT Dfex ; ey ; gxy ; gxz ; gyz gT ;
14
Fig. 2 4. Formulation for reinforced concrete shells
where 2
E1 6 E21 6 6 E31 Dp 6 6 0 6 4 0 0
E12 E2 E32 0 0 0
E13 E23 E3 0 0 0
0 0 0 G12 0 0
0 0 0 0 G13 0
3 0 0 7 7 0 7 7: 0 7 7 0 5 G23
10
The material stiness matrix must be transformed into the local coordinate system for the element where it becomes a fully populated 6 6 matrix:
Fig. 2. Model for transverse and longitudinal reinforcements.
M.A. Polak / Computers and Structures 68 (1998) 17±29
It is based on the proposed three-dimensional formulation for degenerate shell elements. This section describes the formulation for a concrete layer. It uses the secant stiness approach for the nonlinear material analysis. It is assumed that the cracked concrete behaves like a nonlinear elastic, orthotropic material. Furthermore, it is assumed that cracked concrete can carry certain amount of tensile stresses (tension stiening), that shear can be transferred through cracked concrete interface by aggregate interlock and dowel action, and that the shear modulus can be estimated from three-dimensional states of stress and strain. The evaluation of the stiness matrix of an element requires the formulation of the material stiness matrix speci®ed in the element's x,y,z coordinate system. The isoparametric formulation requires numerical integration of the stiness matrix over the area of an element. Due to the material nonlinearity and the nonlinear distribution of stresses, integration through the thickness of an element is achieved by using a layered model. In the reinforced concrete shell element, the element is divided into a series of concrete and steel layers. Steel layers have the total area equal to the area of in-plane reinforcement and having stiness in one directionÐthe direction of reinforcement. The material properties of concrete change depending on the level of stresses and strains. The layered model allows us to evaluate strains, stresses and material properties at the midpoints of each layer. The total material stiness matrix of an element is calculated by adding the contributions of all layers (concrete and steel). The material stiness matrix of the element's concrete layer is formulated ®rst in the principal directions. The principal directions are found from the known strains in x,y,z directions [Eq. (8)]. From principal strains (e1, e2, e3), the principal stresses ( fc1, fc2, fc3) are found using the appropriate constitutive relationships (Fig. 4). The secant Young's moduli are calculated as:
Ei
fci ; i 1; 2; 3 ei
Ei Ej ; i; j 1; 2; 3: Ei Ej
E1 6 0 6 6 0 Dcp 6 6 0 6 4 0 0
0 E2 0 0 0 0
0 0 E3 0 0 0
0 0 0 G12 0 0
0 0 0 0 G13 0
3 0 0 7 7 0 7 7: 0 7 7 0 5 G23
17
The material stiness matrix is then transformed into the local coordinate system x,y,z: Dc 0 Te T Dcp Te
18
where [Te] is the strain transformation matrix between the principal and the local coordinate system. An important feature of the proposed formulation is the possibility of considering the in¯uence of the transverse reinforcement on the behaviour of shells (Fig. 2). The fact that the strain ez is present, allows the stirrups to strain and thus to contribute to the overall stiness of an element. The transverse reinforcement is speci®ed as a property of any concrete layer. It contributes to the stiness in one direction; namely in the direction of the reinforcement. The formulation starts by ®rst ®nding the strain in the direction of reinforcement, es. The stress ss is then found using the constitutive relationship for out-of-plane steel. The material matrix for transverse reinforcement, [D'st], is formulated in the reinforcement directions as: 2
rs efs 6 0s 6 6 0 D 0st 6 6 0 6 4 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
3 0 07 7 07 7: 07 7 05 0
19
The matrix [D'st] is then transformed to x,y,z coordinate system:
15
Dst Tse T D 0st Tse ;
16
where [Tes] is the strain transformation matrix between the reinforcement directions and x, y, z. The total material stiness matrix for a concrete layer which contains transverse reinforcement [Eq. (11)] is found from:
and the shear moduli [13]:
Gij
2
21
Assuming the Poisson's ratios for concrete after cracking, are zero, the material stiness matrix for the concrete layer, with respect to the principal axes, has the form [see also Eq. (10)]:
D* Dc 0 Dst :
20
21
The in-plane steel material stiness matrices take a form, in the reinforcement directions, analogous to
22
M.A. Polak / Computers and Structures 68 (1998) 17±29
Eq. (19). These matrices are then transformed to the element's local axes according to Eq. (20). The material stiness matrices for all concrete and in-plane steel layers are added resulting in the element's material stiness matrix. This matrix is then modi®ed according to Eq. (13). 4.1. Transverse reinforcement When the transverse reinforcement is present in the concrete shell element, it increases the stiness of the element in the direction of this reinforcement. In the case when transverse reinforcement is located in the z direction, it contributes to the D*33 term in the matrix [D*] [Eqs. (11) and (20)] in the following manner: D*33 D*33c Ds3 ;
22
where Ds3=rs3Es3 is the stiness contribution from the transverse steel, rs3 is the transverse reinforcement ratio, and D33c* is the stiness contribution from the concrete. From Eq. (7), it can be seen that, with the contribution from the transverse reinforcement, the strain in the z-direction will be decreased. This will increase the stiness of an element, decrease the strains in the next iteration and, consequently, allow the structure to sustain higher shear stresses before reinforcement yields or concrete crushes. 4.1.1. Strain, stress and strength considerations An analysis using Mohr's Circles illustrates how the transverse reinforcement helps to carry shear stresses. To simplify the discussion, consider a typical shell element simpli®ed to two dimensions (the y-direction has been omitted). The element is subjected to transverse shear only. The conditions that provide the basis for the following discussion are: equilibrium and the compatibility of concrete and reinforcement strains. Consider two intensities of the shear stress , v1 and v2, with v1
Subscript `1' refers to the shear stress v1 and subscript `2' refers to v2. First, consider the element to have x-direction reinforcement only [Fig. 3(a)]. Due to the condition that sz=0, and in order to maintain equilibrium, the normal stresses in the concrete in the z-direction must be zero. Now consider what occurs as the shear stress is increased to v2. Due to the strain compatibility between the concrete and the reinforcement and in order to maintain the equilibrium of an element in Fig. 3(a), the stresses in the longitudinal reinforcement fsx and stresses in concrete fcx increase rapidly. In addition, the angle y2 becomes much smaller than the angle y1, resulting in less stress able to be transferred through cracks [Eq. (23)]. Fig. 3(b) shows the same shell element but with both x- and z-reinforcement. The concrete can now develop compressive stresses in the z-direction, since this stress can be equilibrated by the tensile stresses in the z-reinforcement (due to the compatibility of strains in the z-direction). Consider the element subjected to the shear stress v1. Strains and stresses develop in both the x- and z-reinforcements, consequently, the stress in the x-reinforcement is lower now than it was in the element without z-reinforcement. The increase of shear stress to v2 increases stresses in both the x- and the z-reinforcements, however, the increase in stress in the x-reinforcement is again less than it would have been for the element without z-reinforcement. The angle y2 also remains larger and more tensile concrete stress can be transferred through cracks [Eq. (23)]. The increase in shear stress from v1 to v2 for the element reinforced in two directions [Fig. 3(b)], left the element with much more remaining strength capacity than it did for the element without transverse reinforcement [Fig. 3(a)]. Hence, the element reinforced in both the x- and z-directions can carry much higher shear forces.
5. Numerical results The proposed degenerate shell element formulation was used to develop a heterosis quadratic element for the nonlinear analysis of reinforced concrete shell structures [14]. The formulation has been checked against the experimental results on shells and plates subjected to high shear stresses. In the presented analyses the material model for concrete is as follows [10]. The main assumption of the model is that the directions of principal strains and stresses coincide [15]. The constitutive relationships are written with respect to the principal axes. The material fails when strains in concrete and
M.A. Polak / Computers and Structures 68 (1998) 17±29
23
Fig. 3. Mohr circles of stress for concrete subjected to pure shear. rxÐlongitudinal reinforcement ratio, rzÐtransverse reinforcement ratio. (a) Element without transverse reinforcement; (b) element with transverse reinforcement.
reinforcement achieve values large enough to signi®cantly reduce the stiness matrix of a structure. This can happen due to yielding of reinforcement, crushing of concrete or the inability to transfer tensile stresses through cracks. The stress±strain relationship for compression [9, 16] accounts for the reduction in strength and stiness due to the presence of transverse
cracks [Fig. 4(a)]. The model for concrete in tension includes tension stiening [17] that signi®cantly in¯uences post-cracking shell response [Fig. 4(b)]. The smeared, rotating crack approach is used with the crack orientations assumed to be perpendicular to the principal tensile strain directions. The mechanism of transmitting tensile stresses across cracks is rigorously
24
M.A. Polak / Computers and Structures 68 (1998) 17±29
Fig. 4. Constitutive model for concrete: (a) compression; (b) tension.
considered. This mechanism is very important, not only for modelling in-plane tensile behaviour, but also for the out-of-plane shear model. In order to check that the average concrete tensile stress can be transmitted across the cracks, the reserve of stress available in the reinforcement must satisfy the following relationship: fc1 RSni1 rsi fyi ÿ fsi cos2 ysci ;
23
where y is the angle between the reinforcement and the normal to the crack and n is the number of reinforcement components.
5.1. Shell specimens subjected to in-plane shear The ®rst series of tests presented, are tests on shell panels subjected to pure in-plane shear. The tests examined here were the part of an experimental program which included testing of full size shell elements using the Shell Element Tester at the University of Toronto [18]. The SE specimens were full size shell elements reinforced with deformed reinforcing bars. Specimens SE1, SE5 and SE6 were of the dimensions 1524 1524 mm with the thickness of 285 mm. The specimens were reinforced with orthogonal reinforcement placed in two zones, near each face of the specimen. The general layout of the specimens' reinforcement is shown in Fig. 5. Details regarding the
Table 1 Material properties of SE specimens Reinforcement
Test
Concrete f'c [MPa]
Behaviour
e'c
Strong fy1 [MPa]
Ratio r1 per layer
Weak fy2 [MPa]
Ratio r2 per layer
SE1
42.5
0.00254
492
0.01465
479
0.0049
SE5
25.9
0.0024
492
0.01465
492
0.01465
SE6
40.0
0.0025
492
0.01465
479
0.00163
Observed
Predicted
vcr=2.8 MPa vu=6.77 MPa failed by yielding of weak reinf. vcr=2.15 MPa vu=8.1 MPa failed by crushing of concrete. vcr=2.1 MPa vu=3.51 MPa failed by yielding of weak reinf.
vcr=2.4 MPa vu=7.80 MPa failed by yielding of weak reinf. vcr=2.03 MPa vu=8.5 MPa failed by a concrete failure vcr=2.06 MPa vu=4.5 MPa failed by yielding of weak reinf.
M.A. Polak / Computers and Structures 68 (1998) 17±29
25
material properties can be found in Table 1. The heavier reinforcement is referred to as the strong reinforcement, the lighter reinforcement as the weak reinforcement. Pure in-plane shear loads were applied along the reinforcement directions, by applying equal tension and compression forces in the two orthogonal directions at a 458 angle to the reinforcement. The specimens were modeled by a mesh of four equal-size elements. The number of layers used per element was 10. Although it was not necessary to model shells in pure in-plane shear by any more than one layer, in general cases where the actual stress distribution is unknown, the number of layers used will be approximately 10. The predictions obtained are shown in Fig. 6. The observed and predicted cracking and ultimate loads, as well as failure mode descriptions, are given in Table 1. The results are in close agreement with the experimentally observed responses and, in all cases, the failure mode was predicted correctly. 5.2. Shell elements subjected to transverse shear In order to develop a rational method for the shear design of concrete oshore structures, an experimental program was undertaken at the University of Toronto which included tests on shell panels under dierent combinations of transverse shear and in-plane forces [19]. Five of the specimens, SP3, SP4, SP7, SP8 and SP9 were chosen for the analysis. The specimens were tested in the Shell Element Tester at the University of Toronto. They were of a size of 1524 1524 mm, with a thickness of 310 mm. The in-plane reinforcement consisted of two orthog-
Fig. 6. Shear stress versus shear strain response for SE specimens.
Fig. 5. Dimensions and reinforcement layout for SE and SP specimens.
onal layers of bars. In both reinforcement directions, the amount of reinforcement provided was equal. The reinforcement was oriented at a 458 angle with respect to the specimens' sides (Fig. 5). The specimens were also reinforced in the out-of-plane direction. Shear
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M.A. Polak / Computers and Structures 68 (1998) 17±29
Table 2 Material properties of SP specimens
Specimen SP3 SP4 SP7 SP8 SP9
Concrete properties
Reinforcement properties
f'c [MPa]
e'c
In-plane Ratio* rx=ry
fy [MPa]
fult [MPa]
f [mm]
Transverse Ratio rz
fy [MPa]
f [mm]
0.0022 0.0023 0.0020 0.0021 0.0026
0.0358 0.0358 0.0375 0.0375 0.0375
480 480 536 536 536
660 660 637 637 637
20 20 20 20 20
0.0008 0.0008 0.0008 0.0008 0.0008
460 460 460 460 460
8 8 8 8 8
50 52 54 53 50
Table 3 Observed and predicted shear strength of SP specimen
Specimen
Applied loading±ratio Shear strengthÐtest of in-plane shear to transverse/in-plane transverse shear [MPa]
Shear strengthÐpredicted transverse/in-plane [MPa]
SP3 SP4 SP7 SP8 SP9
transverse shear only 4 -4 in-plane shear only -8
1.6/0 2.0/8.0 1.1/4.4 0.0/16 0.9/7.2
1.64/0.0 1.90/7.6 1.60/6.4 0.0/16.75 1.22/9.76
reinforcement was used in the form of T-headed bars placed in a 120 120 mm grid. The applied loading consisted of uniform, out-of-plane shear, equilibrating bending moments and various combinations of inplane forces. The details concerning the material properties and loadings of the SP-series specimens are given in Tables 2 and 3. Each SP-specimen was modelled using a mesh of eight elements (Fig. 7). Along the line of action of the bending moments, four elements were
Fig. 7. Finite element mesh used for SP-specimens.
chosen (because of the complex de¯ection pro®le); along the line of action of the uniform in-plane load, only two elements were used. The shell was divided into ten concrete layers. The out-of-plane reinforcement was placed in the inner eight layers, leaving the outside concrete layers to simulate concrete cover. The analyses of the SP-series specimens demonstrated the ability of the program to predict out-ofplane shear failures. The predicted and observed shear capacities of the specimens are shown in Table 3 and Fig. 8. The presence of in-plane shear signi®cantly alters the transverse shear strength of a shell element.
Fig. 8. Transverse shear versus in-plane shear for SP shell specimens.
M.A. Polak / Computers and Structures 68 (1998) 17±29
27
Fig. 9. Stirrup strains versus transverse shear stresses for SPshell specimens. Fig. 10. Curved shell specimensÐgeometry and ®nite element mesh.
The interaction diagram between in-plane and out-ofplane shear for a typical shell panel is shown in Fig. 8. The proposed model accurately predicted the capacity of SP3 and SP4, specimens subjected to pure transverse shear and transverse shear plus compressive force. For specimens SP7 and SP8, subjected to transverse shear with in-plane tension, the predicted strengths were lower than observed (Fig. 8). This can suggest, that the material model and the shear modulus formulation underestimate the amount of shear that can be carried by cracked concrete. Future research in this area should address this problem. The model can predict, with reasonable accuracy, the tensile strains in the reinforcement (Fig. 9). For specimen SP7, which was tested in tension and transverse shear, the stirrups carried a large percentage of the applied transverse shear stress. In the case of specimen SP4 (compression plus transverse shear), the results of ®nite element analysis show that stirrups did not undergo any strain and thus did not carry the applied shear. In the test specimen, however, some amount of tensile strain was observed in stirrups.
5.3. Reinforced concrete shells subjected to punching shear The test specimens analyzed in this part of the investigation were curved reinforced concrete shells subjected to concentrated central loading [20]. The specimens were approximately one-sixth-scale representations of typical exterior walls for Arctic oshore structures. These were single span shells, simply supported along two edges and with two dierent curvatures: R/t = 6 and R/t = 12, where R is a specimen's radius of curvature and t is the wall thickness. Table 4 summarizes the specimens' properties and the results of the tests and of the analyses. Fig. 10 shows the geometry and the ®nite element mesh used in the analyses. The ratios of the predicted to the observed shear strengths of all ®ve specimens are presented in Fig. 11. The specimens were reinforced with an out-of-plane reinforcement in the ratio of 0, 0.24 and 0.48%. All specimens failed very suddenly. For the specimens with
Table 4 Properties of the curved shell specimens
Specimen AS1 AS2 AS3 AS5 AS6
R/t 12 12 12 6 6
Shear reinforcement [%] 0.0 0.24 0.48 0.0 0.24
f'c [MPa] 48 54 50 50 41
Vtest [MPa] 553 848 856 676 757
Vtest/b0dGf'c 10.1 14.7 15.4 12.1 14.9
Vpred
Shear ratio Vpred/Vtest
300 650 650 600 600
0.54 0.77 0.76 0.89 0.79
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these three-dimensional degenerate shell elements can be used in the global analyses of plate and shell structures.
Acknowledgements The support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC, Grant no. OGP0138111) is gratefully acknowledged. Fig. 11. Shear ratio versus transverse reinforcement ratio for curved shell specimens.
transverse reinforcement and R/t = 6, there was little bene®t from increasing the amount of transverse reinforcement from 0.24 to 0.48%. Also, for shells with the same amount of stirrups (0.24%) and two dierent curvatures (AS2 and AS6, Table 4), the normalized shear strength was almost the same [Vtest/(bodZf'c), where bo is a perimeter of critical section and d is the distance from the extreme compression ®bre to the centre of the tension reinforcement]. This suggests that these specimens (AS2, AS3 and AS6) failed by reaching the limit in concrete compressive strength rather than by yielding of stirrups. The predicted shear strength capacities suggest the same characteristics. Specimens with 0.24 and 0.48% of transverse reinforcement (AS2, AS3 and AS6) failed at the same load level. For the specimens without shear reinforcement (AS1 and AS5), the presented ®nite element formulation was able to correctly predict lower shear strength of the specimen with higher R/t ratio. 6. Summary and conclusions The formulation presented in this paper utilizes the concepts of degenerate shell elements. The formulation diers from the commonly used nonlinear models for reinforced concrete shells in that it considers concrete to be in three-dimensional states of stress and strain. This allows the formulation of a reinforced concrete shell element which can be reinforced in both in-plane and out-of-plane directions. The proposed ®nite element formulation was implemented in the computer program. It was used for the analysis of shell structures subjected to a combination of in-plane stresses and high transverse shear stresses. The consideration of out-of-plane shear and the implementation of a three-dimensional material model permits accurate predictions of out-of-plane shear failures. The practical advantage of the presented formulation is that the
References [1] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved ®nite elements. International Journal for Numerical Methods in Engineering 1970;:2, 419±51. [2] Kanok-Nukulchai W. A simple and ecient ®nite element for general shell analysis. International Journal for Numerical Methods in Engineering 1979;14:179±200. [3] Hughes TJR, Liu WK. Nonlinear ®nite element analysis of shells, part I, three-dimensional shells. Computer Methods in Applied Mechanics and Engineering ;1981:26, 331±62. [4] Papadopoulos P, Taylor RL. A triangular element based on Reissner±Mindlin plate theory. International Journal for Numerical Methods in Engineering 1990;:30, 1029± 49. [5] Pantazopoulou SJ, Ding Y. Collocation in ®nite elements analysis of constrained problems. Canadian Journal of Civil Engineering 1995;22:1±14. [6] Owen DRJ, Figueiras JA. Anisotropic elasto-plastic ®nite element analysis of thick and thin plates and shells. International Journal for Numerical Methods in Engineering 1983;19:541±66. [7] Di S, Cheung YK. Nonlinear analysis of RC shell structures using laminated element. I. ASCE Journal of Structural Engineering 1993;119(7):2059±73. [8] Di S, Cheung YK. Nonlinear analysis of RC shell structures using laminated element. II. ASCE Journal of Structural Engineerin 199;119(7):2074±94. [9] Thorenfeldt E, Tomaszewicz A, Jensen JJ. Mechanical properties of high-strength concrete and application in design. In: Proceedings of the Symposium Utilization of High-Strength Concrete, Stavanger, Norway. Tapir Trondheim, 1987. [10] Rericha P. Layer model of bending-shear failure in RC plates and beams. ASCE Journal of Structural Engineering 1991;117(10):2865±83. [11] Harmon TG, Zhangyuan N. Shear strength of reinforced concrete plates and shells determined by ®nite element analysis using layered elements. ASCE Journal of Structural Engineering 1989;115(5):1141±57. [12] Malkus DS, Hughes TJR. Mixed ®nite element methodsreduced and selective integration technique: a uni®cation of concepts. Computer Methods in Applied Mechanics and Engineering 1978;63(81):15.
M.A. Polak / Computers and Structures 68 (1998) 17±29 [13] Vecchio FJ. Reinforced concrete membrane element formulations. ASCE Journal of Structural Engineerin 1990;116(3):730±50. [14] Hughes TJR, Cohen M. The `Heterosis' plate bending element. Computers and Structures 1978;9:445±50. [15] Vecchio FJ, Collins MP. The modi®ed compression ®eld theory for reinforced concrete elements subjected to shear. Journal of the American Concrete Institut 198;83(2):219±31. [16] Collins MP, Porasz A. Shear strength for high strength concrete. Bulletin D'Information, no. 193, Design aspects of high strength concrete 1989;CEB:75±83.
29
[17] Izumo J, Shin H, Maekawa K., Okamura H. An analytical model for RC panels subjected to in-plane stresses. In: Concrete Shear in Earthquake. Elsevier Applied Science, London, 1992:206±15. [18] Kirschner U, Collins MP. Investigating the behaviour of reinforced concrete shell elements. University of Toronto, Department of Civil Engineering, 1986;Publication no. 86-09:82. [19] Adebar P, Collins MP. Shear design of concrete oshore structures. ACI Structural Journal 1994;901(3):324±45. [20] McLean DI, Phan LT, Lew HS, White RN. Punching shear behaviour of lightweight concrete slabs and shells. ACI Structural Journal 1990;87(4):386±92.
Computers and Structures 68 (1998) 17±29
Shear analysis of reinforced concrete shells using degenerate elements M.A. Polak Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 Received 31 October 1995; accepted 1 January 1997
Abstract The paper presents a formulation of degenerate shell ®nite elements which take into account three-dimensional states of stress and strain throughout the element. The formulation was developed with the aim of modelling reinforced concrete shell structures subjected to high transverse shear stresses. It requires implementation of a threedimensional constitutive model for concrete and it allows to use out-of-plane reinforcement in the elements. Through the consideration of three-dimensional stress states, the elements can model the in¯uence of in-plane stresses on transverse shear deformations and the in¯uence of transverse shear stresses on the capacity of shell structures. The formulation was veri®ed against experimental results. The specimens analyzed were shell panels subjected to out-of-plane (transverse) and in-plane shear and reinforced concrete shells subjected to punching shear. # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved.
fcx, fcy
Nomenclature [B] [D] [D*] [Dp] [Dcp] [Dc'] [D'st] [Dst] [K] [Te] {d} Ei Gij R
strain±displacement matrix material stiness matrix (modi®ed to 5 5) in element's axes (x,y,z) material stiness matrix (6 6) in element's axes material stiness matrix in principal axes material stiness matrix for concrete, in principal axes material stiness matrix for concrete (6 6), in element's axes transverse reinforcement material stiness matrix in reinforcement directions transverse reinforcement material stiness matrix in element's axes element stiness matrix strain transformation matrix vector of nodal displacements of an element secant Young's moduli for concrete shear moduli for concrete radius
fc1, fc2, fc3, fci fcr f'c fc2max fsx, fsy, fsi fs fy fscr u, v, w v1, v2 t ec e0 ecr e1, e2, e3 es ex, ey, ey gxy, gxz, gyz
concrete normal stresses in x- and ydirections concrete principal stresses cracking stress for concrete compressive cylinder strength of concrete compressive strength of cracked concrete reinforcement stresses in x, y and `i' directions transverse reinforcement stress yield stress of reinforcement reinforcement stress at a crack displacements in global directions shear stresses in concrete thickness concrete strain concrete strain corresponding to peak compressive stress cracking strain for concrete principal strains reinforcement strain normal strains shear strains
0045-7949/98/$19.00 # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 0 2 7 - 3
18
rs rsx, rsz, rsi s1, s2, s3 sx, sy, sz txy, txz, tyz ysci
M.A. Polak / Computers and Structures 68 (1998) 17±29
transverse reinforcement ratio reinforcement ratios in x, z and `i' directions principal stresses normal stresses shear stresses angle between the reinforcement and the normal to the crack
inforced concrete shell structures subjected to high transverse stresses. The model proposed in this paper utilizes the formulation of the displacement-based, degenerate shell elements. The modi®cation is in the approach to constitutive modelling used in the element formulation. The rotating crack model is adopted where the transverse shear strains in¯uence the principal directions of strains and stresses. At every point of the shell element, a three-dimensional state of stress and strain is assumed allowing a model of the in-plane versus out-of-plane shear interaction.
1. Introduction Degenerate shell elements have been used extensively for the global analyses of shell structures ever since they were introduced by Ahmad et al. [1]. These elements were developed because of the diculties in ensuring C1 continuity of displacements in the development of shell ®nite elements based on thin shell theories. Consequently, a dierent type of element was derived which requires only C0 continuity. These C0 elements are able to model the behaviour of shells where transverse shear deformations are signi®cant. Such behaviour is important in thick shell structures, in shells subjected to concentrated transverse loads and in plates supported by columns. In these cases, transverse shear can be the governing failure mechanism. The degenerate shell elements which include transverse shear strain energy in their formulation, have received considerable attention in the recent developments in the ®nite element analysis of shells [2±5]. The elements are based on a simple concept of `degenerating' threedimensional elasticity to a shell `situation' and, although the problems of shear-locking and zero energy modes must be overcome in the successful analysis, the elements perform well for both thick and thin shells. The degenerate shell elements' formulations are based on a theory equivalent to the Mindlin plate bending theory. They include transverse shear strain energy in their formulation. In order to model nonlinear material behaviour (e.g. concrete), the shell elements are divided into layers. However, it is usually assumed that each concrete layer remains in the state of plane stress and the transverse shear eects are decoupled from the in-plane strains and stresses [6±8]. It is also assumed that the transverse shear stresses do not aect the in-plane behaviour of concrete and reinforcement. Such models can predict the magnitude of transverse shear strains and the in¯uence these strains will have on the response (e.g. de¯ections) of a structure, but cannot predict when the transverse shear becomes responsible for the failure mechanism. The emphasis in the presented study is on the ability of shell ®nite elements to model the behaviour of re-
2. Degenerate shell elements The degenerate shell elements are derived from the equations of three-dimensional continuum mechanics [9]. The three-dimensional stress and strain conditions are degenerated to shell behaviour by adopting assumptions representing a typical shell behaviour (Fig. 1). These assumptions are ®rst, normals to the midsurface remain straight, but not necessarily normal to the midsurface after deformation and second, stresses normal to the midsurface are negligible [Fig. 1(b)]. The ®rst assumption has two implications: ®rst, that transverse shear strains exist and are included in the strain energy functional; and second, that these transverse strains are uniformly distributed through the
Fig. 1. Degenerate shell element. (a) reference system and degrees of freedom; (b) assumptions.
M.A. Polak / Computers and Structures 68 (1998) 17±29
thickness of a shell. This is a crude approximation of the real distribution of transverse strains. In the case of homogeneous linear elastic shell, this distribution takes a parabolic shape along the thickness of an element and it can be accounted for in the element formulation (to correctly assess transverse strain energy) by using a shear correction factor equal to 5/6. In the case of nonlinear, fracturing material this distribution of strain along the thickness is more dicult to assess. One way of ensuring the correct magnitude of transverse shear strain energy is the proper formulation of the shear moduli for the material [e.g. Eq. (16)]. The ®rst assumption also results in elements with 5 degrees-of-freedom per node: three translations (u, v and w) and two rotations (yx, yy). The two rotations are not dependent on the translations. The second assumption implies that the strain energy associated with the out-of-plane, axial deformations (ez) is ignored. It is approximately valid in most real situations and in fact most transverse shear theories for beams, slabs and shells make use of this assumption [10, 11]. It implies that there are ®ve nonzero stresses: fsg fsx ; sy ; txy ; txz ; tyz g
1
and ®ve independent strains: feg fex ; ey ; gxy ; gxz ; gyz g:
3. The three-dimensional formulation of the degenerate elements The presented formulation follows the assumptions and derivations of the degenerate ements. The ®ve independent strains [Eq. (1)] culated in the local coordinate system of an (Fig. 1) and they are equal to:
In the presented formulation it is assumed that strain ez is present and can be found from the condition sz=0. The degenerate shell elements use lagrangian or serendipity shape functions (depending on the number of nodes per element) for both translational and rotational degrees-of-freedom. They adopt isoparametric formulation for modelling geometry. This enables modelling of any shape of a shell structure. The stiness matrix is evaluated using gaussian integration. Reduced and selective integration can be used for thin shells where shear locking can become a problem [12]. In the nonlinear formulations, integration through the element's thickness is facilitated using a layered model where each layer has one integration point at the depth of its midsurface. The major advantages of the elements are that they are independent of any complex shell theory, their formulation is inherently simple, and yet they account for transverse shear deformations.
general shell elare calelement
2
3 @u 6 @x 7 6 7 6 @v 7 7 2 3 6 6 ex @y 7 6 7 6 ey 7 6 7 6 7 6 @u @v 7 7 6 7; g feg 6 xy 6 7 6 7 4 gxz 5 6 @y @x 7 6 7 @u @w 6 7 gyz 6 7 6 @z @x 7 6 7 4 @v @w 5 @z @y
3
where u, v and w are displacement components in the local system. Using the appropriate formulations [1], these local strains are calculated from the derivatives of the global displacements. The strain±displacement matrix [B] is formed which relates: feg Bfdg;
2
19
4
where {d} is a vector of element's nodal displacements in the global coordinate system and {ee} is a vector of strain as de®ned by Eq. (3). The stiness matrix of a degenerate shell element is found using the matrix [B] and the material stiness matrix [D], from the wellknown equation: Z K
v
BT DBdv:
5
The description that follows concentrates on the formulation of the material stiness matrix [D] in Eq. (5). The formulation is applicable for any nonlinear material and it takes into account the in¯uence of the transverse shear deformations on the orientation of the material's principal axes. The assumption of the normal stress as equal to zero (sz=0), permits the calculation of the normal strain (ez). It depends on the other ®ve strains [Eq. (1)] and for linear elastic material it is equal to:
20
M.A. Polak / Computers and Structures 68 (1998) 17±29
ez
ÿv ex ey 1 ÿ v
6
In the general case of an anisotropic material and the fully populated material stiness matrix [D*] [see Eq. (11)], the out-of-plane strain can be calculated as:
ez
2
D*11 6 D*21 6 * 6D 31 D* 6 6 D* 6 41 4 D* 51 D*61
D*12 D*22 D*32 D*42 D*52 D*62
D*13 D*23 D*33 D*43 D*53 D*63
D*14 D*24 D*34 D*44 D*54 D*64
D*15 D*25 D*35 D*45 D*55 D*65
3 D*16 * 7 D26 7 D*36 7 7: D*46 7 7 D*56 5 D*66
11
Matrix [D*] relates: fsx ; sy ; sz ; txy ; txz ; tyz gT D* fex ; ey ; ez ; gxy ; gxz ; gyz gT ;
ÿD*31 ex D*32 ey D*34 gxy D*35 gxz D*36 gyz : D*33
12 7
The material principal directions are assumed to coincide with the directions of principal strains. These are calculated from the six strains in the element's coordinate system (Fig. 1):
Before including matrix [D*] in the element stiness matrix equation [Eq. (3)] it must be modi®ed to enforce the zero normal stress condition. Modi®cation of the [D*] requires removal of row 3 and column 3 and the modi®cation of the material stiness coecients:
feg fex ; ey ; ez ; gxy ; gxz ; gyz g:
Dij D*ij ÿ
8
D*i3 D*3j D*33
:
13
Knowing the magnitude of the principal strains, (e1, e2, e3), the principal stresses, (s1, s2, s3), and the material stiness constants can be found using the constitutive model adopted in the formulation. Any rational three-dimensional constitutive model can be used in the element's formulation. The material is assumed to be orthotropic. The stress±strain relationship in the principal directions takes the form:
The new matrix [D] is a 5 5 matrix and it relates strains [Eq. (2)] to stresses [Eq. (1)]
fs1 ; s2 ; s3 ; t12 ; t13 ; t23 gT Dp fe1 ; e2 ; e3 ; g12 ; g13 ; g23 gT ; 9
The formulation presented in this section is a nonlinear procedure for the analysis of reinforced concrete.
fsx ; sy ; txy ; txz ; tyz gT Dfex ; ey ; gxy ; gxz ; gyz gT ;
14
Fig. 2 4. Formulation for reinforced concrete shells
where 2
E1 6 E21 6 6 E31 Dp 6 6 0 6 4 0 0
E12 E2 E32 0 0 0
E13 E23 E3 0 0 0
0 0 0 G12 0 0
0 0 0 0 G13 0
3 0 0 7 7 0 7 7: 0 7 7 0 5 G23
10
The material stiness matrix must be transformed into the local coordinate system for the element where it becomes a fully populated 6 6 matrix:
Fig. 2. Model for transverse and longitudinal reinforcements.
M.A. Polak / Computers and Structures 68 (1998) 17±29
It is based on the proposed three-dimensional formulation for degenerate shell elements. This section describes the formulation for a concrete layer. It uses the secant stiness approach for the nonlinear material analysis. It is assumed that the cracked concrete behaves like a nonlinear elastic, orthotropic material. Furthermore, it is assumed that cracked concrete can carry certain amount of tensile stresses (tension stiening), that shear can be transferred through cracked concrete interface by aggregate interlock and dowel action, and that the shear modulus can be estimated from three-dimensional states of stress and strain. The evaluation of the stiness matrix of an element requires the formulation of the material stiness matrix speci®ed in the element's x,y,z coordinate system. The isoparametric formulation requires numerical integration of the stiness matrix over the area of an element. Due to the material nonlinearity and the nonlinear distribution of stresses, integration through the thickness of an element is achieved by using a layered model. In the reinforced concrete shell element, the element is divided into a series of concrete and steel layers. Steel layers have the total area equal to the area of in-plane reinforcement and having stiness in one directionÐthe direction of reinforcement. The material properties of concrete change depending on the level of stresses and strains. The layered model allows us to evaluate strains, stresses and material properties at the midpoints of each layer. The total material stiness matrix of an element is calculated by adding the contributions of all layers (concrete and steel). The material stiness matrix of the element's concrete layer is formulated ®rst in the principal directions. The principal directions are found from the known strains in x,y,z directions [Eq. (8)]. From principal strains (e1, e2, e3), the principal stresses ( fc1, fc2, fc3) are found using the appropriate constitutive relationships (Fig. 4). The secant Young's moduli are calculated as:
Ei
fci ; i 1; 2; 3 ei
Ei Ej ; i; j 1; 2; 3: Ei Ej
E1 6 0 6 6 0 Dcp 6 6 0 6 4 0 0
0 E2 0 0 0 0
0 0 E3 0 0 0
0 0 0 G12 0 0
0 0 0 0 G13 0
3 0 0 7 7 0 7 7: 0 7 7 0 5 G23
17
The material stiness matrix is then transformed into the local coordinate system x,y,z: Dc 0 Te T Dcp Te
18
where [Te] is the strain transformation matrix between the principal and the local coordinate system. An important feature of the proposed formulation is the possibility of considering the in¯uence of the transverse reinforcement on the behaviour of shells (Fig. 2). The fact that the strain ez is present, allows the stirrups to strain and thus to contribute to the overall stiness of an element. The transverse reinforcement is speci®ed as a property of any concrete layer. It contributes to the stiness in one direction; namely in the direction of the reinforcement. The formulation starts by ®rst ®nding the strain in the direction of reinforcement, es. The stress ss is then found using the constitutive relationship for out-of-plane steel. The material matrix for transverse reinforcement, [D'st], is formulated in the reinforcement directions as: 2
rs efs 6 0s 6 6 0 D 0st 6 6 0 6 4 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
3 0 07 7 07 7: 07 7 05 0
19
The matrix [D'st] is then transformed to x,y,z coordinate system:
15
Dst Tse T D 0st Tse ;
16
where [Tes] is the strain transformation matrix between the reinforcement directions and x, y, z. The total material stiness matrix for a concrete layer which contains transverse reinforcement [Eq. (11)] is found from:
and the shear moduli [13]:
Gij
2
21
Assuming the Poisson's ratios for concrete after cracking, are zero, the material stiness matrix for the concrete layer, with respect to the principal axes, has the form [see also Eq. (10)]:
D* Dc 0 Dst :
20
21
The in-plane steel material stiness matrices take a form, in the reinforcement directions, analogous to
22
M.A. Polak / Computers and Structures 68 (1998) 17±29
Eq. (19). These matrices are then transformed to the element's local axes according to Eq. (20). The material stiness matrices for all concrete and in-plane steel layers are added resulting in the element's material stiness matrix. This matrix is then modi®ed according to Eq. (13). 4.1. Transverse reinforcement When the transverse reinforcement is present in the concrete shell element, it increases the stiness of the element in the direction of this reinforcement. In the case when transverse reinforcement is located in the z direction, it contributes to the D*33 term in the matrix [D*] [Eqs. (11) and (20)] in the following manner: D*33 D*33c Ds3 ;
22
where Ds3=rs3Es3 is the stiness contribution from the transverse steel, rs3 is the transverse reinforcement ratio, and D33c* is the stiness contribution from the concrete. From Eq. (7), it can be seen that, with the contribution from the transverse reinforcement, the strain in the z-direction will be decreased. This will increase the stiness of an element, decrease the strains in the next iteration and, consequently, allow the structure to sustain higher shear stresses before reinforcement yields or concrete crushes. 4.1.1. Strain, stress and strength considerations An analysis using Mohr's Circles illustrates how the transverse reinforcement helps to carry shear stresses. To simplify the discussion, consider a typical shell element simpli®ed to two dimensions (the y-direction has been omitted). The element is subjected to transverse shear only. The conditions that provide the basis for the following discussion are: equilibrium and the compatibility of concrete and reinforcement strains. Consider two intensities of the shear stress , v1 and v2, with v1
Subscript `1' refers to the shear stress v1 and subscript `2' refers to v2. First, consider the element to have x-direction reinforcement only [Fig. 3(a)]. Due to the condition that sz=0, and in order to maintain equilibrium, the normal stresses in the concrete in the z-direction must be zero. Now consider what occurs as the shear stress is increased to v2. Due to the strain compatibility between the concrete and the reinforcement and in order to maintain the equilibrium of an element in Fig. 3(a), the stresses in the longitudinal reinforcement fsx and stresses in concrete fcx increase rapidly. In addition, the angle y2 becomes much smaller than the angle y1, resulting in less stress able to be transferred through cracks [Eq. (23)]. Fig. 3(b) shows the same shell element but with both x- and z-reinforcement. The concrete can now develop compressive stresses in the z-direction, since this stress can be equilibrated by the tensile stresses in the z-reinforcement (due to the compatibility of strains in the z-direction). Consider the element subjected to the shear stress v1. Strains and stresses develop in both the x- and z-reinforcements, consequently, the stress in the x-reinforcement is lower now than it was in the element without z-reinforcement. The increase of shear stress to v2 increases stresses in both the x- and the z-reinforcements, however, the increase in stress in the x-reinforcement is again less than it would have been for the element without z-reinforcement. The angle y2 also remains larger and more tensile concrete stress can be transferred through cracks [Eq. (23)]. The increase in shear stress from v1 to v2 for the element reinforced in two directions [Fig. 3(b)], left the element with much more remaining strength capacity than it did for the element without transverse reinforcement [Fig. 3(a)]. Hence, the element reinforced in both the x- and z-directions can carry much higher shear forces.
5. Numerical results The proposed degenerate shell element formulation was used to develop a heterosis quadratic element for the nonlinear analysis of reinforced concrete shell structures [14]. The formulation has been checked against the experimental results on shells and plates subjected to high shear stresses. In the presented analyses the material model for concrete is as follows [10]. The main assumption of the model is that the directions of principal strains and stresses coincide [15]. The constitutive relationships are written with respect to the principal axes. The material fails when strains in concrete and
M.A. Polak / Computers and Structures 68 (1998) 17±29
23
Fig. 3. Mohr circles of stress for concrete subjected to pure shear. rxÐlongitudinal reinforcement ratio, rzÐtransverse reinforcement ratio. (a) Element without transverse reinforcement; (b) element with transverse reinforcement.
reinforcement achieve values large enough to signi®cantly reduce the stiness matrix of a structure. This can happen due to yielding of reinforcement, crushing of concrete or the inability to transfer tensile stresses through cracks. The stress±strain relationship for compression [9, 16] accounts for the reduction in strength and stiness due to the presence of transverse
cracks [Fig. 4(a)]. The model for concrete in tension includes tension stiening [17] that signi®cantly in¯uences post-cracking shell response [Fig. 4(b)]. The smeared, rotating crack approach is used with the crack orientations assumed to be perpendicular to the principal tensile strain directions. The mechanism of transmitting tensile stresses across cracks is rigorously
24
M.A. Polak / Computers and Structures 68 (1998) 17±29
Fig. 4. Constitutive model for concrete: (a) compression; (b) tension.
considered. This mechanism is very important, not only for modelling in-plane tensile behaviour, but also for the out-of-plane shear model. In order to check that the average concrete tensile stress can be transmitted across the cracks, the reserve of stress available in the reinforcement must satisfy the following relationship: fc1 RSni1 rsi fyi ÿ fsi cos2 ysci ;
23
where y is the angle between the reinforcement and the normal to the crack and n is the number of reinforcement components.
5.1. Shell specimens subjected to in-plane shear The ®rst series of tests presented, are tests on shell panels subjected to pure in-plane shear. The tests examined here were the part of an experimental program which included testing of full size shell elements using the Shell Element Tester at the University of Toronto [18]. The SE specimens were full size shell elements reinforced with deformed reinforcing bars. Specimens SE1, SE5 and SE6 were of the dimensions 1524 1524 mm with the thickness of 285 mm. The specimens were reinforced with orthogonal reinforcement placed in two zones, near each face of the specimen. The general layout of the specimens' reinforcement is shown in Fig. 5. Details regarding the
Table 1 Material properties of SE specimens Reinforcement
Test
Concrete f'c [MPa]
Behaviour
e'c
Strong fy1 [MPa]
Ratio r1 per layer
Weak fy2 [MPa]
Ratio r2 per layer
SE1
42.5
0.00254
492
0.01465
479
0.0049
SE5
25.9
0.0024
492
0.01465
492
0.01465
SE6
40.0
0.0025
492
0.01465
479
0.00163
Observed
Predicted
vcr=2.8 MPa vu=6.77 MPa failed by yielding of weak reinf. vcr=2.15 MPa vu=8.1 MPa failed by crushing of concrete. vcr=2.1 MPa vu=3.51 MPa failed by yielding of weak reinf.
vcr=2.4 MPa vu=7.80 MPa failed by yielding of weak reinf. vcr=2.03 MPa vu=8.5 MPa failed by a concrete failure vcr=2.06 MPa vu=4.5 MPa failed by yielding of weak reinf.
M.A. Polak / Computers and Structures 68 (1998) 17±29
25
material properties can be found in Table 1. The heavier reinforcement is referred to as the strong reinforcement, the lighter reinforcement as the weak reinforcement. Pure in-plane shear loads were applied along the reinforcement directions, by applying equal tension and compression forces in the two orthogonal directions at a 458 angle to the reinforcement. The specimens were modeled by a mesh of four equal-size elements. The number of layers used per element was 10. Although it was not necessary to model shells in pure in-plane shear by any more than one layer, in general cases where the actual stress distribution is unknown, the number of layers used will be approximately 10. The predictions obtained are shown in Fig. 6. The observed and predicted cracking and ultimate loads, as well as failure mode descriptions, are given in Table 1. The results are in close agreement with the experimentally observed responses and, in all cases, the failure mode was predicted correctly. 5.2. Shell elements subjected to transverse shear In order to develop a rational method for the shear design of concrete oshore structures, an experimental program was undertaken at the University of Toronto which included tests on shell panels under dierent combinations of transverse shear and in-plane forces [19]. Five of the specimens, SP3, SP4, SP7, SP8 and SP9 were chosen for the analysis. The specimens were tested in the Shell Element Tester at the University of Toronto. They were of a size of 1524 1524 mm, with a thickness of 310 mm. The in-plane reinforcement consisted of two orthog-
Fig. 6. Shear stress versus shear strain response for SE specimens.
Fig. 5. Dimensions and reinforcement layout for SE and SP specimens.
onal layers of bars. In both reinforcement directions, the amount of reinforcement provided was equal. The reinforcement was oriented at a 458 angle with respect to the specimens' sides (Fig. 5). The specimens were also reinforced in the out-of-plane direction. Shear
26
M.A. Polak / Computers and Structures 68 (1998) 17±29
Table 2 Material properties of SP specimens
Specimen SP3 SP4 SP7 SP8 SP9
Concrete properties
Reinforcement properties
f'c [MPa]
e'c
In-plane Ratio* rx=ry
fy [MPa]
fult [MPa]
f [mm]
Transverse Ratio rz
fy [MPa]
f [mm]
0.0022 0.0023 0.0020 0.0021 0.0026
0.0358 0.0358 0.0375 0.0375 0.0375
480 480 536 536 536
660 660 637 637 637
20 20 20 20 20
0.0008 0.0008 0.0008 0.0008 0.0008
460 460 460 460 460
8 8 8 8 8
50 52 54 53 50
Table 3 Observed and predicted shear strength of SP specimen
Specimen
Applied loading±ratio Shear strengthÐtest of in-plane shear to transverse/in-plane transverse shear [MPa]
Shear strengthÐpredicted transverse/in-plane [MPa]
SP3 SP4 SP7 SP8 SP9
transverse shear only 4 -4 in-plane shear only -8
1.6/0 2.0/8.0 1.1/4.4 0.0/16 0.9/7.2
1.64/0.0 1.90/7.6 1.60/6.4 0.0/16.75 1.22/9.76
reinforcement was used in the form of T-headed bars placed in a 120 120 mm grid. The applied loading consisted of uniform, out-of-plane shear, equilibrating bending moments and various combinations of inplane forces. The details concerning the material properties and loadings of the SP-series specimens are given in Tables 2 and 3. Each SP-specimen was modelled using a mesh of eight elements (Fig. 7). Along the line of action of the bending moments, four elements were
Fig. 7. Finite element mesh used for SP-specimens.
chosen (because of the complex de¯ection pro®le); along the line of action of the uniform in-plane load, only two elements were used. The shell was divided into ten concrete layers. The out-of-plane reinforcement was placed in the inner eight layers, leaving the outside concrete layers to simulate concrete cover. The analyses of the SP-series specimens demonstrated the ability of the program to predict out-ofplane shear failures. The predicted and observed shear capacities of the specimens are shown in Table 3 and Fig. 8. The presence of in-plane shear signi®cantly alters the transverse shear strength of a shell element.
Fig. 8. Transverse shear versus in-plane shear for SP shell specimens.
M.A. Polak / Computers and Structures 68 (1998) 17±29
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Fig. 9. Stirrup strains versus transverse shear stresses for SPshell specimens. Fig. 10. Curved shell specimensÐgeometry and ®nite element mesh.
The interaction diagram between in-plane and out-ofplane shear for a typical shell panel is shown in Fig. 8. The proposed model accurately predicted the capacity of SP3 and SP4, specimens subjected to pure transverse shear and transverse shear plus compressive force. For specimens SP7 and SP8, subjected to transverse shear with in-plane tension, the predicted strengths were lower than observed (Fig. 8). This can suggest, that the material model and the shear modulus formulation underestimate the amount of shear that can be carried by cracked concrete. Future research in this area should address this problem. The model can predict, with reasonable accuracy, the tensile strains in the reinforcement (Fig. 9). For specimen SP7, which was tested in tension and transverse shear, the stirrups carried a large percentage of the applied transverse shear stress. In the case of specimen SP4 (compression plus transverse shear), the results of ®nite element analysis show that stirrups did not undergo any strain and thus did not carry the applied shear. In the test specimen, however, some amount of tensile strain was observed in stirrups.
5.3. Reinforced concrete shells subjected to punching shear The test specimens analyzed in this part of the investigation were curved reinforced concrete shells subjected to concentrated central loading [20]. The specimens were approximately one-sixth-scale representations of typical exterior walls for Arctic oshore structures. These were single span shells, simply supported along two edges and with two dierent curvatures: R/t = 6 and R/t = 12, where R is a specimen's radius of curvature and t is the wall thickness. Table 4 summarizes the specimens' properties and the results of the tests and of the analyses. Fig. 10 shows the geometry and the ®nite element mesh used in the analyses. The ratios of the predicted to the observed shear strengths of all ®ve specimens are presented in Fig. 11. The specimens were reinforced with an out-of-plane reinforcement in the ratio of 0, 0.24 and 0.48%. All specimens failed very suddenly. For the specimens with
Table 4 Properties of the curved shell specimens
Specimen AS1 AS2 AS3 AS5 AS6
R/t 12 12 12 6 6
Shear reinforcement [%] 0.0 0.24 0.48 0.0 0.24
f'c [MPa] 48 54 50 50 41
Vtest [MPa] 553 848 856 676 757
Vtest/b0dGf'c 10.1 14.7 15.4 12.1 14.9
Vpred
Shear ratio Vpred/Vtest
300 650 650 600 600
0.54 0.77 0.76 0.89 0.79
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M.A. Polak / Computers and Structures 68 (1998) 17±29
these three-dimensional degenerate shell elements can be used in the global analyses of plate and shell structures.
Acknowledgements The support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC, Grant no. OGP0138111) is gratefully acknowledged. Fig. 11. Shear ratio versus transverse reinforcement ratio for curved shell specimens.
transverse reinforcement and R/t = 6, there was little bene®t from increasing the amount of transverse reinforcement from 0.24 to 0.48%. Also, for shells with the same amount of stirrups (0.24%) and two dierent curvatures (AS2 and AS6, Table 4), the normalized shear strength was almost the same [Vtest/(bodZf'c), where bo is a perimeter of critical section and d is the distance from the extreme compression ®bre to the centre of the tension reinforcement]. This suggests that these specimens (AS2, AS3 and AS6) failed by reaching the limit in concrete compressive strength rather than by yielding of stirrups. The predicted shear strength capacities suggest the same characteristics. Specimens with 0.24 and 0.48% of transverse reinforcement (AS2, AS3 and AS6) failed at the same load level. For the specimens without shear reinforcement (AS1 and AS5), the presented ®nite element formulation was able to correctly predict lower shear strength of the specimen with higher R/t ratio. 6. Summary and conclusions The formulation presented in this paper utilizes the concepts of degenerate shell elements. The formulation diers from the commonly used nonlinear models for reinforced concrete shells in that it considers concrete to be in three-dimensional states of stress and strain. This allows the formulation of a reinforced concrete shell element which can be reinforced in both in-plane and out-of-plane directions. The proposed ®nite element formulation was implemented in the computer program. It was used for the analysis of shell structures subjected to a combination of in-plane stresses and high transverse shear stresses. The consideration of out-of-plane shear and the implementation of a three-dimensional material model permits accurate predictions of out-of-plane shear failures. The practical advantage of the presented formulation is that the
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