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Nonlinear analysis of reinforced concrete cylindrical shells
Computers and Structures 80 (2002) 2177–2184 www.elsevier.com/locate/compstruc
Nonlinear analysis of reinforced concrete cylindrical shells Aloisio Ernesto Assan
*
Faculty of Civil Engineering, Department of Structures, State University of Campinas, Caixa Postal 6021, CEP 13083-971 Campinas, SP, Brazil Received 16 November 2000; accepted 4 July 2002
Abstract In this paper, analysis of reinforced concrete cylindrical shells is performed using a strain-based finite element. The shell element employed is bidimensional, cylindrical circular and has four-nodes and five nodal degrees of freedom. The nonlinearities due to concrete cracking and yielding of the steel are taken into account. The constitutive models for the materials employ the smeared cracking concept and a finite element layered approach. Concrete is modeled by a strain-induced orthotropic-elastic model under plane state of stress. A bilinear steel model is used and the stress/reversal with Baushinger effect is included. Examples show the good accuracy provided by this analysis. Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical shells; Reinforced concrete; Newton–Raphson iteration; Nonlinear analysis; Strain-based finite element; Smeared cracking
1. Introduction Reinforced concrete shells, due their special characteristics from the economical and strength points of view, have been extensively used to cover large spans as hangars, industrial buildings, exhibition halls and sports grounds. With the arrival of the finite element method and the development of computers with large capacity of memory, the analysis of concrete shell structures has progressed significantly, and most of the problems concerned with the design of such structures was overcome. Simultaneously, the techniques to deal with nonlinear problems and the conception of analytical models to predict a more realistic behaviour of reinforced concrete structures, in special shells, has systematically advanced over the last years. Finite element modelling for use in the analysis of several type of reinforced concrete thin shells can be
*
Fax: +55-19-3788-2411. E-mail address: [email protected] (A.E. Assan).
carried out by using flat finite elements, curved elements formulated on the basis of shell theories and elements derived from three-dimensional elements by applying the degeneration process. Flat elements proved to be adequate in the analysis of shells with no severe gradient in stress variation. One disadvantage of flat elements is the possible presence of discontinuity bending generated at the junction lines due to their geometric shape approximation. Many general curved shell finite elements were developed and are available to be incorporated into finite element codes. Usually, these finite elements have a greater number of nodal unknowns than the flat elements, making the imposition of boundary conditions more difficult, besides increasing the bandwidth of the system of linear equations. Rigid body modes are also more difficult to be reproduced due to the curved geometry. However, they give a better geometrical representation of the shell surface, providing very good results. Degenerate finite elements are computationally efficient, being the Semiloof and Heterosis the most known of these finite elements. An excellent description of such finite elements can be found in Chan [1].
0045-7949/02/$ - see front matter Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 2 4 5 - 6
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A.E. Assan / Computers and Structures 80 (2002) 2177–2184
Fig. 1. Coordinate frame of the shell element (a) and layered element (b).
The material models used for concrete shells has been based on approaches reported by Chen and Chen [2], Murray et al. [3], Buyukozturk [4], Darwin and Pecknold [5], Bazant and Bhat [6] among others. Excellent surveys of research in the field of nonlinear finite element analysis of reinforced concrete shells have been presented by Sordelis [7], Schnobrich [8], Popov [9] and Mang and Meschke [10]. A very complete guide is the state-of-art-report [11] published by the ASCE. In this paper the nonlinear geometric and material analysis of reinforced concrete cylindrical shells using a layered cylindrical circular strain-based finite element is presented. Concrete is modelled by a strain-induced orthotropic-elastic model under plane state of stress, developed by Darwin and Pecknold [5]. Concrete is supposed to behave in tension as a linear elastic brittle material. Cracking and yielding of concrete are considered. A linear unloading curve based on the envelope curve obtained from uniaxial cyclic load tests is used to model the softening effect. Tension stiffening is also taken into account. The reinforced concrete section is modelled as a layered system of concrete and equivalent smeared steel layers, as proposed by Rashid [12], assuming perfect bond between concrete and steel. A bilinear steel model is used and stress reversal with Baushinger effect is included. The nonlinear analysis is conducted by an incremental-iterative procedure based on Newton–RaphsonÕs method. Illustrative examples are presented in order to show the accuracy of the approach adopted in this paper.
The strain-based finite element used herein has five nodal degrees of freedom: ow ox
and
ow v oy R
u ¼ a1 þ Ra2 cosðy=RÞ Ra3 sinðy=RÞ þ a7 x þ a8 xy=R þ 3a11 y=4 Ra17 y 2 =2 þ ðR2 y y 3 =6Þa19 Ra20 y=2
v ¼ xa2 sinðy=RÞ þ xa3 cosðy=RÞ þ a4 þ a5 sinðy=RÞ þ a6 cosðy=RÞ þ a8 x2 =6R þ a11 x=4 þ Ra16 y þ Ra17 xy þ a18 y 2 =2 þ ðxy 2 =2 R2 xÞa19 þ Ra20 x=2
w ¼ a2 x cosðy=RÞ þ a3 x sinðy=RÞ a5 cosðy=RÞ þ a6 sinðy=RÞ þ Ra9 þ Ra10 x a12 x2 =2 a13 x3 =6 a14 x2 y=2R a15 x3 y=6R R2 a16 R2 a17 x Ra18 y Ra19 xy
2. Strain-based finite element
u; v; w;
with u, v and w denoting displacements along the x, y and z directions, respectively, and R is the radius of the shell, as shown in Fig. 1a. This cylindrical shell finite element was originally proposed by Ashwell and Sabir [13] and was later modified, independently, by Charchafchi [14] and Assan [15]. This element is based on simple generalized strain functions satisfying the requirements of constant and independent strains rather than the usual formulation based on independent displacements. The element includes all rigid body displacements, and satisfies the constant strain conditions. It has been used by Assan [16] and Assan and Aliabadi [17] in the analysis of multiple stiffened barrel shell concrete roofs and in the study of pressurized cylinders with circumferential and longitudinal cracks. The displacements u, v and w are represented by the following functions:
ð1Þ
ð2Þ
where the an coefficients are the generalized parameters. The shell finite element was divided into layers, as shown in Fig. 1b, to model more realistically the propagation of the cracks along the thickness of the shell.
A.E. Assan / Computers and Structures 80 (2002) 2177–2184
D11 ¼ E1 c2 þ E2 s2
3. Concrete model 3.1. Concrete constitutive matrix An incremental orthotropic model was proposed by Darwin and Pecknold [5] based on the concept of equivalent uniaxial strain, whereby the effects of biaxial stresses on internal damage of the concrete are represented by equivalent uniaxial stress–strain curves for each of the principal axes. Referring to the principal axes of orthotropy, the incremental constitutive relations are written as: 38 2 pffiffiffiffiffiffiffiffiffiffi 8 9 9 E1 t E1 E2 0 < de1 = < dr1 = 1 6 pffiffiffiffiffiffiffiffiffiffi 7 dr ¼ 0 5 de2 E2 4 t E1 E2 : 2 ; 1 m2 : ; ds12 dc12 0 0 G ð3Þ where Ei ði ¼ 1; 2Þ are the YoungÕs modulii for the principal directions, de1 , de2 and dc12 are increments in the principal strain components, t ¼ ðt1 t2 Þ1=2 is an equivalent PoissonÕs ratio and G is given by:
G ¼
1 4
2179
pffiffiffiffiffiffiffiffiffiffi E1 þ E2 2m E1 E2
ð4Þ
D22 ¼ E1 s2 þ E2 c2 D33 ¼ G pffiffiffiffiffiffiffiffiffiffi D12 ¼ D21 ¼ t E1 E2 D13 ¼ D31 ¼ D23 ¼ D32 ¼ 12ðE1 E2 Þcs
ð6Þ
with c ¼ cos h, s ¼ sin h and h is the angle between the coordinate systems (1,2) and ðx; yÞ. The uniaxial equivalent stress–strain curve considered in this work, proposed by Saenz [18], (Fig. 2a), and also adopted by Darwin and Pecknold [5] is: ri ¼
Ec0 iu 2 Ec0 iu iu 1 2 þ Ec0 ic ic
ð7Þ
being Ec0 the initial YoungÕs modulus for the concrete, iu the uniaxial equivalent strain in the i direction, ic the compression strain corresponding to the maximum compressive uniaxial equivalent stress ric and Ec0 the secant modulus of elasticity as defined in Fig. 2a. YoungÕs modulus Ei is evaluated by deriving Eq. (7), resulting: Ei ¼
dri E ð1 q2 Þ c0 ¼ diu ½1 Ec00 2 q þ q2 2 E
ð8Þ
c
The constitutive matrix is first generated with respect to the principal strain directions (1,2), after which it is transformed to the global coordinate axis ðx; yÞ, as shown below: 8 9 8 9 dex > > < drx = < = 1 dry D dey ¼ 2 : ; 1m > : dc > ; dsxy xy where the coefficients of D matrix are:
where q ¼ iu =ic . To avoid numerical problems, Eq. (8) is not used to calculate Ei where a zero or negative value would result. For tensile stress ri the constitutive relation is linearized as: ri ¼ Ec0 iu
ð5Þ
ð9Þ
The stress–strain relationship used for the concrete in tension is depicted in Fig. 2b. The tension stiffening constants b and em are assumed to be equal to 0.6 and 0.002, respectively, following Figueiras [19].
Fig. 2. Compression (a) and tensile (b) stress–strain relationships for concrete.
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Orthotropic models were criticized by Bazant [20] due their lack of uniqueness and stability and violation of tensorial invariance. However, they have been used successfully by many researchers, mainly in the analysis of thin shells and plates as, for example, Kabir and Scordelis [21], Milford and Schnobrich [22], Chan [1], VanGreunen [23], Rule and Rowlands [24], Elwi and Murray [25] and Sarne [26].
In this region concrete is assumed to yield beyond maximum compressive strength, failing by crushing. (b) Tension–compression: 0:17 6 a 6 0 r2c ¼
1 þ 3:28a ð1 þ aÞ2
fc0
e2c ¼ ec ð4:42 8:38p2 þ 7:54p22 2:58p23 Þ
3.2. Failure envelope
r1t ¼ ar2c
The biaxial strength envelope of Kupfer and Gerstle [27] is used to find the maximum stress ric and the corresponding strain eic . The biaxial strength envelope is divided into four regions, shown in Fig. 3, depending on the biaxial stress ratio a, being a ¼ r1 =r2 , with r1 algebraically greater than r2 . These four regions, shown in Fig. 3, are
e1t ¼ r1t =E0 p2 ¼ r2c =fc0
ð11Þ
In this case concrete fails by yielding and crushing. (c) Tension–compression: 1 < a < 0:17 r2c ¼ 0:65fc0
(a) Biaxial compression: 0 6 a 6 1 r2c ¼
1 þ 3:65a ð1 þ aÞ2
e2c ¼ ec ð4:42 8:38p2 þ 7:54p22 2:58p23 Þ
fc0
r1t ¼ ft0
e2c ¼ ec ð3p2 2Þ e1t ¼ r1t =E0 r1c ¼ ar2c e1c ¼
ec ð1:6p13
þ
2:25p12
p2 ¼ r2c =fc0 þ 0:35p1 Þ
The failure of the concrete will be due to cracking in direction 1 and yielding in direction 2. (d) Tension–tension: 1 6 a 6 1
p2 ¼ r2c =fc0 p1 ¼ r1c =fc0
ð12Þ
ð10Þ
r1t ¼ r2t ¼ ft0 e1t ¼ e2t ¼ ft0 =E0
ð13Þ
Failure occurs by cracking of the concrete.
4. Reinforcing steel model
Fig. 3. Biaxial strength envelop.
Reinforcing steel was modeled as a bilinear material and stress reversal with Baushinger effect was included, as shown in Fig. 4. Four parameters are necessary as input data to define the steel model: the initial YoungÕs modulus Es , the strain-hardening modulus Esh , the yield stress fy and the ultimate strain esu . Steel is considered as a layer embedded into a concrete layer with thickness equivalent to the thickness of the concrete layer where it is embedded. The steel layer is considered as an one-dimensional material in the reinforcement direction. The constitutive relations in the elastic range are given by
A.E. Assan / Computers and Structures 80 (2002) 2177–2184
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In this study the tangential stiffnesses matrices were updated at the beginning of each iteration. The criterion to check the convergence of the solution is based on the accuracy of both the node displacements and the unbalanced nodal forces. Unloading and reloading, opening and closure of cracks were taken into account. Time dependent effects were not considered in this analysis.
6. Examples
Fig. 4. Stress–strain relationship for the steel.
8 92 < drx = px Es dry 4 0 : ; dsxy 0
0 py Es 0
9 38 0 < dx = 0 5 dy : dc ; 0 xy
ð14Þ
where px and py are the relationships (in percentage) between the steel (in x- and y-directions) and the concrete layers.
5. Strategy for the nonlinear analysis An incremental-iterative strategy based on the Newton–RaphsonÕs method updating the stiffness matrix after each iteration was adopted. Nonlinear geometric behaviour was also taken into account. The strains and stresses are evaluated at nine Gaussian points of the finite elements. When the stress in one Gaussian point reaches the ultimate tensile strength of the concrete, it is assumed that one crack opens. The YoungÕs modulus of the concrete in that point is set equal to zero. This procedure is carried out for every Gaussian point. If the tensile stress is still below the ultimate tensile strength the same YoungÕs modulus is maintained. The same approach is adopted when considering the compressive stresses. If the compressive stresses in the Gaussian points do not reach the ultimate strength of the concrete, a new YoungÕs modulus are calculated with Eq. (8) relatively to the strain c . With the new value of the YoungÕs modulus another stiffness matrix is updated and the analysis continues following the conventional Newton–RaphsonÕs method.
Two examples are considered. The first is concerned with an unreinforced cylinder submitted to an external water pressure until explosion occurred. The analyzed cylinder was part of an experimental test program conducted at the Naval Civil Engineering Laboratory, California, where 15 unreinforced concrete cylinders were put into a vessel loaded with external water pressure until explosion occurred. The experimental results were reported by Runge and Haynes [28]. An independent analytical analysis was carried out at the Purdue University by Chen et al. [29], using the NFAP computer program, which is an extended version of the NONSAP-A [30] program. This software is a modified version of the NONSAP program developed by Bathe et al. [31]. The input data for one of the analyzed cylinders is given in Fig. 5. The first analytical results were published by Chen et al. [29] and Chang and Chen [30], who performed the analytical study with the NFAP computer code, considering a plasticity-based model developed by Chen and Chen [2]. Later, Rule and Rowlands [24] analyzed one of those cylinders, shown in Fig. 5, with a flat shell finite element which uses a linear shape function for in-plane displacements and a cubic polynomial for out of plane displacements; they did not consider the geometric nonlinearity in their analysis. These authors adopted the orthotropic-elastic model and the uniaxial strain approach proposed by Darwin and Pecknold [5] with simplified biaxial failure envelope and uniaxial stress– strain curves for concrete. The analysis of this example was performed considering the cylinder discretized by 30 equal finite elements, being 10 finite elements laid out along the height of the cylinder and its thickness divided into 10 layers. The same mesh was considered by Rule and Rowlands [24], Chen et al. [29] and Chang and Chen [30]. Results for the transversal midlength displacement of the cylinder are plotted in Fig. 6. External pressure load increments of 0.172 MPa were used in the analysis. Fig. 6 shows the results obtained by Chen et al. [29], Chang and Chen [30], Rule and Rowlands [24], Rule [32] the experimental values published by Runge and Haynes
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A.E. Assan / Computers and Structures 80 (2002) 2177–2184
Fig. 5. Unreinforced cylinder under external water pressure.
[28] and the ones obtained in this study. The average implosion pressure of three specimens, determined ex-
perimentally, was 3.97 MPa with a standard deviation of 0.17 MPa. The second example comprises a circular reinforced concrete cylindrical shell analyzed numerically and experimentally by Takayama et al. [33]. These authors were investigating the carrying capacities of reinforced concrete cylindrical shells with initial imperfections. The experiments were performed using two cases of specimens: one a nearly perfect circular cylindrical shell and the other a shell which was given artificial geometrical imperfections. In this paper the results obtained by Takayama et al. [33] for the first of the shells, whose geometric configuration and general input data are shown in Fig. 7, were used for comparison. Those authors adopted, in the numerical analysis, a similar model as suggested by Cedolin and Dei Poli [34], also described by Hinton and Owen [35]. The constitutive relationship of concrete was assumed to the HookeÕs law until the equivalent effective stress reached the uniaxial compressive strength of concrete. After this point the material was regarded as yielding and the associated flow rule was considered. The yield function of concrete is as follow:
Fig. 6. Applied pressure versus midlength radial displacements.
A.E. Assan / Computers and Structures 80 (2002) 2177–2184
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Fig. 7. Overall input data for Takayama et al. [33] shell.
n h 2 2 i 2 2 2 r ¼ b r0x þ r0y r0x r0y þ 3 s0xy þ s0xz þ s0yz o1=2 þ aðr0x þ r0y Þ
ð15Þ
where a and b are equal to 0.355r and 1.355, respectively. These constants were determined based on the studies of Kupfer and Gerstle [27] and Kupfer et al. [36].
In tension, Takayama et al. [33] used the same approach as used in this paper to model the behaviour of concrete. The finite element used by Takayama et al. [33] was the nine-node Heterosis element with reduced integration ð2 2Þ and layered approach. In their analysis, these authors divided one fourth of the shell into 45 finite elements. The shell was assumed to be pin supported at the contour and subjected to uniformly vertical load. Each element was divided into eight layers for concrete and the reinforcement, with diameter of 1.2 mm, was distributed into 2 layers in the middle surface of the shell. In this paper the shell was divided into 48 finite elements but the same number of concrete and steel layers as adopted by Takayama et al. [33], was considered. Fig. 8 depicts the vertical displacements in nodes A, B and C showed in Fig. 7.
7. Conclusion
Fig. 8. Load-vertical displacements in points A, B and C of the shell.
In this paper the nonlinear analysis of cylindrical concrete shells by a strain-based circular cylindrical shell finite element was shown. The results of the analysis, compared with values obtained by more sophisticated finite elements, show that the proposed approach can be efficiently used in the design of concrete cylindrical shells with great economy of computing time and reduced programming effort, since the shell finite element used herein is very simple to incorporate into finite element codes.
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References [1] Chan EC. Nonlinear geometric, material and time dependent analysis of reinforced concrete shells with edge beams. Report UCB/SESM-82/08, Department of Civil Engineering, University of California, Berkeley, California, 1982. [2] Chen ACT, Chen WF. Constitutive relations for concrete. J Eng Mech 1975;101(4):465–81. [3] Murray DH, Chitnuyanondh L, Agha KY, Wong C. A concrete plasticity theory for biaxial stress analysis. J Eng Mech 1979;105(6):989–1006. [4] Buyukozturk O. Nonlinear analysis of reinforced concrete structures. Comput Struct 1977;7:149–56. [5] Darwin D, Pecknold DA. Inelastic model for cyclic biaxial loading of reinforced concrete. Report SRS 409, Civil Engineering Studies, University of Illinois at Urbana Champaign, Urbana, Illinois, 1974. [6] Bazant ZP, Bhat P. Endochronic theory of inelasticity and failures of concrete. J Eng Mech 1976;102:701–22. [7] Scordelis AC. Nonlinear material, geometric and timedependent analysis of reinforced and prestressed concrete shells. In: Proceedings of the International Association for Shells and Spatial Structures, Madrid, Spain, 1989. p. 57– 70. [8] Schnobrich WC. Nonlinear analysis of reinforced concrete shells. Preprint 3372, ASCE Convention and Exposition, Chicago, 1978. [9] Popov E. Progress in analysis and design of RC shells. Eng Struct 1991;13:128–43. [10] Mang HA, Meschke G. Nonlinear finite element analysis of reinforced and prestressed concrete structures. Eng Struct 1991;13:211–26. [11] ASCE. Finite element analysis of reinforced concrete, Report Task Committee, New York, 1982. [12] Rashid YR. Ultimate strength analysis of prestressed concrete pressure vessels. Nucl Eng Des 1968;7(4):334– 55. [13] Ashwell DG, Sabir AB. A new cylindrical shell finite element based on independent strain functions. Int J Mech Sci 1972;14:171–83. [14] Charchafchi TA. Curved rectangular and general quadrilateral shell elements for cylindrical shells. In: Mathematics of finite elements and applications. London: Academic Press; 1982. p. 231–9. [15] Assan AE. Reservat orios com Parede Ondulada. Um Enfoque pelo Metodo dos Elementos Finitos. PhD thesis, Escola de Engenharia de S~ao Carlos, USP, S~ao Carlos, SP, Brazil, 1984 (in portuguese). [16] Assan AE. Analysis of multiple stiffened barrel shell structures by strain-based finite elements. Thin-Walled Struct 1999;35(4):233–54. [17] Assan AE, Aliabadi MH. Analysis of cracked cylindrical shells by a strain-based finite element. Adv Eng Software 1993;17:39–47. [18] Saenz LP. In: Desayi P, Krishnan S, editors. Discussion of equation for the stress-strain curve of concrete. J ACI 1964;61(9):1227–39. [19] Figueiras JA. Ultimate load analysis of anisotropic and reinforced concrete plates and shells. PhD thesis, Depart-
[20] [21]
[22]
[23]
[24]
[25] [26]
[27] [28]
[29]
[30]
[31]
[32]
[33]
[34] [35] [36]
ment of Civil Engineering, University College of Swansea, Swansea, Wales, UK, 1983. Bazant ZP. Comment on orthotropic models for concrete and geomaterials. J Eng Mech 1983;109(3):849–65. Kabir AF, Scordelis AC. Analysis of reinforced concrete shells for time dependent effects. Bull Int Assoc Shell and Spatial Struct 1979;XXI:3–13. Milford RV, Schnobrich WC. Nonlinear behavior of reinforced concrete cooling towers. Report SRS 514, Civil Engineering Studies, University of Illinois at Urbana Champaign, Urbana, Illinois, 1984. VanGreunen J. Nonlinear geometric material and time dependent analysis of reinforced and prestressed concrete slabs and panels. Report UCB/SESM-79/3, Department of Civil Engineering, University of California, Berkeley, California, 1979. Rule WK, Rowlands RE. A simple orthotropic elasticitybased constitutive model for reinforced concrete. Expt Mech 1989:448–54. Elwi AA, Murray DW. A 3D hypoelastic concrete constitutive relationship. J Eng Mech 1979;105(4):623–41. Sarne Y. Material nonlinear time dependent three-dimensional finite element analysis for reinforced and prestressed concrete structures. PhD thesis, Department of Civil Engineering, Massachussets Institute of Technology, Cambridge, 1975. Kupfer HB, Gerstle KH. Behavior of concrete under biaxial stresses. J Eng Mech 1979;105(4):623–41. Runge KH, Haynes HH. Experimental implosion study of concrete structures. In Proceedings of the Eighth Congress of the Federation International de la Precontrainte, London, England, May 1978. Chen WF, Suzuki H, Chang TY. Nonlinear analysis of concrete cylinder structures under hydrostatic loading. Comput Struct 1980;12:559–70. Chang TY, Chen WF. Extended NONSAP program for OTEC Structural Systems. ERDA Report COO-2682-7, 1976. Bathe KJ, Wilson EL, Iding RH. NONSAP––A structural analysis program for static dynamic response of nonlinear analysis. SESM Report 73-3, Department of Civil Engineering, University of California, Berkeley, California, 1974. Rule WK. A simple nonlinear constitutive model for finite element investigation of reinforced concrete structures. PhD thesis, Department of Engineering Mechanics, University of Wisconsin, Madison, Wisconsin, 1986. Takayama M, Nakamura H, Hara T, Kato S. Comparison between experiments and analysis for ultimate strength of reinforced concrete cylindrical shells with geometric initial imperfections subjected to uniformly vertical loads. In: Proceedings of the Third Summer Colloquium on Shell and Spatial Structures, Taegue, Korea, 1990. p. 237– 49. Cedolin L, Dei Poli S. Finite element studies of shearcritical r/c beams. J Eng Mech 1977;103(3):395–410. Hinton E, Owen DRJ. Finite element software for plates and shells. Swansea: Pineridge Press; 1984. Kupfer H, Hilsdorf HK, Rush H. Behavior of concrete under biaxial stress. J ACI 1969;66(8):656–66.
Nonlinear analysis of reinforced concrete cylindrical shells Aloisio Ernesto Assan
*
Faculty of Civil Engineering, Department of Structures, State University of Campinas, Caixa Postal 6021, CEP 13083-971 Campinas, SP, Brazil Received 16 November 2000; accepted 4 July 2002
Abstract In this paper, analysis of reinforced concrete cylindrical shells is performed using a strain-based finite element. The shell element employed is bidimensional, cylindrical circular and has four-nodes and five nodal degrees of freedom. The nonlinearities due to concrete cracking and yielding of the steel are taken into account. The constitutive models for the materials employ the smeared cracking concept and a finite element layered approach. Concrete is modeled by a strain-induced orthotropic-elastic model under plane state of stress. A bilinear steel model is used and the stress/reversal with Baushinger effect is included. Examples show the good accuracy provided by this analysis. Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Cylindrical shells; Reinforced concrete; Newton–Raphson iteration; Nonlinear analysis; Strain-based finite element; Smeared cracking
1. Introduction Reinforced concrete shells, due their special characteristics from the economical and strength points of view, have been extensively used to cover large spans as hangars, industrial buildings, exhibition halls and sports grounds. With the arrival of the finite element method and the development of computers with large capacity of memory, the analysis of concrete shell structures has progressed significantly, and most of the problems concerned with the design of such structures was overcome. Simultaneously, the techniques to deal with nonlinear problems and the conception of analytical models to predict a more realistic behaviour of reinforced concrete structures, in special shells, has systematically advanced over the last years. Finite element modelling for use in the analysis of several type of reinforced concrete thin shells can be
*
Fax: +55-19-3788-2411. E-mail address: [email protected] (A.E. Assan).
carried out by using flat finite elements, curved elements formulated on the basis of shell theories and elements derived from three-dimensional elements by applying the degeneration process. Flat elements proved to be adequate in the analysis of shells with no severe gradient in stress variation. One disadvantage of flat elements is the possible presence of discontinuity bending generated at the junction lines due to their geometric shape approximation. Many general curved shell finite elements were developed and are available to be incorporated into finite element codes. Usually, these finite elements have a greater number of nodal unknowns than the flat elements, making the imposition of boundary conditions more difficult, besides increasing the bandwidth of the system of linear equations. Rigid body modes are also more difficult to be reproduced due to the curved geometry. However, they give a better geometrical representation of the shell surface, providing very good results. Degenerate finite elements are computationally efficient, being the Semiloof and Heterosis the most known of these finite elements. An excellent description of such finite elements can be found in Chan [1].
0045-7949/02/$ - see front matter Ó 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 2 4 5 - 6
2178
A.E. Assan / Computers and Structures 80 (2002) 2177–2184
Fig. 1. Coordinate frame of the shell element (a) and layered element (b).
The material models used for concrete shells has been based on approaches reported by Chen and Chen [2], Murray et al. [3], Buyukozturk [4], Darwin and Pecknold [5], Bazant and Bhat [6] among others. Excellent surveys of research in the field of nonlinear finite element analysis of reinforced concrete shells have been presented by Sordelis [7], Schnobrich [8], Popov [9] and Mang and Meschke [10]. A very complete guide is the state-of-art-report [11] published by the ASCE. In this paper the nonlinear geometric and material analysis of reinforced concrete cylindrical shells using a layered cylindrical circular strain-based finite element is presented. Concrete is modelled by a strain-induced orthotropic-elastic model under plane state of stress, developed by Darwin and Pecknold [5]. Concrete is supposed to behave in tension as a linear elastic brittle material. Cracking and yielding of concrete are considered. A linear unloading curve based on the envelope curve obtained from uniaxial cyclic load tests is used to model the softening effect. Tension stiffening is also taken into account. The reinforced concrete section is modelled as a layered system of concrete and equivalent smeared steel layers, as proposed by Rashid [12], assuming perfect bond between concrete and steel. A bilinear steel model is used and stress reversal with Baushinger effect is included. The nonlinear analysis is conducted by an incremental-iterative procedure based on Newton–RaphsonÕs method. Illustrative examples are presented in order to show the accuracy of the approach adopted in this paper.
The strain-based finite element used herein has five nodal degrees of freedom: ow ox
and
ow v oy R
u ¼ a1 þ Ra2 cosðy=RÞ Ra3 sinðy=RÞ þ a7 x þ a8 xy=R þ 3a11 y=4 Ra17 y 2 =2 þ ðR2 y y 3 =6Þa19 Ra20 y=2
v ¼ xa2 sinðy=RÞ þ xa3 cosðy=RÞ þ a4 þ a5 sinðy=RÞ þ a6 cosðy=RÞ þ a8 x2 =6R þ a11 x=4 þ Ra16 y þ Ra17 xy þ a18 y 2 =2 þ ðxy 2 =2 R2 xÞa19 þ Ra20 x=2
w ¼ a2 x cosðy=RÞ þ a3 x sinðy=RÞ a5 cosðy=RÞ þ a6 sinðy=RÞ þ Ra9 þ Ra10 x a12 x2 =2 a13 x3 =6 a14 x2 y=2R a15 x3 y=6R R2 a16 R2 a17 x Ra18 y Ra19 xy
2. Strain-based finite element
u; v; w;
with u, v and w denoting displacements along the x, y and z directions, respectively, and R is the radius of the shell, as shown in Fig. 1a. This cylindrical shell finite element was originally proposed by Ashwell and Sabir [13] and was later modified, independently, by Charchafchi [14] and Assan [15]. This element is based on simple generalized strain functions satisfying the requirements of constant and independent strains rather than the usual formulation based on independent displacements. The element includes all rigid body displacements, and satisfies the constant strain conditions. It has been used by Assan [16] and Assan and Aliabadi [17] in the analysis of multiple stiffened barrel shell concrete roofs and in the study of pressurized cylinders with circumferential and longitudinal cracks. The displacements u, v and w are represented by the following functions:
ð1Þ
ð2Þ
where the an coefficients are the generalized parameters. The shell finite element was divided into layers, as shown in Fig. 1b, to model more realistically the propagation of the cracks along the thickness of the shell.
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D11 ¼ E1 c2 þ E2 s2
3. Concrete model 3.1. Concrete constitutive matrix An incremental orthotropic model was proposed by Darwin and Pecknold [5] based on the concept of equivalent uniaxial strain, whereby the effects of biaxial stresses on internal damage of the concrete are represented by equivalent uniaxial stress–strain curves for each of the principal axes. Referring to the principal axes of orthotropy, the incremental constitutive relations are written as: 38 2 pffiffiffiffiffiffiffiffiffiffi 8 9 9 E1 t E1 E2 0 < de1 = < dr1 = 1 6 pffiffiffiffiffiffiffiffiffiffi 7 dr ¼ 0 5 de2 E2 4 t E1 E2 : 2 ; 1 m2 : ; ds12 dc12 0 0 G ð3Þ where Ei ði ¼ 1; 2Þ are the YoungÕs modulii for the principal directions, de1 , de2 and dc12 are increments in the principal strain components, t ¼ ðt1 t2 Þ1=2 is an equivalent PoissonÕs ratio and G is given by:
G ¼
1 4
2179
pffiffiffiffiffiffiffiffiffiffi E1 þ E2 2m E1 E2
ð4Þ
D22 ¼ E1 s2 þ E2 c2 D33 ¼ G pffiffiffiffiffiffiffiffiffiffi D12 ¼ D21 ¼ t E1 E2 D13 ¼ D31 ¼ D23 ¼ D32 ¼ 12ðE1 E2 Þcs
ð6Þ
with c ¼ cos h, s ¼ sin h and h is the angle between the coordinate systems (1,2) and ðx; yÞ. The uniaxial equivalent stress–strain curve considered in this work, proposed by Saenz [18], (Fig. 2a), and also adopted by Darwin and Pecknold [5] is: ri ¼
Ec0 iu 2 Ec0 iu iu 1 2 þ Ec0 ic ic
ð7Þ
being Ec0 the initial YoungÕs modulus for the concrete, iu the uniaxial equivalent strain in the i direction, ic the compression strain corresponding to the maximum compressive uniaxial equivalent stress ric and Ec0 the secant modulus of elasticity as defined in Fig. 2a. YoungÕs modulus Ei is evaluated by deriving Eq. (7), resulting: Ei ¼
dri E ð1 q2 Þ c0 ¼ diu ½1 Ec00 2 q þ q2 2 E
ð8Þ
c
The constitutive matrix is first generated with respect to the principal strain directions (1,2), after which it is transformed to the global coordinate axis ðx; yÞ, as shown below: 8 9 8 9 dex > > < drx = < = 1 dry D dey ¼ 2 : ; 1m > : dc > ; dsxy xy where the coefficients of D matrix are:
where q ¼ iu =ic . To avoid numerical problems, Eq. (8) is not used to calculate Ei where a zero or negative value would result. For tensile stress ri the constitutive relation is linearized as: ri ¼ Ec0 iu
ð5Þ
ð9Þ
The stress–strain relationship used for the concrete in tension is depicted in Fig. 2b. The tension stiffening constants b and em are assumed to be equal to 0.6 and 0.002, respectively, following Figueiras [19].
Fig. 2. Compression (a) and tensile (b) stress–strain relationships for concrete.
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Orthotropic models were criticized by Bazant [20] due their lack of uniqueness and stability and violation of tensorial invariance. However, they have been used successfully by many researchers, mainly in the analysis of thin shells and plates as, for example, Kabir and Scordelis [21], Milford and Schnobrich [22], Chan [1], VanGreunen [23], Rule and Rowlands [24], Elwi and Murray [25] and Sarne [26].
In this region concrete is assumed to yield beyond maximum compressive strength, failing by crushing. (b) Tension–compression: 0:17 6 a 6 0 r2c ¼
1 þ 3:28a ð1 þ aÞ2
fc0
e2c ¼ ec ð4:42 8:38p2 þ 7:54p22 2:58p23 Þ
3.2. Failure envelope
r1t ¼ ar2c
The biaxial strength envelope of Kupfer and Gerstle [27] is used to find the maximum stress ric and the corresponding strain eic . The biaxial strength envelope is divided into four regions, shown in Fig. 3, depending on the biaxial stress ratio a, being a ¼ r1 =r2 , with r1 algebraically greater than r2 . These four regions, shown in Fig. 3, are
e1t ¼ r1t =E0 p2 ¼ r2c =fc0
ð11Þ
In this case concrete fails by yielding and crushing. (c) Tension–compression: 1 < a < 0:17 r2c ¼ 0:65fc0
(a) Biaxial compression: 0 6 a 6 1 r2c ¼
1 þ 3:65a ð1 þ aÞ2
e2c ¼ ec ð4:42 8:38p2 þ 7:54p22 2:58p23 Þ
fc0
r1t ¼ ft0
e2c ¼ ec ð3p2 2Þ e1t ¼ r1t =E0 r1c ¼ ar2c e1c ¼
ec ð1:6p13
þ
2:25p12
p2 ¼ r2c =fc0 þ 0:35p1 Þ
The failure of the concrete will be due to cracking in direction 1 and yielding in direction 2. (d) Tension–tension: 1 6 a 6 1
p2 ¼ r2c =fc0 p1 ¼ r1c =fc0
ð12Þ
ð10Þ
r1t ¼ r2t ¼ ft0 e1t ¼ e2t ¼ ft0 =E0
ð13Þ
Failure occurs by cracking of the concrete.
4. Reinforcing steel model
Fig. 3. Biaxial strength envelop.
Reinforcing steel was modeled as a bilinear material and stress reversal with Baushinger effect was included, as shown in Fig. 4. Four parameters are necessary as input data to define the steel model: the initial YoungÕs modulus Es , the strain-hardening modulus Esh , the yield stress fy and the ultimate strain esu . Steel is considered as a layer embedded into a concrete layer with thickness equivalent to the thickness of the concrete layer where it is embedded. The steel layer is considered as an one-dimensional material in the reinforcement direction. The constitutive relations in the elastic range are given by
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In this study the tangential stiffnesses matrices were updated at the beginning of each iteration. The criterion to check the convergence of the solution is based on the accuracy of both the node displacements and the unbalanced nodal forces. Unloading and reloading, opening and closure of cracks were taken into account. Time dependent effects were not considered in this analysis.
6. Examples
Fig. 4. Stress–strain relationship for the steel.
8 92 < drx = px Es dry 4 0 : ; dsxy 0
0 py Es 0
9 38 0 < dx = 0 5 dy : dc ; 0 xy
ð14Þ
where px and py are the relationships (in percentage) between the steel (in x- and y-directions) and the concrete layers.
5. Strategy for the nonlinear analysis An incremental-iterative strategy based on the Newton–RaphsonÕs method updating the stiffness matrix after each iteration was adopted. Nonlinear geometric behaviour was also taken into account. The strains and stresses are evaluated at nine Gaussian points of the finite elements. When the stress in one Gaussian point reaches the ultimate tensile strength of the concrete, it is assumed that one crack opens. The YoungÕs modulus of the concrete in that point is set equal to zero. This procedure is carried out for every Gaussian point. If the tensile stress is still below the ultimate tensile strength the same YoungÕs modulus is maintained. The same approach is adopted when considering the compressive stresses. If the compressive stresses in the Gaussian points do not reach the ultimate strength of the concrete, a new YoungÕs modulus are calculated with Eq. (8) relatively to the strain c . With the new value of the YoungÕs modulus another stiffness matrix is updated and the analysis continues following the conventional Newton–RaphsonÕs method.
Two examples are considered. The first is concerned with an unreinforced cylinder submitted to an external water pressure until explosion occurred. The analyzed cylinder was part of an experimental test program conducted at the Naval Civil Engineering Laboratory, California, where 15 unreinforced concrete cylinders were put into a vessel loaded with external water pressure until explosion occurred. The experimental results were reported by Runge and Haynes [28]. An independent analytical analysis was carried out at the Purdue University by Chen et al. [29], using the NFAP computer program, which is an extended version of the NONSAP-A [30] program. This software is a modified version of the NONSAP program developed by Bathe et al. [31]. The input data for one of the analyzed cylinders is given in Fig. 5. The first analytical results were published by Chen et al. [29] and Chang and Chen [30], who performed the analytical study with the NFAP computer code, considering a plasticity-based model developed by Chen and Chen [2]. Later, Rule and Rowlands [24] analyzed one of those cylinders, shown in Fig. 5, with a flat shell finite element which uses a linear shape function for in-plane displacements and a cubic polynomial for out of plane displacements; they did not consider the geometric nonlinearity in their analysis. These authors adopted the orthotropic-elastic model and the uniaxial strain approach proposed by Darwin and Pecknold [5] with simplified biaxial failure envelope and uniaxial stress– strain curves for concrete. The analysis of this example was performed considering the cylinder discretized by 30 equal finite elements, being 10 finite elements laid out along the height of the cylinder and its thickness divided into 10 layers. The same mesh was considered by Rule and Rowlands [24], Chen et al. [29] and Chang and Chen [30]. Results for the transversal midlength displacement of the cylinder are plotted in Fig. 6. External pressure load increments of 0.172 MPa were used in the analysis. Fig. 6 shows the results obtained by Chen et al. [29], Chang and Chen [30], Rule and Rowlands [24], Rule [32] the experimental values published by Runge and Haynes
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Fig. 5. Unreinforced cylinder under external water pressure.
[28] and the ones obtained in this study. The average implosion pressure of three specimens, determined ex-
perimentally, was 3.97 MPa with a standard deviation of 0.17 MPa. The second example comprises a circular reinforced concrete cylindrical shell analyzed numerically and experimentally by Takayama et al. [33]. These authors were investigating the carrying capacities of reinforced concrete cylindrical shells with initial imperfections. The experiments were performed using two cases of specimens: one a nearly perfect circular cylindrical shell and the other a shell which was given artificial geometrical imperfections. In this paper the results obtained by Takayama et al. [33] for the first of the shells, whose geometric configuration and general input data are shown in Fig. 7, were used for comparison. Those authors adopted, in the numerical analysis, a similar model as suggested by Cedolin and Dei Poli [34], also described by Hinton and Owen [35]. The constitutive relationship of concrete was assumed to the HookeÕs law until the equivalent effective stress reached the uniaxial compressive strength of concrete. After this point the material was regarded as yielding and the associated flow rule was considered. The yield function of concrete is as follow:
Fig. 6. Applied pressure versus midlength radial displacements.
A.E. Assan / Computers and Structures 80 (2002) 2177–2184
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Fig. 7. Overall input data for Takayama et al. [33] shell.
n h 2 2 i 2 2 2 r ¼ b r0x þ r0y r0x r0y þ 3 s0xy þ s0xz þ s0yz o1=2 þ aðr0x þ r0y Þ
ð15Þ
where a and b are equal to 0.355r and 1.355, respectively. These constants were determined based on the studies of Kupfer and Gerstle [27] and Kupfer et al. [36].
In tension, Takayama et al. [33] used the same approach as used in this paper to model the behaviour of concrete. The finite element used by Takayama et al. [33] was the nine-node Heterosis element with reduced integration ð2 2Þ and layered approach. In their analysis, these authors divided one fourth of the shell into 45 finite elements. The shell was assumed to be pin supported at the contour and subjected to uniformly vertical load. Each element was divided into eight layers for concrete and the reinforcement, with diameter of 1.2 mm, was distributed into 2 layers in the middle surface of the shell. In this paper the shell was divided into 48 finite elements but the same number of concrete and steel layers as adopted by Takayama et al. [33], was considered. Fig. 8 depicts the vertical displacements in nodes A, B and C showed in Fig. 7.
7. Conclusion
Fig. 8. Load-vertical displacements in points A, B and C of the shell.
In this paper the nonlinear analysis of cylindrical concrete shells by a strain-based circular cylindrical shell finite element was shown. The results of the analysis, compared with values obtained by more sophisticated finite elements, show that the proposed approach can be efficiently used in the design of concrete cylindrical shells with great economy of computing time and reduced programming effort, since the shell finite element used herein is very simple to incorporate into finite element codes.
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