Nonlinear Finite Element Analysis of Concrete Structures

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- 172 - in Reinforced Concrete, Copenhagen, Denmark, May 21-2 3, 1979 (International Association for Bridge and Structural Engineering, Zurich) (Reports of the Working Commissions, Vol. 29) 3-10. WILLAK, K.J. and WARNKE, E.P. (1974). Constitutive Model for the Triaxial Behavior of Concrete. In: Seminar on Concrete Structures subjected to Triaxial Stresses, Bergamo, Italy, May 17-19, 1974 (International Association for Bridge and Structural Engineering, Zurich) (Reports of the Working Commissions, Vol. 19) paper 3/1. WASTLUND, G. (1937). Nye ron angående betongens grundlaggende hållfasthetsegenskaber. Betong _3* Stockholm, Sweden, 189-206. YUZUGULLU, O. and SCHNOBRICH, W.C. (1973). A Numerical Procedure for the Determination of the Behavior of a Shear Wall Frame System. Proc. Am. Concr. Inst. 29.' 474-479. ZIENKIEWICZ, O.C. and CHEUNG, Y.K. (1967). The Finite Element Method in Structural and Continuum Mechanics (McGraw-Hill, London) 274 p. ZIENKIEWICZ, O.C., VALLIAPPAN, S. and KING, I.P. (1969). Elasto- Plastic Solutions of Engineering Problems. Initial-Stress. Finite Element Approach. Int. J. Numer. Methods 1^, 75-100. ZIENKIEWICZ, O.C., OWEN, D.R.J., PHILLIPS, D.V., and NAYAK, G.C. (1972). Finite Element Methods in the Analysis of Reactor Vessels. Nucl. Eng. Des. ^£, 507-541. ZIENKIEWICZ, O.C., PHILLIPS, D.V. and OWEN, D.R.J. (1974). Finite Element Analysis of some Concrete Non-Linearities. Theory and Examples. In: Seminar on Concrete Structures subjected to Triaxial Stresses, Bergamo, Italy, May 17-19, 1974 (International Association for Bridge and Structural Engineering, Zurich) (Reports of the Working Commission, vol. 19) paper 3/2. ZIENKIWICZ, O.C. (1977). The Finite Element Method. 3ed. (McGraw- Hill, London) 787 p.
- 173 - LIST OF SYMBOLS Unless otherwise stated, the following symbols are used in the present report: A = parameter in failure criterion, eq. (2.1-5); A = E./E = parameter in stress-strain equation (2,2-3); B = parameter in failure criterion, eq. (2.1-5); B. . = tensor relating strains and nodal displacements, eq. (4.1-11); B B^ 3 b D = matrix relating strains and nodal displacements, eqs.(4.2-6) and (4.2-7); = matrix in a local coordinate system relating reinforcement strains with reinforcement nodal displacements, eqs. (4.3-4) and (4.3-5); = strain softening parameter in stress-strain equation (2.2-3); D. ... = elasticity tensor, eq. (4.1-3); D = constitutive or material matrix, eq. (4.2-10); D D_ D D = material matrix when circumferential cracks exist, eq. (4.2-21); = material matrix when circumferential and radial cracks exist, eq. (4.2-25); = material matrix when secondary circumferential cracks exist together with radial cracks, eq. (4.2-28); = material matrix when radial cracks exist, eq. (4.2-23); D* = material matrix in a local coordinate system for a reinforcement element, eqs. (4.3-6) to (4.3-9); E = Young's modulus;
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å Risa-R-411 Nonlinear Finite Elem
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RISØ-R-411 NONLINEAR FINITE ELEMEN
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CONTENTS Page PREFACE 5 1. INTRODUC
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- 5 - PREFACE This report is submit
Page 9 and 10:
- 8 - arbitrarily located reinforce
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- 10 - mertal data and ve will then
Page 13 and 14:
- 12 - IONI = £ = OP • e = (a 1
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- 14 - 4) in accordance with 1), th
Page 17 and 18:
- 16 - Fig. 2 shows the comparison
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- 18 - (1965, 1974) and of Cedolin
Page 21 and 22:
- 20 - ^2 i r f~2 *! i ~" 2A " A +
Page 23 and 24:
- 22 - Table 2.1-3: A.-values and t
Page 25 and 26:
tensile and compressive meridians w
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- 26 - ite program. A comparison wi
Page 29 and 30:
- 28 - Figure 9 contains the experi
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- 30 - used for cyclic loading in t
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- 32 - to that of nonlinear elastic
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- 34 - of being proportional to the
Page 37 and 38:
- 36 - affecting the descending cur
Page 39 and 40:
- 33 - E f " I"« 4(A C -1) x {2 -
Page 41 and 42:
- 40 - 1.0 i i 1 r 0.6 0.2 J I I L
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Uniaxial and biaxial (
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- 44 - -300
Page 47 and 48:
- 46 - Using Hooke's law and eq. (1
Page 49 and 50:
- 48 - stress invariant is consider
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- 50 - as the constitutive behaviou
Page 53 and 54:
- 52 - where Hooke's law and eq. (5
Page 55 and 56:
e.. = e e . + eP (3-15) ID i] i] Fr
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- 56 - gram is applicable for axisy
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- p o ized in this formulation f
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-"* — Use of Gauss's divergence t
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- bZ - tensor c., determines the ap
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- 64 - propriate assembly rules usi
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- 66 - for the structure, where the
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- 68 - where the (2 x 6) matrix w c
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- 70 - of E and v depending on stre
Page 73 and 74:
- 72 - where the B-matrix is given
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- 74 - Z Fig- 4.2-3: Cracking in an
Page 77 and 78:
- 76 - retained along the crack pla
Page 79 and 80:
- 78 - stance one circumferential c
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- 80 - vcr v little sensitivit v to
Page 83 and 84:
- 82 - ment is treated here. Howeve
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- 84 - evdn though some features of
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- 86 - at point A, B and C are give
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- 86 - (4.3-6) where the material
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- 91 the reinforcement element, i.e
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- 93 - RZ-reinforcement: =T =, f 2r
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- 95 - identical to those for RZ-re
Page 97 and 98:
- 97 - where the same notation as i
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- 99 - '..-here a again is given by
Page 101 and 102:
- 101 - within the element are stil
Page 103 and 104:
- 103 - now be directed towards nan
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- 103 - total formulation as oppose
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- 107 - question violates the failu
Page 109 and 110:
- 109 - premature failure load. How
Page 111 and 112:
- Ill - inforcement bar modelling.
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- 113 - 600 ^ 500 t- CT uit = 518 M
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- 115 - large failure capacity when
Page 117 and 118:
- 117 - that in all other structure
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- 119 - 40 mild steel ribs with a t
Page 121 and 122: - 121 - Fig. 5.2-4: Uppe^ surface o
Page 123 and 124: - 123 - clined cracks initiate in t
Page 125 and 126: to no softening writer's criterion
Page 127 and 128: - 127 - Returning *-.o the calculat
Page 129 and 130: - 129 - the load point for both bea
Page 131 and 132: - 131 - max. principal ; stress loa
Page 133 and 134: - 133 - lil loading = 21% IMM/I b)
Page 135 and 136: - 135 - loading = 26 % loading = 63
Page 137 and 138: - 137 - terized as diagonal tension
Page 139 and 140: - 139 - program utilizes the values
Page 141 and 142: - 141 - no stiffness of the concret
Page 143 and 144: - 143 - sults in a tendency to diag
Page 145 and 146: - 145 - Fig. 5.4-2: Punched-out pie
Page 147 and 148: '}sai,->(OT aqq 50 jnoxABqaq xean^o
Page 149 and 150: - 149 - tension k-0 . $&&* / t circ
Page 151 and 152: - 151 -
Page 153 and 154: - 153 - of the resulting prediction
Page 155 and 156: - i 5 T - tensile strength. As demo
Page 157 and 158: Section 2 dealt with failure and no
Page 159 and 160: - 15'J - insight into the physical
Page 161 and 162: - IS!. - obtained using the AXIPLAN
Page 163 and 164: - 163 - BAZANT, Z.P. and GAMBAROVA,
Page 165 and 166: - 1G 5 - HAND, F.R., PECKNOLD, D.A.
Page 167 and 168: - 167 - LAUNAY, P., GACHON, H. and
Page 169 and 170: - lb9 - OTTOSEN, N.S. and ANDERSEN,
Page 171: SCHIMKELPFENNIG, K. (1971). Die Fes
Page 175 and 176: - 175 - force vector due to initial
Page 177 and 178: - .17 7 - deviatoric strain tensor,
Page 179 and 180: - 179 - e* = strain in the R 1 -dir
Page 181 and 182: - 131 - ob RZ RZ = initial stress v
Page 183 and 184: — 1 f> -> _ A transformation to p
Page 185 and 186: - 155 - K^IO 10 -!) cos 2 a is adde
Page 187: Sales distributors: Jul. Gjellerup,
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Nonlinear Finite Element Analysis of Concrete Structures

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