Nonlinear Finite Element Analysis of Concrete Structures

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- 55 - sent values of total strains e, and e_ are obtained. The initial stresses in question can then be determined through eq. (12). For uniaxial reinforcement bars the approach is much simpler. As perfect bond is assumed, the axial strain e is directly determined in the finite element program. Through the uniaxial stress-strain curve of fig. 3-1 a) the corresponding stress is determined, i.e., a = a (e) This constitutive equation is equivalent to a = Ee + a_ where the initial stress is given by o Q = a(e) - Ee (3-19) No iteration sequence is involved here. Summarizing, the constitutive equations for usual embedded reinforcement corresponds to nonlinear elasticity. The numerical considerations of plastic deformation are applied using the i- nitiai stress method outlined in section 4.3.2. For a given loading stage, the finite element program determines the total strains in the reinforcement plane. For membrane reinforcement the corresponding initial stresses are given by eq. (12) while the initial stress for uniaxial reinforcement bars is given by eq. (19). 4. FINITE ELEMENT MODELLING In this section the finite element formulation of the AXIPLANEprogram described by the writer (1980) is given. This pro-
- 56 - gram is applicable for axisymmetric, plane stress and plane strain structures. Triangular elements are utilized for simulation of concrete, while one- and two- dlnensional elements simulate arbitrarily located reinforcement bars and membranes- Linear displacement functions are used in all elements resulting in perfect bond between concrete and steel. Figure 1 shows the available axisymmetric elements while figs. 2 and 3 show the plane stress and plane strain elements, respectively. These I Concrete element * Tangential bars ^ = — R l Bars in RZ-plane Membrane i — R — R Fig. 4-1: Axisymmetric elements, I Concrete element | Bars in XY-plane -X • — X Fig. 4-2: Plane stress elements. figures also illustrate bow the discrete reinforcement bars are smeared out to equivalent "shells" possessing identical volumina and stiffness characteristics as the bars. It is apparent that
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å Risa-R-411 Nonlinear Finite Elem
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RISØ-R-411 NONLINEAR FINITE ELEMEN
Page 5 and 6: CONTENTS Page PREFACE 5 1. INTRODUC
Page 7 and 8: - 5 - PREFACE This report is submit
Page 9 and 10: - 8 - arbitrarily located reinforce
Page 11 and 12: - 10 - mertal data and ve will then
Page 13 and 14: - 12 - IONI = £ = OP • e = (a 1
Page 15 and 16: - 14 - 4) in accordance with 1), th
Page 17 and 18: - 16 - Fig. 2 shows the comparison
Page 19 and 20: - 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
Page 27 and 28: - 26 - ite program. A comparison wi
Page 29 and 30: - 28 - Figure 9 contains the experi
Page 31 and 32: - 30 - used for cyclic loading in t
Page 33 and 34: - 32 - to that of nonlinear elastic
Page 35 and 36: - 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
Page 43 and 44: Uniaxial and biaxial (
Page 45 and 46: - 44 - -300
Page 47 and 48: - 46 - Using Hooke's law and eq. (1
Page 49 and 50: - 48 - stress invariant is consider
Page 51 and 52: - 50 - as the constitutive behaviou
Page 53 and 54: - 52 - where Hooke's law and eq. (5
Page 55: e.. = e e . + eP (3-15) ID i] i] Fr
Page 59 and 60: - p o ized in this formulation f
Page 61 and 62: -"* — Use of Gauss's divergence t
Page 63 and 64: - bZ - tensor c., determines the ap
Page 65 and 66: - 64 - propriate assembly rules usi
Page 67 and 68: - 66 - for the structure, where the
Page 69 and 70: - 68 - where the (2 x 6) matrix w c
Page 71 and 72: - 70 - of E and v depending on stre
Page 73 and 74: - 72 - where the B-matrix is given
Page 75 and 76: - 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
Page 81 and 82: - 80 - vcr v little sensitivit v to
Page 83 and 84: - 82 - ment is treated here. Howeve
Page 85 and 86: - 84 - evdn though some features of
Page 87 and 88: - 86 - at point A, B and C are give
Page 89 and 90: - 86 - (4.3-6) where the material
Page 91 and 92: - 91 the reinforcement element, i.e
Page 93 and 94: - 93 - RZ-reinforcement: =T =, f 2r
Page 95 and 96: - 95 - identical to those for RZ-re
Page 97 and 98: - 97 - where the same notation as i
Page 99 and 100: - 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
Page 105 and 106: - 103 - total formulation as oppose
Page 107 and 108:
- 107 - question violates the failu
Page 109 and 110:
- 109 - premature failure load. How
Page 111 and 112:
- Ill - inforcement bar modelling.
Page 113 and 114:
- 113 - 600 ^ 500 t- CT uit = 518 M
Page 115 and 116:
- 115 - large failure capacity when
Page 117 and 118:
- 117 - that in all other structure
Page 119 and 120:
- 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 and 172:
SCHIMKELPFENNIG, K. (1971). Die Fes
Page 173 and 174:
- 173 - LIST OF SYMBOLS Unless othe
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:
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Nonlinear Finite Element Analysis of Concrete Structures

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