Nonlinear Finite Element Analysis of Concrete Structures

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- 120 - O 20 30 40 f%.J 'ig. 5.2-2: Stress-ctrain curves for the concrete and the liner. Fig. 5.2-3: Axisymmetric finite element mesh of LM-3 terized by extensive radial cracking initiating at the centre at a pressure = 3 MPa and extending to the flange when the pressure is around 8.5 MPa. With increasing pressure these radial cracks open considerably and circumferential cracks locaced approximately half-way between the centre and the flange also are initiated. The maximum pressure obtained was 37.0 MPa where severe cracking was present. This is illustrated in fig. 4 showing the upper surface of the LM-1 closure at maximum pressure. This closure is almost similar to the LM-3 closure. The test terminated dramatically by ejection of the central part of the closure A section through the remaining part of the LM-3 closure is shown in fig. 5
- 121 - Fig. 5.2-4: Uppe^ surface of the LM-1 closure at maximum pressure. Fig. 5.2-5: Section through the remaining part of the LM-3 closure after ejection of the central part. Let us now consider the predicted behaviour using the failure criterion of the writer (1977) and assuming softening of the
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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
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- 8 - arbitrarily located reinforce
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- 10 - mertal data and ve will then
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- 12 - IONI = £ = OP • e = (a 1
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- 14 - 4) in accordance with 1), th
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- 16 - Fig. 2 shows the comparison
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- 18 - (1965, 1974) and of Cedolin
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- 20 - ^2 i r f~2 *! i ~" 2A " A +
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- 22 - Table 2.1-3: A.-values and t
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tensile and compressive meridians w
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- 26 - ite program. A comparison wi
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- 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
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- 36 - affecting the descending cur
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- 33 - E f " I"« 4(A C -1) x {2 -
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- 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
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- 46 - Using Hooke's law and eq. (1
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- 48 - stress invariant is consider
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- 50 - as the constitutive behaviou
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- 52 - where Hooke's law and eq. (5
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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
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: - 119 - 40 mild steel ribs with a t
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,
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SCHIMKELPFENNIG, K. (1971). Die Fes
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- 173 - LIST OF SYMBOLS Unless othe
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- 175 - force vector due to initial
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- .17 7 - deviatoric strain tensor,
Page 179 and 180:
- 179 - e* = strain in the R 1 -dir
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- 131 - ob RZ RZ = initial stress v
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— 1 f> -> _ A transformation to p
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- 155 - K^IO 10 -!) cos 2 a is adde
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Nonlinear Finite Element Analysis of Concrete Structures

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