Nonlinear Finite Element Analysis of Concrete Structures

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- 77 - Considering formation of a radial crack where the crack plane follows a radial plane, use of the assumptions just outlined results in B R = % = E 1 - v 2 1 0 V 1 0 0 0 0 0 0 0 (4.2-23) 0 0 0 1-v where index R indicates that this constitutive matrix refers to a radial crack. Obviously no transformation of coordinates is involved. If both a radial and a circumferential crack exist, we derive S CR = E 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2(l+v) (4.2-24) Use of the transformation formula eq. (19) and of eq. (15) infers EU, = E CR 4 2 2 2 2 2 cos a+Msin 2a sin ctcos a-Msin 2a 0 - 1 3sin2acos a+Msin2acos2a . 4 2 0 - 1 :sin2asin a-Msin2acos2a sin a+Msin 2a symmetric 1 2 2 -rsin 2a+Mcos 2a 4 (4.2-25) where M = 2(l+v) (4.2-26) Consideration is also taken to secondary cracks where for in-
- 78 - stance one circumferential crack already exists and according to the crack criterion a new circumferential crack is formed that has a different direction than the first crack. In that situation it is assumed that all load carrying capacity in the RZ-plane is lost, i.e., D cc = D cc = E 0 0 0 0 0 1 0 0 0 (4.2-27) 0 0 0 Obviously no transformation is involved. Similarly, if both two circumferential cracks and a radial crack develop in an element, the material loses its carrying capacity in all directions and the constitutive matrix degenerates to the null matrix, i.e., 0 0 D CCR S CCR 0 0 0 0 (4.2-28) 0 0 The above approach infers that the resistance of the material to carry tensile load normal to the crack plane is exactly zero. However, this may give rise to an ill-conditioned finite element equation system and in the program 0.5% of the stiffness normal to a crack plane and existing just before cracking is therefore retained. In the program it is also assumed that once a crack is developed it will remain open with a fixed direction. This seems to be a reasonable assumption as we are dealing only with structures subject to increasing loadings. However, assuming closing of a crack when the strain normal to the crack plane becomes compressive the problem of crack healing has been dealt with by e.g. Phillips and Zienkiewics (1976) and Marchertas et al. (1978) . In case of cyclic loading, a more refined model has been proposed by Arnesen et al. (1979) .
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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
Page 13 and 14:
- 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
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 and 56: e.. = e e . + eP (3-15) ID i] i] Fr
Page 57 and 58: - 56 - gram is applicable for axisy
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: - 76 - retained along the crack pla
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
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- 137 - terized as diagonal tension
Page 139 and 140:
- 139 - program utilizes the values
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- 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
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'}sai,->(OT aqq 50 jnoxABqaq xean^o
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- 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
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- 15'J - insight into the physical
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- IS!. - obtained using the AXIPLAN
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- 163 - BAZANT, Z.P. and GAMBAROVA,
Page 165 and 166:
- 1G 5 - HAND, F.R., PECKNOLD, D.A.
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- 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
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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