Nonlinear Finite Element Analysis of Concrete Structures

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- 126 - tion of no-softening, where larger and larger regions contribute significantly to the load-carrying mechanism due to stress redistribution. The importance of realistic post-failure behaviours have been demonstrated earlier by Argyris et al. (1976) analyzing among other structures also the LM-3 closure considered here. In fact, the LM-3 closure has been analyzed extensively by Argyris et al. (1974) as well as by Schimmelpfennig (1S75, 1976). 50 -i r 40 i- X30 (- a: 3 20 U Ui (E 0. calculated- »—experimental n / £T -1 -2 -3 -4 -5 RAOIAL STRAIN c r |%o) x experimental J failure load ! _ 3 -6 -7 Fig. 5.2-11: Experimental and calculated radial strains at the centre. calculated experimental B- 0 -1 -2 -3 -4 TANGENTIAL STRAIN c s (%o) Fig. 5.2-12: Experimental and calculated circumferential strains below the ribs.
- 127 - Returning *-.o the calculation where the writer's criterion and softening are assumed, figs. 11 and 12 show a further comparison with experimental data. Fair agreement is obtained. The present section has demonstrated the analysis of a complicated structure where both large triaxial compressive stresses and cracking as well as plastic deformations of the steel parts are involved. It has been shown that suitable analysis of the theoretical data may provide a clear insight in the physical behaviour of a structure. Moreover, the influence of using two different failure criteria has been investigated and the importance of a realistic post-failure behaviour in a constitutive model for concrete has been highlighted. As expected, the use of the writer's failure criterion (1977) and giving consideration to softening effects in the post-failure region result in the closest agreement with experimental data. Deformations and strains are predicted with fair accuracy and the failure load is overestimated by 13%. 5.3. Beams failing in shear Beams failing in shear represent very delicate problems subject in the past to considerable experimental as well as computational efforts. Despite this, the structural behaviour of shear beams is only partly known and computations are generally of semiempirical nature. In this section, the calculations will be compared with the classical test results of Bresler and Scordelis (1963); a beam without shear reinforcement as well as an identical beam, but now including shear reinforcement will be considered. The structural behaviour of the beams is illustrated and of special interest is aggregate interlock, secondary cracks, influence of the magnitude of tensile strength and dowel action. Fig. 1, where all dimensions are in mm, shows the geometry and loading of the beams as well as their reinforcement arrangements. In the tests of Bresler and Scordelis (1963) the beams were labelled OA-2 and A-2 corresponding to no shear reinforcement and shear reinforcement, respectively. The longitudinal tensile reinforcement consists of five #9 bars (nominal area =645 mm )
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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
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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
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- 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: to no softening writer's criterion
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
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- .17 7 - deviatoric strain tensor,
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- 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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