Nonlinear Finite Element Analysis of Concrete Structures

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- 130 - vated by two reasons. Firstly, it is well known that the constant strain element utilized requires a detailed mesh to describe bending. Secondly, the stress state in shear beams are two-dimensional with tensile and compressive stresses prevailing in the critical regions of the beams, and accurate description of these small tensile stresses is mandatory for an accurate analysis of the beam. The element mesh is especially detailed near the load point and also at the supports where large gradients exist. Except for the plane stress assumption, the actual locations of the bars are simulated in the finite element modelling. In the following calculations, the failure criterion of the writer (1977) will be used and softening in the post-failure region as well as the influence of gravity will be considered. In the first place, the program will be used in its standard form, where the shear retention factor is n = 1% and no lateral stiffness of trie bars is considered, i.e., < = 0. Moreover, the OA-2 beam will be considered first. To illustrate the stress distribution in the beam, the isostress curves for the principal stresses as well as their directions in the nodal points are shown in fig. 4. The loading is 51% of the predicted failure load. However, no essential difference in the stress distribution exists for other loadings. The arch-action of the beam is quite obvious from the figure and apart from the regions at the support and at the load point where biaxial compressive stresses exist, biaxial tensile-compressive stress states prevail. The severity of the loading is illustrated in fig. 5 where the development of the contour lines for the nonlinearity index in per cent with increasing loading is shown. The loadings are again expressed in relation to the predicted failure load. It should be recalled that when tensile stresses are present the nonlinearity index is less than unity even when the stress state is located on the failure surface, cf. section 2.2.1. However, it is obvious from fig. 5 that the region adjacent to the load point is severely loaded and strain softening initiates in fact
- 131 - max. principal ; stress loading = 51% min. principal stress loading = 51 % / / * I t ' I I / f * — ' -/-'•-//^-^-///tV / // • - - » / - / / - ' / ' / • - - / / » - / ( —1\} i i . i i - i i * c*sj i / - ' » • - i \ -1 i AY t \-" H*- t » » •» > *• - - > | ' ^ . I % * t » > . *ff* I. M > Fig. 5.3-4: Isostress curves and directions of the principal stresses in the OA-2 beam. Loading = 51% of predicted failure load. Quantities are in KPa. **^—I loading = 21% —10-' loading = 72 % * i loading = 97% Fig. 5.3.-5: Development in the OA-2 beam of contour lines for the nonlinearity index in per cent. Loadings expressed as per cent of predicted failure load.
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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
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- 74 - Z Fig- 4.2-3: Cracking in an
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- 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: - 129 - the load point for both bea
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
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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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