Nonlinear Finite Element Analysis of Concrete Structures

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- 81 - ments normal to the crack plane. Using the cracking approach followed here, the stiffness related to these two displacement fields would be identical, but in reality much less stiffness would be related tc shear parallel to the crack plane than to shear normal to the crack plane. Terminating this section, attention should be drawn to a new cracking concept proposed very recently by Bazant and Gamborova (1979,a,b) and termed the "rough crack approach". This procedure evades r.uch of the drawbacks of the method adopted here and encompasses apparently most of the essential features of cracking. 4.3. Reinforcement elements This section deals with the finite element formulation of the three types of axisymmetric reinforcement elements shown in fig. 1, namely, tangential reinforcement where the reinforcement bars are located circumferentially, RZ-reinforcement where the bars are located in the RZ-plane and êmbran reinforcement with dimensions both in the circumferential direction and in the RZplane. It is to be emphazised that arbitrarily located reinforce- Z • Tangential bars Z f Bars in RZ-plane I -R '• -R Membrane Fig. 4.3-1: Axisymmetric reinforcement elements
- 82 - ment is treated here. However, before proceeding further it is of interest to reveiw various concepts that are employed when considering the effect of reinforcement. As reinforcement and concrete follow very different constitutive equations a separate treatment of the two materials is necessary in the finite element formulation. However, different approaches to consider the effect of reinforcement exist. Assuming perfect bond between concrete and reinforcement the "smeared" approach assumes that the reinforcement bars are distributed uniformly all over the region occupied by the involved elements. From these anisotropic homogeneous elements and taking advantage of concrete strains being equal to reinforcement strains, constitutive matrices for the reinforced concrete elements can be derived that consider the directional effect of the reinforcement. This approach has been employed e.g. by Cervenka and Gerstle (1971) and by Suidan and Schnobrich (197 3). For inhomogeneous reinforcement arrangements problems may arise when determining the concrete elements involved. Moreover, the specific location of tê reinforcement is accounted for only within certain limits that depend on the type and size of the concrete elements. Another approach which often is termed the discrete idealization employs different elements for concrete and reinforcement. Moreover, reinforcement elements and concrete elements share nodal points. This approach considers the specific reinforcement location, and additional "link" elements that connect concrete and reinforcement may be applied. The use of such link element as proposed by Ngo and Sccrdelis (1967) opens for consideration to slip and bond failure. Ngo and Scordelis used a linear relation while nonlinear slip relations have been utilized by e.g. Nilson (1968) and Cedoli;i =*nd Dei Poli (1977). When slip effects are ignored the considered approach is very common in various computer programs, cf. for instance Bathe and Ramaswamy (1979) , but an obvious drawback is the requirement that reinforcement elements and concrete elements have to share nodal points. This places severe restrictions on the finite element mesh. A third approach in considering the effect of reinforcement is
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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
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 and 78: - 76 - retained along the crack pla
Page 79 and 80: - 78 - stance one circumferential c
Page 81: - 80 - vcr v little sensitivit v to
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
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- 141 - no stiffness of the concret
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- 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 xeanô
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- 149 - tension k-0 . $&&* / t circ
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- 151 -
Page 153 and 154:
- 153 - of the resulting prediction
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- i 5 T - tensile strength. As demo
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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,
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- 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
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 - KÎO 10 -!) cos 2 a is adde
Page 187:
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Nonlinear Finite Element Analysis of Concrete Structures

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