Nonlinear Finite Element Analysis of Concrete Structures

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- 94 - Membrane reinforcement: =T =, = _ 2irr*td E L \ L — ; 2~ l-v 2 2 c h_ sc vs c_ _l _ sc vs „2 2 lWSC) „2~2r*d ' 2 A J. 2 2r*d h , 2, h , 2 . d d d d 4r* d —(sc-c ) —(s -sc) d C vc ^ 1 Ah . 2 2 % A h_. 2, h , 2. "r-51 + ~2 *~2
- 95 - identical to those for RZ-reinforcement. r d 2r* 1 d sina L T F* . = 27rr*dt ^~ e b 1-v o a AT 1 d 1 d cosa + •=—* 2r* sina 0 0 These expressions together with eqs. (21) and (22) constitute the final finite element formulation when elastic behaviour of reinforcement is assumed. The next section deals with the finite element formulation when the reinforcement is loaded into the plastic region. 4.3.2. Plastic deformation of reinforcement As already mentioned in section 3, the initial stress method is employed in the finite element program when considering plastic deformation of the reinforcement. In essence this method reformulates the original constitutive equation o = DU) e (4.3-23) where D(e) is the nonlinear material matrix that depends on the strain state, into the equivalent equation o = 5 e + a. (4.3-24) Here D is the usual constant material matrix while o are the initial stresses determined so that eqs. (23) and (24) result in identical stresses when the strains are identical. The initial stresses are therefore determined by o o = (D(e) - D) e
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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
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 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: - 93 - RZ-reinforcement: =T =, f 2r
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
Page 141 and 142: - 141 - no stiffness of the concret
Page 143 and 144: - 143 - sults in a tendency to diag
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- 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
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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.
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- 167 - LAUNAY, P., GACHON, H. and
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- 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,
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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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