Nonlinear Finite Element Analysis of Concrete Structures

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- 11 - where J^ is defircu by J 3 = 3 (S 1 + S 2 + S 3 } 1 •Z S . . S ., 3, . 3 i] jk ki and s.. is the stress deviator tensor defined by ID s.. = a. . --r 6.. a., ID ID 3 ID kk where the usual tensor notation is employed with indices running from 1 to 3. The principal values of the stress deviator tensor are termed s,, s_ and s.,.I., is the first invariant of the stress tensor; J 2 and J^ are the second and third invariants of the stress deviator tensor. The often applied octahedral normal stress a and shear stress T are related to the preceding ino 2° variants by a o = I,/3 1 and T o =2 J->/3. 2 The invariants of eq. (2) have a simple geometrical interpretation when eq. (1) is considered as a surface in a Cartesian coordinate system with axes a , ø 2 and o, - the Haigh-Westergaard coordinate system - and the necessary symmetry properties of the failure surface appear explicitly when use is made of these invariants. For this purpose, any point, P(o,, o 2 > o^t in the stress space is described by the coordinates (£, p, 6), in which C is the projection on the unit vector e= (1, 1, 1)/ \/Ton the hydrostatic axis, and )) are polar coordinates in the deviatoric plane, which is orthogonal to (1, 1, 1) , cf. fig. 1. The length P|ff 1 ,ff 2 ' ff 3' •» ffi Fig. 2.1.1: (a) Haigh-Westergaard coordinate system; (b) Deviatoric plane b) of ON is
- 12 - IONI = £ = OP • e = (a 1 , a 2 , a 3 > and ON is therefore determined by ^ ? I Ij - v* ON = (1, 1, 1) lj/3 The component NP is given by NP = OP - ON = ((*]_, o 2 , a 3 ) - (1, 1, 1) 3^/3 = (s 1 , 3 2 , s 3 ) and the length of NP is INPI = p = (sj 5 + s 2 + s 3 ) 1/2 = /2X To obtain an interpretation of J, consider the deviatoric plane, fig. 1 b). The unit vector i, located along the projection of the a.-axis on the deviatoric plane is easily shown to be determined by i = (2, -1, -D/V6. The angle 8 is measured from the unit vector i and we have p cosB = NP • i i.e. cose = vfc (s i' v s 3 ) -å 2 -1 -1 2/3J2~ (2s l' " s 2' " s 3 } Using s, + s. + s = 0 we obtain 3s, cosO = 2VOTJ 2a l ~ a 2 " °3 As a. - a- - o 3 is assumed throughout th*=» text, 0 - 8 - 60 3 holds. Using the identity cos38 = 4 cos 8-3 cos8, the invariant J in eq. (2) is after some algebra found to be given by J = cos38 (2.1-3) The failure criterion eq. (1) can therefore be stated more conveniently using only invariants as
Page 1 and 2: å Risa-R-411 Nonlinear Finite Elem
Page 3 and 4: RISØ-R-411 NONLINEAR FINITE ELEMEN
Page 5 and 6: CONTENTS Page PREFACE 5 1. INTRODUC
Page 7 and 8: - 5 - PREFACE This report is submit
Page 9 and 10: - 8 - arbitrarily located reinforce
Page 11: - 10 - mertal data and ve will then
Page 15 and 16: - 14 - 4) in accordance with 1), th
Page 17 and 18: - 16 - Fig. 2 shows the comparison
Page 19 and 20: - 18 - (1965, 1974) and of Cedolin
Page 21 and 22: - 20 - ^2 i r f~2 *! i ~" 2A " A +
Page 23 and 24: - 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 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 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
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
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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