Nonlinear Finite Element Analysis of Concrete Structures

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- ?7 - istence of tensile cracks. Following the proposal of the writer (1979) we assume that the cracking occurs, firstly, if the failure criterion is violated and secondly, if a, > a./2 holds. Note that this crack criterion may be applied to any smooth failure surface, The other failure criterion implemented in the finite element program - the modified Coulomb criterion - already includes a cracking criterion determined by a, > a.. 1 1 r 1 Concrete 1. o c = 18.7 MPa ot/Oc,= 0.10usfd in criteria a y h c I 0.2-j -pr—^gracc*"*"—
- 28 - Figure 9 contains the experimental results of Kupfer et al. (1969, 1973) for biaxial tensile-compressive loading of three different types of concrete. Both failure stresses and failure modes are indicated In addition, the figure shows the corresponding failure curves together with their failure mode criteria using the two failure criteria implemented in the finite element program. It appears that the two failure mode criteria and the two failure criteria are in close agreement with the experimental evidence. In accordance with earlier conclusions the proposals of the writer are favourable when considering accuracy. The modified Coulomb criterion , on the other hand, possesses an attractive simplicity. For both failure mode criteria it is assumed that the orientation of the crack plane is normal to the principal direction of o.. This assumption is also in good agreement with the aforementioned tests. 2.2. Stress-strain relations Having discussed the strength of concrete in some detail, the stress-strain behaviour will now be dealt with. Ideally, a constitutive model for concrete should reflect the strain hardening before failure, the failure itself as well as the strain softening in the post-failure region. The post-failure behaviour has received considerable attention in the last years especially, where it has become evident that the calculated load capacity of a structure may be strongly influenced by the particular postfailure behaviour employed for the concrete; for example ideal plasticity with its infinite ductility might be an over-simplified model. This is just to say that redistribution o i stresses in a structure must be dealt with in a proper way. These aspects will be considered in some detail in section 5. Moreover, the constitutive model should ideally be simple and flexible, i.e. different assumptions can easily be incorporated. The numerical performance of the model in a computer program should also be considered. Moreover, it should be applicable to all stress states and both loading and unloading should ideally be dealt
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 and 12: - 10 - mertal data and ve will then
Page 13 and 14: - 12 - IONI = £ = OP • e = (a 1
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: - 26 - ite program. A comparison wi
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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