Inelastic Analysis of Geometrically Exact Rods

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Inelastic Analysis of Geometrically Exact Rods

Engineering Analysis
Engineering Mathematics

Inelastic Analysis of Geometrically Exact Rods
P.L. Mata
A. H. Barbat
S. Oller
R. Boroschek
281 pages

Open: Inelastic Analysis of Geometrically Exact Rods
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Abstract

In this work a formulation for rod structures able to consider coupled geometric and constitutive sources of nonlinearity in both the static and the dynamic range is developed. Additionally, it is extended for allowing the inclusion of passive energy dissipating elements as a special rod element and geometric irregularities as a full three-dimensional body connected to the framed structure by means of a two-scale model.

The proposed formulation is based on the Reissner-Simo geometrically exact formulation for rods considering an initially curved reference configuration and extended to include arbitrary distribution of composite materials in the cross sections. Each material point of the cross section is assumed to be composed of several simple materials with their own thermodynamically consistent constitutive laws. The simple mixing rule is used for treating the resulting composite.

Cross sections are meshed into a grid of quadrilaterals, each of them corresponding to a fiber directed along the axis of the beam. A mesh independent response is obtained by means of the regularization of the energy dissipated at constitutive level considering the characteristic length of the mesh and the fracture energy of the materials. Local and global damage indices have been developed based on the ratio between the visco elastic and nonlinear stresses.

The consistent linearization of the weak form of the momentum balance equations is performed considering the effects of rate dependent inelasticity. Due to the fact that the deformation map belongs to a nonlinear manifold, an appropriated version of Newmark's scheme and of the iterative updating procedure of the involved variables is developed. The space discrimination of the linearized problem is performed using the standard Galerkin finite element approach. A Newton-Raphson type of iterative scheme is used for the step-by-step solution of the discrete problem. A specific element for energy dissipating devices is developed, based on the rod model but releasing the rotational degrees of freedom. Appropriated constitutive relations are given for a wide variety of possible dissipative mechanisms.

Several numerical examples have been included for the validation of the proposed formulation. The examples include elastic and inelastic finite deformation response of framed structures with initially straight and curved beams. Comparisons with existing literature is performed for the case of plasticity and new results are presented for degrading and composite materials. Those examples show how the present formulation is able to capture different complex mechanical phenomena such as the uncoupling of the dynamic response from resonance due to inelastic incursions and suppression of the high frequency content. The study of realistic flexible pre-cast and cast in place reinforced concrete framed structures subjected to static and dynamic actions is also carried out. Detailed studies regarding to the evolution of local damage indices, energy dissipation and ductility demands are presented. The studies include the seismic response of concrete structures with energy dissipating devices. Advantages of the use of passive control are verified.

TOC

1 Introduction 1
1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 State of the art review 9
2.1 Geometric nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Large rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Research related to the Reissner{Simo rod theory . . . . . . . . . . 12
2.1.3 Time{stepping schemes on the rotational manifold . . . . . . . . . . 14
2.2 Constitutive nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Inelasticity in rod elements . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Cross sectional analysis . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Concrete structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Geometrically exact formulation for rods 23
3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1 Initially curved and twisted reference rod . . . . . . . . . . . . . . . 24
3.1.2 Straight reference rod . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.3 Current rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.4 Geometric interpretation of elongation and shearing . . . . . . . . . 31
3.1.5 Time derivatives, angular velocity and acceleration . . . . . . . . . 32
3.1.6 Curvature vectors and tensors . . . . . . . . . . . . . . . . . . . . . 33
3.2 Strain measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Co{rotated derivative of the orientation triads . . . . . . . . . . . . 35
3.2.2 Deformation gradient tensor . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Other strain measurements . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.4 Material time derivative of Fn and strain rates . . . . . . . . . . . . 44
3.3 Stress measures and stress resultants . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 Cauchy stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.2 First Piola Kirchhof stress tensor . . . . . . . . . . . . . . . . . . . 47
3.3.3 Second Piola Kirchhoff stress tensor . . . . . . . . . . . . . . . . . . 50
3.3.4 Stress resultants and stress couples . . . . . . . . . . . . . . . . . . 50
3.4 Power balance condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Internal power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Virtual work forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6.1 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6.2 Weak form of the balance equations . . . . . . . . . . . . . . . . . . 62
3.6.3 Reduced form virtual work principle . . . . . . . . . . . . . . . . . 64
3.7 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7.1 Hyperelastic materials . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7.2 General formulation for the linear elastic case . . . . . . . . . . . . 68
3.8 External loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8.1 Point loads and concentrated moments . . . . . . . . . . . . . . . . 70
3.8.2 Distributed loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.8.3 Body loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8.4 Seismic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Constitutive nonlinearity 75
4.1 Softening materials and strain localization . . . . . . . . . . . . . . . . . . 77
4.2 Constitutive laws simple materials . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1 Degrading materials: damage model . . . . . . . . . . . . . . . . . . 79
4.2.2 Rate dependent efects . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2.3 Plastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 Mixing theory for composite materials . . . . . . . . . . . . . . . . . . . . 91
4.3.1 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.2 Free energy density of the composite . . . . . . . . . . . . . . . . . 92
4.3.3 Secant constitutive relation and mechanical dissipation . . . . . . . 92
4.3.4 Tangent constitutive tensor . . . . . . . . . . . . . . . . . . . . . . 93
4.3.5 Rate dependent efects . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.4 Stress resultant, couples and related reduced tensors . . . . . . . . . . . . . 94
4.4.1 Cross sectional tangential tensors . . . . . . . . . . . . . . . . . . . 94
4.5 Damage indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.1 Cross sectional damage index . . . . . . . . . . . . . . . . . . . . . 98

5 Linearization of the virtual work principle 99
5.1 Consistent linearization: admissible variations . . . . . . . . . . . . . . . . 100
5.1.1 Basic linearized forms . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1.2 Linearization of the strain measures . . . . . . . . . . . . . . . . . . 102
5.1.3 Linearization of the spin variables . . . . . . . . . . . . . . . . . . . 106
5.1.4 Linearization of the strain rates . . . . . . . . . . . . . . . . . . . . 108
5.2 Linearization of the stress resultants and couples . . . . . . . . . . . . . . . 112
5.2.1 Elastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.2 Inelastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.3 Equivalence between Gm and Gs . . . . . . . . . . . . . . . . . . . . 117
5.3 Linearization of the virtual work functional . . . . . . . . . . . . . . . . . . 118
5.3.1 Linearization of Gint . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3.2 Linearization of Gine . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.3 Linearization of Gext . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4 Material updating rule of the rotational ¯eld . . . . . . . . . . . . . . . . . 122
5.4.1 Linearization of Gint . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.2 Linearization of Gine . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.3 Linearization of Gext . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Time-steeping schemes and con¯guration update 127
6.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1.1 Newmark algorithm on the rotational manifold . . . . . . . . . . . . 129
6.1.2 Con¯guration update . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.3 Updating procedure for the angular velocity and acceleration . . . . 132
6.1.4 Iterative strain and strain rate updating procedure . . . . . . . . . 134
6.2 Discrete form of the linearized functional . . . . . . . . . . . . . . . . . . . 137
6.2.1 Discrete form of the out of balance forces . . . . . . . . . . . . . . . 139
6.2.2 Discrete tangential stifness . . . . . . . . . . . . . . . . . . . . . . 140

7 Finite element implementation 145
7.1 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.1.1 Spatial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Out of balance force vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.1 Internal force vector . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.2 External force vector . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2.3 Inertial force vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3 Tangential stifness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3.1 Internal contribution to the tangential stifness . . . . . . . . . . . . 150
7.3.2 Inertial contribution to the tangential stifness . . . . . . . . . . . . 151
7.3.3 External contribution to the tangential stifness . . . . . . . . . . . 152
7.4 Material updating of the rotational ¯eld . . . . . . . . . . . . . . . . . . . 153
7.5 Out of balance force vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.1 Internal force vector . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5.2 External force vector . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.5.3 Inertial force vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.6 Tangential stifness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.6.1 Internal contribution to the tangential stifness . . . . . . . . . . . . 156
7.6.2 Inertial contribution of the tangential stifness . . . . . . . . . . . . 158
7.6.3 External contribution of the tangential stifness . . . . . . . . . . . 159
7.7 Iterative Newton{Raphson scheme . . . . . . . . . . . . . . . . . . . . . . . 159
7.7.1 Cross sectional analysis . . . . . . . . . . . . . . . . . . . . . . . . . 160

8 Numerical Examples 167
8.1 Validation examples: elastic case . . . . . . . . . . . . . . . . . . . . . . . 167
8.1.1 Unrolling and rerolling of a circular beam . . . . . . . . . . . . . . 167
8.1.2 Flexible beam in helicoidal motion . . . . . . . . . . . . . . . . . . 168
8.2 Nonlinear static examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.2.1 Mesh independent response of a composite cantilever beam . . . . . 169
8.2.2 Framed dome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2.3 Nonlinear response of a forty five degree cantilever bend . . . . . . 172
8.2.4 Nonlinear analysis of a right angle frame . . . . . . . . . . . . . . . 174
8.3 Nonlinear dynamic examples . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.3.1 Visco elastic right angle cantilever beam . . . . . . . . . . . . . . . 176
8.3.2 Near resonance response of a composite cantilever beam . . . . . . 178
8.3.3 Nonlinear vibration of a spatially curved structure . . . . . . . . . . 183
8.4 Reinforced concrete structures . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.4.1 Experimental numerical comparative study of a scaled RC building model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.4.2 Study of a RC plane frame . . . . . . . . . . . . . . . . . . . . . . . 186
8.4.3 Dynamic study of a RC beam structure . . . . . . . . . . . . . . . . 189
8.4.4 Seismic response of a precast RC building with EDDs . . . . . . . . 193
9 Conclusions and further research 197
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.1.1 Summary of conclusions . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2 Further lines of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

A Introduction to nite rotations 205
A.1 Large non-commutative rotations . . . . . . . . . . . . . . . . . . . . . . . 206
A.2 Parametrization of the rotational manifold . . . . . . . . . . . . . . . . . . 209
A.2.1 The Euler's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.2.2 Obtention the rotation pseudo{vector from rotation tensor . . . . . 213
A.2.3 Tangent space to the rotational manifold . . . . . . . . . . . . . . . 213
A.2.4 The exponential form of the rotation tensor . . . . . . . . . . . . . 216
A.2.5 Diferential map associated to exp[²] . . . . . . . . . . . . . . . . . 217
A.2.6 General minimal vectorial parametrization . . . . . . . . . . . . . . 218
A.2.7 Non minimal vectorial parameterizations: quaternions . . . . . . . 219
A.3 Congurational description of motion . . . . . . . . . . . . . . . . . . . . . 221
A.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
A.3.2 Current and initial reference placements . . . . . . . . . . . . . . . 225
A.4 Congurational description of compound rotations . . . . . . . . . . . . . . 227
A.4.1 Material description of the compound rotation . . . . . . . . . . . . 228
A.4.2 Spatial description of the compound rotation . . . . . . . . . . . . . 229
A.4.3 Material tangent space to SO(3) . . . . . . . . . . . . . . . . . . . 229
A.4.4 Incremental additive rotation vectors . . . . . . . . . . . . . . . . . 230
A.4.5 Vector spaces on the rotational manifold . . . . . . . . . . . . . . . 233
A.5 Variation, Lie derivative and Lie variation . . . . . . . . . . . . . . . . . . 234
A.5.1 Variation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
A.5.2 Pullback operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
A.5.3 Push forward operator . . . . . . . . . . . . . . . . . . . . . . . . . 237
A.5.4 Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
A.5.5 Lie variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238