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tdoc.tex - slidergrid - grid of elastic sliders on a frictional surface |
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tdoc.tex (14294B) |
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1 \documentclass[11pt,a4paper]{article} |
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2 |
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3 \usepackage{a4wide} |
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4 |
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5 %\usepackage[german, english]{babel} |
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6 %\usepackage{tabularx} |
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7 %\usepackage{cancel} |
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8 %\usepackage{multirow} |
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9 %\usepackage{supertabular} |
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10 %\usepackage{algorithmic} |
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11 %\usepackage{algorithm} |
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12 %\usepackage{amsthm} |
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13 %\usepackage{float} |
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14 %\usepackage{subfig} |
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15 %\usepackage{rotating} |
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16 \usepackage{amsmath} |
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17 \setcounter{MaxMatrixCols}{20} % allow more than 10 matrix columns |
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18 |
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19 \usepackage[T1]{fontenc} % Font encoding |
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20 \usepackage{charter} % Serif body font |
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21 \usepackage[charter]{mathdesign} % Math font |
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22 \usepackage[scale=0.8]{sourcecodepro} % Monospaced fontenc |
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23 \usepackage[lf]{FiraSans} % Sans-serif font |
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24 |
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25 \usepackage{listings} |
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26 |
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27 \usepackage{hyperref} |
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28 |
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29 \usepackage{soul} % for st strikethrough command |
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30 |
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31 %\usepackage[round]{natbib} |
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32 \usepackage[natbib=true, style=authoryear, bibstyle=authoryear-comp, |
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33 maxbibnames=10, |
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34 maxcitenames=2, backend=bibtex8]{biblatex} |
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35 \bibliography{/home/ad/articles/own/BIBnew.bib} |
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36 |
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37 |
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38 \begin{document} |
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39 |
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40 \title{Lagrangian model of the elastic, viscous and plastic deformation … |
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41 series of bonded nodes moving on a frictional surface} |
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42 |
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43 \author{Anders Damsgaard} |
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44 \date{{\small Institute of Geophysics and Planetary Physics\\Scripps Ins… |
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45 of Oceanography\\University of California, San Diego}\\[3mm] Last revisi… |
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46 \today} |
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47 |
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48 \maketitle |
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49 |
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50 \section{Methods} |
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51 Our approach treats the short-temporal scale behavior of stick-slip as a |
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52 rigid-body dynamics problem. The material is represented as a discrete … |
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53 of Lagrangian points (\emph{nodes}) which are mechanically interacting w… |
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54 other and the boundary conditions. |
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55 |
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56 The Lagrangian nodes are connected with visco-elastic beam elements. |
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57 The bonds are resistive to tension and compression, shearing, twisting, … |
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58 bending, which ensures elastic uniformity regardless of geometric node |
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59 arrangement \citep{Bolander1998, Radjai2011}. Alternatively, the node |
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60 interaction could be parameterized as simple springs which exclusively p… |
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61 resistance to tension and compression, and resistance to shearing, bendi… |
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62 twisting would be introduced by discretizing the elastic material into a… |
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63 irregular network of many more nodes \citep[e.g.][]{Topin2007, Topin2009… |
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64 we chose the former approach which allows us to keep the number of nodes… |
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65 |
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66 The kinematic degrees of freedom are determined by explicit integration … |
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67 Newton's second law of motion for translation and rotation. For a point… |
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68 with bonded interactions to nodes $j\in N_c$, the translational accelera… |
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69 ($\boldsymbol{a}$) are found from the sums of forces: |
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70 \begin{equation} |
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71 \boldsymbol{a}_i = |
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72 \frac{ |
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73 \boldsymbol{f}_i^\text{d} |
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74 + \boldsymbol{f}_i^\text{f} |
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75 + \sum^{N_c}_j \left[ |
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76 \boldsymbol{f}_{i,j}^\text{p} + |
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77 \boldsymbol{f}_{i,j}^\text{s} |
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78 \right] |
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79 }{m_i} |
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80 + \boldsymbol{g} |
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81 \label{eq:n2-tran} |
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82 \end{equation} |
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83 where $\boldsymbol{f}_i^\text{d}$ is the gravitational driving stress du… |
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84 surface slope, $\boldsymbol{g}$ is the gravitational acceleration, and |
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85 $\boldsymbol{f}_i^\text{f}$ is the frictional force provided if the poin… |
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86 resting on the lower surface. Bonded interaction with another point $j$ |
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87 contributes to translational acceleration through bond-parallel and bond… |
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88 shear forces, $\boldsymbol{f}_{i,j}^\text{p}$ and |
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89 $\boldsymbol{f}_{i,j}^\text{s}$, respectively. |
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90 |
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91 The angular accelerations ($\boldsymbol{\alpha}$) are found from the sum… |
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92 torques: |
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93 \begin{equation} |
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94 \boldsymbol{\alpha}_i = |
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95 \sum^{N_c}_j |
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96 \left[ |
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97 \frac{\boldsymbol{t}^{i,j}_{\bar{x}}}{I^\text{p}_{i,j}} + |
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98 \frac{\boldsymbol{t}^{i,j}_{\bar{y}}}{I^\text{n}_{i,j}} + |
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99 \frac{\boldsymbol{t}^{i,j}_{\bar{z}}}{I^\text{n}_{i,j}} |
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100 \right] |
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101 \label{eq:n2-ang} |
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102 \end{equation} |
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103 here, $\boldsymbol{t}^\text{s}$ is the torque resulting from shearing mo… |
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104 the bond, while the torque $\boldsymbol{t}^{t}$ results from relative tw… |
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105 $I^\text{n}_{i,j}$ is the bond-normal mass moment of inertia at the poin… |
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106 $I^\text{p}$ is polar mass moment of inertia of the bond. The above equ… |
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107 implies the simplifying assumption that the nodes are bonded in a config… |
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108 with geometric symmetry, which is a good approximation inside the grid b… |
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109 slightly worse at the grid edges. |
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110 |
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111 |
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112 \subsection{Visco-elastic interaction between nodes} |
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113 The total force ($\boldsymbol{f}$) and torque ($\boldsymbol{t}$) on two … |
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114 ($i$ and $j$) is found by determining the stress response of a three-dim… |
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115 elastic Timoshenko beam to strain \citet{Schlangen1996, Austrell2004, |
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116 Aastroem2013}. |
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117 In the following the forces and torques are described for node $i$ but a… |
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118 of equal magnitude and with opposite sign for node $j$. The interaction |
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119 accounts for resistance to tension and compression, shear, torsion, and … |
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120 The equations below are derived from the stiffness matrix in |
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121 \citet{Austrell2004}. The components for the three-dimensional force ve… |
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122 node $i$ are: |
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123 \begin{equation} |
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124 \begin{split} |
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125 f_{\bar{x}}^i & = \frac{EA}{L} |
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126 \left( p_{\bar{x}}^{*,i} - p_{\bar{x}}^{*,j} \right)\\ |
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127 % |
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128 f_{\bar{y}}^i & = \frac{12EI_A}{L^3} |
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129 \left( p_{\bar{y}}^{*,i} - p_{\bar{y}}^{*,j} \right) |
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130 + \frac{6EI_A}{L^2} |
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131 \left( \Omega_{\bar{z}}^{*,i} + \Omega_{\bar{z}}^{*,j} \right)\\ |
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132 % |
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133 f_{\bar{z}}^i & = \frac{12EI_A}{L^3} |
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134 \left( p_{\bar{z}}^{*,i} - p_{\bar{z}}^{*,j} \right) |
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135 - \frac{6EI_A}{L^2} |
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136 \left( \Omega_{\bar{y}}^{*,i} + \Omega_{\bar{y}}^{*,j} \right) |
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137 \end{split} |
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138 \end{equation} |
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139 where $\bar{x}, \bar{y}, \bar{z}$ are the bond-relative coordinates. |
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140 The linear and angular relative displacement of the nodes is described b… |
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141 and $\Omega^*$. $E$ is Young's modulus, $G$ is the shear modulus, $A$ i… |
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142 beam cross-sectional area, and $L$ is the original beam length. $I_A$ is… |
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143 area moment of inertia of the beam ($I_A = a^4 12^{-1}$ where $a$ is the… |
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144 side length). The torque on node $i$ is: |
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145 \begin{equation} |
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146 \begin{split} |
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147 t_{\bar{x}}^i & = \frac{GJ}{L} |
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148 \left( \Omega_{\bar{x}}^{*,i} + \Omega_{\bar{x}}^{*,j} \right)\\ |
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149 % |
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150 t_{\bar{y}}^i & = \frac{6EI_A}{L^2} |
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151 \left(p_{\bar{z}}^{*,j} - p_{\bar{z}}^{*,i} \right) |
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152 + \frac{4EI_A}{L} |
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153 \left( \Omega_{\bar{y}}^{*,i} + \frac{\Omega_{\bar{y}}^{*,j}}{2} |
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154 \right)\\ |
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155 % |
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156 t_{\bar{z}}^i & = \frac{6EI_A}{L^2} |
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157 \left(p_{\bar{y}}^{*,i} - p_{\bar{y}}^{*,j} \right) |
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158 + \frac{4EI_A}{L} |
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159 \left( \Omega_{\bar{z}}^{*,i} + \frac{\Omega_{\bar{z}}^{*,j}}{2} |
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160 \right)\\ |
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161 \end{split} |
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162 \end{equation} |
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163 $GJ$ is the Saint-Venant torsional stiffness, where $J$ called the torsi… |
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164 rigidity multiplier or torsion constant. For a beam with a solid square |
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165 cross-sectional shape it can be approximated as $J \approx 0.140577 a^4$… |
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166 $a$ is the side length \citep{Timoshenko1951, Roark1954, Weisstein2016}… |
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167 |
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168 % Torsional constant: |
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169 % https://en.wikipedia.org/wiki/Torsion_constant |
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170 % http://mathworld.wolfram.com/TorsionalRigidity.html (K: K_v) |
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171 % http://physics.stackexchange.com/questions/83148/where-i-can-find-a-to… |
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172 % St Venant torsion: K_v = 1/G (Austrell et al. 2004, table 3) Does it … |
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173 % chapter4.pdf: K_v = J |
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174 % https://skyciv.com/free-moment-of-inertia-calculator/ |
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175 |
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176 |
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177 The deformation and reactive forces are determined relative to the orien… |
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178 of the bond. Common geometrical vectors include the inter-distance vect… |
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179 $\boldsymbol{d}$ between nodes $\boldsymbol{p}_i$ and $\boldsymbol{p}_j$: |
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180 \begin{equation} |
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181 \boldsymbol{d}_{i,j} = \boldsymbol{p}_i - \boldsymbol{p}_j |
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182 \end{equation} |
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183 which in normalized form constitutes the bond-parallel unit vector: |
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184 \begin{equation} |
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185 \boldsymbol{n}_{i,j} = \frac{\boldsymbol{d}_{i,j}}{||\boldsymbol{d}_… |
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186 \end{equation} |
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187 |
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188 The nodes move by translational and rotational velocities. The combined |
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189 relative velocity between the nodes is found as \citep[e.g.][]{Hinrichse… |
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190 Luding2008}: |
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191 \begin{equation} |
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192 \boldsymbol{v}_{i,j} = \boldsymbol{v}_i - \boldsymbol{v}_j + |
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193 \frac{\boldsymbol{d}_{i,j}}{2} \times \boldsymbol{\omega}_i + |
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194 \frac{\boldsymbol{d}_{i,j}}{2} \times \boldsymbol{\omega}_j |
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195 \end{equation} |
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196 The velocity can be decomposed into spatial components relative to the b… |
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197 orientation, e.g.\ the bond-parallel and bond-shear velocity, respective… |
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198 \begin{equation} |
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199 v^\text{p}_{i,j} = \boldsymbol{v}_{i,j} \cdot \boldsymbol{n}_{i,j} |
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200 \end{equation} |
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201 \begin{equation} |
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202 \boldsymbol{v}^\text{s}_{i,j} = \boldsymbol{v}_{i,j} - \boldsymbol{n… |
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203 \left( |
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204 \boldsymbol{v}_{i,j} |
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205 \cdot |
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206 \boldsymbol{n}_{i,j} |
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207 \right) |
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208 \end{equation} |
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209 |
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210 |
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211 |
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212 \subsection{Temporal integration} |
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213 Once the force and torque sum components at time $t$ have been determine… |
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214 kinematic degrees of freedom at time $t+\Delta t$ can be found by explic… |
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215 temporal integration of moment balance equations~\ref{eq:n2-tran} |
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216 and~\ref{eq:n2-ang}. |
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217 We use an integration scheme based on the third-order Taylor expansion, … |
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218 results in a truncation error on the order of $O(\Delta t^4)$ for positi… |
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219 $O(\Delta t^3)$ for velocities. This scheme includes changes in acceler… |
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220 the highest order term, which are approximated by backwards differences.… |
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221 the translational degrees of freedom: |
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222 \begin{equation} |
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223 \boldsymbol{p}^i_{t+\Delta t} = |
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224 \boldsymbol{p}^i_{t} + |
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225 \boldsymbol{v}^i_{t} \Delta t + |
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226 \frac{1}{2} \boldsymbol{a}^i_{t} \Delta t^2 + |
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227 \frac{1}{6} \frac{\boldsymbol{a}^i_{t} |
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228 - \boldsymbol{a}^i_{t - \Delta t}}{\Delta t} \Delta t^3 |
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229 \end{equation} |
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230 \begin{equation} |
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231 \boldsymbol{v}^i_{t+\Delta t} = |
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232 \boldsymbol{v}^i_{t} + |
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233 \boldsymbol{a}^i_{t} \Delta t + |
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234 \frac{1}{2} \frac{\boldsymbol{a}^i_{t} |
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235 - \boldsymbol{a}^i_{t - \Delta t}}{\Delta t} \Delta t^2 |
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236 \end{equation} |
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237 At $t=0$ the acceleration change term is defined as zero. The angular d… |
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238 of freedom are found correspondingly: |
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239 \begin{equation} |
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240 \boldsymbol{\Omega}^i_{t+\Delta t} = |
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241 \boldsymbol{\Omega}^i_{t} + |
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242 \boldsymbol{\omega}^i_{t} \Delta t + |
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243 \frac{1}{2} \boldsymbol{\alpha}^i_{t} \Delta t^2 + |
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244 \frac{1}{6} \frac{\boldsymbol{\alpha}^i_{t} |
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245 - \boldsymbol{\alpha}^i_{t - \Delta t}}{\Delta t} \Delta t^3 |
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246 \end{equation} |
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247 \begin{equation} |
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248 \boldsymbol{\omega}^i_{t+\Delta t} = |
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249 \boldsymbol{\omega}^i_{t} + |
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250 \boldsymbol{\alpha}^i_{t} \Delta t + |
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251 \frac{1}{2} \frac{\boldsymbol{\alpha}^i_{t} |
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252 - \boldsymbol{\alpha}^i_{t - \Delta t}}{\Delta t} \Delta t^2 |
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253 \end{equation} |
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254 |
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255 The numerical time step $\Delta t$ is found by considering the largest e… |
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256 stiffness in the system relative to the smallest mass: |
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257 \begin{equation} |
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258 \Delta t = |
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259 \epsilon |
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260 \left[ |
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261 \min (m_i)^{-1} |
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262 \max \left( |
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263 \max \left( |
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264 \frac{E A_{i,j}}{||\boldsymbol{d}_{0}^{i,j}||} |
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265 \right) |
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266 , |
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267 \max \left( |
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268 \frac{G A_{i,j}}{||\boldsymbol{d}^{i,j}||} |
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269 \right) |
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270 \right) |
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271 \right]^{-1/2} |
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272 \end{equation} |
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273 where $\epsilon$ is a safety factor related to the geometric structure o… |
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274 bonded network. We use $\epsilon = 0.07$. |
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275 |
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276 |
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277 \appendix |
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278 \section{Stiffness matrix} |
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279 \begin{equation} |
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280 \begin{bmatrix} |
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281 f_{\bar{x}}^i\\[0.6em] |
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282 f_{\bar{y}}^i\\[0.6em] |
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283 f_{\bar{z}}^i\\[0.6em] |
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284 t_{\bar{x}}^i\\[0.6em] |
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285 t_{\bar{y}}^i\\[0.6em] |
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286 t_{\bar{z}}^i\\[0.6em] |
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287 f_{\bar{x}}^j\\[0.6em] |
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288 f_{\bar{y}}^j\\[0.6em] |
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289 f_{\bar{z}}^j\\[0.6em] |
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290 t_{\bar{x}}^j\\[0.6em] |
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291 t_{\bar{y}}^j\\[0.6em] |
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292 t_{\bar{z}}^j\\ |
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293 \end{bmatrix} |
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294 = |
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295 \begin{bmatrix} |
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296 \frac{EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-EA}{L} & 0 & 0 & 0 & 0… |
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297 0 & \frac{12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & \frac{6EI_{\bar{z}}… |
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298 0 & \frac{-12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & |
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299 \frac{6EI_{\bar{z}}}{L^2}\\[0.5em] |
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300 0 & 0 & \frac{12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L… |
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301 & 0 & 0 & \frac{-12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}… |
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302 0\\[0.5em] |
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303 0 & 0 & 0 & \frac{GJ}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-GJ}{L} & 0… |
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304 0\\[0.5em] |
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305 0 & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0 & \frac{4EI_{\bar{y}}}{L}… |
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306 & 0 & \frac{6EI_{\bar{y}}}{L^2} & 0 & \frac{2EI_{\bar{y}}}{L} & |
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307 0\\[0.5em] |
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308 0 & \frac{6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{4EI_{\bar{z}}}… |
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309 & \frac{-6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & |
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310 \frac{2EI_{\bar{z}}}{L}\\[0.5em] |
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311 \frac{-EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{EA}{L} & 0 & 0 & 0 & 0… |
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312 0 & \frac{-12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & \frac{-6EI_{\bar{z… |
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313 & 0 & \frac{12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & |
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314 \frac{-6EI_{\bar{z}}}{L^2}\\[0.5em] |
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315 0 & 0 & \frac{-12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{… |
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316 & 0 & 0 & \frac{12EI_{\bar{y}}}{L^3} & 0 & \frac{6EI_{\bar{y}}}{… |
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317 0\\[0.5em] |
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318 0 & 0 & 0 & \frac{-GJ}{L} & 0 & 0 & 0 & 0 & 0 & \frac{GJ}{L} & 0… |
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319 0\\[0.5em] |
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320 0 & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0 & \frac{2EI_{\bar{y}}}{L}… |
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321 & 0 & \frac{6EI_{\bar{y}}}{L^2} & 0 & \frac{4EI_{\bar{y}}}{L} & |
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322 0\\[0.5em] |
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323 0 & \frac{6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{2EI_{\bar{z}}}… |
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324 & \frac{-6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{4EI_{\bar{z}}}{L}\\ |
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325 \end{bmatrix} |
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326 \begin{bmatrix} |
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327 p_{\bar{x}}^i\\[0.6em] |
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328 p_{\bar{y}}^i\\[0.6em] |
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329 p_{\bar{z}}^i\\[0.6em] |
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330 \Omega_{\bar{x}}^i\\[0.6em] |
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331 \Omega_{\bar{y}}^i\\[0.6em] |
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332 \Omega_{\bar{z}}^i\\[0.6em] |
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333 p_{\bar{x}}^j\\[0.6em] |
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334 p_{\bar{y}}^j\\[0.6em] |
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335 p_{\bar{z}}^j\\[0.6em] |
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336 \Omega_{\bar{x}}^j\\[0.6em] |
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337 \Omega_{\bar{y}}^j\\[0.6em] |
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338 \Omega_{\bar{z}}^j\\ |
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339 \end{bmatrix} |
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340 \end{equation} |
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341 |
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342 |
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343 |
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344 \printbibliography{} |
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345 |
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346 \end{document} |
