Theoretical model and finite element analysis for restoring moment at column foot during rocking

He, Junxiao; Wang, Juan

doi:10.1007/s10086-017-1677-5

Journal of Wood Science

Official Journal of the Japan Wood Research Society

Table 2 Restoring force and moment of column foot

From: Theoretical model and finite element analysis for restoring moment at column foot during rocking

Sub-states	\(R_{i}^{+}\)	\(M_{{{\text{b}}{{\text{c}}_i}}}^{+}\)
\(S_{2}^{{\text{+}}}\)
Elastic	\(R_{{2{\text{e}}}}^{+}=\int_{{ - r}}^{r} {2w_{{2{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } {\text{d}}x\qquad (50)\)	\(M_{{{\text{b}}{{\text{c}}_{2{\text{e}}}}}}^{+}=\int_{{ - r}}^{r} {2w_{{2{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } x{\text{d}}x\qquad (60)\)
Elasto-plastic	\(\begin{aligned} R_{{2p}}^{ + } = & \int_{{ - r}}^{{L_{{\theta _{{\text{e}}} }} - r}} {2w_{{2p_{{\text{I}}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} - r}}^{r} {2w_{{2{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (51)\)	\(\begin{aligned} M_{{{\text{bc}}_{{2p}} }}^{ + } = & \int_{{ - r}}^{{L_{{\theta _{{\text{e}}} }} - r}} {2w_{{2p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } x{\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} - r}}^{r} {2w_{{2{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (61)\)
\(S_{3}^{{\text{+}}}\)
Elastic	\(R_{{3{\text{e}}}}^{+}=\int_{{ - r}}^{r} {2w_{{3{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } {\text{d}}x\qquad (52)\)	\(M_{{{\text{b}}{{\text{c}}_{3{\text{e}}}}}}^{+}=\int_{{ - r}}^{r} {2w_{{3{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } x{\text{d}}x\qquad (62)\)
Elasto-plastic	\(\begin{aligned} R_{{3{\text{e}}}}^{ + } = & \int_{{ - r}}^{{L_{{\theta _{{\text{e}}} }} - r}} {2w_{{3p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} - r}}^{r} {2w_{{3{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (53)\)	\(\begin{aligned} M_{{{\text{bc}}_{{3p}} }}^{ + } = & \int_{{ - r}}^{{L_{{\theta _{{\text{e}}} }} - r}} {2w_{{3p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } x{\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} - r}}^{r} {2w_{{3{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (63)\)
\(S_{4}^{{\text{+}}}\)
Elastic	\(R_{{4{\text{e}}}}^{+}=\int_{{r - {L_\theta }}}^{r} {2w_{{4{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } {\text{d}}x\qquad (54)\)	\(M_{{{\text{b}}{{\text{c}}_{4{\text{e}}}}}}^{+}=\int_{{r - {L_\theta }}}^{r} {2w_{{4{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } x{\text{d}}x\qquad (64)\)
Elasto-plastic	\(\begin{aligned} R_{{4p}}^{ + } = \int_{{r - L_{\theta } }}^{{r - L_{{\theta _{{\text{y}}} }} }} {2w_{{4p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ + \int_{{r - L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{4{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (55)\)	\(\begin{aligned} M_{{bc_{{4p}} }}^{ + } = & \int_{{r - L_{\theta } }}^{{r - L_{{\theta _{{\text{y}}} }} }} {2w_{{4p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } x{\text{d}}x \\ & + \int_{{r - L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{4{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (65)\)
\(S_{5}^{{\text{+}}}\)
Elastic	\(R_{{5{\text{e}}}}^{+}=\int_{0}^{r} {2w_{{5{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } {\text{d}}x\qquad (56)\)	\(M_{{{\text{b}}{{\text{c}}_{5{\text{e}}}}}}^{+}=\int_{0}^{r} {2w_{{5{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } x{\text{d}}x\qquad (66)\)
Elasto-plastic	\(\begin{aligned} R_{{5p}}^{ + } = & \int_{0}^{{L_{{\theta _{{\text{e}}} }} }} {2w_{{5p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} }}^{r} {2w_{{5{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (57)\)	\(\begin{aligned} M_{{{\text{bc}}_{{5p}} }}^{ + } = & \int_{0}^{{L_{{\theta _{{\text{e}}} }} }} {2w_{{5p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } x{\text{d}}x \\ & + \int_{{L_{{\theta _{{\text{e}}} }} }}^{r} {2w_{{5{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } x} {\text{d}}x \\ \end{aligned} \qquad (67)\)
\(S_{6}^{{\text{+}}}\)
Elastic	\(R_{{6{\text{e}}}}^{+}=\int_{{r - {L_\theta }}}^{r} {2w_{{6{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } {\text{d}}x\qquad (58)\)	\(M_{{{\text{b}}{{\text{c}}_{6{\text{e}}}}}}^{+}=\int_{{r - {L_\theta }}}^{r} {2w_{{6{\text{e}}}}^{+}(x)\sqrt {{r^2} - {x^2}} } x{\text{d}}x\qquad (68)\)
Elasto-plastic	\(\begin{aligned} R_{{6p}}^{ + } = & \int_{{r - L_{\theta } }}^{{r - L_{{\theta _{{\text{y}}} }} }} {2w_{{6p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ & + \int_{{r - L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{6{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } } {\text{d}}x \\ \end{aligned} \qquad (59)\)	\(\begin{aligned} M_{{{\text{bc}}_{{6p}} }}^{ + } = & \int_{{r - L_{\theta } }}^{{r - L_{{\theta _{{\text{y}}} }} }} {2w_{{6p_{{\rm I}} }}^{ + } (x)\sqrt {r^{2} - x^{2} } } x{\text{d}}x \\ & + \int_{{r - L_{{\theta _{{\text{y}}} }} }}^{r} {2w_{{6{p_\Pi } }}^{ + } (x)\sqrt {r^{2} - x^{2} } } x{\text{d}}x \\ \end{aligned} \qquad (69)\)

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