Nonlinear vibrations of bolted flange joint plate system considering the stick–slip–separation state: theory and experiment

Abstract

The contact states including stick, slip, and separation can occur at the bolted joint interface of bolted flange joint structures, which significantly affects and complicates the vibration characteristics of connected structures. However, the mechanism of the influence of these three contact states on structural vibration is still not well understood. This paper establishes a universal theoretical model of bolted flange joint plate systems considering the stick, slip, and separation states simultaneously, which can be extended to a structure with an arbitrary number of bolts and plates. The bi-linear hysteretic model is adopted to model the three contact states. The contact pressure between the bolts and flanges is considered as two-dimensional distribution form, which is closer to real contact situation. Based on the Kirchhoff plate theory, the energy functions of plate and flange are obtained. Further, the nonlinear dynamic equation is derived using Lagrange equations, and solved by the Newmark-beta method. The experimental studies are performed on a plate system connected with bolted flange joint to illustrate the effectiveness of the theoretical model. The results highlight the influence of separation state and friction coefficient on the vibration response. It is found that as the tightening torque decreases, the resonance amplitude increases due to the occurrence of interface separation. The slip state can cause the nonlinear softening property, but this property will weaken when the separation occurs at bolted joint interfaces.

Appendices

Appendix 1

For an isotropic plate, the relationships between stress and strain for i-th plate are expressed as

$$ \left\{ {\begin{array}{*{20}l} {\sigma_{xi} = \frac{E}{{1 - \nu^{2} }}\left( {\varepsilon_{xi} + \nu \varepsilon_{yi} } \right)} \hfill & {} \hfill \\ {\sigma_{yi} = \frac{E}{{1 - \nu^{2} }}\left( {\nu \varepsilon_{xi} + \varepsilon_{yi} } \right){,}} \hfill & {(i = 1,2, \ldots ,n)} \hfill \\ {\tau_{xyi} = G\gamma_{xyi} = \frac{{E\gamma_{xyi} }}{{2\left( {1 + \nu } \right)}}} \hfill & {} \hfill \\ \end{array} } \right. $$

(35)

where E is the Young’s modulus, G is the shear modulus, and v is the Poisson’s ratio of the plate.

The kinetic energy of n plates can be expressed as

$$ T_{plate} = \frac{1}{2}\rho h\sum\limits_{i = 1}^{n} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {\left( {\frac{{\partial u_{0,i}^{p} }}{\partial t}} \right)^{2} + \left( {\frac{{\partial v_{0,i}^{p} }}{\partial t}} \right)^{2} + \left( {\frac{{\partial w_{0,i}^{p} }}{\partial t}} \right)^{2} } \right]{\text{d}}x_{i} {\text{d}}y_{i} } } } $$

(36)

The kinetic energy of bolts can be written as

$$ T_{bolts} = \frac{1}{2}m_{b} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{{N_{B} }} {\left[ {\frac{{\partial w_{0,i}^{p} \left( {x_{bj} ,y_{bj} } \right)}}{\partial t}} \right]^{2} } } $$

(37)

The kinetic energy of bolts can be written as

$$ T_{sensor} = \frac{1}{2}m_{c} \left[ {\frac{{\partial w_{0,n}^{p} \left( {x_{c} ,y_{c} } \right)}}{\partial t}} \right]^{2} $$

(38)

where ρ, m_b, and m_c represent the density of the plate, bolt’s mass, and acceleration sensor’s mass, respectively. In this study, it is assumed that the accelerometer is placed on the rightmost rectangular plate, which position coordinate is (x_c, y_c). The position coordinate of j-th bolt is represented as (x_bj, y_bj).

The potential energy of n plates can be expressed as

$$ \begin{aligned} U_{plate} & = \frac{1}{2}\sum\limits_{i = 1}^{n} {\int_{{ - \frac{{h_{i} }}{2}}}^{{\frac{{h_{i} }}{2}}} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left( {\sigma_{xi} \varepsilon_{xi} + \sigma_{yi} \varepsilon_{yi} + \tau_{xyi} \gamma_{xyi} } \right){\text{d}}x_{i} {\text{d}}y_{i} {\text{d}}z_{i} } } } } \\ & \; = \sum\limits_{i = 1}^{n} {\left\{ {\frac{{D_{f} }}{2}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {\left( {\frac{{\partial^{2} w_{0,i} }}{{\partial x_{i}^{2} }}} \right)^{2} + 2v\left( {\frac{{\partial^{2} w_{0,i} }}{{\partial x_{i}^{2} }}} \right)\left( {\frac{{\partial^{2} w_{0,i} }}{{\partial y_{i}^{2} }}} \right) + \left. {\left( {\frac{{\partial^{2} w_{0,i} }}{{\partial y_{i}^{2} }}} \right)^{2} + 2\left( {1 - v} \right)\left( {\frac{{\partial^{2} w_{0,i} }}{{\partial x_{i} \partial y_{i} }}} \right)^{2} } \right]{\text{d}}x_{i} {\text{d}}y_{i} } \right.} } } \right.} \\ & \;\left. { + \frac{{D_{e} }}{2}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {\left( {\frac{{\partial u_{0,i} }}{{\partial x_{i} }}} \right)^{2} + \left( {\frac{{\partial v_{0,i} }}{{\partial y_{i} }}} \right)^{2} + 2v\frac{{\partial u_{0,i} }}{{\partial x_{i} }}\frac{{\partial v_{0,i} }}{{\partial y_{i} }} + \frac{1 - v}{2}\left( {\frac{{\partial u_{0,i} }}{{\partial y_{i} }} + \frac{{\partial v_{0,i} }}{{\partial x_{i} }}} \right)^{2} } \right]} } {\text{d}}x_{i} {\text{d}}y_{i} } \right\} \\ \end{aligned} $$

(39)

where D_f and D_e denote the flexural rigidity and extensional rigidity of the plate, respectively, and their expressions can be written as

$$ D_{f} = \frac{{Eh^{3} }}{{12\left( {1 - v^{2} } \right)}}{, }D_{e} = \frac{Eh}{{1 - v^{2} }} $$

(40)

The potential energy of the 2n-2 flanges can be written as

$$ \begin{aligned} U_{flange} & = \frac{1}{2}\sum\limits_{i = 1}^{2n - 2} {\int_{{ - \frac{{f_{{\text{k}}} }}{2}}}^{{\frac{{f_{{\text{k}}} }}{2}}} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {\left( {\sigma_{xi} \varepsilon_{xi} + \sigma_{yi} \varepsilon_{yi} + \tau_{xyi} \gamma_{xyi} } \right){\text{d}}x_{f,i} {\text{d}}y_{f,i} {\text{d}}z_{f,i} } } } } \\ & \quad = \sum\limits_{i = 1}^{2n - 2} {\left\{ {\frac{{D_{f}^{flange} }}{2}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{g} }}{2}}}^{{\frac{{f_{g} }}{2}}} {\left[ {\left( {\frac{{\partial^{2} w_{0,i}^{f} }}{{\partial x_{f,i}^{2} }}} \right)^{2} + 2v\left( {\frac{{\partial^{2} w_{0,i}^{f} }}{{\partial x_{f,i}^{2} }}} \right)\left( {\frac{{\partial^{2} w_{0,i}^{f} }}{{\partial y_{f,i}^{2} }}} \right)} \right.} } } \right.} \\ & \quad + \left. {\left( {\frac{{\partial^{2} w_{0,i}^{f} }}{{\partial y_{f,i}^{2} }}} \right)^{2} + 2\left( {1 - v} \right)\left( {\frac{{\partial^{2} w_{0,i}^{f} }}{{\partial x_{f,i} \partial y_{f,i} }}} \right)^{2} } \right]{\text{d}}x_{f,i} {\text{d}}y_{f,i} \\ & \quad \left. { + \frac{{D_{e}^{flange} }}{2}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{g} }}{2}}}^{{\frac{{f_{g} }}{2}}} {\left[ {\left( {\frac{{\partial u_{0,i}^{f} }}{{\partial x_{f,i} }}} \right)^{2} + \left( {\frac{{\partial v_{0,i}^{f} }}{{\partial y_{f,i} }}} \right)^{2} + 2v\frac{{\partial u_{0,i}^{f} }}{{\partial x_{f,i} }}\frac{{\partial v_{0,i}^{f} }}{{\partial y_{f,i} }} + \frac{1 - v}{2}\left( {\frac{{\partial u_{0,i}^{f} }}{{\partial y_{f,i} }} + \frac{{\partial v_{0,i}^{f} }}{{\partial x_{f,i} }}} \right)^{2} } \right]} } {\text{d}}x_{f,i} {\text{d}}y_{f,i} } \right\} \\ \end{aligned} $$

(41)

where ${\text{D}}_{\text{f}}^{\text{flange}}$ and ${\text{D}}_{\text{e}}^{\text{flange}}$ represent the flexural rigidity and extensional rigidity of the flange, respectively, and their expressions can be expressed as

$$ D_{f}^{flange} = \frac{{Ef_{{\text{k}}}^{3} }}{{12\left( {1 - v^{2} } \right)}}{, }D_{e}^{flange} = \frac{{Ef_{{\text{k}}} }}{{1 - v^{2} }} $$

(42)

The expressions of the ${\text{U}}_{\text{coupling}}^{1}$, ${\text{U}}_{\text{coupling}}^{\text{middle}}$, and ${\text{U}}_{{\text{couplin}}{\text{g}}}^{\text{n}}$ are, respectively, written as

$$ \begin{aligned} U_{coupling}^{1} & = \frac{1}{2}\int_{{ - \frac{{b_{1} }}{2}}}^{{\frac{{b_{1} }}{2}}} {\left[ {k_{x}^{f} \left( {\left. {u_{0,1} } \right|_{{x_{1} = \frac{{L_{1} }}{2}}} + \left. {w_{0,1}^{f} } \right|_{{x_{f1} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{y}^{f} \left( {\left. {v_{0,1} } \right|_{{x_{1} = \frac{{L_{1} }}{2}}} - \left. {v_{0,1}^{f} } \right|_{{x_{f1} = - \frac{{f_{g} }}{2}}} } \right)^{2} } \right]} \\ & \quad \left. { + k_{z}^{f} \left( {\left. {w_{0,1} } \right|_{{x_{1} = \frac{{L_{1} }}{2}}} - \left. {u_{0,1}^{f} } \right|_{{x_{f1} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{\theta }^{f} \left( {\frac{{\left. {\partial w_{0,1} } \right|_{{x_{1} = \frac{{L_{1} }}{2}}} }}{{\partial x_{1} }} - \frac{{\left. {\partial w_{0,1}^{f} } \right|_{{x_{f1} = - \frac{{f_{g} }}{2}}} }}{{\partial x_{f1} }}} \right)^{2} } \right]dy_{1} \\ \end{aligned} $$

(43)

$$ \begin{aligned} U_{coupling}^{middle} = & \frac{1}{2}\sum\limits_{i = 2}^{2n - 3} {\left\{ {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\left[ {k_{x}^{f} \left( {\left. {u_{0,i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} + \left. {w_{0,2i - 2}^{f} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{y}^{f} \left( {\left. {v_{0,i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} - \left. {v_{0,2i - 2}^{f} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} } \right)^{2} } \right]} } \right.} \\ & \;\left. { + k_{z}^{f} \left( {\left. {w_{0,i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} - \left. {u_{0,2i - 2}^{f} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{\theta }^{f} \left( {\frac{{\left. {\partial w_{0,i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} }}{{\partial x_{i} }} - \frac{{\left. {\partial w_{0,2i - 2}^{f} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} }}{{\partial x_{f,2i - 2} }}} \right)^{2} } \right] \\ & \; \times dy_{i} \int_{{ - \frac{{b_{n} }}{2}}}^{{\frac{{b_{n} }}{2}}} {\left[ {k_{x}^{f} \left( {\left. {u_{0,i} } \right|_{{x_{i} = \frac{{L_{i} }}{2}}} + \left. {w_{0,2i - 1}^{f} } \right|_{{x_{f,2i - 1} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{y}^{f} \left( {\left. {v_{0,i} } \right|_{{x_{i} = \frac{{L_{i} }}{2}}} - \left. {v_{0,2i - 1}^{f} } \right|_{{x_{f,2i - 1} = - \frac{{f_{g} }}{2}}} } \right)^{2} } \right]} \\ & \;\left. { + k_{z}^{f} \left( {\left. {w_{0,i} } \right|_{{x_{i} = \frac{{L_{i} }}{2}}} - \left. {u_{0,2i - 1}^{f} } \right|_{{x_{f,2i - 1} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{\theta }^{f} \left( {\frac{{\left. {\partial w_{0,i} } \right|_{{x_{i} = \frac{{L_{i} }}{2}}} }}{{\partial x_{i} }} - \frac{{\left. {\partial w_{0,2i - 1}^{f} } \right|_{{x_{f,2i - 1} = - \frac{{f_{g} }}{2}}} }}{{\partial x_{f,2i - 1} }}} \right)^{2} } \right]dy_{i} \\ \end{aligned} $$

(44)

$$ \begin{gathered} U_{{{\text{co}}upling}}^{n} { = }\frac{1}{2}\int_{{ - \frac{{b_{n} }}{2}}}^{{\frac{{b_{n} }}{2}}} {\left[ {k_{x}^{f} \left( {\left. {u_{0,n} } \right|_{{x_{n} = - \frac{{L_{n} }}{2}}} + \left. {w_{0,2n - 2}^{f} } \right|_{{x_{f,2n - 2} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{y}^{f} \left( {\left. {v_{0,n} } \right|_{{x_{n} = - \frac{{L_{n} }}{2}}} - \left. {v_{0,2n - 2}^{f} } \right|_{{x_{f,2n - 2} = - \frac{{f_{g} }}{2}}} } \right)^{2} } \right]} \\ \left. { + k_{z}^{f} \left( {\left. {w_{0,n} } \right|_{{x_{n} = - \frac{{L_{n} }}{2}}} - \left. {u_{0,2n - 2}^{f} } \right|_{{x_{f,2n - 2} = - \frac{{f_{g} }}{2}}} } \right)^{2} + k_{\theta }^{f} \left( {\frac{{\left. {\partial w_{0,n} } \right|_{{x_{n} = - \frac{{L_{n} }}{2}}} }}{{\partial x_{n} }} - \frac{{\left. {\partial w_{0,2n - 2}^{f} } \right|_{{x_{f,2n - 2} = - \frac{{f_{g} }}{2}}} }}{{\partial x_{f,2n - 2} }}} \right)^{2} } \right]{\text{d}}y_{n} \\ \end{gathered} $$

(45)

The Lagrangian energy function L is written as

$$ L = T_{{{\text{total}}}} + W_{{{\text{total}}}} - U_{{{\text{total}}}} $$

(46)

where ${\text{T}}_{\text{total}}$ denotes the total kinetic energy of the whole structure, ${\text{U}}_{\text{total}}$ denotes the total potential energy of the whole structure, and ${\text{W}}_{\text{total}}$ denotes the work done by the bolts and base excitation. The expressions of the above three parts are, respectively, given by

$$ T_{{{\text{total}}}} = T_{b} + T_{{{\text{flange}}}} $$

(47)

$$ U_{{{\text{total}}}} = U_{plate} + U_{flange} + U_{coupling} + U_{f,bolt} + U_{b,bolt} $$

(48)

$$ W_{{{\text{total}}}} = W_{{{\text{total}}}}^{{{\text{bolt}}}} + W_{{{\text{base}}}} $$

(49)

Appendix 2

C is the damping matrix, which is given by

$$ {\mathbf{C}} = \alpha {\mathbf{M}} + \beta {\mathbf{K}} $$

(50)

where α and β are damping parameters, and they are written as

$$ \left\{ {\begin{array}{*{20}l} {\alpha = \frac{{2\omega_{1} \omega_{2} \left( {\omega_{2} \xi_{1} - \omega_{1} \xi_{2} } \right)}}{{\omega_{2}^{2} - \omega_{1}^{2} }}} \hfill \\ {\beta = \frac{{2\left( {\omega_{2} \xi_{2} - \omega_{1} \xi_{1} } \right)}}{{\omega_{2}^{2} - \omega_{1}^{2} }}} \hfill \\ \end{array} } \right. $$

(51)

where ω₁ and ω₂ denote the first two natural frequencies, and the damping coefficients are denoted by ξ₁ and ξ₂, respectively.

The mass matrix is given by

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{1}^{{{\text{plate}}}} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {{\mathbf{M}}_{1}^{{{\text{flange}}}} } & {\mathbf{0}} & \ddots & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & \ddots & \ddots & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {{\mathbf{M}}_{i}^{{{\text{plate}}}} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {} & {{\mathbf{M}}_{2i - 1}^{{{\text{flange}}}} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {} & {} & {{\mathbf{M}}_{2i}^{{{\text{flange}}}} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} \\ {} & {} & {} & {} & {} & {} & {{\mathbf{M}}_{i + 1}^{{{\text{plate}}}} } & \ddots & \ddots & {\mathbf{0}} \\ {} & {} & {{\mathbf{Sym}}} & {} & {} & {} & {} & \ddots & {\mathbf{0}} & \vdots \\ {} & {} & {} & {} & {} & {} & {} & {} & {{\mathbf{M}}_{2n - 2}^{{{\text{flange}}}} } & {\mathbf{0}} \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {{\mathbf{M}}_{n}^{{{\text{plate}}}} } \\ \end{array} } \right] $$

(52)

The specific compositions of matrices ${\mathbf{M}}_{i}^{{{\text{plate}}}}$ and ${\mathbf{M}}_{2i - 1}^{{{\text{flange}}}}$ are

$$ {\mathbf{M}}_{i}^{{{\text{plate}}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{i,u}^{{{\text{plate}}}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{M}}_{i,v}^{{{\text{plate}}}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{i,w}^{{{\text{plate}}}} } \\ \end{array} } \right],{\mathbf{M}}_{2i - 1}^{{{\text{flange}}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{2i - 1,u}^{{{\text{flange}}}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{M}}_{2i - 1,v}^{{{\text{flange}}}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{2i - 1,w}^{{{\text{flange}}}} } \\ \end{array} } \right](i = \, 1, \, 2, \ldots ,n) $$

(53)

The expressions of the elements in mass matrices ${\mathbf{M}}_{i}^{{{\text{plate}}}}$ and ${\mathbf{M}}_{2i - 1}^{{{\text{flange}}}}$ are expressed as

$$ {\mathbf{M}}_{i,u}^{{{\text{plate}}}} = \rho h\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {{\mathbf{U}}_{i} {\mathbf{U}}_{i}^{{\text{T}}} {\text{d}}x_{i} {\text{d}}y_{i} } } $$

(54)

$$ {\mathbf{M}}_{i,v}^{{{\text{plate}}}} = \rho h\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {{\mathbf{V}}_{i} {\mathbf{V}}_{i}^{{\text{T}}} {\text{d}}x_{i} {\text{d}}y_{i} } } $$

(55)

$$ {\mathbf{M}}_{i,w}^{{{\text{plate}}}} = \rho h\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {{\mathbf{W}}_{i} {\mathbf{W}}_{i}^{{\text{T}}} {\text{d}}x_{i} {\text{d}}y_{i} } } + \frac{1}{2}m_{b} \sum\limits_{j = 1}^{{N_{B} }} {\left. {{\mathbf{W}}_{i} {\mathbf{W}}_{i}^{{\text{T}}} } \right|_{{x_{j} = x_{bj} ,y_{j} = y_{bj} }} } $$

(56)

$$ {\mathbf{M}}_{2i - 1,u}^{{{\text{flange}}}} = \rho f_{{\text{k}}} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {{\mathbf{U}}_{2i - 1} {\mathbf{U}}_{2i - 1}^{{\text{T}}} {\text{d}}x_{f,2i - 1} {\text{d}}y_{f,2i - 1} } } $$

(57)

$$ {\mathbf{M}}_{i,v}^{{{\text{flange}}}} = \rho f_{{\text{k}}} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {{\mathbf{V}}_{i} {\mathbf{V}}_{i}^{{\text{T}}} {\text{d}}x_{f,i} {\text{d}}y_{f,i} } } $$

(58)

$$ {\mathbf{M}}_{i,w}^{{{\text{flange}}}} = \rho f_{{\text{k}}} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {{\mathbf{W}}_{i} {\mathbf{W}}_{i}^{{\text{T}}} {\text{d}}x_{f,i} {\text{d}}y_{f,i} } } $$

(59)

The stiffness matrix K is associating with the potential energy of the plates, flanges, boundary bolts, and the bolts between the flanges, which is expressed as

$$ {\mathbf{K}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{{\text{boundary}}}} + {\mathbf{K}}_{1}^{{{\text{plate}}}} } & {{\mathbf{K}}_{12} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {{\mathbf{K}}_{1}^{{{\text{flange}}}} } & {{\mathbf{K}}_{23} } & \ddots & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & \ddots & \ddots & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {{\mathbf{K}}_{i}^{{{\text{plate}}}} } & {{\mathbf{K}}_{i,2i - 1}^{{{\text{spring}}}} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {} & {{\mathbf{K}}_{2i - 1}^{{{\text{flange}}}} } & {{\mathbf{K}}_{2i - 1,2i}^{{{\text{bolt}}}} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {} & {} & {{\mathbf{K}}_{2i}^{{{\text{flange}}}} } & {{\mathbf{K}}_{2i,i + 1}^{{{\text{spring}}}} } & {\mathbf{0}} & \cdots & {\mathbf{0}} \\ {} & {} & {} & {} & {} & {} & {{\mathbf{K}}_{i + 1}^{{{\text{plate}}}} } & \ddots & \ddots & {\mathbf{0}} \\ {} & {} & {{\mathbf{Sym}}} & {} & {} & {} & {} & \ddots & {{\mathbf{K}}_{2n - 3,2n - 2}^{{{\text{bolt}}}} } & \vdots \\ {} & {} & {} & {} & {} & {} & {} & {} & {{\mathbf{K}}_{2n - 2}^{{{\text{flange}}}} } & {{\mathbf{K}}_{2n - 2,n}^{{{\text{spring}}}} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {{\mathbf{K}}_{n}^{{{\text{plate}}}} } \\ \end{array} } \right] $$

(60)

where K_boundary is the stiffness matrix related to the potential energy of boundary spring, and its expression is

$$ {\mathbf{K}}_{{{\text{boundary}}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{11} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{22} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{33} } \\ \end{array} } \right] $$

(61)

where

$$ {\mathbf{K}}_{11} = \sum\limits_{\zeta = 1}^{2} {\int_{{y_{\zeta }^{b} - D}}^{{y_{\zeta }^{b} + D}} {k_{\zeta }^{u} \left( {0,y} \right)\left. {{\mathbf{U}}_{1} {\mathbf{U}}_{1}^{{\text{T}}} } \right|_{{x = - \frac{{L_{1} }}{2}}} } } {\text{d}}x{\text{d}}y $$

(62)

$$ {\mathbf{K}}_{22} = \sum\limits_{\zeta = 1}^{2} {\int_{{y_{\zeta }^{b} - D}}^{{y_{\zeta }^{b} + D}} {k_{\zeta }^{v} \left( {0,y} \right)\left. {{\mathbf{V}}_{1} {\mathbf{V}}_{1}^{{\text{T}}} } \right|_{{x = - \frac{{L_{1} }}{2}}} } } {\text{d}}x{\text{d}}y $$

(63)

$$ {\mathbf{K}}_{33} = \sum\limits_{\zeta = 1}^{2} {\int_{{y_{\zeta }^{b} - D}}^{{y_{\zeta }^{b} + D}} {k_{\zeta }^{w} \left( {0,y} \right)\left. {{\mathbf{W}}_{1} {\mathbf{W}}_{1}^{{\text{T}}} } \right|_{{x = - \frac{{L_{1} }}{2}}} } } {\text{d}}x{\text{d}}y $$

(64)

${\mathbf{K}}_{i}^{{{\text{plate}}}}$ is the stiffness matrix related to the strain energy of i-th plate, given by

$$ {\mathbf{K}}_{i}^{{{\text{plate}}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{i,11}^{{{\text{plate}}}} } & {{\mathbf{K}}_{i,12}^{{{\text{plate}}}} } & {\mathbf{0}} \\ {{\mathbf{K}}_{i,21}^{{{\text{plate}}}} } & {{\mathbf{K}}_{i,22}^{{{\text{plate}}}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{i,33}^{{{\text{plate}}}} } \\ \end{array} } \right] $$

(65)

where

$$ {\mathbf{K}}_{i,11}^{{{\text{plate}}}} = D_{e} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {\frac{{\partial {\mathbf{U}}_{i} }}{{\partial x_{i} }}\frac{{\partial {\mathbf{U}}_{i}^{{\text{T}}} }}{{\partial x_{i} }} + \frac{1 - v}{2}\frac{{\partial {\mathbf{U}}_{i} }}{{\partial y_{i} }}\frac{{\partial {\mathbf{U}}_{i}^{{\text{T}}} }}{{\partial y_{i} }}} \right]} } {\text{d}}x_{i} {\text{d}}y_{i} $$

(66)

$$ {\mathbf{K}}_{i,22}^{{{\text{plate}}}} = D_{e} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {\frac{{\partial {\mathbf{V}}_{i} }}{{\partial y_{i} }}\frac{{\partial {\mathbf{V}}_{i}^{{\text{T}}} }}{{\partial y_{i} }} + \frac{1 - v}{2}\frac{{\partial {\mathbf{V}}_{i} }}{{\partial x_{i} }}\frac{{\partial {\mathbf{V}}_{i}^{{\text{T}}} }}{{\partial x_{i} }}} \right]} } {\text{d}}x_{i} {\text{d}}y_{i} $$

(67)

$$ \begin{aligned} {\mathbf{K}}_{i,33}^{{{\text{plate}}}} = & D_{f} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {\frac{{\partial^{2} {\mathbf{W}}_{i} }}{{\partial x_{i}^{2} }}\frac{{\partial^{2} {\mathbf{W}}_{i}^{{\text{T}}} }}{{\partial x_{i}^{2} }} + 2v\left( {\frac{{\partial^{2} {\mathbf{W}}_{i} }}{{\partial x_{i}^{2} }}\frac{{\partial^{2} {\mathbf{W}}_{i}^{{\text{T}}} }}{{\partial y_{i}^{2} }} + \frac{{\partial^{2} {\mathbf{W}}_{i} }}{{\partial y_{i}^{2} }}\frac{{\partial^{2} {\mathbf{W}}_{i}^{{\text{T}}} }}{{\partial x_{i}^{2} }}} \right)} \right.} } \\ & \; + \left. {\frac{{\partial^{2} {\mathbf{W}}_{i} }}{{\partial y_{i}^{2} }}\frac{{\partial^{2} {\mathbf{W}}_{i}^{{\text{T}}} }}{{\partial y_{i}^{2} }} + 2\left( {1 - v} \right)\frac{{\partial^{2} {\mathbf{W}}_{i} }}{{\partial x_{i} \partial y_{i} }}\frac{{\partial^{2} {\mathbf{W}}_{i}^{{\text{T}}} }}{{\partial x_{i} \partial y_{i} }}} \right]{\text{d}}x_{i} {\text{d}}y_{i} \\ \end{aligned} $$

(68)

$$ {\mathbf{K}}_{i,12}^{{{\text{plate}}}} = \left( {{\mathbf{K}}_{i,21}^{{{\text{plate}}}} } \right)^{{\text{T}}} = D_{e} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {\left[ {2v\frac{{\partial {\mathbf{U}}_{i} }}{{\partial x_{i} }}\frac{{\partial {\mathbf{V}}_{i}^{{\text{T}}} }}{{\partial y_{i} }} + \frac{1 - v}{2}\frac{{\partial {\mathbf{U}}_{i} }}{{\partial y_{i} }}\frac{{\partial {\mathbf{V}}_{i}^{{\text{T}}} }}{{\partial x_{i} }}} \right]} } {\text{d}}x_{i} {\text{d}}y_{i} $$

(69)

${\mathbf{K}}_{i,2i - 1}^{{{\text{spring}}}}$ is the stiffness matrix related to the potential energy of spring between i-th plate and (2i-1)-th flange, whose expression is

$$ {\mathbf{K}}_{i,2i - 1}^{{{\text{spring}}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{i,2i - 1,1,3}^{{{\text{spring}}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{i,2i - 1,2,5}^{{{\text{spring}}}} } & {\mathbf{0}} \\ {{\mathbf{K}}_{i,2i - 1,3,1}^{{{\text{spring}}}} } & {\mathbf{0}} & {{\mathbf{K}}_{i,2i - 1,3,6}^{{{\text{spring}}}} } \\ \end{array} } \right] $$

(70)

where

$$ {\mathbf{K}}_{i,2i - 1,1,3}^{{{\text{spring}}}} = k_{x}^{f} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\left( {\left. {{\mathbf{U}}_{i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} \left. {{\mathbf{W}}_{2i - 2}^{{\text{T}}} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} } \right){\text{d}}y_{i} } $$

(71)

$$ {\mathbf{K}}_{i,2i - 1,2,5}^{{{\text{spring}}}} = k_{y}^{f} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\left( {\left. {{\mathbf{V}}_{i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} \left. {{\mathbf{V}}_{2i - 2}^{{\text{T}}} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} } \right){\text{d}}y_{i} } $$

(72)

$$ {\mathbf{K}}_{i,2i - 1,3,1}^{{{\text{spring}}}} = k_{z}^{f} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\left( {\left. {{\mathbf{W}}_{i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} \left. {{\mathbf{U}}_{2i - 2}^{{\text{T}}} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} } \right){\text{d}}y_{i} } $$

(73)

$$ {\mathbf{K}}_{i,2i - 1,3,6}^{{{\text{spring}}}} = k_{\theta }^{f} \int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\left( {\frac{{\left. {\partial {\mathbf{W}}_{i} } \right|_{{x_{i} = - \frac{{L_{i} }}{2}}} }}{{\partial x_{i} }}\frac{{\partial \left. {{\mathbf{W}}_{2i - 2}^{{\text{T}}} } \right|_{{x_{f,2i - 2} = - \frac{{f_{g} }}{2}}} }}{{\partial x_{f,2i - 2} }}} \right){\text{d}}y_{i} } $$

(74)

${\mathbf{K}}_{2i - 1,2i}^{{{\text{bolt}}}}$ is the stiffness matrix related to the potential energy of spring between (2i-1)-th flange and 2i-th flange, which expression is

$$ {\mathbf{K}}_{2i - 1,2i}^{{{\text{bolt}}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{2i - 1,2i,2,2}^{{{\text{bolt}}}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{2i - 1,2i,3,3}^{{{\text{bolt}}}} } \\ \end{array} } \right] $$

(75)

where

$$ {\mathbf{K}}_{2i - 1,2i,2,2}^{{{\text{bolt}}}} = \sum\limits_{\zeta = 1}^{{N_{b} }} {\int_{{y_{\zeta } - D}}^{{y_{\zeta } + D}} {\int_{ - D}^{D} {k_{\zeta }^{v} \left( {x,y} \right){\mathbf{V}}_{i} {\mathbf{V}}_{i + 1}^{{\text{T}}} } } } {\text{d}}x{\text{d}}y $$

(76)

$$ {\mathbf{K}}_{2i - 1,2i,3,3}^{{{\text{bolt}}}} = \sum\limits_{\zeta = 1}^{{N_{b} }} {\int_{{y_{\zeta } - D}}^{{y_{\zeta } + D}} {\int_{ - D}^{D} {k_{\zeta }^{\theta } \left( {x,y} \right)\frac{{\partial {\mathbf{W}}_{i} }}{\partial x}\frac{{\partial {\mathbf{W}}_{i}^{{\text{T}}} }}{\partial x}} } } {\text{d}}x{\text{d}}y $$

(77)

F_e is the force vector related to base excitation, given by

$$ \begin{aligned} {\mathbf{F}}_{e} = & \left[ {{\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,1}^{{{\text{plate}}}} \, {\mathbf{F}}_{u,1}^{{{\text{flange}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{u,2}^{{{\text{flange}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,2}^{{{\text{plate}}}} \, \cdots } \right. \\ & \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,i}^{{{\text{plate}}}} \, {\mathbf{F}}_{u,2i - 1}^{{{\text{flange}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{u,2i}^{{{\text{flange}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,i + 1}^{{{\text{plate}}}} \, \cdots \\ & \left. { \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,n - 1}^{{{\text{plate}}}} \, {\mathbf{F}}_{u,2n - 3}^{{{\text{flange}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{u,2n - 2}^{{{\text{flange}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,n}^{{{\text{plate}}}} } \right]^{{\text{T}}} \\ \end{aligned} $$

(78)

where

$$ {\mathbf{F}}_{w,i}^{{{\text{plate}}}} \left( i \right) = \rho h\tilde{A}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{L_{i} }}{2}}}^{{\frac{{L_{i} }}{2}}} {P_{m} \left( {x_{i} } \right)P_{n} \left( {y_{i} } \right)} } {\text{d}}x_{i} {\text{d}}y_{i} $$

(79)

$$ {\mathbf{F}}_{w,i + 1}^{{{\text{plate}}}} \left( i \right) = \rho h\tilde{A}\int_{{ - \frac{{b_{i + 1} }}{2}}}^{{\frac{{b_{i + 1} }}{2}}} {\int_{{ - \frac{{L_{i + 1} }}{2}}}^{{\frac{{L_{i + 1} }}{2}}} {P_{m} \left( {x_{i + 1} } \right)P_{n} \left( {y_{i + 1} } \right)} } {\text{d}}x_{i + 1} {\text{d}}y_{i + 1} $$

(80)

$$ {\mathbf{F}}_{u,2i - 1}^{{{\text{flange}}}} \left( i \right) = \rho h\tilde{A}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {P_{m} \left( {x_{f,2i - 1} } \right)P_{n} \left( {y_{f,2i - 1} } \right){\text{d}}x_{f,2i - 1} {\text{d}}y_{f,2i - 1} } } $$

(81)

$$ {\mathbf{F}}_{u,2i}^{{{\text{flange}}}} \left( i \right) = \rho h\tilde{A}\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {P_{m} \left( {x_{f,2i} } \right)P_{n} \left( {y_{f,2i} } \right){\text{d}}x_{f,2i} {\text{d}}y_{f,2i} } } $$

(82)

F_bolt is the force vector related to constraining forces of the bolts, which is expressed as

$$ \begin{aligned} {\mathbf{F}}_{{{\text{bolt}}}} = & \left[ {{\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{u,1}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,1}^{{{\text{bolt}}}} \, {\mathbf{F}}_{u,2}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,2}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, \cdots } \right. \\ & \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{u,2i - 1}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,2i - 1}^{{{\text{bolt}}}} \, {\mathbf{F}}_{u,2i}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,2i}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, \cdots \\ & \left. { \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{F}}_{u,2n - 3}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,2n - 3}^{{{\text{bolt}}}} \, {\mathbf{F}}_{u,2n - 2}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{F}}_{w,2n - 2}^{{{\text{bolt}}}} \, {\mathbf{0}} \, {\mathbf{0}} \, {\mathbf{0}}} \right]^{{\text{T}}} \\ \end{aligned} $$

(83)

where

$$ {\mathbf{F}}_{u,2i - 1}^{{{\text{bolt}}}} \left( i \right) = - \sum\limits_{\zeta = 1}^{{N_{b} }} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {\delta \left( {x_{f,2i - 1} - x_{2i - 1}^{\zeta } } \right)\delta \left( {y_{f,2i - 1} - y_{2i - 1}^{\zeta } } \right)f_{u}^{\zeta } u_{i}^{\zeta } \left( {x_{f,2i - 1} ,y_{f,2i - 1} } \right){\text{d}}x_{f,2i - 1} {\text{d}}y_{f,2i - 1} } } } $$

(84)

$$ {\mathbf{F}}_{u,2i}^{{{\text{bolt}}}} \left( i \right) = \sum\limits_{\zeta = 1}^{{N_{b} }} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {\delta \left( {x_{f,2i} - x_{2i}^{\zeta } } \right)\delta \left( {y_{f,2i} - y_{2i}^{\zeta } } \right)f_{u}^{\zeta } u_{i}^{\zeta } \left( {x_{f,2i} ,y_{f,2i} } \right){\text{d}}x_{f,2i} {\text{d}}y_{f,2i} } } } $$

(85)

$$ {\mathbf{F}}_{w,2i - 1}^{{{\text{bolt}}}} \left( i \right) = - \sum\limits_{\zeta = 1}^{{N_{b} }} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {\delta \left( {x_{f,2i - 1} - x_{2i - 1}^{\zeta } } \right)\delta \left( {y_{f,2i - 1} - y_{2i - 1}^{\zeta } } \right)f_{w}^{\zeta } w_{i}^{\zeta } \left( {x_{f,2i - 1} ,y_{f,2i - 1} } \right){\text{d}}x_{f,2i - 1} {\text{d}}y_{f,2i - 1} } } } $$

(86)

$$ {\mathbf{F}}_{w,2i}^{{{\text{bolt}}}} \left( i \right) = \sum\limits_{\zeta = 1}^{{N_{b} }} {\int_{{ - \frac{{b_{i} }}{2}}}^{{\frac{{b_{i} }}{2}}} {\int_{{ - \frac{{f_{{\text{g}}} }}{2}}}^{{\frac{{f_{{\text{g}}} }}{2}}} {\delta \left( {x_{f,2i} - x_{2i}^{\zeta } } \right)\delta \left( {y_{f,2i} - y_{2i}^{\zeta } } \right)f_{w}^{\zeta } w_{i}^{\zeta } \left( {x_{f,2i} ,y_{f,2i} } \right){\text{d}}x_{f,2i} {\text{d}}y_{f,2i} } } } $$

(87)