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The application of the geometric offset method to the rigid joint modeling in the differential quadrature element model updating of frame structures

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Abstract

The following paper deals with the differential quadrature element (DQE) model updating of frame structures when the linear vibration behavior is of interest. To model the rigid L and T joints of the frame, a geometric offset method is employed to define a rigid region around each joint. The kinematic constraints due to the rigid joints, the equilibrium of axial and transverse forces, and the bending moments acting on the joint are utilized to model the joints in the DQE model of the frame. Then, to update the DQE model using the experimental natural frequencies, a minimization problem is defined to reduce an objective function based on the residuals between a measurement set obtained from modal testing on the frame and the corresponding DQE model predictions. Using the proposed approach, the DQE model of a three-story steel frame for in-plane vibrations is updated. To do so, several parameters of the model including the Young’s modulus, the density, the geometric offsets and mass parameters of the joints, and the stiffness of the rotational spring used to model the foundation are considered as the design parameters. The optimum values of the design parameters are then found by employing the particle swarm inspired multi-elitist artificial bee colony algorithm. The results of the model updating indicate a good coincidence of the modal parameters of the updated DQE and the experimental models. The sensitivity analysis also reveals that the highest eigenvalue sensitivities are to the joints’ parameters.

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Acknowledgments

The authors would like to thank Yi Xiang, a researcher from the Guangdong Baiyun University of China for providing the MATLAB code of the PS-MEABC algorithm.

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Authors and Affiliations

  1. Mechanical Engineering Department, Shahid Chamran University, 61357-42237, Ahvaz, Iran

    Laleh Fatahi, Shapour Moradi & Afshin Ghanbarzadeh

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  1. Laleh Fatahi
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  2. Shapour Moradi
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Corresponding author

Correspondence to Laleh Fatahi.

Appendices

Appendix 1

$$ {\mathbf{I}}_{\text{u}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right]_{{N_{e} - 2,2N_{e} }} $$
(37)
$$ {\mathbf{I}}_{\text{v}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right]_{{N_{e} - 4,2N_{e} }} $$
(38)
$$ {\mathbf{C}}_{v}^{(4)} = \left[ {\begin{array}{*{20}c} 0 & {C_{31}^{(4)} } & 0 & {C_{32}^{(4)} } & 0 & \cdots & 0 & {C_{{3N_{e} }}^{(4)} } \\ 0 & {C_{41}^{(4)} } & 0 & {C_{42}^{(4)} } & 0 & \cdots & 0 & {C_{{4N_{e} }}^{(4)} } \\ 0 & {C_{51}^{(4)} } & 0 & {C_{52}^{(4)} } & 0 & \cdots & 0 & {C_{{5N_{e} }}^{(4)} } \\ \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 0 & {C_{{(N_{e} - 2\;)1}}^{(4)} } & 0 & {C_{{(N_{e} - 2)\;2}}^{(4)} } & 0 & \cdots & 0 & {C_{{(N_{e} - 2\,)\,N_{e} }}^{(4)} } \\ \end{array} } \right]_{{N_{e} - 4,2N_{e} }} $$
(39)
$$ {\mathbf{C}}_{u}^{(2)} = \left[ {\begin{array}{*{20}c} {C_{21}^{(2)} } & 0 & {C_{22}^{(2)} } & 0 & {C_{23}^{(2)} } & \cdots & {C_{{2N_{e} }}^{(2)} } & 0 \\ {C_{31}^{(2)} } & 0 & {C_{32}^{(2)} } & 0 & {C_{33}^{(2)} } & \cdots & {C_{{3N_{e} }}^{(2)} } & 0 \\ {C_{41}^{(2)} } & 0 & {C_{42}^{(2)} } & 0 & {C_{43}^{(2)} } & \cdots & {C_{{4N_{e} }}^{(2)} } & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ {C_{{\left( {N_{e} - 1} \right)\,\,1}}^{(2)} } & 0 & {C_{{\left( {N_{e} - 1} \right)\,2}}^{(2)} } & 0 & {C_{{\left( {N_{e} - 1} \right)\,3}}^{(2)} } & \cdots & {C_{{\left( {N_{e} - 1} \right)\,\,N_{e} }}^{(2)} } & 0 \\ \end{array} } \right]_{{N_{e} - 2,2N_{e} }} $$
(40)
$$ \overline{{\mathbf{k}}}_{e} = \frac{{E_{e} A_{e} }}{{l_{e}^{2} }}\left[ {\begin{array}{*{20}c} {C_{21}^{(2)} } & 0 & {C_{22}^{(2)} } & 0 & \cdots & {C_{{2N_{e} }}^{(2)} } & 0 \\ {C_{31}^{(2)} } & 0 & {C_{32}^{(2)} } & 0 & \cdots & {C_{{3N_{e} }}^{(2)} } & 0 \\ 0 & {\frac{1}{{l_{e}^{2} }}C_{31}^{(4)} } & 0 & {\frac{1}{{l_{e}^{2} }}C_{32}^{(4)} } & \cdots & 0 & {\frac{1}{{l_{e}^{2} }}C_{{3N_{e} }}^{(4)} } \\ \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ {C_{{\left( {N_{e} - 2} \right)1}}^{(2)} } & 0 & {C_{{\left( {N_{e} - 2} \right)2}}^{(2)} } & 0 & \cdots & {C_{{\left( {N_{e} - 2} \right)N_{e} }}^{(2)} } & 0 \\ 0 & {\frac{1}{{l_{e}^{2} }}C_{{\left( {N_{e} - 2} \right)1}}^{(4)} } & 0 & {\frac{1}{{l_{e}^{2} }}C_{{\left( {N_{e} - 2} \right)2}}^{(4)} } & \cdots & 0 & {\frac{1}{{l_{e}^{2} }}C_{{\left( {N_{e} - 2} \right)N_{e} }}^{(4)} } \\ {C_{{\left( {N_{e} - 1} \right)1}}^{(2)} } & 0 & {C_{{\left( {N_{e} - 1} \right)2}}^{(2)} } & 0 & \cdots & {C_{{\left( {N_{e} - 1} \right)N_{e} }}^{(2)} } & 0 \\ \end{array} } \right]_{{2N_{e} - 6,2N_{e} }} $$
(41)
$$ \overline{{\mathbf{m}}}_{e} = \rho_{e} A_{e} \left[ {\begin{array}{*{20}c} 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - 1} & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 & 0 & { - 1} & 0 & 0 & 0 \\ \end{array} } \right]_{{2N_{e} \text{ - }6,2N_{e} }} $$
(42)
$$ \overline{{\varvec{\updelta}}}_{i} = \left\{ {\overline{u}_{i}^{1} \quad \overline{v}_{i}^{1} \quad \overline{u}_{i}^{2} \quad \overline{v}_{i}^{2} \;\; \ldots \;\;\overline{u}_{i}^{{\text{N}_{i} \text{ - }1}} \quad \overline{v}_{i}^{{\text{N}_{i} \text{ - }1}} \quad \overline{u}_{i}^{{\text{N}_{i} }} \quad \overline{v}_{i}^{{\text{N}_{i} }} } \right\} $$
(43)

Appendix 2

2.1 The transformation matrices

As mentioned in Equation (44), \( {\text{T}}_{e} \) is the required transformation matrix that transforms the displacement vector of the eth element from global coordinate system to the local one. Since the continuity conditions are employed on the boundary points of each element, only the displacements of these points are expressed in the global coordinate system.

$$ \overline{{\varvec{\updelta}}}_{e} = {\mathbf{T}}_{e} {\varvec{\updelta}}_{e} $$
(44)

where

$$ {\mathbf{T}}_{e} = \left[ {\begin{array}{*{20}c} {{\mathbf{t}}_{e}^{1} } & {} & {} & {} & {} & {} \\ {} & {{\mathbf{t}}_{e}^{2} } & {} & {} & {} & {} \\ {} & {} & {{\mathbf{t}}_{e}^{3} } & {} & {} & {} \\ {} & {} & {} & \ddots & {} & {} \\ {} & {} & {} & {} & {{\mathbf{t}}_{e}^{{N_{e} - 1}} } & {} \\ {} & {} & {} & {} & {} & {{\mathbf{t}}_{e}^{{N_{e} }} } \\ \end{array} } \right]\;,{\mathbf{t}}_{e}^{1} = {\mathbf{t}}_{e}^{{N_{e} }} = \left[ {\begin{array}{*{20}c} {\cos \theta_{e} } & {\sin \theta_{e} } \\ { - \sin \theta_{e} } & {\cos \theta_{e} } \\ \end{array} } \right]\;,\;{\mathbf{t}}_{e}^{2} = \cdots = {\mathbf{t}}_{e}^{{N_{e} - 1}} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right] $$
(45)
$$ {\varvec{\updelta}}_{e} = \left\{ {\,u_{e}^{1} \;\,v_{e}^{1} \;\,u_{e}^{2} \,\,v_{e}^{2} \,\,\,\,\overline{u}_{e}^{3} \,\,\overline{v}_{e}^{3} \,\,\, \ldots \,\overline{u}_{e}^{{N_{e} - 2}} \,\overline{v}_{e}^{{N_{e} - 2}} \,\;\,u_{e}^{{N_{e} - 1}} \,v_{e}^{{N_{e} - 1}} \,\,u_{e}^{{N_{e} }} \;\,v_{e}^{{N_{e} }} } \right\}^{\text{T}} $$
(46)
$$ \overline{{\varvec{\updelta}}}_{e} = \left\{ {\,\overline{u}_{e}^{1} \;\,\overline{v}_{e}^{1} \;\,\overline{u}_{e}^{2} \,\,\overline{v}_{e}^{2} \,\,\,\,\overline{u}_{e}^{3} \,\,\overline{v}_{e}^{3} \,\,\, \ldots \,\overline{u}_{e}^{{N_{e} - 2}} \,\overline{v}_{e}^{{N_{e} - 2}} \,\;\,\overline{u}_{e}^{{N_{e} - 1}} \,\overline{v}_{e}^{{N_{e} - 1}} \,\,\overline{u}_{e}^{{N_{e} }} \;\,\overline{v}_{e}^{{N_{e} }} } \right\}^{\text{T}} $$
(47)

Moreover, the global to local coordinate transformation of the force vector (see Fig. 13) is achieved by Equation (48).

$$ {\bar{\mathbf{F}}}_{e} = {\hat{\mathbf{T}}}_{e} {\mathbf{F}}_{e} $$
(48)

where

$$ {\hat{\mathbf{T}}}_{e} = \left[ {\begin{array}{*{20}c} {{\hat{\mathbf{t}}}_{e}^{1} } & {} & {} & {} & {} & {} \\ {} & {{\hat{\mathbf{t}}}_{e}^{2} } & {} & {} & {} & {} \\ {} & {} & {{\hat{\mathbf{t}}}_{e}^{3} } & {} & {} & {} \\ {} & {} & {} & \ddots & {} & {} \\ {} & {} & {} & {} & {{\hat{\mathbf{t}}}_{e}^{{N_{e} - 1}} } & {} \\ {} & {} & {} & {} & {} & {{\hat{\mathbf{t}}}_{e}^{{N_{e} }} } \\ \end{array} } \right]\;,{\hat{\mathbf{t}}}_{e}^{1} = {\hat{\mathbf{t}}}_{e}^{{N_{e} }} = \left[ {\begin{array}{*{20}c} {\cos \theta_{e} } & {\sin \theta_{e} } \\ {\sin \theta_{e} } & { - \cos \theta_{e} } \\ \end{array} } \right]\;,\;{\hat{\mathbf{t}}}_{e}^{2} = \cdots = {\hat{\mathbf{t}}}_{e}^{{N_{e} - 1}} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right] $$
(49)
$$ {\mathbf{F}}_{e} = \left\{ {\,Fx_{e}^{1} \;\,Fy_{e}^{1} \;\,Fx_{e}^{2} \,\,Fy_{e}^{2} \,\;\overline{F} x_{e}^{3} \,\,\overline{F} y_{e}^{3} \,\, \ldots \,\,\overline{F} x_{e}^{{N_{e} - 2}} \,\,\overline{F} y_{e}^{{N_{e} - 2}} \,\,Fx_{e}^{{N_{e} - 1}} \,Fy_{e}^{{N_{e} - 1}} \,\,Fx_{e}^{{N_{e} }} \;\,Fy_{e}^{{N_{e} }} } \right\}^{\text{T}} $$
(50)
$$ \overline{{\mathbf{F}}}_{e} = \left\{ {\,\overline{F} x_{e}^{1} \;\,\overline{F} y_{e}^{1} \;\,\overline{F} x_{e}^{2} \,\,\overline{F} y_{e}^{2} \,\;\overline{F} x_{e}^{3} \,\,\overline{F} y_{e}^{3} \,\, \ldots \,\,\overline{F} x_{e}^{{N_{e} - 2}} \,\,\overline{F} y_{e}^{{N_{e} - 2}} \,\,\overline{F} x_{e}^{{N_{e} - 1}} \,\overline{F} y_{e}^{{N_{e} - 1}} \,\,\overline{F} x_{e}^{{N_{e} }} \;\,\overline{F} y_{e}^{{N_{e} }} } \right\}^{\text{T}} $$
(51)
Fig. 13
figure 13

The displacements, the forces and the moment in the local coordinate system of the eth element

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Fatahi, L., Moradi, S. & Ghanbarzadeh, A. The application of the geometric offset method to the rigid joint modeling in the differential quadrature element model updating of frame structures. Meccanica 50, 1509–1525 (2015). https://doi.org/10.1007/s11012-015-0103-6

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