Deformation Performance Analysis of a Truss Structure Based on the Deformation Decomposition Method

Trusses are among the basic components of large-span bridges and large-space structures. A method is proposed to conduct a comprehensive deformation analysis of a truss in terms of the basic rigid body displacements and the tension and compression deformation based on complete mathematical orthogonality and mechanical equilibrium. The correctness of the proposed method is verified by comparison with a traditional strain analysis. Furthermore, a relative deformation decomposition of the mode shape is proposed to analyse in detail its relative displacement and deformation. The correctness and superiority of the proposed method are verified by comparison with the modal mass participation coefficient method and the animation from observation method. Additionally, the relative deformation decomposition of a plane truss structure is realized under any load conditions based on the superposition of mode shapes. The quantitative analysis of the basic deformation performance of a plane truss structure can also be conducted by countable mode shapes, which do not involve load conditions. Finally, the number of mode shapes that must be considered differs when using the maximum displacement and the tension and compression deformation analysis indicators.


Introduction
Trusses provide the advantages of a single internal force and uniform deformation, and thus they are ideal and often used as the basic or substructure form in large-span bridges and large-space structures.
Deformation performance is the basis for the analysis and design of truss structures. Some studies have evaluated the deformation performance of truss structures under static and dynamic conditions. Static load tests [1][2][3] and finite element simulations [4][5][6] are the main methods applied to examine the static deformation characteristics of truss structures. For truss bridges, field load tests and strain measurements are commonly used to evaluate the mechanical properties [7,8]. The dynamic deformation characteristics of truss structures are more complex than those of static structures. The harmonic response spectrum method [9] and shaking table test [10,11] are used to study the dynamic deformation of truss bridges and truss bridge piers. As wooden structures were the primary structures in ancient architecture, some scholars are studying the seismic deformation of traditional Chinese timber trusses by shaking table tests [12,13] and time history analysis [14,15]. The abovementioned studies on the deformation performance of truss structures are all based on specific load conditions. Frequency and mode shape are the basic dynamic characteristics of a structure that can be solved without loaded conditions. Finite element analysis [16,17] and modal experiments [18][19][20] are the methods most commonly used to obtain structural modal information, and some researchers are studying the relationship between mode shape and structural design parameters. The Bayesian method [21,22], frequency domain decomposition [23] and local mean decomposition [24] are used to identify the mode shape. Mode shape is closely related to structural deformation performance. Some studies identified structural damage based on mode shape [25][26][27], and other studies analysed structural deformation performance in terms of local mode shapes [28,29]. The stress state identification method was proposed based on the measured modals and frequencies of truss structures [30]. Gao et al. [31] found that the randomness of truss structure frequencies and mode parameters had a significant impact on the dynamic characteristics of the structure.
At present, animation from observation is the main method used to analyse mode shapes; however, mode shapes cannot be described quantitatively. Wilson [32] proposed the modal mass participation coefficient method to quantitatively analyse mode shapes but ignored the relative deformation information in mode shapes. At present, there are few quantification methods for analysing mode shapes and considering relative deformations.
The quantitative evaluation of structural deformation performance is particularly crucial. The deformation decomposition method for analysing plane truss structures is proposed based on complete mathematical orthogonality and mechanical equilibrium. Then, the quantitative analysis of a mode shape is solved. Finally, the maximum node displacement and the tension and compression deformation of a rod element are taken as indicators to study the basic deformation of a truss by the quantitative analysis of the countable mode shapes. Optimal design and performance analysis of truss structures can be guided by the proposed method.

Deformation Decomposition of a Plane Truss Structure
The displacement and deformation of a 2-node plane rod element (as shown in Figure 1a) can be expressed by the displacement of element nodes (as shown in Figure 1b). The rod element is the basic element of truss, and rotations are permitted at ends 1 and 2 such that the rod element is subjected only to axial force [33].

Complete Orthogonal Mechanical Base Vector of a Plane Rod Element
There are four degrees of freedom in a 2-node planar rod element; therefore, the comprehensive deformation of a plane rod element can be decomposed into 4 basic rigid body displacements and a deformation mode, based on complete mathematical orthogonality and mechanical equilibrium. The basic rigid body displacements and deformation mode include rigid body displacement in the X-axis direction, rigid body displacement in the Y-axis direction, rigid body rotation in the XY-plane, and tensile and compressive deformation in the axial direction.
The base vector of the basic rigid body displacements and deformation can be defined as P j = u j,1x u j,1y u j,2x u j,2y where u j,1x is the displacement of node 1 in X-axis direction for the j-th base vector, u j,1y is the displacement of node 1 in Y-axis direction for the j-th base vector, u j,2x is the displacement of node 2 in X-axis direction for the j-th base vector, u j,2y is the displacement of node 2 in Y-axis direction for the j-th base vector. The projection coefficients of base vectors can be comparable, and the base vectors have to satisfy uniformization, namely The base vector P j (j = 1, 2, 3, 4) can represent the basic displacement and deformation. There is no coupling between different basic displacements and deformations. The dot product of different base vectors is zero, namely P 1 is defined as the rigid body displacement base vector in the X-axis direction (as show in The base vector P 4 is obtained by orthogonalizing with the other three base vectors, which can be expressed as the approximate base vector of rigid body rotation displacement in the XY-plane (as shown in Figure 2d), namely The rigid body rotation displacement is a nonlinear displacement, and P 4 can be linearly approximated base vector in small elastic deformation range.
The complete orthogonal mechanical base matrix P of a 2-node plane rod element is composed of the base vector of basic rigid body displacements and deformation. The matrix P can be expressed as follows:

Deformation Decomposition Method of a Plane Rod Element
The comprehensive displacement vector of a rod element node d e can be obtained from node coordinates before and after deformation (as shown in Figure 1) Projecting d e onto the complete orthogonal mechanical base matrix P is given by: Then, Equation (10) is equivalent to: where P T is the transposed matrix of P and (P T ) −1 is the inverse matrix of P T .
The projection coefficient vector p can be expressed as follows: where p 1 represents the projection coefficient of the rigid body displacement base vector in the X-axis direction; p 2 represents the projection coefficient of the rigid body displacement base vector in the Y-axis direction; p 3 represents the projection coefficient of the tensile and compressive deformation base vector in the axial direction, the positive value indicates tensile deformation and the negative values indicate compression deformation; p 4 represents the projection coefficient of the rigid body rotation base vector in the XY-plane. In summary, the deformation decomposition of a plane truss structure can be realized based on the complete set of orthogonal mechanical base vectors of a rod element.

Error Analysis of a Rigid Body Rotation Base Vector
The base vector of rigid body rotation in the XY-plane (as shown in Equation (7)) is obtained by orthogonalizing with the other three base vectors. The displacements of node 1 in X-axis direction and Y-axis direction are zero. As a linearly approximated base vector, the rotation displacement can be projected onto P 4 , and will produce certain calculation error. Therefore, an error analysis of the rotation displacement base vector is needed.
Assuming that the length of the rod element is 2a, the rod element rotates clockwise around its midpoint, and the displacement vector of the element node can be expressed as d e = a − a cos θ a sin θ −a + a cos θ −a sin θ Projecting the displacement vector onto the complete orthogonal mechanical base matrix P, the projection coefficients of each basic displacement and deformation vector can be given by The two projection coefficients of the rigid body displacement base vectors in the Xand Y-axis directions are equal to 0 from Equation (14). However, the projection coefficient of tensile and compressive deformation is not equal to 0. Then, expanding p 3 and p 4 with the Taylor series and ignoring infinitesimal terms gives: In the case of a small deformation, p 4 is a high-order infinitesimal relative to p 3 . Therefore, the projection coefficient of the plane rigid body rotation on the base vector of tension and compression deformation can be ignored.

Verification of the Deformation Decomposition Method
The finite element method is used commonly for structural deformation analysis. Now, the finite element method is compared with the deformation decomposition method.
The static equilibrium equations of a 2-node rod element can be expressed as Ku=F (16) where K is the stiffness matrix and F is the load displacement vector. Based on the geometric equation, the axial normal strain can be obtained as: where u is the displacement function. If the linear shape function is adopted, then the axial normal strain of the rod element can be obtained as follows: where u 1 is the axial displacement of node 1; u 2 is the axial displacement of node 2; and l is the length of the rod element. Based on the deformation decomposition of the plane truss structure, the projection coefficient of the axial tension and compression deformation of the rod element can be obtained as follows: From Equations (18) and (19), the relationship between the axial normal strain and the projection coefficient of axial tension and compression deformation can be obtained as follows: A simple plane truss model (model A) is shown in Figure 3. The horizontal rod and vertical rod are 500 mm; the diagonal rod is 500 √ 2 mm; the cross-sectional area of the rod is 300 mm 2 ; and the modulus of elasticity is 2 × 10 5 MPa. A horizontal load of 50 kN is applied to the top node c of model A. The node displacements of the plane truss can be calculated by Equation (16), as shown in Table 1.  Table 1. Displacements of truss joints.

Node Number
Node a Node b Node c Node d The normal strain can be calculated by Equation (18), and the projection coefficient of the axial tension and compression deformation can be calculated by Equation (19). A comparison between the normal strain and the projection coefficient of the axial tension and compression deformation is shown in Figure 4. As shown in Figure 4, the variation trend of the normal strain is consistent with the projection coefficient of the axial tension and compression deformation, and conforms to Equation (20).

Relative Deformation Decomposition of a Plane Truss Structure
The deformation decomposition method is directly applied to the relative deformation decomposition of the mode shape to realize the quantitative analysis of the mode shape of the plane truss.
The relative node displacement vector d e of the mode shape can be obtained by modal analysis. Projecting d e onto the complete orthogonal mechanical base matrix P gives: The projection coefficient vector p can be expressed as: where p 1 represents the projection coefficient of relative rigid body displacement in the X-axis direction, p 2 represents the projection coefficient of relative rigid body displacement in the Y-axis direction, p 3 represents the projection coefficient of relative tensile and compressive deformation in the axial direction, and p 4 represents the projection coefficient of the relative rigid body rotation in the XY-plane.
In summary, the relative deformation decomposition of mode shape of the plane truss can be realized based on Equation (21).
The absolute values of the relative rigid body displacement and deformation are added to obtain the cumulative value of the relative displacement and deformation of the mode shape, respectively. The cumulative value of the projection coefficient was used to quantitatively analyse the mode shape. Mode shapes of model A are shown in Figure 5. The modal mass participation coefficient method (method 1), the mode shape relative deformation decomposition method (method 2), and animation from observation method (method 3) are used to identify and analyse the mode shape of model A. The analysis results are shown in Tables 2-4.

Mode Shape Description
1st mode shape Translation of cross rod; rotation of vertical and inclined rod 2nd mode shape Compression of the right vertical rod; rotation of cross and inclined rod 3rd mode shape Compression of the left vertical rod; rotation of cross rod 4th mode shape Compression of cross rod; rotation of vertical rod Annotation: method 3 is the animation of observation method, and the mode shape is identified by subjective observation.
The correctness of the mode shape relative deformation decomposition method can be verified by the modal mass participation coefficient method and the animation of observation method, as shown in Tables 2-4. Meanwhile, in a comparison with the modal mass participation coefficient method, the proposed method provides not only relative displacement information but also relative deformation information in the proposed method. Compared with the animation from observation method, the quantitative analysis of the mode shape is realized.
Therefore, the quantitative analysis of any mode shape of the plane truss structure can be realized based on the relative deformation decomposition method.

Modal Superposition Method Based on Relative Deformation Decomposition
The node displacement response of a structure under any load conditions can be expressed as follows: where U is the node displacement of the structure, Φ n is the nth-order mode shape of the structure, and Y n is the modal coordinate value corresponding to the mode shape. The node displacement of the rod element can also be obtained by the mode shape superposition, as shown in Equation (24): where d e,n is the relative displacement vector of the rod element in the nth order mode shape of the structure and d e is the displacement response of the rod element under any load conditions. Based on relative deformation decomposition, Equation (24) is equivalent to: The displacement of the rod element under any load conditions can be superimposed by the relative displacement and deformation of the mode shape from Equation (25). The first few mode shapes are considered sufficient to give accurate results when analysing the deformation performance of a truss structure by the modal superposition method. Therefore, the deformation performance of a plane truss can be analysed by the relative tension and compression deformation projection coefficient of the first few mode shapes.

Case Analysis
For this analysis, the plane signal tower truss (model B) is shown in Figure 6. The height of the truss is 2500 mm, the width is 500 mm, the cross-sectional area of the vertical rod is 8000 mm 2 , the cross-sectional area of the cross and diagonal rod is 4000 mm 2 , the elastic modulus is 2.2 × 10 4 MPa, and the density is 5.2 × 10 −5 kg/mm 3 . In addition, the seismic fortification intensity is 8 degrees and class II site conditions.

Mode Shape and Relative Deformation Analysis of Model B
The finite element software ANSYS is used to establish the plane truss of model B, and the first five mode shapes are extracted. Then, the relative deformation decomposition is performed, and the analysis of the mode shape is shown in Table 5. The relative tension and compression deformation cloud diagram of the first five mode shapes is shown in Figure 7. The three types of rod element numbers with the largest relative tension and compression deformations of the first five mode shapes are shown in Table 6. Cloud diagram of the relative tension and compression (a) 1st mode shape, (b) 2nd mode shape (c) 3rd mode shape, (d) 4th mode shape, (e) 5th mode shape. Table 6. Three types of rod element numbers with the largest relative tension and compression deformations of the first five mode shapes.

Mode Shape
The Vertical Rod The Cross Rod The Inclined Rod The rod elements in which the relative tension and compression deformation is largest for the first three mode shapes are mainly concentrated in the middle and bottom of the truss structure. The rod elements in which the relative tension and compression deformation is largest of the fourth and fifth mode shapes are mainly concentrated in the top of the truss structure. The deformation of the truss under any load conditions can be obtained by superposition of the mode shapes. Therefore, the relative tension and compression deformation of the first five mode shapes of the truss can reflect the basic deformation characteristics of the truss structure under different load conditions.

Deformation Performance Analysis of a Truss under a Static Load
Two equivalent static load cases in the horizontal direction are applied to the truss structure: the top concentrated load based on the equivalent static method and inverted triangular load based on the bottom shear method.
Finite element analysis of the truss structure under the two load conditions is carried out. Under the two load conditions, the maximum displacement and strain appear at the same position, which are shown in Figure 6 (Node a and No. 1 rod). The maximum relative tension and compression deformation of the first-order mode shape is also the No. 1 rod (as shown in Table 6). The rod with large relative tension and compression deformation of a single mode shape is still large under the static load condition. Therefore, the basic static deformation characteristics of the truss can be analysed based on the relative deformation of the countable mode shape.
Then, the displacement vector of the truss is projected onto the mode shape matrix by considering the first five mode shapes. The projection coefficients and proportions of the first five mode shapes are shown in Table 7. The proportions of projection coefficients of the first-order mode shapes are 95.40% and 99.28% under the two static loads (as shown in Table 7). Therefore, the participation of the first mode shape is dominant in the displacement response under static loading.
The relative error of the maximum relative displacement of the node (node a) and tension-compression deformation of the rod element (No. 1 rod) by modal superposition under the two static loads are shown in Table 8. The cloud diagram of the relative tensioncompression deformation under static load is shown in Figure 8.  For the concentrated load condition, the relative error of the maximum relative displacement of the node is less than 5% considering the superposition of the first two mode shapes, and the relative error of the maximum relative tension-compression deformation of the rod element is less than 5%, considering the superposition of the first mode. In addition, for the inverted triangle load condition, the relative error of the maximum displacement of the node and tension-compression deformation of the rod element are both less than 5%, considering the superposition of the first mode shape (as shown in Table 8).
According to Figure 8, the tensile-compression deformation of model B under static loading is basically consistent with the relative tensile-compression deformation by modal superposition, and the relative error of the maximum tension-compression deformation of the rod element is less than 5%.
Three conclusions can be drawn by comparison. (1) There are relatively large tensile and compression deformation characteristics of the bottom vertical rod and inclined rod under two static conditions, which is consistent with the modal analysis results. (2) The tensile and compression deformation of the inverted triangle load condition is more consistent with the relative tension and compression deformation of the first mode shape. For concentrated load conditions, the analysis of the first mode shape may not meet the requirements, and it may be necessary to consider a higher-order mode shape. (3) The number of model shapes under consideration differs when taking the maximum displacement and tension-compression deformation of the rod element as indicators.

Deformation Performance Analysis of a Truss Structure under a Seismic Load
Three seismic waves-the Ninghe wave (N-H wave), Northridge wave (N-G wave) and Duzce Turkey wave (D-T wave)-are selected. The basic information about the three seismic waves is shown in Table 9. The damping ratio is 0.02, and the natural frequencies of the first five mode shapes are 0.28, 1.22, 1.54, 2.68 and 4.13 Hz. The dynamic response time history of model B under three seismic waves is calculated. The members with the maximum tensile and compressive deformation of model B under the three seismic waves are No. 16 rod elements (as shown in Figure 6). The No. 16 rod elements are also the members with the maximum relative tensile and compressive deformation of the first five mode shapes in model B. Therefore, the basic dynamic deformation characteristics of the truss can be analysed based on the relative deformation of the countable mode shapes.
Then, the displacement vector of model B is projected onto the mode shape matrix, taking into account the first five mode shapes when the displacement and acceleration response are maximum for the three seismic waves. The projection coefficients and proportions of the first five mode shapes are shown in Tables 10 and 11.
The first mode shape is not completely dominant when the displacement and acceleration response are maximum for the three seismic waves. The proportions of the higher-order mode shape projection coefficient increase (as shown in Tables 10 and 11).  Tables 12 and 13 show the relative error of the maximum displacement of the node and tension-compression deformation of the rod element by modal superposition when the displacement and acceleration response are maximum for the three seismic waves. The cloud diagram of the relative tension-compression deformation of the first five mode shapes under N-H seismic waves are shown in Figure 9.
The tension and compression deformation of No. 5 and No. 16 rod elements are the largest, when the displacement and acceleration response is maximum under the three seismic waves. Meanwhile, the No. 5 and No. 16 rod elements are also the largest relative tension and compression deformations of the first five mode shapes (as shown in Table 6).  When the displacement response is maximum for the three waves, the relative error in the maximum displacement of the node is less than 5% considering the superposition of the first two mode shapes. However, the relative error in the maximum relative tension and compression deformation of the rod element is less than 5% considering the superposition of the first five mode shapes under the N-H wave, and the relative error is less than 5% considering the superposition of the first four mode shapes under the N-G and D-T waves.
When the acceleration response is maximum under the N-H wave, the relative error of the maximum displacement of the node is less than 5% considering the superposition of the first four mode shapes, and the relative error of the maximum tension and compression deformation of the rod element is less than 5% considering the superposition of the first five mode shapes. When the acceleration response is maximum under the N-G wave, the relative errors of maximum displacement and tension-compression deformation are both less than 5%, considering the superposition of the first two mode shapes. In addition, when the acceleration response is maximum under the D-T wave, the relative errors of the maximum displacement and tension-compression deformation are both less than 5%, considering the superposition of the first four mode shapes.
As shown in Figure 9, the tensile and compression deformation of model B under the N-H seismic waves is basically consistent with the relative tensile and compression deformation by modal superposition, and the relative error of the maximum tensioncompression deformation of the rod element is less than 5%.
For the analysis of structural deformation performance under dynamic conditions, the number of model shapes to be considered is different when taking the maximum displacement and tension-compression deformation of the rod element as indicators. Therefore, the modal truncation method with the maximum displacement as the indication is no longer applicable when using the maximum tensile and compressive deformation as the indication, and special research and analysis are needed in this case. After determining the number of modes, the relative deformation decomposition method can be used to analyse the deformation performance of truss structures.

Discussion
Base vectors are constructed to realize the deformation decomposition of a plane truss structure based on complete mathematical orthogonality and mechanical equilibrium. The correctness of the proposed method is verified by comparison with the traditional strain analysis method.
The deformation decomposition method is applied to mode shape analysis, and the relative deformation decomposition of a plane truss is further proposed to quantitatively analyse the mode shape. Compared with the modal mass participation coefficient method and animation from observation method, the proposed method provides more complete information and more accurate conclusions.
The relative deformation superposition method is studied for a plane truss structure under two equivalent static conditions. The results show that the deformation performance of the plane truss structure can be analysed based on the first mode shape under an inverted triangle load condition, and the relative error is less than 5%. For general static load conditions, the analysis of the first mode shape may not meet the requirements, and it may be necessary to consider a higher-order mode shape.
Verification is provided that the analysis of the basic deformation characteristics of a truss structure can also be realized by the study of the relative tension and compression deformation of the countable mode shape. The proposed method is especially suitable for the study of the deformation of truss structures in the conceptual design stage. In addition, the number of mode shapes to be considered differs when taking the maximum displacement and the tension-compression deformation of the rod element as indicators under static and dynamic load conditions. The modal truncation indicator of maximum tension and compression deformation deserves special study.  Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Some or all data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare that they have no conflict of interest.