Effects of temperature on modal characteristics of non-uniform rigid-frame bridges

Abstract

The effect of temperature on structural modal characteristics plays an important role in structural health monitoring and evaluation. Herein, the free vibration analysis of rigid-frame bridge is conducted by taking into account the effect of temperature. Firstly, the non-uniform rigid-frame bridge is divided into several members to develop the flexural and axial vibration equations for members under the effect of temperature in accordance with Hamilton’s principle. Secondly, each member is further divided into several segments, and the transfer relationship between segments is established to obtain the generalized dynamic stiffness matrix of the member. Then, the generalized dynamic stiffness matrix of each member is constructed by using the numerical assembly method similar to the finite element method so as to obtain the characteristic equation of rigid-frame bridge. Furthermore, the natural frequency and mode shape of the whole rigid-frame bridge are calculated by using the algorithm of Wittrick-Williams. Finally, the dynamic characteristic test of Yong-an bridge and the numerical calculation of the rigid-frame bridge are carried out to verify the generality and accuracy of the proposed method. The results indicate that the change of temperature will lead to the secondary internal force and the change of Young’s modulus, which together affect the modal characteristics of the structure, and the change of natural frequency caused by the change of Young’s modulus is greater than that caused by the secondary internal forces. In addition, there is a negative correlation between temperature and natural frequency of the rigid-frame bridge, and different temperature gradient modes also have certain influence on the mode shape of rigid-frame bridge.

Keywords

temperature effect rigid-frame bridge dynamic stiffness matrix frequency mode shape

Introduction

In recent years, the bridge condition assessment technology based on modal characteristics has become a research hotspot in the field of bridge detection and health monitoring (Cruz and Salgado, 2010; Guo et al., 2021; Moughty and Casas, 2017). However, the tested modal characteristics are always inevitably affected by environmental factors in practical tests. As revealed in plenty of engineering tests, temperature has a significant impact on modal characteristics, and the temperature-induced frequency change may be more impactful than the frequency change caused by structural damage in some cases. As a consequence, the results of bridge state assessment results based on dynamic characteristics are rendered inaccurate or even ineffective (Guo et al., 2021; Talebinejad et al., 2011; Xia et al., 2012; Zhou and Yi 2014). Therefore, it is of great significance to analyze the influence mechanism of temperature on structural modal characteristics for the bridge reliability evaluation. Currently, rigid frame bridge has been widely constructed as part of the railway and highway infrastructures due to its attractive appearance and convenient construction (Hong et al., 2014; Hu et al., 2009; Song et al., 2016). However, the change of temperature will inevitably exert a secondary internal force on the rigid frame bridge, thus affecting the modal characteristics of the bridge. Consequently, the modal characteristics of rigid frame bridge are made more complex under the action of temperature. Therefore, it is essential to understand the effect of temperature on the modal characteristics of rigid frame bridges.

In the past 30 years, there have been plenty of studies conducted to explore the effect of temperature on structural modal characteristics through bridge health monitoring or testing. Askegaard and Mossing (1988) monitored a three-span reinforced concrete pedestrian bridge for 3 years from 1986 to 1988, which led to the results showing that temperature caused the vibration frequency of bridges to vary by about 10%. Since then, more and more researchers (Cornwell et al., 2010; Jin et al., 2016; Peeters and Roeck, 2001; Zhou et al., 2020) have recognized that temperature has obvious effect on structural modal characteristics. For example, Peeters et al. (2001) continuously monitored the modal characteristics of Z24 bridge in Switzerland for 1 year to analyze the effect of temperature on structural modal characteristics, and the monitoring results show that the first-order frequency changed by 14%–18% due to the effect of temperature. To sum up, there is a close relationship between frequency and temperature, despite some clarity on the underlying mechanism of temperature impacting on modal characteristics.

To solve this issue, the variation trend analysis of bridge modal characteristics under the effect of temperature has been widely carried out through numerical simulation, experimental study and field test. In terms of numerical simulation, Tian et al. (2017) proposed a finite element method to study the temperature effect of train-bridge coupling dynamics by considering temperature fields and deformation fields. Balmes et al. (2009) established the finite element model of the whole bridge span to better describe the effect of temperature on modal characteristics. Miao et al. (2011) adopted the spatial FEM to analyze the modal characteristics of a suspension bridge at different temperatures, and then applied the equivalent cable internal force method to simulate the temperature effect. The above methods involve various hypotheses and simplified methods, which is inevitable to result in a potential deviation between the calculation results and the actual situation. In response to this, experimental studies have been carried out to reveal the variations in structural modal characteristics at different ambient temperatures. By installing accelerometers and excitation-inducing shaker on the surface of the test beam, Kim et al. (2007) conducted a series of forced vibration tests on a single span stainless steel plate beam at different temperatures ranging from −3°C to 23°C. Balmes et al. (2007) observed a 16% increase in the first-order frequency of an aluminum-clamped beam during a laboratory test when the temperature was reduced by 17°C. With the development of computer technology and the combined application of sensor technology, the continuous field tests on various bridge structures have been widely conducted to explore the relationship between temperature and modal characteristics. Ni et al. (2005a, 2005b) investigated the effect of temperature on modal characteristics through the measurement data of cable-stayed Ting Kau bridge as collected at different temperatures ranging from 3°C to 53°C throughout 1 year. Mosavi et al. (2012) conducted a field test on vibration responses, deflections and temperatures for a steel-concrete composite bridge, with the test results showing that the changes in temperature from the night to noon caused modal frequency to very.

Despite the above results showing that the modal characteristics of the bridge is negatively correlated with temperature change, the results obtained directly from numerical simulations or experimental data are still determined by the type of bridge and the duration of test. It is thus necessary to further establish a quantitative model between modal frequency and temperature, which is crucial to addressing the environmental impact in vibration-based damage detection. For this reason, there has been attempt made by scholars to conduct research through linear model, nonlinear model and learning model. In terms of linear models, it has been concluded from a series of modal tests that the modal frequency of bridges is linearly related to temperature (Sun et al., 2018; Wang and Gao 2021). Liu et al. (2016) proposed multiple linear regression models by taking into consideration non-uniform temperature gradient. It was suggested that the change of concrete Young’s modulus with temperature was the primary reason for the variations in modal characteristics. Tan et al. (2021) and Zhang et al. (2021) argued that the non-uniform temperature gradient mode can affect the mechanical properties of the structure to a significant extent. In terms of nonlinear models, Ding and Li (2011) determined the relationship between modal frequency and temperature by using the six-order polynomial regression model based on the long-term monitoring data of Runyang suspension bridge located in China. As for learning models, the support vector machine (SVM) technique (Ni et al., 2005a) and principle component analysis (PCA) (Yan et al., 2005a, 2005b) were involved to quantify the effect of temperature on natural frequencies. Hua et al. (2007) adopted a method combining PCA and support vector regression (SVR) to model and analyze the natural frequency change caused by temperature for Ting Kau Bridge. According to the results of quantitative model analysis, the variation in modal characteristics is largely attributed to the change of elastic modulus.

However, these models and methods are only applicable to some particular bridges, which is adverse to revealing the general mechanism. Therefore, it remains necessary to explore the effect of temperature on modal characteristics from the theoretical perspective, so as to minimize the adverse effect of temperature on damage detection. However, it is very difficult to describe the modal characteristics of bridges in theory at different temperatures, and there is little research paying attention to this. Herein, a theoretical method of the simply supported beam under the effect of temperature is proposed by our research group, which considers the temperature-induced change in elastic modulus, deflection and shear stiffness of bearings (Liu et al., 2017). In addition, a discussion is conducted around the effects of different temperature modes on the modal characteristics of Timoshenko beam, which leads to exact solution in closed form (Liu et al., 2018). However, the rigid-frame bridge is prone to secondary internal forces when the temperature changes, which may affect the structural modal characteristics. As a result, the above method cannot be applied directly to the rigid frame bridges.

To solve this problem, a semi-analytical analysis is conducted in this study to explore the modal characteristics of non-uniform rigid-frame bridges at different temperatures. The most important innovation of this paper is that each main girder and pier of a multi-span rigid-frame bridge is treated as a member, which is then divided into several segments. This has two highlights compared with the traditional method: On the one hand, the undetermined coefficients are used to establish the modal characteristic equation for the member in the calculation process, which reduces the dimension of the characteristic equation significantly, thus improving the calculation efficiency; On the other hand, the member is divided into several segments by means of segmentation, with the local characteristics of members considered, thus ensuring the high accuracy of calculation. Figure 1 shows the process of the method proposed in this paper in detail.

Figure 1.

Flow chart of the calculation method of modal characteristics proposed in this paper.

Modal characteristics of rigid-frame bridges under the effect of temperature

Dynamic equations of the rigid-frame bridge members

A typical n-span continuous rigid-frame bridge is shown in Figure 2. The top of the pier is rigidly connected to the main girder, the bottom of the pier is fixed, and the sliding hinge supports are used at both ends of the main girder. The continuous rigid frame bridge is regarded as a structural system composed of 2n-1 members, which are numbered as 1, …, i-1, i, i+1, …, n, n+1, …, n + i-2, n + i-1, n + i, n + i+1, …, 2n-1, respectively. It should be noted that 1 to n are the main girder numbers, that is, each span main girder is one member; from n+1 to 2n-1 are the numbers of piers, corresponding to n-1 piers, each of which is one member. Therefore, there are a total of 2n nodes corresponding to 2n-1 members, whose numbers are represented by the numbers with brackets in Figure 2, namely (1) to (2n).

Figure 2.

A diagram of a continuous rigid-frame bridge.

For a particular cross-section, the temperature distribution along the height of beam can be divided into three parts, that is uniform temperature distribution, linear temperature gradient distribution and nonlinear temperature gradient distribution (Liu et al., 2018). However, the rigid-frame bridge is a multi-statically indeterminate structure, whose temperature variation will result in the secondary internal force, leading to the existence of stress in the structure analogue to the pre-stress load applied on the main girder and pier. Both uniform and linear temperature gradient can cause internal forces, while nonlinear temperature gradient produce self-stress. Therefore, the effect of temperature will cause the additional bending moments, shear and axial forces and temperature self-stress for the rigid-frame bridge, these internal forces and stresses can be obtained through structural static analysis.

The ith member in Figure 2 is taken as an example to illustrate the establishment process of dynamic equations for any member in continuous rigid-frame bridge. The force mode of the ith member with a length of l is provided in Figure 3. The local coordinate system of the ith member is defined as follows: let node (i) be the local coordinate origin, and the direction from node (i) to node (i+1) is deemed to the positive direction of the $\bar{x}$ axis. Turning the $\bar{x}$ axis 90°counterclockwise, that is the positive direction of the $\bar{y}$ axis, and then the $\bar{z}$ axis could be obtained by right-hand rule. The parameters ${\bar{M}}_{(i)}^{i}$ , ${\bar{M}}_{(i + 1)}^{i}$ , ${\bar{N}}_{(i)}^{i}$ , ${\bar{N}}_{(i + 1)}^{i}$ , ${\bar{Q}}_{(i)}^{i}$ and ${\bar{Q}}_{(i + 1)}^{i}$ are bending moment, axial force and shear force of the both ends of the ith member, respectively. Similarly, the parameters ${\bar{θ}}_{(i)}^{i}$ , ${\bar{θ}}_{(i + 1)}^{i}$ , ${\bar{u}}_{(i)}^{i}$ , ${\bar{u}}_{(i + 1)}^{i}$ , ${\bar{y}}_{(i)}^{i}$ and ${\bar{y}}_{(i + 1)}^{i}$ are rotational displacement, axial displacement and transverse displacement, respectively. In fact, the temperature distribution is time-varying, whose rate of variation is extremely slow compared with the rate of deformation caused by free vibration. Therefore, the dynamic equations do not consider the temperature varying with time, so $T (\bar{x}, \bar{y}, \bar{z})$ is adopted to represent the temperature distribution function throughout the member.

Figure 3.

The ith member with displacements and forces at both end.

The variation of temperature will not only produce the secondary internal force and self-stress of continuous rigid frame bridge, but also lead to the change of its material properties, especially the influence of temperature variation on Young’s modulus of material. Here the quantitative expression of Young’s modulus as a function of temperature (Xia et al., 2011) could be denoted as

E = E_{0} (1 - θ_{E} T_{p})

(1)

where

E_{0}

is Young’s modulus of concrete at 0°C, T_p is Celsius temperature,

θ_{E}

is temperature coefficient (or thermal coefficient) of Young’s modulus, which is recommended to be 0.0045 by Baldwin and North’s report (1973). Aiming at this problem, the relationship between the Young’s modulus and temperature within −20°C–60°C was studied by the authors’ research group (Jiao et al., 2014), which was similar to equation (1), verifying its applicability.

Substituting the temperature distribution function $T (\bar{x}, \bar{y}, \bar{z})$ into equation (1),

E (\bar{x}, \bar{y}, \bar{z}) = E_{0} [1 - θ_{E} T (\bar{x}, \bar{y}, \bar{z})] = E_{0} f (\bar{x}, \bar{y}, \bar{z})

(2)

where

f (\bar{x}, \bar{y}, \bar{z}) = 1 - θ_{E} T (\bar{x}, \bar{y}, \bar{z})

The bending stiffness $B S (\bar{x})$ and tension-compression stiffness $C S (\bar{x})$ of arbitrary cross section at $\bar{x}$ under varying temperature are defined as

B S (\bar{x}) = \int_{A} E_{0} f (\bar{x}, \bar{y}, \bar{z}) {\bar{y}}^{2} d A = E_{0} I_{T} (\bar{x})

(3)

C S (\bar{x}) = \int_{A} E_{0} f (\bar{x}, \bar{y}, \bar{z}) d A = E_{0} A_{T} (\bar{x})

(4)

where

I_{T} (\bar{x}) = \int_{A} f (\bar{x}, \bar{y}, \bar{z}) {\bar{y}}^{2} d A

and

A_{T} (\bar{x}) = \int_{A} f (\bar{x}, \bar{y}, \bar{z}) d A

The flexural vibration and axial vibration of the ith member are considered, the governing differential equations of motion can be expanded by extended Hamilton principle

δ \int_{t_{1}}^{t_{2}} (T - V) d t + δ \int_{t_{1}}^{t_{2}} W d t = 0

(5)

where T stands for kinetic energy, V represents potential energy and W is the nonconservative virtual work, namely W contains the virtual work W_T done by temperature load and virtual work W_b done by boundary force, then one has

δ \int_{t_{1}}^{t_{2}} W d t = δ \int_{t_{1}}^{t_{2}} W_{b} d t + δ \int_{t_{1}}^{t_{2}} W_{T} d t

(6)

Without considering shear deformation and moment of inertia, the kinetic energy variation can be written as

δ \int_{t_{1}}^{t_{2}} T d t = δ \int_{t_{1}}^{t_{2}} \int_{0}^{l} \frac{1}{2} m (\bar{x}) ({\dot{\bar{y}}}^{2} + {\dot{\bar{u}}}^{2}) d \bar{x} d t

(7)

where

\bar{y}

and

\bar{u}

are the respective transverse displacement and axial displacement for the ith member.

The variation of potential energy is expanded as

δ \int_{t_{1}}^{t_{2}} V d t = δ \int_{t_{1}}^{t_{2}} \int_{0}^{l} \frac{1}{2} E_{0} [I_{T} (\bar{x}) {\bar{y}}^{″}^{2} + A_{T} (\bar{x}) {\bar{u}}^{'}^{2}] d \bar{x} d t

(8)

And the variation of the virtual work W_b by boundary force is

δ \int_{t_{1}}^{t_{2}} W_{b} d t = \int_{t_{1}}^{t_{2}} ({\bar{M}}_{(i)}^{i} δ {\bar{y}}^{'}_{(i)}^{i} + {\bar{M}}_{(i + 1)}^{i} δ {\bar{y}}^{'}_{(i + 1)}^{i} + {\bar{Q}}_{(i)}^{i} δ {\bar{y}}_{(i)}^{i} + {\bar{N}}_{(i)}^{i} δ {\bar{u}}_{(i)}^{i} + {\bar{Q}}_{(i + 1)}^{i} δ {\bar{y}}_{(i + 1)}^{i} + {\bar{N}}_{(i + 1)}^{i} δ {\bar{u}}_{(i + 1)}^{i}) d t

(9)

The change of temperature will produce both secondary internal forces and self-stress on the structure at the same time. Moreover, the work done by the secondary internal force contains the work done by the axial force W_N, bending moment W_M and shearing force W_Q, one has

δ \int_{t_{1}}^{t_{2}} W_{T} d t = δ \int_{t_{1}}^{t_{2}} W_{M} d t + δ \int_{t_{1}}^{t_{2}} W_{N} d t + δ \int_{t_{1}}^{t_{2}} W_{Q} d t + δ \int_{t_{1}}^{t_{2}} W_{E} d t

(10)

where

W_{E}

is the virtual work done by self-stress,

δ \int_{t_{1}}^{t_{2}} W_{M} d t

and

δ \int_{t_{1}}^{t_{2}} W_{N} d t

can be written as

δ \int_{t_{1}}^{t_{2}} W_{M} d t = δ \int_{t_{1}}^{t_{2}} \int_{0}^{l} M_{T} (\bar{x}) {\bar{y}}^{″} d \bar{x} d t

(11)

δ \int_{t_{1}}^{t_{2}} W_{N} d t = δ \int_{t_{1}}^{t_{2}} \int_{0}^{l} [- \frac{1}{2} N_{T} (\bar{x}) {\bar{y^{'}}}^{2} + N_{T} (\bar{x}) {\bar{u}}^{'}] d \bar{x} d t

(12)

where M_T represents the bending moment generated by temperature, whose direction is specified to be positive when the lower edge of the structure is subjected to tension; N_T is the axial force due to the effect of temperature, whose direction is positive in axial tension.

W_{E}

is the work done by self-stress in the vibration displacement

\bar{y} (\bar{x}, t)

and

\bar{u} (\bar{x}, t)

, and one has

W_{E} = \int_{v} σ_{E} (\bar{z} {\bar{y}}^{″} + {\bar{u}}^{'}) d v = \int_{0}^{l} (\int_{A} σ_{E} \bar{z} d A {\bar{y}}^{″} + \int_{A} σ_{E} d A {\bar{u}}^{'}) d \bar{x} = \int_{0}^{l} [M_{E} (\bar{x}) {\bar{y}}^{″} + N_{E} (\bar{x}) {\bar{u}}^{'}] d \bar{x}

(13)

where

M_{E} (\bar{x}) = \int_{A} σ_{E} \bar{z} d A

N_{E} (\bar{x}) = \int_{A} σ_{E} d A

According to the equation (13), one can obtain the variation of $W_{E}$ as follows

δ \int_{t_{1}}^{t_{2}} W_{E} d t = δ \int_{t_{1}}^{t_{2}} \int_{0}^{l} [M_{E} (\bar{x}) {\bar{y}}^{″} + N_{E} (\bar{x}) {\bar{u}}^{'}] d \bar{x} d t

(14)

W_T could be obtained after substituting equations (11–13) into equation (10). Then substitute equations (7–10) into equation (5), the following equations of motion can be derived:

\begin{array}{l} - \int_{t_{1}}^{t_{2}} \int_{0}^{l} {m (\bar{x}) \ddot{\bar{y}} + {[E_{0} I_{T} (\bar{x}) {\bar{y}}^{″}]}^{″} - {[N_{T} (\bar{x}) {\bar{y}}^{'}]}^{'} - M_{E}^{″} (\bar{x}) - {[M_{T} (\bar{x})]}^{″}} δ \bar{y} d \bar{x} d t \\ - \int_{t_{1}}^{t_{2}} \int_{0}^{l} {m (\bar{x}) \ddot{\bar{u}} - {[E_{0} A_{T} (\bar{x}) {\bar{u}}^{'}]}^{'} + {[N_{T} (\bar{x})]}^{'} + N_{E}^{'} (\bar{x})} δ \bar{u} d \bar{x} d t - \int_{t_{1}}^{t_{2}} [E_{0} A_{T} (\bar{x}) {\bar{u}}^{'} |_{l} - {\bar{N}}_{(i + 1)}^{i} - N_{T} (l) - N_{E} (l)] δ {\bar{u}}_{(i + 1)}^{i} d t \\ - \int_{t_{1}}^{t_{2}} [E_{0} I_{T} (\bar{x}) {\bar{y}}^{″} |_{l} - {\bar{M}}_{(i + 1)}^{i} - M_{T} (l) - M_{E} (l)] δ {\bar{y}}^{'}_{(i + 1)}^{i} d t + \int_{t_{1}}^{t_{2}} [E_{0} I_{T} (\bar{x}) {\bar{y}}^{″} |_{0} + {\bar{M}}_{(i)}^{i} - M_{T} (0) - M_{E} (0)] δ {\bar{y}}^{'}_{(i)}^{i} d t \\ + \int_{t_{1}}^{t_{2}} [{[E_{0} I_{T} (\bar{x}) {\bar{y}}^{″}]}^{'} |_{l} + {\bar{Q}}_{(i + 1)}^{i} - M_{T}^{'} (l) - N_{T} (l) {\bar{y}}^{'} |_{l} - M_{E}^{'} (l)] δ {\bar{y}}_{(i + 1)}^{i} d t \\ - \int_{t_{1}}^{t_{2}} [{[E_{0} I_{T} (\bar{x}) {\bar{y}}^{″}]}^{'} |_{0} - {\bar{Q}}_{(i)}^{i} - M_{T}^{'} (0) - N_{T} (0) y^{'} |_{0} - M_{E}^{'} (0)] δ {\bar{y}}_{(i)}^{i} d t \\ + \int_{t_{1}}^{t_{2}} [E_{0} A_{T} (\bar{x}) {\bar{u}}^{'} |_{0} + {\bar{N}}_{(i)}^{i} - N_{T} (0) - N_{E} (0)] δ {\bar{u}}_{(i)}^{i} d t + \int_{t_{1}}^{t_{2}} δ W_{Q} d t + \int_{0}^{l} m (\bar{x}) \dot{\bar{y}} δ \bar{y} |_{t_{1}}^{t_{2}} d \bar{x} + \int_{0}^{l} m (\bar{x}) \dot{\bar{u}} δ \bar{u} |_{t_{1}}^{t_{2}} d \bar{x} = 0 \end{array}

(15)

Since the effect of structural shear deformation is not considered, there is

\int_{t_{1}}^{t_{2}} δ W_{Q} d t = 0

(16)

Moreover, the variation of displacement is zero at the certain moment, one can also have

\int_{0}^{l} m (\bar{x}) \dot{\bar{y}} δ \bar{y} |_{t_{1}}^{t_{2}} d \bar{x} = 0, \int_{0}^{l} m (\bar{x}) \dot{\bar{u}} δ \bar{u} |_{t_{1}}^{t_{2}} d \bar{x} = 0

(17)

$δ {\bar{y}}_{(i)}^{i}$ , $δ {\bar{y}}_{(i + 1)}^{i}$ , $δ {\bar{y}}^{'}_{(i)}^{i}$ , $δ {\bar{y}}^{'}_{(i + 1)}^{i}$ , $δ {\bar{u}}_{(i)}^{i}$ and $δ {\bar{u}}_{(i + 1)}^{i}$ are all zero for displacement boundary conditions, which can be arbitrary for the force boundary condition. In order for equation (15) to hold, the following equations must be satisfied

m (\bar{x}) \ddot{\bar{y}} + {[E_{0} I_{T} (\bar{x}) {\bar{y}}^{″}]}^{″} - {[N_{T} (\bar{x}) {\bar{y}}^{'}]}^{'} = M_{E}^{″} (\bar{x}) + {[M_{T} (\bar{x})]}^{″}

(18)

m (\bar{x}) \ddot{\bar{u}} + {[E_{0} A_{T} (\bar{x}) {\bar{u}}^{'}]}^{'} = {[N_{T} (\bar{x})]}^{'} - N_{E}^{'} (\bar{x})

(19)

The equations (18) and (19) are the transverse bending and axial vibration equations for the ith member under the effect of temperature. The equations of end force balance condition of the ith member are put in the Appendix A.1.

Establishment of generalized dynamic stiffness matrix for the ith member

Similarly, taking the ith member as an example, the establishment method of the generalized dynamic stiffness matrix for arbitrary member is then discussed. It should be noted that the formation method, meaning and matrix form of element stiffness matrix are different from those of FEM. Therefore, it is called the generalized dynamic stiffness matrix.

It can been seen from Figure 4 that the ith member is divided into h segments based on the idea of segmentation, each of which is connected through internal nodes, numbering s₁, …, s_k-1, s_k, …., s_h above the $\bar{x}$ axis, with length of l₁, … l_k-1, l_k,…., l_h, respectively. There are a total of h-1 internal nodes corresponding to h segments, whose numbers are represented by 1, …, k-2, k-1, k, ….., h-1 below the $\bar{x}$ axis. Namely, ${\bar{x}}_{k}$ is the coordinate of the right end of the segment s_k on the $\bar{x}$ axis.

Figure 4.

Discretization of the ith member.

Each segment can be approximated as a uniform beam when the number of divisions are adequate. Therefore, the equivalent linear density m_k, bending rigidity BS_k, tensile stiffness CS_k and axial load $N_{T, k}$ under the effect of temperature for the segment s_k can be defined as follows:

m_{k} = \frac{\int_{{\bar{x}}_{k - 1}}^{{\bar{x}}_{k}} m (\bar{x}) d \bar{x}}{l_{k}}

(20)

B S_{k} = \frac{\int_{{\bar{x}}_{k - 1}}^{{\bar{x}}_{k}} B S (\bar{x}) d \bar{x}}{l_{k}}

(21)

C S_{k} = \frac{\int_{{\bar{x}}_{k - 1}}^{{\bar{x}}_{k}} C S (\bar{x}) d \bar{x}}{l_{k}}

(22)

N_{T, k} = \frac{\int_{{\bar{x}}_{k - 1}}^{{\bar{x}}_{k}} N_{T} (\bar{x}) d \bar{x}}{l_{k}}

(23)

Let $ξ_{k} = \bar{x} - x_{k - 1}$ , one has $0 \leq ξ_{k} \leq l_{k}$ for the segment s_k of the ith member. Substituting equations (20–23) into equations (18) and (19) gives the free vibration equations for segment s_k as follows

m_{k} \frac{\partial^{2} {\bar{y}}_{k} (ξ_{k}, t)}{\partial t^{2}} + B S_{k} \frac{\partial^{4} {\bar{y}}_{k} (ξ_{k}, t)}{\partial {ξ_{k}}^{4}} - N_{T, k} \frac{\partial^{2} {\bar{y}}_{k} (ξ_{k}, t)}{\partial {ξ_{k}}^{2}} = 0

(24)

\frac{\partial^{2} {\bar{u}}_{k} (ξ_{k}, t)}{\partial {ξ_{k}}^{2}} = \frac{m_{k}}{C S_{k}} \frac{\partial^{2} {\bar{u}}_{k} (ξ_{k}, t)}{\partial t^{2}}

(25)

where

{\bar{y}}_{k} (ξ_{k}, t)

and

{\bar{u}}_{k} (ξ_{k}, t)

are transverse and axial displacement at

ξ_{k}

of time t, respectively.

Assuming that the segment s_k vibrates in simple harmonic mode near the equilibrium position, ${\bar{y}}_{k} (ξ_{k}, t)$ and ${\bar{u}}_{k} (ξ_{k}, t)$ can be gained in the following

{\bar{y}}_{k} (ξ_{k}, t) = {\bar{Y}}_{k} (ξ_{k}) e^{i ω t}

(26)

{\bar{u}}_{k} (ξ_{k}, t) = {\bar{U}}_{k} (ξ_{k}) e^{i ω t}

(27)

where

{\bar{Y}}_{k} (ξ_{k})

and

{\bar{U}}_{k} (ξ_{k})

are the amplitudes of

{\bar{y}}_{k} (ξ_{k}, t)

and

{\bar{u}}_{k} (ξ_{k}, t)

in free vibration,

i = \sqrt{- 1}

Substituting equations (26) and (27) into equations (24) and (25) gives

B S_{k} {\bar{Y}}_{k}^{(4)} (ξ_{k}) - N_{T, k} {\bar{Y}}_{k}^{″} (ξ_{k}) - m_{k} ω^{2} {\bar{Y}}_{k} (ξ_{k}) = 0

(28)

{\bar{U}}_{k}^{″} (ξ_{k}) = - \frac{m_{k} ω^{2}}{C S_{k}} {\bar{U}}_{k} (ξ_{k})

(29)

Thus, the solutions for ${\bar{Y}}_{k} (ξ_{k})$ and ${\bar{U}}_{k} (ξ_{k})$ can be written as

{\bar{Y}}_{k} (ξ_{k}) = D_{k, 1} \sin (δ_{k} ξ_{k}) + D_{k, 2} \cos (δ_{k} ξ_{k}) + D_{k, 3} \sinh (ε_{k} ξ_{k}) + D_{k, 4} \cosh (ε_{k} ξ_{k})

(30)

{\bar{U}}_{k} (ξ_{k}) = D_{k, 5} \cos (η_{k} ξ_{k}) + D_{k, 6} \sin (η_{k} ξ_{k})

(31)

where D_k, i (i = 1, 2, …, 6) are six constants for the segment s_k,

δ_{k} = \sqrt{\sqrt{{α_{k}}^{4} + \frac{{g_{k}}^{4}}{4}} + \frac{{g_{k}}^{2}}{2}}, ε_{k} = \sqrt{\sqrt{{α_{k}}^{4} + \frac{{g_{k}}^{4}}{4}} - \frac{{g_{k}}^{2}}{2}}, η_{k} = \sqrt{\frac{m_{k}}{C S_{k}}} ω

(32)

in which

α_{k}^{4} = ω^{2} m_{k} / B S_{k}

and

g_{k}^{2} = N_{T, k} / B S_{k}

Amplitude of rotational could be then obtained as follows:

{\bar{Φ}}_{k} (ξ_{k}) = {\bar{Y}}_{k}^{'} (ξ_{k})

(33)

According to equations (30)–(33), one could obtain:

{\bar{Y}}_{k, L} = {\bar{Y}}_{k} (0), {\bar{U}}_{k, L} = {\bar{U}}_{k} (0), {\bar{Φ}}_{k, L} = {\bar{Φ}}_{k} (0)

(34)

{\bar{Y}}_{k, R} = {\bar{Y}}_{k} (l_{k}), {\bar{U}}_{k, R} = {\bar{U}}_{k} (l_{k}), {\bar{Φ}}_{k, R} = {\bar{Φ}}_{k} (l_{k})

(35)

where the subscript k, L and k, R represent left and right end for the segment s_k, respectively.

Equations (34) and (35) can be expressed in matrix form as:

{\bar{δ}}_{k} = R_{k} D_{k}

(36)

where

R_{k} = [\begin{array}{c} 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ δ_{k} & 0 & ε_{k} & 0 & 0 & 0 \\ \sin δ_{k} l_{k} & \cos δ_{k} l_{k} & \sinh ε_{k} l_{k} & \cosh ε_{k} l_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & \cos η_{k} l_{k} & \sin η_{k} l_{k} \\ δ_{k} \cos δ_{k} l_{k} & - δ_{k} \sin δ_{k} l_{k} & ε_{k} \cosh ε_{k} l_{k} & ε_{k} \sinh ε_{k} l_{k} & 0 & 0 \end{array}]

(37)

{\bar{δ}}_{k} = {[\begin{array}{c} {\bar{Y}}_{k, L} & {\bar{U}}_{k, L} & {\bar{Φ}}_{k, L} & {\bar{Y}}_{k, R} & {\bar{U}}_{k, R} & {\bar{Φ}}_{k, R} \end{array}]}^{T}

(38)

D_{k} = {[\begin{array}{c} D_{k, 1} & D_{k, 2} & D_{k, 3} & D_{k, 4} & D_{k, 5} & D_{k, 6} \end{array}]}^{T}

(39)

Amplitude of shear force ${\bar{Q}}_{k} (ξ_{k})$ , axial force ${\bar{N}}_{k} (ξ_{k})$ and bending moment ${\bar{M}}_{k} (ξ_{k})$ for the segment s_k are as follows:

{\bar{Q}}_{k} (ξ_{k}) = B S_{k} {\bar{Y}}^{‴}, {\bar{M}}_{k} (ξ_{k}) = B S_{k} {\bar{Y}}^{″}, {\bar{N}}_{k} (ξ_{k}) = C S_{k} {\bar{U}}^{'}

(40)

Similar to the end displacements of segment s_k, ${\bar{Q}}_{k, L}$ , ${\bar{N}}_{k, L}$ , ${\bar{M}}_{k, L}$ , ${\bar{Q}}_{k, R}$ , ${\bar{N}}_{k, R}$ and ${\bar{M}}_{k, R}$ are adopted to represent the end forces, whose positive directions are also shown in Figure 3. It is worth noting that these end force amplitudes are generated by free vibration and do not include the end force caused by temperature. According to equation (40), one could obtain:

{\bar{Q}}_{k, L} = {\bar{Q}}_{k} (0), {\bar{N}}_{k, L} = - {\bar{N}}_{k} (0), {\bar{M}}_{k, L} = - {\bar{M}}_{k} (0)

(41)

{\bar{Q}}_{k, R} = - {\bar{Q}}_{k} (l_{k}), {\bar{N}}_{k, R} = {\bar{N}}_{k} (l_{k}), {\bar{M}}_{k, R} = {\bar{M}}_{k} (l_{k})

(42)

Equations (41) and (42) are expressed in matrix form as follows:

{\bar{F}}_{k} = Q_{k} D_{k}

(43)

where

Q_{k} = [\begin{array}{c} - B S_{k} {δ^{k}}_{3} & 0 & B S_{k} {ε^{k}}_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - C S_{k} η_{k} \\ 0 & B S_{k} {δ^{k}}_{2} & 0 & - B S_{k} {ε^{k}}_{2} & 0 & 0 \\ B S_{k} {δ^{k}}_{3} \cos δ_{k} l_{k} & - B S_{k} {δ^{k}}_{3} \sin δ_{k} l_{k} & - B S_{k} {ε^{k}}_{3} \cosh ε_{k} l_{k} & - B S_{k} {ε^{k}}_{3} \sinh ε_{k} l_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & - C S_{k} η_{k} \sin η_{k} l_{k} & C S_{k} η_{k} \cos η_{k} l_{k} \\ - B S_{k} {δ^{k}}_{2} \sin δ_{k} l_{k} & - B S_{k} {δ^{k}}_{2} \cos δ_{k} l_{k} & B S_{k} {ε^{k}}_{2} \sinh ε_{k} l_{k} & B S_{k} {ε^{k}}_{2} \cosh ε_{k} l_{k} & 0 & 0 \end{array}]

(44)

{\bar{F}}_{k} = {[\begin{array}{c} {\bar{Q}}_{k, L} & {\bar{N}}_{k, L} & {\bar{M}}_{k, L} & {\bar{Q}}_{k, R} & {\bar{N}}_{k, R} & {\bar{M}}_{k, R} \end{array}]}^{T}

(45)

Eliminating the undetermined coefficient vector $D_{k}$ in equation (43) according to equation (36), one obtains

{\bar{F}}_{k} = Q_{k} R_{k}^{- 1} {\bar{δ}}_{k} = {\bar{K}}_{k} {\bar{δ}}_{k}

(46)

where

{\bar{K}}_{k} = Q_{k} R_{k}^{- 1}

, and it expresses the relationship between the end force and displacement of segment s_k, which is similar to the element stiffness matrix in the FEM, so it is called the generalized dynamic stiffness matrix.

To write equation (46) in block matrix form gives

{\begin{array}{c} {\bar{F}}_{k}^{L} \\ {\bar{F}}_{k}^{R} \end{array}} = [\begin{array}{c} {\bar{K}}_{k}^{L L} & {\bar{K}}_{k}^{L R} \\ {\bar{K}}_{k}^{R L} & {\bar{K}}_{k}^{R R} \end{array}] {\begin{array}{c} {\bar{δ}}_{k}^{L} \\ {\bar{δ}}_{k}^{R} \end{array}}

(47)

where

{\bar{F}}_{k}^{L} = {[\begin{array}{c} {\bar{Q}}_{k, L} & {\bar{N}}_{k, L} & {\bar{M}}_{k, L} \end{array}]}^{T}

{\bar{F}}_{k}^{R} = {[\begin{array}{c} {\bar{Q}}_{k, R} & {\bar{N}}_{k, R} & {\bar{M}}_{k, R} \end{array}]}^{T}

{\bar{δ}}_{k}^{L} = {[\begin{array}{c} {\bar{Y}}_{k, L} & {\bar{U}}_{k, L} & {\bar{Φ}}_{k, L} \end{array}]}^{T}

and

{\bar{δ}}_{k}^{R} = {[\begin{array}{c} {\bar{Y}}_{k, R} & {\bar{U}}_{k, R} & {\bar{Φ}}_{k, R} \end{array}]}^{T}

Hence from equation (47), one can obtain

{\bar{δ}}_{k}^{R} = {\bar{K}}_{k}^{L R}^{- 1} ({\bar{F}}_{k}^{L} - {\bar{K}}_{k}^{L L} {\bar{δ}}_{k}^{L})

(48)

The substitution of equation (48) into equation (47) leads to

{q_{k}^{R}} = {[T]}_{k} {q_{k}^{L}}

(49)

where

{q_{k}^{R}} = {[\begin{array}{c} {\bar{δ}}_{k}^{R} & {\bar{F}}_{k}^{R} \end{array}]}^{T}, {q_{k}^{L}} = {[\begin{array}{c} {\bar{δ}}_{k}^{L} & {\bar{F}}_{k}^{L} \end{array}]}^{T}

(50)

{[T]}_{k} = [\begin{array}{c} - {K_{k}^{L R}}^{- 1} K_{k}^{L L} & {K_{k}^{L R}}^{- 1} \\ K_{k}^{R L} - K_{k}^{R R} {K_{k}^{L R}}^{- 1} K_{k}^{L L} & K_{k}^{R R} {K_{k}^{L R}}^{- 1} \end{array}]

(51)

and

{[T]}_{k}

is the transfer matrix for the segment s_k, which denotes the transfer relationship between the vibration amplitude and deformation vibration amplitude of the right and left ends of the segment s_k.

The derivation of the relationship between force and displacement of both ends (i.e. node (i) and node (i + 1)) for the ith member is presented in Appendix A.2. Then, one can obtain

{\bar{F}}^{i} = {\bar{K}}^{i} {\bar{δ}}^{i}

(52)

where

{\bar{F}}^{i} = {[\begin{array}{c} {\bar{F}}_{(i)}^{i} & {\bar{F}}_{(i + 1)}^{i} \end{array}]}^{T}, {\bar{δ}}^{i} = {[\begin{array}{c} δ_{(i)}^{i} & δ_{(i + 1)}^{i} \end{array}]}^{T}

(53)

{\bar{K}}^{i} = [\begin{array}{c} - {\bar{H}}_{12}^{i - 1} {\bar{H}}_{11}^{i} & {\bar{H}}_{12}^{i - 1} \\ {\bar{H}}_{21}^{i} - {\bar{H}}_{22}^{i} {\bar{H}}_{12}^{i - 1} {\bar{H}}_{11}^{i} & {\bar{H}}_{22}^{i} {\bar{H}}_{12}^{i - 1} \end{array}]

(54)

Equation (52) represents the relationship between force and displacement of both ends (i.e. node (i) and node (i+1)) for the ith member, and ${\bar{K}}^{i}$ can be called the generalized dynamic stiffness matrix of the ith member similarly.

Characteristic equations of the rigid-frame bridge under the effect of temperature

Due to the fact that equation (52) does not contain messages on the force and displacement of internal nodes, the relationship of force and displacement between left end and right end of the ith member similar to equation (52) can still be obtained, no matter how many segments the ith member is divided into. In this case, the ith member and the generalized dynamic stiffness matrix ${\bar{K}}^{i}$ can be considered as an element and the element stiffness matrix in FEM.

It should be noted that ${\bar{F}}^{i}$ , ${\bar{K}}^{i}$ and ${\bar{δ}}^{i}$ in equation (52) are all obtained in the local coordinate system shown in Figure 3. Thus, the appropriate transformation should be used to convert them to global coordinates. As shown in Figure 5, ${\bar{Q}}_{f}_{(i + 1)}^{i}$ , ${\bar{N}}_{f}_{(i + 1)}^{i}$ , ${\bar{M}}_{f}_{(i + 1)}^{i}$ , ${\bar{Q}}_{f}_{(i)}^{i}$ , ${\bar{N}}_{f}_{(i)}^{i}$ and ${\bar{M}}_{f}_{(i)}^{i}$ of the end forces in local coordinate system $\bar{x} o \bar{y}$ are replaced with $F_{x (i)}^{i}$ , $F_{y (i)}^{i}$ , $M_{(i)}^{i}$ , $F_{x (i + 1)}^{i}$ , $F_{y (i + 1)}^{i}$ and $M_{(i + 1)}^{i}$ in global coordinate xoy. $α_{i}$ , the angle between the two coordinate systems, is measured from the x axis counter clockwise to $\bar{x}$ axis. Then, the shear force, axial force and bending moments at each end of the ith member can be resolved from global to local co-ordinates, see Appendix A.3 for the equations.

Figure 5.

Internal forces at the ends of the ith member shown in local and global coordinates.

Writing equations (70)–(75) in matrix form, we obtain

{\bar{F}}^{i} = T T_{i} F

(55)

where

T T_{i} = [\begin{array}{c} - \sin α_{i} & \cos α_{i} & 0 & 0 & 0 & 0 \\ \cos α_{i} & \sin α_{i} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \sin α_{i} & \cos α_{i} & 0 \\ 0 & 0 & 0 & \cos α_{i} & \sin α_{i} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]

(56)

F = {[\begin{array}{c} F_{x (i)}^{i} & F_{y (i)}^{i} & M_{(i)}^{i} & F_{x (i + 1)}^{i} & F_{y (i + 1)}^{i} & M_{(i + 1)}^{i} \end{array}]}^{T}

(57)

Then dynamic stiffness matrix K _i of the ith member in global coordinate is

K^{i} = T T_{i}^{T} {\bar{K}}^{i} T T_{i}

(58)

Similarly, the displacement at both ends of the local coordinate system and the global coordinate system can also be transformed by the transformation matrix TT_i in equation (56). Since each member of the rigid frame bridge can be regarded as an element in FEM, the generalized dynamic stiffness matrix of each member can be assembled according to the numerical assembly method similar to the FEM, so as to obtain the characteristic equation of the entire rigid frame bridge

K (ω) U = 0

(59)

where

K (ω)

is coefficient matrix with boundary conditions and U is nodal vibration amplitude.

Then $ω$ can be evaluated by solving the nonlinear equation

\det (K (ω)) = 0

(60)

A reliable method to solve the natural frequencies in equation (58) is the well-known Wittrick and Williams algorithm (Wittrick and Williams 1971). It has been widely used in numerous papers because the use of this algorithm is very simple unlike the complexity of its proof (Wittrick and Williams, 1983; Williams, 1993). If the frequency is obtained, the displacement of each node can be obtained by equation (59), and then the corresponding vibration mode can be gained by equations (30) and (31), (33) and (36). It should be noted that all the algorithms proposed in this paper are implemented by MATLAB software.

Engineering verification and numerical examples

Several examples are presented to illustrate the accuracy and generality of the proposed method in this section. The physical properties of the material used in the examples are as follows: Young’s modulus $E_{0}$ at 0°C is $3.25 \times 10^{10} P a$ , mass density ρ = 2550 kg/m³, temperature coefficient of Young’s modulus $θ_{E} = - 4.5 \times 10^{- 3}$ and thermal coefficient of linear expansion $θ_{T} = 1 \times 10^{- 5}$ .

Engineering verification

A T-type rigid-frame bridge located in Changchun of China, namely Yong-an bridge shown in Figure 6(a), was tested for a series of dynamic characteristics in September 2010. The bridge is 23.42 m width with two 9m spans, as illustrated in Figure 6(b). The pier consists of six reinforced concrete columns with a diameter of 150 cm and two ends of bridge are supported by elastomeric bearings.

Figure 6.

The Yong-an Bridge, Changchun, China: (a) The Yong-an Bridge, (b) The elevation of the bridge (unit: m).

Three tests were conducted on the bridge on September 21, 2010. The first test was conducted at 0:00 a.m. midnight, the second test happened at 7:00 a.m. in the following morning, and the third test was recorded at 12:00 noon. During each test, two accelerometers were symmetrically placed according to the positions shown in Figure 6(b). The data from the two acceleration channels was collected by a DH5922 system with sampling frequency of 512 Hz under ambient vibration, and the signal stacking method and low-pass filter built in DH5922 system were used to remove noise. Since the effect was sufficient to meet the accuracy requirements of subsequent spectral analysis, no additional noise reduction analysis was performed. On this basis, the DHDAS-2013 software platform was selected for data processing, and the frequency spectrum was analyzed by fast Fourier transform to calculate and extract the modal frequency. Considering the symmetry of bridge shape and the symmetrical arrangement of sensors, the test results of two accelerometers are very close, so only one set of frequency results is given, as shown in Figure 7. The experimental and the calculated results are both listed in Table 1, in which the results in bold are the change rates of the morning and noon frequencies relative to the night frequencies. The air temperature both under the bridge (Tu) and over the bridge (To) during the test were also recorded and listed in Table 1, and the temperature distribution of the whole structure can be calculated by the widely used thermal analysis model (Liu et al., 2017; Zhang et al., 2021).

Figure 7.

Amplitude spectrums of 1# accelerometer at different times of day: (a) Spectrums graph, (b) Local enlarging graph, (c) Local enlarging graph.

Table 1.

Frequency and corresponding temperature at different times of the day

Time	Temperature (°C)		Measured frequencies (Hz)			Propose method (Hz)
Time	To	Tu	1st	2nd	3rd	1st	2nd	3rd
Night	14	16	8.49	26.37	28.34	8.08	24.62	26.20
Morning	19	18	8.43	26.19	28.14	8.01	24.41	25.97
			−0.7%	−0.6%	−0.9%	−0.8%	−0.8%	−0.8%
Noon	28	19	8.32	25.76	27.72	7.92	24.11	25.65
			−2.0%	−2.3%	−2.2%	−2.1%	−2.1%	−2.1%

It is obvious from Figure 7 and Table 1 that temperature variation between morning and night is slight, but increased noticeably at noon, which leads to the test frequencies between night and morning closely and the noon ones changed significantly. What’s more, the maximum temperature difference can even exceed 40°C due to short-term climate change in the seasonal frozen area where the authors located (Wang et al., 2015), so it needs to be noted that the effect of temperature on frequency could be much greater under such actual situation than that under the test conditions in this paper.

The differences between the tested and the calculated results of frequencies in Table 1 may be due to the following: (1) the lower end of the pier is completely restrained in theoretical calculation, while it is not the case in actual structure; (2) the surface temperature of the structure is considered to be the same as the air temperature in the theoretical calculation, while there are certain differences between the of the two in actual structure. The point is that the measured frequencies variation at different times are close to the theoretical ones, which proves the correctness of the proposed method to some extent.

Numerical example of single span rigid-frame bridge with uniform section

In this example, the modal characteristics of the plane rigid-frame bridge with single span and uniform section under the effect of temperature are calculated by the proposed method and FEM respectively in order to verify the accuracy of the proposed method.

The 2D finite element model are established by plane stress element PLANE 182 in ANSYS, and the dimensional parameters of rigid structure are presented in Figure 8(a). Two temperature variation modes are defined: Mode A. temperature of the structure rises by 30°C; Mode B. the inner surface temperature rises by T₁ = 20°C while exterior surface temperature rises by T₂ = 30°C, and the temperature changes linearly from the inner surface to the exterior surface (see Figure 8(b)). It should be pointed out that the linear temperature gradient is simplified in the numerical example, but the method is also suitable for complex nonlinear temperature gradients.

Figure 8.

The 2D finite element model of rigid-frame structural: (a) Dimension parameters of finite element model, (b) Linear variation of internal temperature.

The analysis steps of the FEM are as follows: Firstly, the temperature is applied to each node for thermal analysis, and the temperature distribution of the structure is calculated. Then the temperature at the center of the element is extracted and substituted into equation (1) to calculate the Young’s modulus of each element; Secondly, the structural static analysis is carried out to calculate secondary internal force caused by the temperature of each element; Finally, the secondary internal force is regarded as the prestressing force of the structure, and the modal analysis is carried out to calculate the natural frequency and mode shape of the plane rigid-frame under the effect of temperature.

The first three natural frequencies of the rigid-frame bridge under different temperature modes are shown in Table 2, and the corresponding first three order mode shapes under the effect of temperature mode A are calculated, as shown in Figure 9.

Table 2.

Frequencies of the rigid-frame.

Temperature mode	Mode	Natural frequencies (Hz)		Difference (%)
Temperature mode	Mode	FEM	Present theory	Difference (%)
Reference temperature	1st	17.62	17.69	0.43
	2nd	46.86	47.15	0.61
	3rd	94.00	94.80	0.85
Mode A	1st	16.02	16.09	0.47
	2nd	43.07	43.36	0.68
	3rd	86.83	87.63	0.92
Mode B	1st	16.28	16.36	0.49
	2nd	43.71	44.01	0.69
	3rd	88.02	88.86	0.95

Figure 9.

The first three mode shapes under the effect of temperature mode (a) (a) The 1st mode shapes, (b) The 2nd mode shapes, (c) The 3rd mode shapes.

It can be seen from Table 2 and Figure 9 that the first three frequencies calculated by the present theory are close to the results calculated by the FEM. Although the error increases with the order, the maximum error does not exceed 1%, which is within an acceptable range in actual engineering. Furthermore, under the effect of temperature mode A, the first three mode shapes calculated by the present theory and the FEM are in good agreement, which verifies the correctness of the proposed method.

The temperature variation will not only lead to the change of Young’s modulus of material, but also cause the secondary internal forces of structure, which jointly affect the modal characteristics of structure. In order to investigate the influence law of these two parts on the structural modal characteristics thoroughly, the following three cases are set: Case 1, both the Young’s modulus and the secondary internal force are considered; Case 2, only the Young’s modulus is considered; Case 3, only the secondary internal force is considered. The first four natural frequencies of the rigid-frame bridge shown in Figure 8 are calculated under the above three cases, whose results are listed in Table 3. Variation ratios shown in bold in Table 3 represent the rates of change between the frequencies of the structure under each case and that when the temperature is 0°C. Annotating: when T = 0°C, there is no variation of Young’s modulus, and the structure has no secondary internal forces.

Table 3.

Natural frequencies of the rigid-frame frequencies (Hz).

Temperature (°C)	Mode	Case 1	Variation ratio (%)	Case 2	Variation ratio (%)	Case 3	Variation ratio (%)
T = 0	1st	17.69	/	/	/	/	/
	2nd	47.15	/	/	/	/	/
	3rd	94.80	/	/	/	/	/
	4th	116.38	/	/	/	/	/
T = −40	1st	19.77	11.74	19.22	8.63	18.20	2.87
	2nd	51.96	10.20	51.21	8.63	47.83	1.45
	3rd	103.80	9.50	102.98	8.62	95.56	0.80
	4th	126.61	8.78	126.42	8.63	116.55	0.14
T = −20	1st	18.74	5.91	18.47	4.40	17.95	1.44
	2nd	49.58	5.16	49.22	4.40	47.49	0.73
	3rd	99.37	4.82	98.97	4.40	95.18	0.40
	4th	121.59	4.48	121.51	4.41	116.47	0.07
T = 20	1st	16.63	−6.00	16.88	−4.61	17.43	−1.47
	2nd	44.64	−5.31	44.97	−4.61	46.80	−0.73
	3rd	90.06	−5.00	90.43	−4.61	94.42	−0.40
	4th	110.94	−4.68	111.02	−4.61	116.30	−0.07
T = 40	1st	15.55	−12.13	16.02	−9.45	17.17	−2.96
	2nd	42.06	−10.78	42.69	−9.45	46.45	−1.48
	3rd	85.15	−10.18	85.84	−9.45	94.03	−0.82
	4th	105.24	−9.58	105.39	−9.45	116.21	−0.15

The following conclusions can be drawn from Table 3: (1) The temperature and frequency are negatively correlated; (2) Variation ratios of each order of frequency at the same temperature are basically the same in Case 2; (3) The effect of temperature on each order of frequency is different in Case 3, which has the largest effect on the first-order frequency of the structure and decreases gradually with the increase of the order; (4) The variation of natural frequency caused by the change of Young’s modulus due to temperature is larger than that caused by secondary internal force due to temperature. However, the effect of the secondary internal forces caused by temperature can not be ignored for the first few order natural frequencies.

Numerical example of three-span rigid-frame bridge with non-uniform section

In this numerical example, a three-span rigid-frame bridge with non-uniform section as exhibited in Figure 10 is selected to calculate its model characteristics under the effect of temperature. The superstructure and piers of the rigid-frame bridge are both solid rectangular sections with the width of 5 m, the height of the side span varies linearly and that of middle span varies in quadratic parabola.

Figure 10.

Three-span continuous rigid-frame bridge with non-uniform section.

The following three temperature modes are defined in this example: Mode A. temperature of the overall structure rises by 30°C; Mode B. temperature of the overall structure drops by 30°C; Mode C. the inner surface temperature of the second span rises by 20°C, the other surfaces rises by 30°C and the temperature changes linearly from the inner surface to the exterior surface.

The calculated results of frequencies under three temperature modes as well as the reference temperature are listed in Table 4. The first order mode shapes under different temperature modes are also illustrated in Figure 11.

Table 4.

Frequencies of a rigid-frame bridge under different temperature modes.

Temperature mode	Frequency (Hz)
Temperature mode	1st	2nd	3rd
Reference temperature	7.93	8.22	17.32
Mode A	7.36 (−7%)	7.64 (−7%)	16.10 (−7%)
Mode B	8.45 (7%)	8.76 (7%)	18.45 (7%)
Mode C	7.47 (−6%)	7.75 (−6%)	16.32 (−6%)

Figure 11.

Comparison of first order mode shapes of rigid-frame bridge under different temperature modes.

As can be seen from Table 4, the frequencies decrease by 7% when the temperature overall structure rises by 30°C, while the frequencies increase by about 7% when the temperature overall structure is reduced by 30°C. The main reason for this phenomenon is that the secondary internal forces of the structure caused by temperature rise are mainly axial pressures, which will reduce the frequency of the structure. And the Young’s modulus of the material will decrease with the increase of temperature at the same time, as a consequence, the increase of structure temperature will inevitably lead to the decrease of frequency, and vice versa. In addition, the frequency of the structure changes by about 6% when the cross section has a temperature gradient (i.e., Mode C), indicating that the effect of temperature gradient on the natural frequency cannot be ignored. Furthermore, it’s apparent from Figure 11 that different temperature modes also have a certain effect on the mode shape of the continuous rigid-frame bridge.

Conclusions

This paper presents a semi-analytical theoretical analysis method for modal characteristics of non-uniform rigid-frame bridges under the effect of temperature, which can easily analyze the inherent mechanism of influences of temperature on the modal characteristics of rigid-frame bridges. The effectiveness of the method is verified by engineering examples and FEM, and the effect of different temperature modes on the modal characteristics is discussed by numerical examples. The following conclusions are drawn based on the results of this study:

(1) The main innovation of this paper is that every span and pier of a multi-span bridge are treated as a member, and a member is divided into several segments, only one member need to considered to obtain the dynamic characteristic equation of the whole bridge, which leads to high computational efficiency and accuracy.

(2) The theoretical results of modal characteristics are compared with the test results and the FEM results, respectively, which both proves the correctness and reliability of the present method.

(3) The change of Young’s modulus and secondary internal force induced by temperature will both affect the modal characteristics of rigid-frame bridges, in which the change of the frequency caused by Young’s modulus is greater than that of secondary internal force. However, the frequency variation caused by the secondary internal force cannot be neglected for the low-order frequencies.

(4) The effect of different temperature modes on the modal characteristics is different. As temperature increases, the Young’s modulus of the material will decrease and the secondary internal force caused by temperature is mainly axial pressure, which together leads to the frequency of the structure will decrease when the temperature increases.

At present, there are a large number of rigid-frame bridges in China, such as Guangxi Province, and damage identification methods based on bridge vibration characteristics are more and more widely used in the field of bridge health monitoring and condition assessment. However, the obtained modal characteristics often include the influence of temperature, structural damage and other factors and the changes in modal characteristics caused by environmental temperature sometimes even exceed the changes in structural modal characteristics caused by structural damage. The method in this paper can provide technical and theoretical support for data analysis in bridge health monitoring and eliminating the effect of environmental factors, which is of great significance to practical engineering. Based on this, the author will continue to conduct subsequent research and practical application on the modal characteristics of different forms of rigid-frame bridges, so as to further verify and improve the reliability and universality of the method.

Footnotes

Acknowledgement

The research was supported by the National Key R&D Program of China [grant numbers 2021YFB2600604, 2021YFB2600600], the National Natural Science Foundation of China (No.51978309), Key Project of Department of Transportation of Heilongjiang Province (2022-1); Scientific and Technological Planning Project of Jilin Province (No.20190303052SF), and Transportation innovation and Development Supporting Project of Jilin Province (No.2020-1-3, No.2020-3-2).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key R&D Program of China [grant numbers 2021YFB2600604, 2021YFB2600600], the National Natural Science Foundation of China (No. 51978309), Key Project of Department of Transportation of Heilongjiang Province (2022-1); Scientific and Technological Planning Project of Jilin Province (No. 20190303052SF), and Transportation innovation and Development Supporting Project of Jilin Province (No. 2020-1-3, 2020-3-2).

ORCID iD

Xin He

Appendix

A.1 The equations of end force balance condition of the ith member. (A1)

E_{0} I_{T} (\bar{x}) {\bar{y}}^{″} |_{l} - {\bar{M}}_{(i + 1)}^{i} - M_{T} (l) - M_{E} (l) = 0

(A2)

E_{0} I_{T} (\bar{x}) {\bar{y}}^{″} |_{0} - {\bar{M}}_{(i)}^{i} - M_{T} (0) - M_{E} (0) = 0

(A3)

{[E_{0} I_{T} (\bar{x}) {\bar{y}}^{″}]}^{'} |_{l} + {\bar{Q}}_{(i + 1)}^{i} - M_{T}^{'} (l) - N_{T} (l) {\bar{y}}^{'} |_{l} - M_{E}^{'} (l) = 0

(A4)

{[E_{0} I_{T} (\bar{x}) {\bar{y}}^{″}]}^{'} |_{0} - {\bar{Q}}_{(i)}^{i} - M_{T}^{'} (0) - N_{T} (0) {\bar{y}}^{'} |_{0} - M_{E}^{'} (0) = 0

(A5)

{\bar{Q}}_{f}_{(i + 1)}^{i} = - F_{x (i + 1)}^{i} \sin α_{i} + F_{y (i + 1)}^{i} \cos α_{i}

(A6)

E_{0} A_{T} (\bar{x}) {\bar{u}}^{'} |_{0} + {\bar{N}}_{(i)}^{i} - N_{T} (0) - N_{E} (0) = 0

where equations (A1) and (A2) represent moment equilibrium, equations (A3) and (A4) represent shear equilibrium and equations (A5) and (26) represent axial force equilibrium. It is important to note that

{\bar{M}}_{(i)}^{i}

{\bar{N}}_{(i)}^{i}

{\bar{Q}}_{(i)}^{i}

{\bar{M}}_{(i + 1)}^{i}

{\bar{N}}_{(i + 1)}^{i}

and

{\bar{Q}}_{(i + 1)}^{i}

contain the end force induced by temperature and free vibration, in which

{\bar{M}}_{f}_{(i)}^{i}

{\bar{N}}_{f}_{(i)}^{i}

{\bar{Q}}_{f}_{(i)}^{i}

{\bar{M}}_{f}_{(i + 1)}^{i}

{\bar{N}}_{f}_{(i + 1)}^{i}

and

{\bar{Q}}_{f}_{(i + 1)}^{i}

are force induced by free vibration.

A.2 The derivation of the relationship between force and displacement of both ends for the ith member

Using the recursive relation and equation (49), one can have the relationship of force and deformation between right end of segment s_h (i.e., node (i+1) in Figure 3) and left end of segment s₁ (i.e., node (i) in Figure 3) for the ith member as follows: (A7)

{[T]}_{k} = [\begin{array}{c} - {K_{k}^{L R}}^{- 1} K_{k}^{L L} & {K_{k}^{L R}}^{- 1} \\ K_{k}^{R L} - K_{k}^{R R} {K_{k}^{L R}}^{- 1} K_{k}^{L L} & K_{k}^{R R} {K_{k}^{L R}}^{- 1} \end{array}]

(A8)

[{\bar{H}}^{i}] = {[T]}_{h} {[T]}_{h - 1} \dots {[T]}_{2} {[T]}_{1}

(A9)

{{\bar{q}}_{(i + 1)}^{i}} = {[\begin{array}{c} {\bar{δ}}_{(i + 1)}^{i} & {\bar{F}}_{(i + 1)}^{i} \end{array}]}^{T}, {{\bar{q}}_{(i)}^{i}} = {[\begin{array}{c} {\bar{δ}}_{(i)}^{i} & {\bar{F}}_{(i)}^{i} \end{array}]}^{T}

where

{\bar{δ}}_{(i + 1)}^{i} = {[\begin{array}{c} {\bar{Y}}_{(i + 1)}^{i} & {\bar{U}}_{(i + 1)}^{i} & {\bar{Φ}}_{(i + 1)}^{i} \end{array}]}^{T}

{\bar{δ}}_{(i)}^{i} = {[\begin{array}{c} {\bar{Y}}_{(i)}^{i} & {\bar{U}}_{(i)}^{i} & {\bar{Φ}}_{(i)}^{i} \end{array}]}^{T}

{\bar{F}}_{(i)}^{i} = {[\begin{array}{c} {\bar{Q}}_{f}_{(i)}^{i} & {\bar{N}}_{f}_{(i)}^{i} & {\bar{M}}_{f}_{(i)}^{i} \end{array}]}^{T}

{\bar{F}}_{(i + 1)}^{i} = {[\begin{array}{c} {\bar{Q}}_{f}_{(i + 1)}^{i} & {\bar{N}}_{f}_{(i + 1)}^{i} & {\bar{M}}_{f}_{(i + 1)}^{i} \end{array}]}^{T}

To write equation (A7) in block matrix form gives (A10)

{\begin{array}{c} {\bar{δ}}_{(i + 1)}^{i} \\ {\bar{F}}_{(i + 1)}^{i} \end{array}} = [\begin{array}{c} {\bar{H}}_{11}^{i} & {\bar{H}}_{12}^{i} \\ {\bar{H}}_{21}^{i} & {\bar{H}}_{22}^{i} \end{array}] {\begin{array}{c} {\bar{δ}}_{(i)}^{i} \\ {\bar{F}}_{(i)}^{i} \end{array}}

(3) Relationship transformation of force and displacement at the end of member in two coordinate systems

(A11)

{\bar{Q}}_{f}_{(i)}^{i} = - F_{x (i)}^{i} \sin α_{i} + F_{y (i)}^{i} \cos α_{i}

(A12)

{\bar{N}}_{f}_{(i)}^{i} = F_{x (i)}^{i} \cos α_{i} + F_{y (i)}^{i} \sin α_{i}

(A13)

{\bar{M}}_{f}_{(i)}^{i} = M_{(i)}^{i}

(A14)

{\bar{Q}}_{f}_{(i + 1)}^{i} = - F_{x (i + 1)}^{i} \sin α_{i} + F_{y (i + 1)}^{i} \cos α_{i}

(A15)

{\bar{N}}_{f}_{(i + 1)}^{i} = F_{x (i + 1)}^{i} s \cos α_{i} + F_{y (i + 1)}^{i} \sin α_{i}

(A16)

{\bar{M}}_{f}_{(i + 1)}^{i} = M_{(i + 1)}^{i}

References

Askegaard

Mossing

(1988) Long term observation of RC-bridge using changes in natural frequency. Nordic Concrete Research 7: 20–27.

Balmes

Basseville

Mevel

, et al. (2007) Merging sensor data from multiple temperature scenarios for vibration monitoring of civil structures. Journal of Civil Structural Health Monitoring 7(2): 129–142.

Baldwin

North

(1973): Stress-strain relationship for concrete at high temperatures. Magazine of Concrete Research 25(85): 208–212.

Balmès

Basseville

Mevel

, et al. (2009) Handling the temperature effect in vibration monitoring of civil structures: a combined subspace-based and nuisance rejection approach. Control Engineering Practice 17(1): 80–87.

Cornwell

Farrar

Doebling

, et al. (2010) Environmental variability of modal properties. Experimental Techniques 23(6): 45–48.

Cruz

Salgado

(2010) Performance of vibration-based damage detection methods in bridges. Computer-Aided Civil and Infrastructure Engineering 24(1): 62–79.

Ding

(2011) Temperature-induced variations of measured modal frequencies of steel box girder for a long-span suspension bridge. International Journal of Steel Structures 11(2): 145–155.

Guo

Zhu

, et al. (2021) Monitoring-based evaluation of dynamic characteristics of a long span suspension bridge under typhoons. Journal of Civil Structural Health Monitoring 11(2): 397–410.

Hong

Liu

Lei

, et al. (2014) Elastic-plastic Seismic Response Analysis of the Long Span Combined Highway and Railway Continuous Rigid Frame Bridge, 2014 International Conference of Logistics Engineering and Management, Shanghai, China.

10.

Deng

Zhou

, et al. (2009) Study of closure scheme of multi-span continuous rigid frame bridge. Journal of China & Foreign Highway 29(3): 109–114.

11.

Hua

, et al. (2007) Modeling of temperature-frequency correlation using combined principal component analysis and support. Journal of Computing in Civil Engineering 21(2): 122–135.

12.

Jiao

Liu

Wang

, et al. (2014) Temperature effect on mechanical properties and damage identification of concrete structure. Advances in Materials Science and Engineering 2014: 1–10.

13.

Jin

Jang

Sun

, et al. (2016) Damage detection of a highway bridge under severe temperature changes using extended Kalman filter trained neural network. Journal of Civil Structural Health Monitoring 6(3): 1–16.

14.

Kim

Park

Lee

(2007) Vibration-based damage monitoring in model plate-girder bridges under uncertain temperature conditions. Engineering Structures 29(7): 1354–1365.

15.

Liu

Wang

Jiao

(2016) Effect of temperature variation on modal frequency of reinforced concrete slab and beam in cold regions. Shock and Vibration 2016(6): 1–17.

16.

Liu

Wang

Tan

, et al. (2017) Vibration analysis of reinforced concrete simply supported beam versus variation temperature. Shock and Vibration 2017(5): 1–20.

17.

Liu

Wang

Tan

, et al. (2018) Effect of temperature and spring-mass systems on modal properties of Timoshenko concrete beam, Structural Engineering & Mechanics 65(4): 389–400.

18.

Miao

Chen

Feng

(2011) Study on effects of environmental temperature on dynamic characteristics of Taizhou Yangtze River Bridge. Engineering Sciences 9(2): 78–82.

19.

Mosavi

Seracino

Rizkalla

(2012) Effect of temperature on daily modal variability of a steel-concrete composite bridge. Journal of Bridge Engineering 17(17): 979–983.

20.

Moughty

Casas

(2017) A state of the art review of modal-based damage detection in bridges: development, challenges, and solutions. Applied Sciences 7(5): 510.

21.

Hua

Fan

, et al. (2005a) Correlating modal properties with temperature using long-term monitoring data and support vector machine technique. Engineering Structures 27(12): 1762–1773.

22.

Fan

Zheng

, et al. (2005b) Automatic modal identification and variability in measured modal vectors of a cable-stayed bridge. Structural Engineering & Mechanics 19(2): 123–139.

23.

Peeters

Roeck

(2001) One-year monitoring of the Z24-Bridge: environmental effects versus damage events. Earthquake Engineering & Structural Dynamics 30(2): 149–171.

24.

Peeters

Maeck

Roeck

(2001) Vibration-based damage detection in civil engineering: excitation sources and temperature effects. Smart Materials and Structures 10(3): 518–527.

25.

Song

Melhem

Cheng

, et al. (2016) Optimization of closure jacking forces in multispan concrete rigid-frame bridges. Journal of Bridge Engineering 22(3): 04016122.

26.

Sun

Zhou

Min

(2018) Experimental study on the effect of temperature on modal frequencies of bridges. International Journal of Structural Stability and Dynamics 18(12): 1850155.

27.

Talebinejad

Fischer

Ansari

(2011) Numerical evaluation of vibration-based methods for damage assessment of cable-stayed bridges. Computer-Aided Civil and Infrastructure Engineering 26(3): 239–251.

28.

Tan

Chen

Wang

, et al. (2021) The impact of uneven temperature distribution on stability of concrete structures using data analysis and numerical approach. Advances in Structural Engineering 24(2): 279–290.

29.

Tian

Zhang

Xia

(2017) Temperature effect on service performance of high-speed railway concrete bridges. Advances in Structural Engineering 20(6): 865–883.

30.

Wang

Gao

(2021) Modal Frequency Analysis of Suspension Bridge under the Action of Temperature. 2021 World Transportation Engineering Technology Forum. Xi’an: World Transport Convention.

31.

Wang

Tai

Liu

, et al. (2015) Study on the sunny-shady slope effect on the subgrade of a high-speed railway in a seasonal frozen region. Sciences in Cold and Arid Regions 7(5): 513–519.

32.

Wittrick

Williams

(1971) A general algorithm for computing natural frequencies of elastic structures. Quarterly Journal of Mechanics & Applied Mathematics 24(3): 263–284.

33.

Williams

Wittrick

(1983) Exact buckling and frequency calculations surveyed. Journal of Structural Engineering 109(1): 169–187.

34.

Williams

(1993) Review of exact buckling and frequency calculations with optional multi-level substructuring. Computers & Structures 48(3): 547–552.

35.

Xia

Wei

, et al. (2011) Variation of structural vibration characteristics versus non-uniform temperature distribution. Engineering Structures 33(1): 146–153.

36.

Xia

Chen

Weng

, et al. (2012) Temperature effect on vibration properties of civil structures: a literature review and case studies. Journal of Civil Structural Health Monitoring 2(1): 29–46.

37.

Yan

Kerschen

, et al. (2005a) Structural damage diagnosis under varying environmental conditions -- part I: a linear analysis. Mechanical Systems and Signal Processing 19(4): 847–864.

38.

Yan

Kerschen

, et al. (2005b) Structural damage diagnosis under varying environmental conditions -- part II: local PCA for non-linear cases. Mechanical Systems and Signal Processing 19(4): 865–880.

39.

Zhang

Liu

Deng

(2021) Temperature gradient modeling of a steel box-girder suspension bridge using Copulas probabilistic method and field monitoring. Advances in Structural Engineering 24(5): 947–961.

40.

Zhou

(2014) A summary review of correlations between temperatures and vibration properties of long-span bridges. Mathematical Problems in Engineering 2014(1): 1–19.

41.

Zhou

Xia

Chen

, et al. (2020) Analytical solution to temperature-induced deformation of suspension bridges. Mechanical Systems and Signal Processing 139.