A weak solution of free vibration is developed for multi-span beams, which can adapt general elastic boundary and coupling conditions. Firstly, create the energy functional of the multi-span beam system based on the small deformation theory. Then, adopt the modified Fourier series method to rewrite the displacement functions. Compared with the traditional Fourier series method, the present series representations provide a solution for general elastic restrains. Lastly, combined with the Rayleigh-Ritz technique, all the series expansion coefficients can be obtained as the generalized coordinates. Numerical results demonstrate that the current weak solution has good convergence and high accuracy compared with the existing results in literature and FEM results

Dongyan Shi

Kwang Nam Choe

Qingshan Wang

Yue Tian

English

Vibroengineering PROCEDIA

 298 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
A weak solution for free vibration of multi-span beams 
with general elastic boundary and coupling conditions 
Dongyan Shi1, Yue Tian2, Kwang Nam Choe3, Qingshan Wang4 
1, 2Harbin Engineering University, Harbin, P. R. China 
3Pyongyang University of Mechanical Engineering, Pyongyang, North Korea 
4College of Mechanical and Electrical Engineering, Central South University, Changsha, P. R. China 
4Corresponding author 
E-mail: 1shidongyan@hrbeu.edu.cn, 2tianyue@hrbeu.edu.cn, 3choekn@hrbeu.edu.cn, 
4wangqingshanxlz@hotmail.com 
(Received 3 November 2016; accepted 7 November 2016) 
Abstract. A weak solution of free vibration is developed for multi-span beams, which can adapt 
general elastic boundary and coupling conditions. Firstly, create the energy functional of the 
multi-span beam system based on the small deformation theory. Then, adopt the modified Fourier 
series method to rewrite the displacement functions. Compared with the traditional Fourier series 
method, the present series representations provide a solution for general elastic restrains. Lastly, 
combined with the Rayleigh-Ritz technique, all the series expansion coefficients can be obtained 
as the generalized coordinates. Numerical results demonstrate that the current weak solution has 
good convergence and high accuracy compared with the existing results in literature and FEM 
results.  
Keywords: weak solution, free vibration, multi-span beams, general boundary conditions, elastic 
coupling conditions. 
1. Introduction 
Beams have been applied in many filed such as aerospace, marine, construction industry due 
to their excellent engineering features. And in some specific field, the multi-span beam structures 
are also used widely, such as bridge engineering, water conservancy projects, etc. As a result, over 
the years, the vibration of the beam and multi span beam structure characteristics has always been 
the hot spot of the scholars’ concerned problem. 
As so far, many scholars have proposed a number of computational techniques for multi-span 
beams. Zheng et al. [1] apply the assumed modes method to study the vibration of a multi-span 
non-uniform beam subjected to a moving load. Wang [2] presents a method of modal analysis to 
investigate the forced vibration of multi-span Timoshenko beams with clamped boundary 
condition and rigid joint. The vibration of a multi-span non-uniform bridge subjected to a moving 
vehicle is analyzed by Cheung et al. [3] using the assumed modes method. Based on the modal 
analysis method and the direct integration method, Dugush et al. [4] and Ichikawa [5] study the 
dynamic behavior of multi-span non-uniform beams. Lin and Chang [6] deal with the free 
vibration analysis of a multi-span beam with an arbitrary number of flexible constraints by using 
the transfer matrix method. Li and Xu [7] present an exact Fourier Series method to study the 
vibration analysis of multi-span beam systems. Lin and Tsai [8] use the finite element method to 
analyze free vibration analysis of a uniform multi-span beam carrying multiple spring-mass 
systems. Hong and Kim [9] present an exact modelling and modal analysis method for 
non-uniform, multi-span beam-type structure supported and/or connected by resilient joints with 
damping. Marchesiello [10] presents an analytical approach to study the dynamics of multi-span 
continuous straight bridges subject to multi-degrees of freedom moving vehicle excitation. 
Johansson et al. [11] present a closed-form solution for evaluating the dynamical behavior of a 
general multi-span Bernoulli–Euler beam by using mode superposition method. Lin [12] employs 
the numerical assembly method (NAM) to determine dynamic behavior of a multi-span uniform 
beam carrying a number of various concentrated elements.  
In this paper, the authors present a weak solution for free vibration of multi-span beams 
A WEAK SOLUTION FOR FREE VIBRATION OF MULTI-SPAN BEAMS WITH GENERAL ELASTIC BOUNDARY AND COUPLING CONDITIONS.  
DONGYAN SHI, YUE TIAN, KWANG NAM CHOE, QINGSHAN WANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 299 
subjected to general elastic boundary and coupling conditions. The general elastic boundary and 
coupling constraints of the multi-span beams are realized by applying the artificial stiffness-like 
spring technique. Based on the modified Fourier series method, each of displacements is written 
as a form of a standard one-dimensional Fourier cosine series with several auxiliary functions. 
The series expansion coefficients are obtained by using the Rayleigh-Ritz technique. The accuracy 
and convergence of the current weak solution is checked by comparing with the existing results 
in literature and FEM results. 
2. Theoretical formulations 
A multi-span beam system is shown as Figure.1. The system includes multiple beams which 
are coupled with joints. Each joint consists of two group springs which are linear and rotational 
springs respectively. The effects of non-rigid or resilient connectors are allowed in using the 
coupling springs between two adjacent beams. When the stiffness of these springs become 
considerably larger than the bending rigidities of the involved beams, it turns to be a conventional 
rigid connector. Each of beams are supported by a set of elastic restraints at both ends. By adjusting 
the spring stiffness values from zero to infinite, different boundary conditions including traditional 
intermediate supports and classical boundary conditions (i.e., the combinations of the simply 
supported (S), free (F), and clamped end conditions (C)) can be obtained. 
 
Fig. 1. Multi-span beam system with general boundary conditions 
The differential equation for the vibration of the ݅th beam can expressed as: 
ܦ௜݀ସݓ௜ሺݔሻ/݀ݔସ − ߩ௜ܣ௜߱ଶݓ௜ሺݔሻ = 0, (1)
where ݓ is the angular frequency of the beam, ݓ௜, ܦ௜, ܣ௜ and ߩ௜ are the displacement function, 
bending rigid, the cross-sectional area and the density of the beams, respectively.  
From the previous reviews, in this study, the artificial stiffness-like spring technique is adopted 
to simulate the arbitrary boundary conditions and continuity conditions. With this method, the 
boundary and continuity conditions can be expressed as follows:  
At ݔ௜ = 0: 
݇௜,௜ିଵሺݓ௜ − ݓ௜ିଵሻ + ෨݇௜଴ݓ௜ = −ܦ௜ݓ௜ᇱᇱᇱ, (2)
ܭ௜,௜ିଵሺݓ௜ᇱ − ݓ௜ିଵᇱ ሻ + ܭ෩௜଴ݓ௜ᇱ = ܦ௜ݓ௜ᇱᇱ. (3)
At ݔ௜ = ܮ௜: 
݇௜,௜ାଵሺݓ௜ − ݓ௜ାଵሻ + ෨݇௜ଵݓ௜ = ܦ௜ݓ௜ᇱᇱᇱ, (4)
ܭ௜,௜ାଵሺݓ௜ᇱ − ݓ௜ାଵᇱ ሻ + ܭ෩௜ଵݓ௜ᇱ = −ܦ௜ݓ௜ᇱᇱ. (5)
At the left end (of the first beam): 
෨݇ଵ଴ݓଵ = −ܦଵݓଵᇱᇱᇱ, (6)
10K
10k 1nk
1nK
11K 11k 0ik0iK 1iK 1ik 0nk0nK
1,2K 1,2k , 1i i
K  , 1i ik 
, 1i iK  , 1i ik 
, 1n nK  , 1n nk 
1 1 1,EI A ,i i iEI A ,n n nEI A
x ix
A WEAK SOLUTION FOR FREE VIBRATION OF MULTI-SPAN BEAMS WITH GENERAL ELASTIC BOUNDARY AND COUPLING CONDITIONS.  
DONGYAN SHI, YUE TIAN, KWANG NAM CHOE, QINGSHAN WANG 
300 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
ܭ෩ଵ଴ݓଵᇱ = ܦଵݓଵᇱᇱ. (7)
At the right end (of the ܰth beam): 
෨݇௡ଵݓ௡ = ܦ௡ݓ௡ᇱᇱᇱ, (8)
ܭ෩௡ଵݓ௡ᇱ = −ܦ௡ݓ௡ᇱᇱ, (9)
where refer to Fig. 1, ݇௜,௝ denote the stiffnesses of the linear coupling springs in ݖ௜ –directions, 
and ܭ௜,௝ denote the stiffnesses of the rotational coupling springs at the junction of beams ݅ and ݆, 
respectively; ෨݇௜଴ and ෨݇௜ଵ are the stiffnesses of linear boundary springs, and ܭ෩௜଴, ܭ෩௜ଵ the stiffnesses 
of the rotational boundary springs at the left and right ends of beam ݅, respectively.  
For the multi-span beams, the total strain energy ( ܸ ) and kinetic energy ( ܶ ) can be  
expressed as: 
ܸ = ෍ ௕ܸ,௜
ே
௜ୀଵ
+ ෍ ௜ܸ,௜ାଵ௦
ேିଵ
௜ୀଵ
, (10)
ܶ = ෍ ௕ܶ,௜
ே
௜ୀଵ
, (11)
where ௕ܸ,௜ and ௕ܶ,௜ are represent the strain energy and kinetic energy of the ݅th beams, and ௜ܸ,௜ାଵ௦  
is the potential energy expression in the connective springs related to ݅th and ݅ + 1th beams. The 
detailed expression of the ௕ܸ,௜, ௜ܸ,௜ାଵ௦  and ௕ܶ,௜ can be written as: 
௕ܸ,௜ =
1
2 න ቊන ܦ௜ሺ߲
ଶݓ௜ሺݔሻ ߲ݔଶ⁄ ሻ
௅೔
଴
݀ݔቋ
௅೔
଴
݀ߠ௜
       + 12 ቀ൫ ෨݇௜଴ݓ௜ሺݔሻ
ଶ + ܭ෩௜଴ሺ߲ݓ௜ሺݔሻ ߲ݔ⁄ ሻଶ൯௫೔ୀ଴ + ൫ ෨݇௜ଵݓ௜ሺݔሻ
ଶ + ܭ෩௜ଵሺ߲ݓ௜ሺݔሻ ߲ݔ⁄ ሻଶ൯௫೔ୀ௅೔ቁ, 
(12)
௜ܸ,௜ାଵ௦ =
1
2
ە
ۖۖ
۔
ۖۖ
ۓ ݇௜,௜ିଵ൫ݓ௜ሺݔሻ|௫೔ୀ଴ − ݓ௜ିଵሺݔሻ|௫೔షభୀ௅೔షభ൯
ଶ
+ܭ௜,௜ିଵ ቀ߲ݓ௜ሺݔሻ ߲ݔ⁄ |௫೔ୀ଴ − ߲ݓ௜ିଵሺݔሻ ߲ݔ⁄ |௫೔షభୀ௅೔షభቁ
ଶ
݇௜,௜ାଵ൫ݓ௜ሺݔሻ|௫೔ୀ௅೔ − ݓ௜ାଵሺݔሻ|௫೔శభୀ଴൯
ଶ
+ܭ௜,௜ାଵ ቀ߲ݓ௜ሺݔሻ ߲ݔ⁄ |௫೔ୀ௅೔ − ߲ݓ௜ାଵሺݔሻ ߲ݔ⁄ |௫೔శభୀ଴ቁ
ଶ
ۙ
ۖۖ
ۘ
ۖۖ
ۗ
, (13)
௕ܶ,௜ =
1
2 ߩ௜ܣ௜߱
ଶ න ሼݓ௜ሺݔሻሽ
థ೔
଴
ܴ௜݀ߠ௜. (14)
The Lagrangian for the multi-span beams can be generally expressed as: 
ܮ = ܶ − ܸ. (15)
The traditional Fourier series is a well-known form of admissible function for its excellent 
convergence. However, it is only available to some very simple boundary conditions and would a 
modified Fourier series technique proposed by Li [13, 14] is widely used in the vibrations of plates 
and shells with different boundary conditions by Rayleigh-Ritz method. Thus, in this formulation, 
the modified Fourier series technique is adopted and extended to investigate the free vibrations of 
multi-span beams with general elastic boundary and coupling conditions: 
A WEAK SOLUTION FOR FREE VIBRATION OF MULTI-SPAN BEAMS WITH GENERAL ELASTIC BOUNDARY AND COUPLING CONDITIONS.  
DONGYAN SHI, YUE TIAN, KWANG NAM CHOE, QINGSHAN WANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 301 
ݓ௜ሺݔ௜ሻ = ෍ ܣ௜,௠cosߣ௜,௠ݔ
ஶ
௠ୀ଴
+ ෍ ܤ௜,௡sinߣ௜,௡
ସ
௡ୀଵ
ݔ,   ሺߣ௠ = ݉ߨ/ܮ௜ሻ, (16)
where ߣ௠ = ݉ߨ ܮ௜⁄ , ܣ௜,௠  and ܤ௜,௠  are the unknown Fourier coefficients of one-dimensional 
Fourier series expansions for the displacements functions, respectively.  
Then, substituting Eq. (16) into the Lagrangian function Eq. (15) and taking its derivatives 
with respect to each of the undetermined coefficients and making them equal to zero: 
߲ܮ
߲ߌ = 0,     ൝
ߌ = ܣ௜,௠, ܤ௜,௠,
݅ = 1,2, ⋯ , ܰ,
݉ = 1,2, ⋯ , ܯ.
(17)
The problem will be transformed into an eigenvalue and eigenvector problem, and the 
following governing eigenvalue equation in the matrix form can be achieved: 
ሺ۹ − ߱ଶۻሻ۵ = 0, (18)
where ۹ is the stiffness matrix for the beam, and ۻ is the mass matrix.  
For conciseness, the detailed expression for sub-stiffness and sub-mass matrices will not be 
shown here. By solving the Eq. (18), the frequencies (or eigenvalues) of multi-span beams can be 
readily obtained and the mode shapes can be yielded by substituting the corresponding 
eigenvectors into series representations of displacement components.  
3. Case studies 
Free vibration of three-span beams with elastic boundary and coupling conditions are 
researched through the present weak solutions. The related geometrical and material parameters 
of the following case are given in Table 1.  
Table 1. A list of beam parameters and material properties 
Variables Beam 1 Beam 2 Beam 3 
ܮ (m) 1.0 1.5 2.0 
ܣ (m2) 5×10-5 1.5×10-5 5×10-5 
ܫ (m4) 10-10 5×10-11 10-10 
ܧ (GPa) 207 207 207 
ߩ (kg/m3) 7800 7800 7800 
The schematic diagram, geometrical and material parameters of three-span beam are shown in 
Fig. 2 and Table 1, respectively. Table 2 gives the corresponding stiffness of the boundary and 
coupling springs with refer to the Fig. 2. Also, the Table 3 gives the convergence and validation 
study. In the FEM mode, the element type and mesh size are the beam 188 and 0.1 m, respectively. 
The mode shapes for the first four modes are plotted in Fig. 3. From the Table 3 and Fig. 3, it is 
obvious that the present weak solution is not only available to solve the multi-span beams with 
elastic boundary and coupling conditions, but also has a good accuracy and reliability.  
Table 2. Stiffness values for the boundary and coupling springs of a three-beam system 
Spring constants  
for beam 1 
Spring constants 
for beam 2 
Spring constants 
for beam 3 
Spring constants 
for joints 
෨݇ଵ଴ = 1010 N/m ෨݇ଵଵ = 5000 N/m 
෨݇ଶ଴ = 4000 N/m ෨݇ଶଵ = 4000 N/m 
෨݇ଷ଴ = 5000 N/m ෨݇ଷଵ = 1010 N/m 
݇ଵ,ଶ = 1000 N/m 
݇ଶ,ଷ = 1000 N/m 
ܭ෩ଵ଴ = 1010 Nm/rad ܭ෩ଶ଴ = 1000 Nm/rad ܭ෩ଷ଴ = 2000 Nm/rad ܭଵ,ଶ = 200 Nm/rad 
ܭ෩ଵ଴ = 2000 Nm/rad ܭ෩ଶଵ = 1000 Nm/rad ܭ෩ଷଵ = 0 Nm/rad ܭଶ,ଷ = 200 Nm/rad 
A WEAK SOLUTION FOR FREE VIBRATION OF MULTI-SPAN BEAMS WITH GENERAL ELASTIC BOUNDARY AND COUPLING CONDITIONS.  
DONGYAN SHI, YUE TIAN, KWANG NAM CHOE, QINGSHAN WANG 
302 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
 
Fig. 2. Schematic diagram of three-span beam 
Table 3. The six lowest natural frequencies for various numbers  
of terms in Fourier series under the three-span beam 
Mode Present Method Exact solution FEM ܯ = 6 ܯ = 8 ܯ = 10 ܯ = 12 ܯ = 15 
1 4.3676 4.3675 4.3675 4.3675 4.3675 4.3675 4.3675 
2 13.640 13.640 13.639 13.639 13.639 13.646 13.646 
3 13.833 13.833 13.833 13.833 13.833 13.848 13.848 
4 21.690 21.690 21.690 21.690 21.688 21.689 21.689 
5 26.698 26.696 26.696 26.696 26.695 26.697 26.697 
6 34.327 34.323 34.323 34.322 34.322 34.325 34.325 
 
 
Fig. 3. The four-mode shape for the three-span beam 
4. Conclusions 
In this paper, a weak solution of free vibration is developed for multi-span beams, which can 
be applied with general elastic boundary and coupling conditions. Each of displacements is written 
as a form of a standard one-dimensional Fourier cosine series with several auxiliary functions. 
The introducing of these functions is to eliminate the discontinuities of all the related 
displacements and their derivatives at the ends and to improve the convergence speed of the series 
representations. Combined with the Rayleigh-Ritz technique, all the series expansion coefficients 
can be obtained as the generalized coordinates. The numerical results show that the current weak 
solution has good convergence and high accuracy compared with existing results in literature and 
10K
10k 31k
31K
11K
11k
1,2K 1,2k
2,3K 2,3k
20K
20k
30K
30k21K
21k
1L 2L 3L
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1
-0.8
-0.6
-0.4
-0.2
0
x/L
 
 
ANSYS
Present method
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1
-0.5
0
0.5
1
x/L
 
 
ANSYS
Present method
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
x/L
 
 
ANSYS
Present method
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.2
0
0.2
0.4
0.6
0.8
1
x/L
 
 
ANSYS
Present method
A WEAK SOLUTION FOR FREE VIBRATION OF MULTI-SPAN BEAMS WITH GENERAL ELASTIC BOUNDARY AND COUPLING CONDITIONS.  
DONGYAN SHI, YUE TIAN, KWANG NAM CHOE, QINGSHAN WANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 303 
FEM results. 
References 
[1] Zheng D., et al. Vibration of multi-span non-uniform beams under moving loads by using modified 
beam vibration functions. Journal of Sound and Vibration, Vol. 3, Issue 212, 1998, p. 455-467. 
[2] Wang R.-T. Vibration of multi-span Timoshenko beams to a moving force. Journal of Sound and 
Vibration, Vol. 5, Issue 207, 1997, p. 731-742. 
[3] Cheung Y., et al. Vibration of multi-span non-uniform bridges under moving vehicles and trains by 
using modified beam vibration functions. Journal of Sound and Vibration, Vol. 3, Issue 228, 1999, 
p. 611-628. 
[4] Dugush Y., Eisenberger M. Vibrations of non-uniform continuous beams under moving loads. 
Journal of Sound and Vibration, Vol. 5, Issue 254, 2002, p. 911-926. 
[5] Ichikawa M., Miyakawa Y., Matsuda A. Vibration analysis of the continuous beam subjected to a 
moving mass. Journal of Sound and Vibration, Vol. 3, Issue 230, 2000, p. 493-506. 
[6] Lin H.-P., Chang S. Free vibration analysis of multi-span beams with intermediate flexible 
constraints. Journal of Sound and Vibration, Vol. 1, Issue 281, 2005, p. 155-169. 
[7] Li W. L., Xu H. An exact Fourier series method for the vibration analysis of multispan beam systems. 
Journal of Computational and Nonlinear Dynamics, Vol. 2, Issue 4, 2009, p. 021001. 
[8] Lin H.-Y., Tsai Y.-C. Free vibration analysis of a uniform multi-span beam carrying multiple 
spring-mass systems. Journal of Sound and Vibration, Vol. 3, Issue 302, 2007, p. 442-456. 
[9] Hong S.-W., Kim J.-W. Modal analysis of multi-span Timoshenko beams connected or supported by 
resilient joints with damping. Journal of Sound and Vibration, Vol. 4, Issue 227, 1999, p. 787-806. 
[10] Marchesiello S., et al. Dynamics of multi-span continuous straight bridges subject to multi-degrees 
of freedom moving vehicle excitation. Journal of Sound and Vibration, Vol. 3, Issue 224, 2999, 
p. 541-561. 
[11] Johansson C., Pacoste C., Karoumi R. Closed-form solution for the mode superposition analysis of 
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[12] Lin H.-Y. Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated 
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[13] Li W. L. Free vibrations of beams with general boundary conditions. Journal of Sound and Vibration, 
Vol. 4, Issue 237, 2000, p. 709-725. 
[14] Li W. L. Comparison of Fourier sine and cosine series expansions for beams with arbitrary boundary 
conditions. Journal of Sound and Vibration, Vol. 1, Issue 255, 2002, p. 185-194. 


A weak solution for free vibration of multi-span beams with general elastic boundary and coupling conditions

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A weak solution for free vibration of multi-span beams with general elastic boundary and coupling conditions

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