STRUCTURAL ANALYSIS USING SPECTRAL FINITE ELEMENT … · Finite Element Method (FEM) ... used in the analysis substantially increase which ends up with considerable computational

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11-13 Ekim 2017 – ANADOLU ÜNİVERSİTESİ – ESKİŞEHİR

STRUCTURAL ANALYSIS USING

SPECTRAL FINITE ELEMENT METHOD (SFEM)

N.M. Çağlar1

and E. Şafak2

1PhD candidate, Kandilli Observatory and Earthquake Research Institute, Boğaziçi University, Istanbul, Turkey

2Professor, Kandilli Observatory and Earthquake Research Institute, Boğaziçi University, Istanbul, Turkey

Email: [email protected]

ABSTRACT:

Spectral Finite Element Method (SFEM) is a methodology to analyze the dynamic response and to identify the

dynamic characteristics of engineering structures in frequency domain. SFEM is highly accurate in capturing the

high-frequency response, since it uses the exact wave solution of governing differential equations (dynamic shape

functions) to construct the stiffness matrices of the elements. High-frequency wave modes need to be captured for

an accurate dynamic response, and these modes are more sensitive to small changes in the dynamic characteristics

of the structure. Therefore, SFEM also allows development of more powerful tools for system identification and

damage detection. The conventional finite element method (FEM) is not capable of capturing high-frequency wave

modes unless the mesh size used in the modelling is smaller than the lowest wavelength. Complex structures can

be handled using SFEM. SFEM does not need to divide an element into smaller segments, and consequently

requires smaller number of degrees of freedom and lower computation cost than FEM. In this study, we outline

the basic theory for SFEM, and as an example, calculate the dynamic response of a plane frame structure subjected

to high-frequency excitation. We compare the responses obtained from the FEM and SFEM, and comment on their

accuracy at high frequencies.

KEY WORDS: Spectral Finite Element Method, Dynamic Stiffness Matrix

1. INTRODUCTION

Finite Element Method (FEM) is the most commonly used methodology to analyze the dynamic response and to

identify the dynamic characteristics of engineering structures. However, in medium to high frequency range, this

method loses its accuracy unless the mesh size used in the modelling is smaller than the lowest wavelength. This

problem arises from the fact that the formulation of FEM is based on frequency independent shape functions which

are obtained from the weak form of the governing differential equation and concentration of the inertial loads at

the member ends. In order to define inertial effects more properly and to capture short wavelengths, the element

size used in FEM has to be decreased. Therefore, number of the elements and number of degrees of freedom (DOF)

used in the analysis substantially increase which ends up with considerable computational costs (Gopalakrishnan,

1992).

Spectral Finite Element Method (SFEM)1 is a finite element method, which is based on the exact solution to the

governing differential equation of an element and is entirely in the frequency domain. By virtue of the validity of

matrix assembly procedure in SFEM, several spectral elements can be assembled. Thus, complex structures can

be handled using SFEM. Since the method uses exact solution to the governing differential equation, distribution

of the inertial forces is modeled accurately. Unlike the conventional FEM, high-frequency wave modes, which are

1 In literature, this method is also referred to Spectral Element Method (SEM). However, there is also another class of finite

element methods, which is based on the strong form of the governing differential equation in the time domain. In order to

distinguish between two different SEM approaches, the terminology SFEM is preferred.



the modes more sensitive to small changes in the dynamic characteristics of the structure, can be captured without

dividing elements into small sized segments. In this fashion, the number of elements and the number of DOFs are

decreased substantially (Gopalakrishnan, 2000). Furthermore, SFEM also allows development of more powerful

tools for system identification and damage detection. The frequency response function (FRF) of the structural

system is directly derived if the structural elements are formulated in the frequency domain.

SFEM can be considered as a combination of Dynamic Stiffness Method (DSM), Spectral Analysis Method (SAM)

and FEM (Lee, 2009). The dynamic stiffness matrix (spectral element matrix) of an element is derived in the same

manner as derived in DSM. The dynamic stiffness matrix of an element is constructed using frequency dependent

dynamic shape functions, which are derived from the exact solution of the governing differential equations. Since

the frequency dependent dynamic shape functions are derived directly from the governing differential equation of

the element, this method is referred as an exact method (Banerjee, 1997), and the accuracy of the method depends

on the accuracy of the adopted governing differential equation. In SFEM, a similar procedure to SAM is adopted

to evaluate the dynamic response of an element. Derivation of the dynamic response of an element is based on the

superposition of the wave modes at different frequencies, which are limited to the highest frequency of interest.

The procedure followed in SFEM and FEM is almost similar except some differences in the formulation of the

shape functions. In FEM, shape functions are derived from the displacement polynomials, which satisfy the weak

form of governing differential equations. These shape functions yield mass and stiffness matrices. However, in

SFEM, the dynamic stiffness matrix is derived from the exact solution of the governing differential equation,

which is the combination of inertia, stiffness and damping properties of an element, and is frequency dependent.

In this study, spectral element matrices for rod and beam elements are constructed based on elementary rod and

beam theories. The spectral element matrix for a plane frame is derived assuming the deflections of the elements

are small. In this manner, interaction between axial and flexural loading is neglected. Thus, spectral element

matrix of a plane frame member is formed by the superposition of the spectral element matrices of a beam and a

rod. The dynamic response of a two-story plane frame is obtained using the SFEM and FEM and the results are

compared with each other.

2. FORMULATION OF SPECTRAL ELEMENT FOR A ROD

The elementary rod theory assumes that only axial force F(x,t) is acting through each section, and considers only

the axial deformation and longitudinal wave motion. The lateral motion due to the Poisson’s ratio effect is

neglected. The governing differential equation of the motion of a rod, in accordance with the elementary rod

theory, is obtained using the compatibility and equilibrium concepts as given in equation ( 1 ).

𝜕

𝜕𝑥[𝐸𝐴

𝜕𝑢

𝜕𝑥] = 𝜌𝐴

𝜕2𝑢

𝜕𝑡2− 𝑞(𝑥, 𝑡) ( 1 )

EA is the axial stiffness of the rod, ρA is the mass density per unit length, q(x,t) is the externally applied body force

per unit volume and u(x,t) is the displacement in the x direction. Spectral representation of the governing

differential equation is given in the Eq. ( 2 ).

𝑑

𝑑𝑥[𝐸𝐴

𝑑�̂�

𝑑𝑥] + 𝜌𝐴𝜔2�̂� = −�̂� ( 2 )

The longitudinal displacement at any arbitrary point of a rod element using the wave solution obtained for

elementary rod theory is derived as,

�̂�(𝑥) = 𝐴𝑒−𝑖𝑘𝑅𝑥 + 𝐵𝑒−𝑖𝑘𝑅(𝐿−𝑥) ( 3 )



The first term in Eq. ( 3 ) stands for the forward-moving (incident) wave and the second term for the backward-

moving (reflected) wave, L is the length of the member, 𝐴 and 𝐵 are the constants to be determined and 𝑘𝑅 is the

wavenumber which is obtained from the characteristic equation as,

𝑘𝑅 = √𝜔2𝜌𝐴

𝐸𝐴 ( 4 )

Frequency dependent shape functions are derived by substituting the nodal displacements at the both ends of the

element into Eq. ( 3 ) , and given as follows,

�̂�(𝑥) = [𝑔𝑅1(𝑥) 𝑔𝑅2(𝑥)] {�̂�1�̂�2} = [𝑒−𝑖𝑘𝑅𝑥 𝑒−𝑖𝑘𝑅(𝐿−𝑥)] [

1 𝑒−𝑖𝑘𝑅𝐿

𝑒−𝑖𝑘𝑅𝐿 1]−1

{�̂�1�̂�2} ( 5 )

Displacement at any arbitrary point can be obtained using the shape functions given in Eq. ( 5 ). The spectral

element matrix for a rod element based on elementary rod theory is obtained using the relation between force and

deformation as follows,

{�̂�1�̂�2} = {

−�̂�(0)

�̂�(𝐿)} = [�̂�]{�̂�} = 𝐸𝐴 [

−𝑔𝑅1′(0) −𝑔𝑅2′(0)

𝑔𝑅1′(𝐿) 𝑔𝑅2′(𝐿)] {�̂�1�̂�2} = [

�̂�𝑅11 �̂�𝑅12�̂�𝑅21 �̂�𝑅22

] {�̂�1�̂�2} ( 6 )

3. FORMULATION OF SPECTRAL ELEMENT FOR A BEAM

Bernoulli-Euler beam theory assumes that only bending moment M(x,t) and shear forces V(x,t) are acting through

each section. The theory considers only flexural deformations and flexural wave motions. Existence of transverse

shear force is adopted in this theory. However, any shear deformation due to transverse shear force is neglected.

Since shear deformation through the thickness is neglected, vertical displacement v(x,t) can be assumed to be

nearly constant and the stress state is uniaxial along the thickness. Thus, the assumption of “plane sections remain

plane” is valid in this theory, and the slope 𝜙(𝑥, 𝑡) of the section can be obtained by differentiating the transverse

displacement v(x,t). The governing differential equation of the motion for a Bernoulli-Euler beam is given in

equation ( 7 ).

𝜕2

𝜕𝑥2[𝐸𝐼

𝜕2𝑣

𝜕𝑥2] + 𝜌𝐴

𝜕2𝑣

𝜕𝑡2= 𝑞 ( 7 )

EI is the flexural stiffness of the beam, ρA is the mass density per unit length, q(x,t) is the externally applied body

force per unit volume and v(x,t) is the deflection of the centerline. Spectral representation of the governing

differential equation is,

𝑑2

𝑑𝑥2[𝐸𝐼

𝑑2𝑣

𝑑𝑥2] − 𝜔2𝜌𝐴v̂ = �̂� ( 8 )

By the use of wave solution obtained for the flexural wave propagation through a Bernoulli-Euler beam, transverse

displacement at any arbitrary point is derived as in equation ( 9 ), and the slope is simply derived by differentiating

with respect to the vertical displacement.

𝑣(𝑥) = 𝐴𝑒−𝑖𝑘𝐵𝑥 + 𝐵𝑒−𝑘𝐵𝑥 + 𝐶𝑒−𝑖𝑘𝐵(𝐿−𝑥) +𝐷𝑒−𝑘𝐵(𝐿−𝑥) ( 9 )

�̂�(𝑥) = −𝑖𝑘𝐵𝐴𝑒−𝑖𝑘𝐵𝑥 − 𝑘𝐵𝐵𝑒

−𝑘𝐵𝑥 + 𝑖𝑘𝐵𝐶𝑒−𝑖𝑘𝐵(𝐿−𝑥) + 𝑘𝐵𝐷𝑒

−𝑘𝐵(𝐿−𝑥)



The first two terms of the equation ( 9 ) are the incident waves and the last two terms are the reflected waves. In

addition, the first and the third terms are the wave propagation solution while the remaining terms are related to

spatially damped vibrations (Doyle, 1997). A, B, C and D are the constants, L is the length of the member, and 𝑘𝐵

is the wave-number which is obtained as,

𝑘𝐵 = (𝜔2𝜌𝐴

𝐸𝐼)

1/4

( 10 )

The dynamic shape functions are developed analogously as in the elementary rod. The dynamic shape

functions are formed in pairs, and each component of the pairs are symmetric to each other.

𝑣(𝑥) = [𝑔𝐵1(𝑥) 𝑔𝐵2(𝑥) 𝑔𝐵3(𝑥) 𝑔𝐵4(𝑥)]

{

𝑣1�̂�1𝑣2�̂�2}

( 11 )

= [𝑒−𝑖𝑘𝑅𝑥 𝑒−𝑘𝑅𝑥 𝑒−𝑖𝑘𝑅(𝐿−𝑥) 𝑒−𝑘𝑅(𝐿−𝑥) ]

[

1 1 𝑒−𝑖𝑘𝑅𝐿 𝑒−𝑘𝑅𝐿

−𝑖𝑘𝑅 −𝑘𝑅 𝑖𝑘𝑅𝑒−𝑖𝑘𝑅𝐿 𝑘𝑅𝑒

−𝑘𝑅𝐿

𝑒−𝑖𝑘𝑅𝐿 𝑒−𝑘𝑅𝐿 1 1−𝑖𝑘𝑅𝑒

−𝑖𝑘𝑅𝐿 −𝑘𝑅𝑒−𝑘𝑅𝐿 𝑖𝑘𝑅 𝑘𝑅 ]

−1

{

𝑣1�̂�1𝑣2�̂�2}

The spectral element matrix for a beam according to the Bernoulli Euler beam theory is constructed using the

force-displacement relationship as given below,

{

�̂�1�̂�1�̂�2�̂�2}

=

{

−�̂�(0)

−�̂�(0)

�̂�(𝐿)

�̂�(𝐿) }

= [�̂�]{�̂�} = 𝐸𝐼

[ 𝑔𝐵1′′′(0) 𝑔𝐵2

′′′(0) 𝑔𝐵3′′′(0) 𝑔𝐵4

′′′(0)

−𝑔𝐵1′′ (0) −�̂�𝐵2

′′ (0) −𝑔𝐵3′′ (0) −�̂�𝐵4

′′ (0)

−𝑔𝐵1′′′(𝐿) −�̂�𝐵2

′′′ (𝐿) −𝑔𝐵3′′′(𝐿) −�̂�𝐵4

′′′ (𝐿)

𝑔𝐵1′′(𝐿)(𝐿) 𝑔𝐵2

′′ (𝐿) 𝑔𝐵3′′ (𝐿) 𝑔𝐵4

′′ (𝐿) ]

{

𝑣1�̂�1𝑣2�̂�2}

= 𝐸𝐼

[ �̂�𝐵11 �̂�𝐵12 �̂�𝐵13 �̂�𝐵14�̂�𝐵21 �̂�𝐵22 �̂�𝐵23 �̂�𝐵24�̂�𝐵31 �̂�𝐵32 �̂�𝐵33 �̂�𝐵34�̂�𝐵41 �̂�𝐵42 �̂�𝐵43 �̂�𝐵44]

{

𝑣1�̂�1𝑣2�̂�2}

( 12 )

4. NUMERICAL EXAMPLE

Dynamic response of a plane frame given in Figure 1 under the effect of a white noise base excitation is obtained

using SFEM and FEM. The base input has 1024 points and the sampling rate is 400 Hz. In FEM analysis,

“SAP2000 V.16.0.0” software is utilized to obtain the dynamic response and characteristics of the system. In order

to see the effect of increased number of elements in FEM analysis, two different models are used. In the first

model, each element is modeled as a single element. In the second FEM model, each element is divided into

smaller segments 10 cm long. The mass of the system is lumped to its structural nodes in both models. The plane

frame consists of 2 stories having 3 m story height, and 2 bays with 5 m and 4 m span lengths. The elements’

dimensions and the material properties are given in Table 1.



Table 1. Material properties and dimensions of the plane frame elements

Young’s modulus 27.4 GPa Dimensions of the beams b/h 25/30 cm

Mass density 2400 kg/m3 Dimensions of the columns b/h 30/30 cm

(a)

(b)

Figure 1. Geometry of the structure (a) and FAS of the base excitation (b)

Dynamic stiffness matrices of the elements are assembled and the waveguide resultants at the structural nodes are

obtained in an analogous way as in FEM. “Penalty method” is adopted to impose the displacements produced by

the excitation to the related degrees of freedom. This method is implemented by adding a spring with a large

stiffness to the related degrees of freedom and defining a load that produces the required displacement at that

degree of freedom (Bathe, 1996). Displacements at an arbitrary point of a member at any frequency can be found

by the multiplication of the frequency dependent shape functions to the nodal displacements of the concerned

element at the related frequency.

The damping is introduced using the hysteretic damping. It can be applied by replacing the elasticity modulus of

the system with a complex value as follows,

𝐸𝑑 = 𝐸(1 + 2𝑖𝜁) ( 13 )

Hysteretic damping is also referred as rate-independent damping or structural damping and it can easily be applied

in frequency domain analysis. It represents both of the elastic and damping forces at the same time (Chopra, 1995).

The following figures present the displacement response of the structure at node 9 both in the frequency and time

domains. The mentioned node can be seen from Figure 1a. Figure 2 and Figure 3 illustrate the longitudinal

displacement in frequency and time domain, respectively. Fourier amplitude spectrum (FAS) and time history of

the transverse displacement at node 9 are presented in Figure 4 and Figure 5. The last two figures stand for the

rotation about z axis. In these figures, the black line stands for the response obtained from the first FEM analysis,

the blue one is for the second FEM analysis and the red one is for the SFEM analysis. Following figures reveal

that, increment in the number of elements and number of DOFs make dynamic response equivalent to the response

obtained from the SFEM analysis. When the response obtained from first and second FEM analysis are compared,

they differentiate from each other at higher frequencies. As a result, it can be said that FEM analysis is not capable

enough to capture high-frequency wave modes unless a very small mesh size used in the modeling. When the

second FEM and SFEM analysis results are compared, total number of 1077 DOFs are used in the FEM analysis,



whereas only 27 DOFs used in SFEM analysis. However, this increment in the number of DOFs is not sufficient

to capture the effects of mode conversions at the angled joints.

Figure 2. FAS of the longitudinal displacements at node 9

Figure 3. Time history of the longitudinal displacements at node 9

Figure 4. FAS of the transverse displacements at node 9



Figure 5. Time history of the transverse displacements at node 9

Figure 6. FAS of the rotation about z axis at node 9

Figure 7. Time history of the rotation about z axis at node 9



5. CONCLUSION

SFEM (Spectral Finite Element Method) uses the exact wave solution of governing differential equations (dynamic

shape functions) to construct the stiffness matrices of the elements, and is highly accurate in capturing the high-

frequency response of structures. High-frequency modes are important to calculate the dynamic response more

accurately, as these modes are more sensitive to small changes in the dynamic characteristics of structures.

Therefore, SFEM also allows development of more powerful tools for system identification and damage detection.

The conventional finite element method (FEM) is not capable of capturing high-frequency modes unless the mesh

size used in the modeling is smaller than the lowest wavelength.

When compared to standard FEM (Finite Element Method), SFEM analysis requires a coarse mesh size to capture

adequately the dynamic response, whereas the FEM analysis needs a much higher number of elements to capture

the response with the same precision. An increase in the number of elements results in considerable computational

costs. Complex structures can be handled using SFEM by virtue of the validity of matrix assembly procedure.

Since the method uses exact solution of the governing differential equations, the distribution of inertial forces is

modeled accurately, including the medium and high frequency ranges. Moreover, damping can be modeled in a

more realistic way in the SFEM approach, since the damping properties are generally frequency dependent.

REFERENCES

Gopalakrishnan, S. (1992). Spectral Analysis Of Wave Propagation In Connected Waveguides. Ph.D. Thesis,

Purdue University, IN, USA.

Gopalakrishnan, S. (2000). A deep rod finite element for structural dynamics and wave propagation problems.

International Journal for Numerical Methods in Engineering 48:5, 731-744.

Lee, U. (2009). Spectral Element Method in Structural Dynamics, John Wiley & Sons (Asia), Singapore.

Banerjee, J. R. (1997). Dynamic stiffness formulation for structural elements: a general approach. Computers &

structures 63:1, 101-103.

Doyle, J. F. (1997). Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier

Transformations, Springer Science and Business Media, New York, USA.

Bathe, K. J. (1996). Finite Element Procedures, Prentice Hall, New Jersey, USA.

Chopra, A. K. (1995). Dynamics of structures, Prentice Hall, New Jersey, USA.

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STRUCTURAL ANALYSIS USING SPECTRAL FINITE ELEMENT … · Finite Element Method (FEM) ... used in the analysis substantially increase which ends up with considerable computational