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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERINGInt. J. Numer. Meth. Biomed. Engng. 2010; 26:587–596Published online 22 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1152COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

Exact solution of vibration problems of frame structures

Haitao Ma∗,†

State Key Laboratory of Subtropical Building Science, Department of Civil Engineering, South ChinaUniversity of Technology, Guangzhou 510641, China

SUMMARY

Although exact solutions for linear static analysis of most frame structures can be obtained by the finiteelement method, it is very difficult to get exact solutions for free vibration and harmonic analyses for non-trivial cases. This paper extends an earlier study on exact solutions of axial vibration problems of elasticbars to dynamic analyses of elastic frame structures. New shape functions for the transverse displacementfield are constructed by using the homogeneous governing equations and then a novel beam element isformulated. Combining the new (bending) beam element and the one developed earlier for elastic barsyields a new element for general frame structures. The new frame element can be used to get exactsolutions for both natural frequency and undamped harmonic analyses of frame structures. Illustrativeexamples are presented to demonstrate the effectiveness of the new element and the algorithm. Copyrightq 2008 John Wiley & Sons, Ltd.

Received 26 January 2008; Revised 1 May 2008; Accepted 27 May 2008

KEY WORDS: finite element method; structural dynamics; exact solution; natural frequency; harmonicresponse

1. INTRODUCTION

It is well known that, even with a coarse mesh, exact solutions for linear static analysis of framestructures can be obtained by the finite element method. Theoretical proofs have been made forboth problem with single variable [1] and that with multiple variables [2]. Elements that can be usedto produce exact solutions for linear static analysis have also been developed based on differentformulations; see, for example, [2–5]. For dynamic analysis, on the other hand, performance ofthese elements is not as good as desired. Quite often, meshes good for linear static solution aretoo coarse and require substantial refinement to model structural dynamic behaviours, and usuallyfurther refinement is required to get accurate results for high-order modes.

Various algorithms have been proposed for dynamic analysis of frame structures, including thosebased on continuous mass method of Ovunc [6], the integral method of Antes and co-workers [7],and the solution of transcendental eigenvalue problem of Williams and co-workers [8–11].Recently, this author presented a new approach for developing elements for dynamic analysis[12]. By constructing shape functions based on solutions to the homogeneous governing equationsof the motion, he developed a new element for axial vibration of elastic bars, which can giveexact solutions when employed together with the proposed algorithms for natural frequency andharmonic response analyses. Compared with other algorithms, one of the major advantages of the

∗Correspondence to: Haitao Ma, State Key Laboratory of Subtropical Building Science, Department of CivilEngineering, South China University of Technology, Guangzhou 510641, China.

†E-mail: [email protected]

Contract/grant sponsor: South China University of Technology

Copyright q 2008 John Wiley & Sons, Ltd.

588 H. MA

proposed algorithm for natural frequency analysis is that no special numerical method is required,and as a result it is simple to integrate this algorithm into conventional finite element analysisprograms.

In this paper, the study reported in [12] is extended to the analysis of bending beam and a newelement for frame structures is formulated by including the axial, torsional and transverse bendingcomponents. We start by summarizing the governing equation for transverse vibration of thinelastic beams. Based on general solutions to the homogeneous governing equation, new elementshape functions are constructed for a predefined vibration frequency. Then element stiffness matrix,mass matrix and dynamic stiffness matrix are formulated. Combining the new formulation withthe one for axial vibration in [12], the new element for frame structures is derived. Illustrativeexamples are presented to show the effectiveness of the new element and algorithms from [12].Finally, concluding remarks are made.

2. ELEMENT FORMULATION

2.1. Governing equation

The motion of transverse vibration of an elastic beam can be described by the following equation(see, e.g. [13]):

�2

�x2

[E(x)I (x)

�2W (x, t)

�x2

]+�(x)A(x)

�2W (x, t)

�t2=Q(x, t) (1)

where x is the coordinate of an arbitrary point, t is the time, E(x) and �(x) are material modulusand mass density, respectively, I (x) is the second moment of inertia, A(x) is the area of crosssection, W (x, t) is the transverse displacement and Q(x, t) is the distributed force in the transversedirection (Figure 1).

Considering a uniform bar and denoting

E(x) ≡ E=Constant

I (x) ≡ I =Constant

A(x) ≡ A=Constant

�(x) ≡ �=Constant

we may write Equation (1) as follows:

E I�4W (x, t)

�x4+�A

�2W (x, t)

�t2=Q(x, t) (2)

Limiting the loading to be harmonic and letting

Q(x, t)=q(x)cos(� t+�) (3)

where � is the loading frequency, � is the phase angle, and q(x) is the magnitude of the distributedload, the displacement can be expressed assumed to be in the following form:

W (x, t)=w(x)cos(�t+�) (4)

Figure 1. Transverse vibration of a beam.

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. 2010; 26:587–596DOI: 10.1002/cnm

EXACT SOLUTION OF VIBRATION PROBLEMS OF FRAME STRUCTURES 589

Substituting Equations (3) and (4) into Equation (2) yields the following governing equation forthe steady-state harmonic response:

E Id4w(x)

dx4−�A�2w(x)=q(x) (5)

Note that both the variables q(x) and w(x) are now functions of coordinate only, and the equationis time-independent except for the presence of frequency �.

2.2. General solutions to the homogeneous governing equation

For undamped free vibration, Equation (5) can be written as

d4w(x)

dx4−�4w(x)=0 (6)

where � is a parameter defined as

�= 4

√�2 �A

E I(7)

The general solutions to Equation (6) can be expressed as

w(x) = ax3+bx2+cx2+d, �=0

w(x) = a sin�x+bcos�x+c sinh�x+d cosh�x, �>0(8)

where a, b, c and d are arbitrary constants. When �=0 (i.e. static case or dynamic case withmassless beam), the general solution is a cubic function of the coordinate; otherwise, it is acombination of harmonic and hyperbolic functions of the coordinate.

2.3. Shape functions satisfying homogeneous governing equation

The cubic shape functions for conventional thin-beam element satisfy the governing equation forstatic analysis when �=0 and should be kept. Therefore, � is assumed to be greater than zeroin the following discussion. We choose to describe the displacement within an element usingthe expression for the general solutions given in Equation (8b). Let the transverse deflection beexpressed as

w= ∑i=1,2

(N0i (x)wi +N1i (x)�i )

where i (i=1,2) denotes element node, wi and �i nodal deflection and rotation, respectively, andN0i and N1i are shape functions for nodal deflection and rotation, respectively.Introducing the shape function matrix N as

N=[N01(x) N11(x) N02(x) N12(x)] (9)

and the nodal displacement vector d as

d=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

w1

�1

w2

�2

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(10)

the transverse deflection field can be expressed as

w=N ·d (11)


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Let the shape functions be a linear combination of the homogeneous solutions

NT=CV (12)

where C is a constant coefficient matrix

C=

⎡⎢⎢⎢⎢⎣c11 c12 c13 c14

c21 c22 c23 c24

c31 c32 c33 c34

c41 c42 c43 c44

⎤⎥⎥⎥⎥⎦ (13)

and V is a vector with the four harmonic and hyperbolic functions in the homogeneous solutionsas its components:

V=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

V1(x)

V2(x)

V3(x)

V4(x)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

sin�x

cos�x

sinh�x

cosh�x

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(14)

Using the conditions that the shape functions must satisfy at the two ends and letting the lengthof the element be l, we have ⎡

⎢⎢⎢⎢⎣N|x=0

N′|x=0

N|x=l

N′|x=l

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎣1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤⎥⎥⎥⎥⎦ (15)

Substituting the expression of shape function matrix into the above expression, we can get thefollowing equation:

CP=I4 (16)

where I4 is an identity matrix of order 4 and matrix P is defined as

P=[V|x=0 V′|x=0 V|x=l V′|x=l ]=

⎡⎢⎢⎢⎢⎣0 � s �c

1 0 c −�s

0 � sh �ch

1 0 ch �sh

⎤⎥⎥⎥⎥⎦ (17)

in which

s = sin�l

c = cos�l

sh = sinh�l

ch = cosh�l

(18)

Thus, C=P−1 and the inversion of matrix P yields

C= 1

a

⎡⎢⎢⎢⎢⎣

s ch+c sh −s sh+cch−1 −s ch−c sh s sh+cch−1

(s sh+cch−1)/� (−s ch+c sh)/� (−s sh+cch−1)/� (s ch−c sh)/�

−s−sh −c+ch s+sh c−ch

(−c+ch)/� (s−sh)/� (c−ch)/� (−s+sh)

⎤⎥⎥⎥⎥⎦ (19)



where

a=2(cch−1)=2(cos�l cosh�l−1) (20)

Generally speaking, once values of � and l are given, the shape function coefficient matrix C canbe formed and shape functions constructed. However, when the parameter a is exactly or nearlyzero, matrix P is singular or nearly singular and matrix C cannot be accurately calculated. In thiscase, the solution can still proceed after the element length (l) is modified by a small fraction.

The curvature at any point within the element can be expressed as

�= d2w

dx2=N′′d=(CV′′)Td=(V′′)TCTd (21)

where V′′ are shape function derivatives

V′′ =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

V ′′1 (x)

V ′′2 (x)

V ′′3 (x)

V ′′4 (x)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=�2

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−sin�x

−cos�x

sinh�x

cosh�x

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(22)

2.4. Element stiffness and mass matrices

By following the standard finite element procedure, the element stiffness matrix can now beexpressed as

Ke=∫ l

0(N′′)TE IN′′ dx=

∫ l

0CV′′E I (V′′)TCT dx=(E I )CHCT (23)

where H is the following integration:

H=∫ l

0V ′′(V ′′)T dx=

∫ l

0

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

V ′′1 (x)

V ′′2 (x)

V ′′3 (x)

V ′′4 (x)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

{V ′′1 (x) V ′′

2 (x) V ′′2 (x) V ′′

4 (x)}dx (24)

Noting that ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

V ′′1 (x)

V ′′2 (x)

V ′′2 (x)

V ′′2 (x)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=�2

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−V1(x)

−V2(x)

V3(x)

V4(x)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(25)

matrix H can be expressed as

H=�4

⎡⎢⎢⎢⎢⎣

S11 S12 −S13 −S14

S21 S22 −S23 −S24

−S31 −S32 S33 S34

−S41 −S42 S43 S44

⎤⎥⎥⎥⎥⎦ (26)

in which Si j (i, j =1,4) are entries in matrix S, which is defined as

S=∫ l

0VVT dx (27)


592 H. MA

Substituting Equation (14) into the above equation and integrating all the elements in the matrixgive the following expression:

S= 1

2�

⎡⎢⎢⎢⎢⎢⎢⎣

�l−s c s2 s ch−c sh s sh−cch+1

s2 �l+s c s sh+cch−1 s ch+c sh

s ch−c sh s sh+cch−1 sh ch−�l sh2

s sh−cch+1 s ch+c sh sh2 sh ch+�l

⎤⎥⎥⎥⎥⎥⎥⎦

(28)

The element mass matrix can be expressed as

Me(�)=∫ l

0�ANTNdx

Substituting the expression of shape functions and then making use of Equation (27), we have

Me(�)=∫ l

0�ACVVTCT dx=(�A)CSCT (29)

2.5. Discussions

First let us compare the new shape functions with the conventional shape functions for the conven-tional 2-node thin-beam element, which can be expressed as

N̄01(x) = 1−3( xl

)2+2( xl

)3

N̄11(x) = l

(( xl

)−2

( xl

)2+( xl

)3)

N̄02(x) = 3( xl

)2−2( xl

)3

N̄12(x) = l

(−( xl

)2+( xl

)3)(30)

In the extreme case of �→0, we have �→0. Let �=�l and when �→0, matrix C can beapproximated as

C= − 3

�3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2− �4

15−�−�3

6−2+ �4

15�

(1−�2

6− �4

90

)

�

�

(1−�2

6− �4

90

)�2

�

(−2

3+ �2

315

)�

�

(−1−�2

6+ �4

90

)�2

�

(2

3− �4

315

)

−2− �4

60�

(1+ �4

360

)2+ �4

60−�

(1+ �4

360

)

�

�

(1+ �4

360

)− �2

3�

(1+ �4

8400

)−�

�

(1+ �4

360

)�2

3�

(1+ �4

840

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+O(�4)

(31)



As for vector V, let y=�x (0�x�l) and when �→0, we have y→0. In this case, vector V canbe approximated as

V=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

sin y

cos y

sinh y

cosh y

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

y

(1− y2

6+ y4

120− y6

5040

)

1− y2

2+ y4

24− y6

720

y

(1+ y2

6+ y4

120+ y6

5040

)

1+ y2

2+ y4

24+ y6

720

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

+O(y8) (32)

Based on Equations (31) and (32), the following expression for shape functions can be derived:

lim�→0

NT= lim�→0

1

�3

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

�3−3�y2+2y3

(�3y−2�2y2+�y3)/�

3�y2−2y3

(−�2y2+�y3)/�

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1−3( xl

)2+2( xl

)3l

(x

l−2

( xl

)2+( xl

)3)

3( xl

)2−2( xl

)3l

(−( xl

)2+( xl

)3)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(33)

Thus, it is shown that the new shape functions given in Equation (12) approach the conventionalshape functions given in Equation (30), i.e.

lim�→0

Npi (x)= lim�→0

Npi = N̄pi (p=0,1, i=1,2) (34)

indicating that the conventional shape functions are good approximations of the new shape functionswhen �l�1. It is obvious that the differences between the new and the conventional shape functionsvanish when �=0 and become significant when �l increases. In other words, when � is a fixednon-zero value, the difference is smaller for shorter elements; and when the element length l isfixed, the difference is smaller for lower frequency (or equivalently smaller � value).

Now turn to the element stiffness and mass matrices. As the new shape functions approach theconventional ones when � approaches zero, it can be shown that the element matrices given inEquations (23) and (29) will also approach those for the conventional beam element.

When the element in under harmonic motion, its contribution to the dynamic equilibrium canbe expressed as

f=Kd(�)u (35)

where f is the element node force vector, u is the displacement vector and Kd(�) is the dynamicstiffness matrix defined as

Kd(�)=Ke(�)−�2Me(�) (36)

which can be derived by substituting expressions for element stiffness and mass matrices into theabove equation.

2.6. New beam element

The governing equation for the torsional vibration of a beam is similar to that for the axial vibration;therefore, ‘exact’ stiffness and mass matrices for torsion can be easily formulated as in [12].

With the available formulations for axial, torsional and bending vibration, a new beam elementcan be developed by combining deformation in axial, torsional and two transverse directions.


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Further, the iterative algorithm proposed for natural frequency analysis and that for the harmonicresponse analysis can be used together with this new element. The algorithms are described indetail in [12] and will not be presented here. In the following section, numerical results will bepresented using the new element and the iterative algorithm to calculate natural frequencies offrame structures.

3. EXAMPLES

3.1. Natural frequency analysis of cantilever beam

Consider the simply supported beam shown in Figure 2. The natural frequencies of transversevibration of the beam are [13]

�n = n2�2

L2

√E I

�A, n=1,2,3, . . . (37)

The beam is modelled with two elements of equal length (0.5) and the mesh has four degreesof freedom (DOFs). Start with initial frequency �0=0; the first natural frequency converges tothe exact solution in four iterations. Using the result for �2 at the fourth iteration as the initialvalue, the solution proceeds for the second mode shape and converges to the exact solution in fouriterations. The solution history is summarized in Table I.

Using the proposed algorithm, the FEM solutions with two elements are identical to the analyticalsolutions with nine significant digits. In contrast to this, a much more refined mesh is neededto get a solution of similar accuracy when the conventional element is used. Solutions withdifferent numbers of conventional elements of equal length are summarized in Table II. It canbe seen that with the mesh consisting of 256 elements of equal length, the results converge tothe exact solutions. Note that with the new algorithm, eigenvalue problem with four DOFs issolved 8 times and with the conventional beam element, eigenvalue problems with hundreds ofDOFs must be solved to get results of similar accuracy for the first two modes. As the numberof numerical operations in eigenvalue solution is proportional to the number of DOFS cubed, thesaving in computational cost achieved with the new algorithm is considerable and can be muchmore significant for larger models.

Figure 2. A simply supported beam.

Table I. Solution history for simply supported beam with two elements—new element.

Iteration �0 �1 �2 Difference (%) Note

1 0.00000000 9.90855871 43.8178046 — Start iteration for �12 9.90855871 9.86960686 43.3150902 3.93E−013 9.86960686 9.86960440 43.3189325 2.49E−054 9.86960440 9.86960440 43.3189328 0.00E+00 �1 converged5 43.3189328 19.8757435 39.6716615 8.42E+00 Start iteration for �26 39.6716615 17.2607382 39.4788647 4.86E−017 39.4788647 17.1344262 39.4784176 1.13E−038 39.4784176 17.1338772 39.4784176 0.00E+00 �2 convergedExact 9.86960440 39.4784176



Table II. Solutions with different meshes for simplysupported beam—conventional element.

Mode �∗1 �∗

2

2 9.90855871 43.81780464 9.87216716 (3.673E−01) 39.6342348 (9.548E+00)8 9.86976668 (2.432E−02) 39.4886687 (3.673E−01)16 9.86961458 (1.541E−03) 39.4790667 (2.432E−02)32 9.86960504 (9.666E−05) 39.4784583 (1.541E−03)64 9.86960444 (6.079E−06) 39.4784202 (9.651E−05)128 9.86960440 (4.053E−07) 39.4784178 (6.079E−06)256 9.86960430 (1.013E−06) 39.4784176 (5.066E−07)

∗Values in brackets are percentage changes from coarse mesh result.

Figure 3. A plane frame of pin-ended cross: (a) mesh with sevenDOFs and (b) NAFEMS benchmark mesh.

Table III. Frequency analysis results for pin-ended cross.

New algorithm

Frequency (Hz)

Mode Starting value Converged valueNumber ofiterations

Target frequencyin [14] (Hz)

1 0.0 11.33626 5 11.3362 and 3 19.99231 17.68079 5 17.7094 17.70948 17.70940 2 17.7095 54.57786 45.34504 5 45.3456 and 7 59.71113 57.07463 4 57.3908 57.39541 57.38980 3 57.390

3.2. Natural frequency of in-plane vibration of pin-ended cross

The in-plane vibration frequencies of the frame structure shown in Figure 3 are calculated. Thematerial Young’s modulus and mass density are

E = 200×109 Pa

� = 8000kg/m3

This test problem of pin-ended cross is one of NAFEMS’ benchmarks for natural frequencyanalysis [14]. The results are summarized in Table III.

The mesh used for the new algorithm (Figure 3(a)) consists of four elements and has just sevenDOFs. Not more than five iterations are needed for each mode or pair of modes to get natural


596 H. MA

frequency results with seven significant digits. The target value from Reference [14] is for themesh with 16 elements and 43 DOFs (Figure 3(b)), each member of the frame being modelledwith four elements of equal length. It should be noted that, using the new algorithm, both elementrotational inertia and transverse shear deformation are ignored.

4. CONCLUSIONS

Based on general solutions to the homogeneous dynamic equilibrium equation for undampedvibration, new element shape functions are derived for transverse displacement field and a newbeam element is formulated for undamped vibration analysis of elastic frame structures. This studyillustrates, once again, that the element performance in dynamic analysis can be improved byincluding the effect of element inertial force on shape functions. The proposed approach to the exactsolution of vibration problems is also applicable to other types of analyses for the improvement offinite element performance. For example, this research work has already been extended to linearbuckling analysis of frame structures and related results will be reported in a separate paper.

ACKNOWLEDGEMENTS

Financial support under ‘Xinghua Talents Project’ from South China University of Technology is gratefullyacknowledged.

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