The principle of virtual force is employed to derive the exact shape function for a taperedcurved frame element in space and these shape functions ca n be employed to calculate the exact consistent mass and geometric stiffness matrixes for curved-tapered frame elements. The lack of any displacement shape function with exact fulfilment of equilibrium equations by an accurate force interpolation is the salient feature of this approach. The formulation adopts the linear elastic behaviour of the material and the strain compatibility is satisfied based on the Nevier-Bernoulli hypothesis. Shear deformations are considered and the Saint-Venat hypothesis for torsion is adopted. The efficiency and accuracy of the formulation are verified using some numerical examples

Foster, S. J.

Vali pour, H. R.

University of Queensland eSpace

5th Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
Consistent Mass and Exact Displacement Shape Function for a Tapered 
Curved Frame Element 
 
H. R. Vali pour1 and S. J. Foster2 
 
1Research student, School of Civil and Environmental Engineering, University of New South Wales, 
NSW, Australia  
2A/Prof., School of Civil and Environmental Engineering, University of New South Wales, NSW, 
Australia 
Abstract: The principle of virtual force is employed to derive the exact shape function for a tapered-
curved frame element in space and these shape functions ca n be employed to calculate the exact 
consistent mass and geometric stiffness matrixes for curved-tapered frame elements. The lack of any 
displacement shape function with exact fulfilment of equilibrium equations by an accurate force 
interpolation is the salient feature of this approach. The formulation adopts the linear elastic behaviour 
of the material and the strain compatibility is satisfied based on the Nevier-Bernoulli hypothesis. Shear 
deformations are considered and the Saint-Venat hypothesis for torsion is adopted. The efficiency and 
accuracy of the formulation are verified using some numerical examples.  
Keywords: force interpolation, Navier-Bernoulli hypothesis, Saint-Venant torsion, tapered element. 
1 Introduction 
The finite element method can be formulated in the framework of stiffness, flexibility or mixed (hybrid) 
methods [1]. Generally, stiffness and flexibility formulations have the same degree of approximations 
due to coupling between compatibility and equilibrium equations. However, the stiffness method or 
mixed methods rooted from stiffness-based techniques are more popular than flexibility formulation, 
because the stiffness formulation is straightforward while the compatibility fulfilment in a flexibility 
method is somewhat more complex. However, this is not the case always and in specific types of 
structures, such as for a frame, by adopting some simplifying assumptions the equilibrium equations 
become uncoupled with compatibility and they can be satisfied exactly [2]. In such a case the 
formulation accuracy is improved without increasing the number of degrees of freedom per element. 
Tapered-curved frame elements are used in various types of structures subjected to dynamic and 
static loadings. Analysing these types of systems in an efficient way is one of the structural engineers 
concerns. Mostly the present finite element formulations use the direct stiffness method to establish 
the stiffness, mass and geometric stiffness matrix of the frame elements [3]. The concept of stiffness-
based elements with more degrees of accuracy has been utilized for limited specific cases to derive 
the mass and stability matrices more accurately [4-6]. Moreover in free vibration analysis the accuracy 
of results can be improved by taking higher order terms in the eigenvalue expansion and application of 
Eulerian stress in the moveable coordinate system is another approach to improve the efficiency of the 
stability analysis [7, 8]. The flexibility method for frame element formulation has extensively been used 
by some researchers for static, cyclic and dynamic analysis of reinforced concrete frames including 
material nonlinearities [9-11]. However, less attention has been paid to the capability of method for 
deriving consistent mass and geometric stiffness matrices of element [12].  
In this paper, the flexibility approach is adopted for deriving the exact stiffness matrix of the element 
[12]. The exact elastic curves of the element (exact in the sense of the adopted assumptions) are 
obtained by using the principle of virtual force for different degrees of freedom with respect to the 
exact element stiffness sub-matrices. These elastic curves are collected in a matrix of element shape 
functions and are used for calculating the consistent mass. The possible application of the method for 
deriving the stability matrix is discussed briefly and the accuracy of the formulation is verified by some 
simple numerical examples for free vibration and bucking analysis. 
2 Flexibility and stiffness matrices of element 
The flexibility and stiffness matrix of elements are derived by direct fulfilment of the equilibrium, 
compatibility and the constitutive law of the material which is taken to be linear elastic in this study. 
  
2.1 Equilibrium equations at element level  
The axis of the curved element in global coordinate system , , XYZ  can be represented by, 
[ ],)()()()( sZsYsXs =G  where s  denotes the arc length in the curvilinear coordinate system (Figure 
1a). The section local coordinate system is denoted by xyz  where the -x axis is perpendicular to 
section plane (Figure 1b).  
G(s i )
G(s j )
i 
j 
G(s) 
X
Y
Z
s
(a)
i 
j 
G(s) 
X
Y
Z
(b)
x
y
z
 
Figure 1 (a) element axis in global coordinate, (b) section local coordinate system. 
Satisfying the equilibrium equations for a differential element of the curved member without element 
loads and then integrating, the resulting equations yields [12], 
jjsss RNR ),()( =  (1)  
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ë
é
---
---
---
=
´
´´
33
3333
0)()(
)(0)(
)()(0
),( I
0I
N
jj
jj
jj
j
XXYY
XXZZ
YYZZ
ss  (2)  
where, [ ]Tzyxzyx MMMFFFs =)(R and [ ]TZjYjXjZjYjXjj MMMFFF=R are the 
generalized internal force vector at section s  and end node j,  respectively, ),( jssN  is the exact force 
interpolation function, 33´I  and 33´0  denote the 3 by 3 identity and zero matrices, respectively. 
2.2 Compatibility equations 
The Navier-Bresse strain-displacement relationships for a spatially curved element (see Figure 1a), 
clamped at end  i,  leads to the following compatibility relationship 
dssss XY
j
i j
T
j )(),( N ò=  (3)  
where, [ ]TZYXZYXXY s fffeeee =)(  is the generalized strain vector at section s  and 
[ ] TZjYjXjZjYjXjj uuu qqq=  denotes vector of generalized displacement at end .j  
2.3 Main assumptions and section stiffness matrix 
The section stiffness matrix is derived by adopting the following assumptions: (i) the material is 
assumed to be linear elastic; (ii) the mechanical properties of the material are constant over each 
section; (iii) Saint-Venant hypothesis for free warping of sections subjected to torsion is adopted; and 
(iv) Navier hypothesis for compatibility of axial strains at section level is accepted. With regards to 
assumption (iv) the tangential and normal strains (stresses) are uncoupled.  
Equilibrium between stresses acting over the section and generalized internal forces (i.e. axial force, 
shear forces and bending moments) in a section free of initial strains (stresses) leads to 
)()( ss sxy rf =  (4)  
  
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
é
--
--
-
-+-
-+
=
)()(
0000
)()(
0000
0010
0010
00
1
0
000001
22
22
2
2
yzzy
y
yzzy
yz
yzzy
yz
yzzy
z
cc
cc
sz
cc
cccc
sy
s
IIIE
I
IIIE
I
IIIE
I
IIIE
I
JGJG
y
JG
z
JG
y
JG
y
AGJG
zy
JG
z
JG
zy
JG
z
AG
EA
f  (5)  
where, )(sr  is the section generalized internal force vector, )(s sf  is the section flexibility matrix, 
)(sxy  is the section generalized strain vector, ),( cc zy  is the position of the section shear centre in 
the local system, )(zyI  and J  are the second moment of inertia and warping constant of the section, 
respectively. If sT  and jjF  denote the local to global coordinate transformation matrix at s  and the 
flexibility sub-matrix of end j  in the configuration ij  clamped at end i,  then virtual force principle 
yields 
dsssss j
T
sss
j
i j
T
jj ),(),( NTfTNF ò=  (6)  
The flexibility sub-matrix of end i,  is obtained by a similar procedure for the configuration clamped at 
end j,  then by mathematical transformation the element stiffness matrix eK  is obtained as 
[ ] [ ]
[ ] [ ] ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
-
=
--
--
11
11
),(
),(),(),(
jjji
T
jj
jjjiji
T
jjji
e
ss
ssssss
FNF
FNNFN
K     (7)  
This stiffness matrix is the simplified form of the stiffness matrix obtained by Molins et al. [12] and it is 
the exact stiffness matrix of the element for the adopted assumptions. 
3 Consistent mass matrix of element 
In the displacement-based formulation the mass matrix of t he element eM  is calculated as follows 
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
òò
òò
j
i
Tj
i
T
j
i
Tj
i
T
jj
T
ij
ijii
e
dssmdssm
dssmdssm


mm
mm
M
)()(
)()(
    (8)  
where, )(sm  represents the mass distribution with respect to the arc length ,s    and   are shape 
functions matrices corresponding to degrees of freedom at node i  and j,  respectively, and are: 
[ ]654321  =     (9)  
[ ]654321  =     (10)  
  
In (9) and (10) matrices, [ ]Tzzyyxxzyxk yyyyyy= denotes the exact shape function 
corresponding to thk  degree of freedom at end node i  and [ ]Tzzyyxxzyxk llllll= is the 
exact shape function corresponding to thk  degree of freedom at end node j.  These exact shape 
functions can be obtained by superimposing the elastic curves of the cantilever configurations 
subjected to nodal unit forces at different degrees of freedom at the free end (Figure 2). 
i 
s
X
Y
Z
x
Translational DOF Rotational DOF
1
+ 1 a
i 
sx
1
+ 2 a
i 
j sx
1
+ 3 a
j 
j 
i 
sx
1
+ 4 a
i 
sx
1
+ 5 a
i 
j sx
1
+ 6 a
j 
j 
k a : is a column vector which contains the generalised displacement of the section s when a unit force is imposed at k     DOF at end  j.  
+
th
 
Figure 2 Outline of the clamped configuration subjected to unit virtual forces at end i.  
It is assumed that [ ]654321 âââââââ =  and [ ]654321 ááááááá =  are matrices of 
elastic curves corresponding to DOFs at end i  and j,  respectively; where kâ  is a column vector 
containing the generalized displacement of the section at s,  for a cantilever clamped at end j  and 
subjected to unit force at the thk  DOF at end i,  and ká  is a column vector containing the generalized 
displacement of the section at s,  for a cantilever clamped at end i  and subjected to unit force at thk  
DOF at end j  (Figure 2). Application of the displacement compatibility within superposition of the 
elastic curves leads to the following relationships between elastic curves and shape function matrices: 
Ëká =jj     (11)  
Økâ =ii     (12)  
Rewriting (3) for the section at s  of the element clamped at end i  (see Figure 2) leads to  
xxx ds XY
s
i
T
s )(),( åNÄ ò=     (13)  
Using local to global coordinate transformation along with (1) and (4) in the recent relationship yields 
js RÖÄ =     (14)  
xxx xxx dss j
Ts
i
T ),(),( NTfTNÖ ò=     (15)  
Matrix Ö  relates the vector of generalized displacements, sÄ , to the vector of nodal forces imposed 
at end .j Thus, regarding the definition of matrix ,á  it is concluded that Öá =  and 
xxx xxx dss j
Ts
i
T ),(),( NTfTNá ò=     (16)  
Matrix â  can be obtained by following a similar procedure for the member clamped at end j  and 
subjected to nodal forces at end i, as 
  
xxx xxx dss i
Ts
j
T ),(),( NTfTNâ ò=     (17)  
where xT  denotes the transformation matrix from local curvilinear coordinate, ,x  to global coordinate 
system at any arbitrary section along the element. Having the matrices á  and â  determined, the 
mass matrix of the element is obtained by substituting (11) and (12) into (8), giving 
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ï
þ
ï
ý
ü
ï
î
ï
í
ì
ú
ú
ú
û
ù
ê
ê
ê
ë
é
ú
ú
ú
û
ù
ê
ê
ê
ë
é
= ò
jj
iij
i TT
TT
jj
ii
e dssm
k0
0k
ââáâ
âááá
k0
0k
M )(     (18)  
4 Stability matrix of the element 
In a displacement-based formulation, the geometric stiffness (stability) matrix of the element eGK  is 
calculated as follows: 
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
¢¢¢¢
¢¢¢¢
=
òò
òò
j
i
Tj
i
T
j
i
Tj
i
T
e
G
dsNdsN
dsNdsN
ËËØË
ËØØØ
K     (19)  
where, N  is the axial force function along the element and  Ø¢  and Ë¢  denote the first derivative of 
the shape function matrices with respect to the coordinate system. Since dealing with an analytical 
form of the curves in a general FE code is cumbersome, the present method can be used with a 
super-parametric FE formulation. In this case, the complex spatial geometry of the element axis is 
approximated by a shape function (polynomial) and the integrals in the formulation are analytically or 
numerically estimated within this shape function.   
5 Numerical examples 
5.1 Buckling load of a simply supported tapered column 
Figure 3 shows a tapered simply supported column. The second moment of inertia varies parabolically 
along the length with .)/1()( 20 LxIxI +=  The axial and shear deformations are neglected.  
L
0I4 
E : Modulus of Elasticity
I0
x
P
 
Figure 3 Geometry of the tapered simply supported column. 
The buckling load of the column obtained from the method presented in this paper with a single 
element is ;/25.20 20 LIEPcr =  where as the analytical solution by Timoshenko [13] is 
,/75.20 20 LIEPcr @  and where a single element based on a stiffness method with a hermitian 
displacement shape function gives ./52.23 20 LIEPcr =  Thus the method based on the flexibility 
formulation developed in this study provides an improved accuracy, and efficiency, compared to the 
stiffness method. It is also worthy of mention that the flexibility method in buckling analysis converges 
to the exact critical load from the lower side, which is conservative from a designers point of view. 
5.2 Free vibration of a circular arch 
Consider a 90o cantilever (clamped-end) circular arch with radius of mm200=r  and square cross 
section mm10== hb  that can vibrate freely in the XY  plane. The material properties are, 
  
MPa105=E  (elasticity modulus), MPa105 4´=G  (shear modulus) and 3Ton/m50=r (mass density). 
By the method presented in this paper, the circular arch is approximated by a super-parametric 
function which includes 8 equal distance points along the arch. The resulting first and second natural 
frequencies of the in-plane vibration modes are given in Table 1 which clearly shows the efficiency of 
the formulation.  
Table 1 The two first natural frequency of the in-plane vibration modes in (Hz). 
Hermitian displacement-based formulation 
Formulation type 
Present formulation 
(1 super-parametric element) 5 element 10 element 20 element 
1st natural frequency 0.7568 0.7509 0.7535 0.7542 
2nd natural frequency 0.7675 0.7632 0.7677 0.7688 
 
6 Conclusions and recommendations 
The exact displacement shape function of the curved-tapered frame element is derived with respect to 
the exact stiffness and implemented in a concise matrix form suitable for finite element programming. 
These shape functions can be used independently or in combination with ordinary displacement-
based codes to calculate the consistent mass and geometric stiffness (stability) matrices within a static 
or dynamic linear elastic analysis. The procedure is not limited, however, to just linear elastic cases 
and can be used for nonlinear dynamic as well as stability analysis of frames using an appropriate 
iterative procedure. The computer implementation of the present method can be facilitated by 
recasting it in the super-parametric FE formulation framework without significant loss of accuracy. 
References 
[1] Zienkiewicz, O. C. & Taylor, R. L. (2000) The Finite Element Method, Vol. 1: The Basis ed.,Vol. Oxford, 
London, Butterworth-Heinemann. 
[2] Carol, I. & Murcia, J. (1989) Nonlinear time-dependent analysis of planar frames using an 'exact' 
formulation. I. Theory. Computers and Structures, Vol. 33, No. 1, 79-87. 
[3] Bathe, J. K. (1997) Finite element procedures, 4th ed.,Vol. New York, McGraw Hill. 
[4] Blandford, G. E. & Glass, G. C. (1987) Static/Dynamic analysis of locally buckled frames. Journal of 
Structural Engineering, Vol. 113, No. 2 , 363-380. 
[5] Eisenberger, M. & Reich, Y. (1989) Static, vibration and stability analysis of non-uniform beams. Computers 
and Structures, Vol. 31, No. 4, 567-573. 
[6] Gupta, A. K. (1985) Vibration of tapered beams. Journal of Structural Engineering, Vol. 111, No. 1 , 19-36. 
[7] Gupta, A. K. (1986) Frequency-dependent matrices for tapered beams. Journal of Structural Engineering, 
Vol. 112, No. 1, 85-103. 
[8] Ronagh, H. R., Bradford, M. A. & Attard, M. M. (2000) Nonlinear analysis of thin-walled members of variable 
cross-section. Part I: Theory. Computers and Structures, Vol. 77, No. 3, 285-299. 
[9] Marini, A. & Spacone, E. (2006) Analysis of reinforced concrete elements including shear effects. ACI 
Structural Journal, Vol. 103, No. 5 , 645-655. 
[10] Petrangeli, M., Pinto, P. E. & Ciampi, V. (1999) Fibre element for cyclic bending and shear of RC structures. 
I: theory. Journal of Engineering Mechanics, Vol. 125, No. 9 , 994-1001. 
[11] Spacone, E., Filippou, F. C. & Taucer, F. F. (1996) Fibre beam-column model for non-linear analysis of R/C 
frames: Part I. Formulation. Earthquake Engineering & Structural Dynamics, Vol. 25, No. 7, 711-725. 
[12] Molins, C., Roca, P. & Barbat, A. H. (1998) Flexibility-based linear dynamic analysis of complex structures 
with curved-3D members. Earthquake Engineering & Structural Dynamics, Vol. 27, No. 7, 731-747. 
[13] Timoshenko, S. (1941) strength of materials, 2nd ed.,Vol. 2, Van Nostrand. 
 
 
 


Consistent Mass and Exact Displacement Shape Function for a Tapered Curved Frame Element

http://espace.library.uq.edu.au/view/UQ:131940/B3.5.pdf

Consistent Mass and Exact Displacement Shape Function for a Tapered Curved Frame Element

Abstract

Similar works

Full text

Available Versions

University of Queensland eSpace