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An efficient 3D Timoshenko beam element with consistant shape functions

Article in Advances in Theoretical and Applied Mechanics · October 2008

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 Adv. Theor. Appl. Mech., Vol. 1, 2008, no. 3, 95 - 106

An Efficient 3D Timoshenko Beam Element

with Consistent Shape Functions
Yunhua Luo

Department of Mechanical & Manufacturing Engineering

University of Manitoba, Winnipeg, R3T 5V6, Canada
luoy@cc.umanitoba.ca

Abstract
An efficient three-dimensional (3D) Timoshenko beam element is
presented in this paper. First, the homogeneous Euler-Lagrangian equa-
tions governing a 3D Timoshenko beam are derived by introducing plane
cross-section assumption into the kinematic description of a 3D solid
continuum; then, consistent shape functions for a 3D 2-node Timo-
shenko beam element are constructed from the general solutions to the
homogeneous Euler-Lagrangian equations. Numerical tests show that
the developed 3D Timoshenko beam element is completely free from
shear locking, and furthermore the performance of the element in con-
vergence is superior to the isoparametric Timoshenko beam element
with reduced integration. For the benchmark used in the test, using
one of the developed beam element can produce exact solution.

Keywords: 3D Timoshenko beam, Euler-Lagrangian equation, consistent

shape function

1 Introduction
Under the assumption that a plane cross-section remains plane after deforma-
tion, there are basic two categories of beam theories: Euler-Bernoulli beam
and Timoshenko beam. The difference between them is whether, or not, a
cross-section plane of the beam is always perpendicular to the beam axis in
deformation. Experiments have demonstrated that Euler-Bernoulli assump-
tion is more accurate for thin beams; while for a deep beam, a cross-section
plane is not necessarily perpendicular to the beam axis, and shear force is a
more dominant factor in the damage of material. As it is difficult to put a strict
line between the two beam theories in practice, it is highly desired that the two
beam theories can be unified [3, 2], both theoretically and in implementation.
96 Y. Luo

Theoretically, Timoshenko beam theory is more general, and Euler-Bernoulli

theory can be considered as a special case of Timoshenko assumption by en-
forcing the constraint condition between deflection and cross-section rotation.
But in implementation, the story is different. It is well known that the con-
ventional 2-node isoparametric Timoshenko beam element [1, 8] suffers from
the so-called shear locking. Although reduced integration [12] can alleviate
shear locking to some extent, the performance of the element is still not op-
timal. It is noticed that a very efficient 2D Timoshenko beam element was
developed independently in [4, 5, 6] and in [11] at nearly the same time period
and based on a very similar way, i. e. by constructing shape functions from
the general solutions to the homogeneous Euler-Lagrange equations. It was
further demonstrated in [6] that shear locking is an extreme case of numerical
deficiency arising from neglecting inter-dependence between field functions.
In this paper, based on the work of [4, 5, 6], an efficient three-dimensional
2-node Timoshenko beam element is developed. The layout of this paper is as
follows. In Section 2, the homogeneous Euler-Lagrangian equations governing
a 3D Timoshenko beam is revisited based on the degeneration of a 3D solid
continuum; In Section 3, consistent shape functions of a 2-node 3D Timoshenko
beam element are constructed; Numerical tests are presented in Section 4 and
concluding remarks are given in Section 5.

2 Homogeneous Euler-Lagrangian equations gov-

erning 3D Timoshenko beam
In this section, homogeneous Euler-Lagrangian equations governing a three-
dimensional Timoshenko beam is derived from the kinematic description of a
3D solid continuum by introducing the assumption of plane cross-section of
beam in deformation.
Consider a three-dimensional 2-node beam element depicted in Fig. 1. The

ψ1 v1
ψ2 v2
ϕ1
u1 ϕ2 u2 x

θ1 θ2
w1 w2
z

Figure 1: A 3-D 2-node beam element

An efficient 3D Timoshenko beam element 97

element has two nodes, each node has six degrees of freedom (DOF), i.e., three
translations and three rotations. There is no external force acting on the beam
in between the two nodes. To describe the motion of a typical particle p in the
beam, we split its motion into two parts as shown in Fig. 2, the translation
y

p’
t’
r’ c’

rc’

p
r
t x
rc c

Figure 2: Kinematics of a typical particle in a 3D beam

of point c at the beam axis and the rotation of vector t defined by p and c.
Vectors t and r c are orthogonal to each other in the undeformed configuration.
The motion of point c is described by three translations uc , vc and wc along the
three axes, respectively. According to Euler-Chasles’ theorem, the rotation of
vector t can be realized by a rotational operator R [9, 10], i. e.

t = R t (1)

where
1 2
R = exp(Ψ̃) = I 3 + Ψ̃ + Ψ̃ + · · · (2)
2
⎡ ⎤
0 −θ ψ
Ψ̃ = ⎣ θ 0 −ϕ ⎦ (3)
−ψ ϕ 0
ϕ, ψ and θ are the rotational angles of vector t around the three axes x, y and
z, respectively, as shown in Fig. 2.
The position vectors of particle p before and after deformation are, respec-
tively,
r = rc + t
    (4)
r = rc + t = r c + R t
98 Y. Luo

So the displacement vector of p is


u = r − r = uc + (R − I 3 ) t (5)

where I 3 is a 3 × 3 unit matrix; u = [ u v w ]T and uc = [ uc vc wc ]T .

If it is assumed that the rotation angles ϕ, ψ and θ are small, and in the
Taylor series in Eq. (2), only the first two terms are retained. Therefore, the
displacements of particle p in Eq. (5) are now expressed as
⎧
⎨ u = uc − θ y + ψ z
v = vc − ϕ z (6)
⎩
w = wc + ϕ y

If it is further assumed that the strains are small so that a linear strain-
displacement relation can be adopted, the three non-trivial strains for a 3D
beam have the following expressions
⎧
⎪ ∂u   
⎪
⎪ xx = = uc − θ y + ψ z
⎪
⎨ ∂x
∂v ∂u  
xy = + = vc − ϕ z − θ (7)
⎪
⎪ ∂x ∂y
⎪
⎪ ∂w ∂u
⎩  = +
 
= wc + ϕ y + ψ
xz
∂x ∂z

 d
where ( ) = .
dx
From now on, the subscript c is dropped, and u, v and w are now used to
represent the displacements of a point at the beam axis.
Six generalized strains are introduced. They are, respectively, axial strain
(¯), curvature in xoy plane (κ̄y ), curvature in xoz plane (κ̄z ), shear strains (γy
and γz ), and torsional rate (κ̄x ),
⎧ 
⎪
⎪ 
¯ = u
⎪
⎪ 
⎪
⎪ κ̄y = −θ
⎨ 
κ̄z = ψ
 (8)
⎪
⎪ γ̄y = v − θ
⎪
⎪ 
⎪
⎪ γ̄z = w + ψ
⎩ 
κ̄x = ϕ

With (8), Eq. (7) is reformed as

⎧
⎨ xx = ¯ + κ̄y y + κ̄z z
xy = γ̄y − κ̄x z (9)
⎩
xz = γ̄z + κ̄x y
An efficient 3D Timoshenko beam element 99

The virtual strain energy in the beam body is calculated by

δV = (σxx δxx + σxy δxy + σxz δxz ) dΩ

Ω
L (10)
= (N δ¯ + Mz δκ̄y + My δκ̄z + Qy δγ̄y + Qz δγ̄z + Mx δκ̄x ) dx
0

where L is the length of the beam element; N , My , Mz , Qy , Qz and Mx are

internal forces that are conjugate, in energy, with the generalized strains, ¯,
κ̄y , κ̄z , γ̄y , γ̄z and κ̄x , respectively,
⎧
⎪
⎪ N= σxx dA = EA ¯ + EAy κ̄y + EAz κ̄z
⎪
⎪
⎪
⎪ A
⎪
⎪
⎪
⎪ Mz = σxx y dA = EAy ¯ + EIy κ̄y + EJ κ̄z
⎪
⎪
⎪
⎪ A
⎪
⎪
⎨ My = σxx z dA = EAz ¯ + EJ κ̄y + EIz κ̄z
A
⎪
⎪ σxy dA = k GA γ̄y − kGAz κ̄x
⎪
⎪ Qy =
⎪
⎪ A
⎪
⎪
⎪
⎪ Qz = σxz dA = k GA γ̄z + kGAy κ̄x
⎪
⎪
⎪
⎪ A
⎪
⎪
⎩ Mx = (σxz y − σxz z) dA = −kGAz γ̄y + kGAy γ̄z + kG(Iy + Iz )κ̄x
A
(11)
In the above expressions, A, Ay , Az , Iy , Iz and J are beam cross-section
parameters; k is the coefficient accounting for different shapes of beam cross-
section.
A= dA, J= y z dA
A A

Ay = y dA, Az = z dA
A A

Iy = y 2 dA, Iz = z 2 dA
A A
Equation (11) can be put in a compact form
p = Dρ (12)
in which
p = [ N My Mz Qy Qz M x ]T
ρ = [ ¯ κ̄y κ̄z γ̄y γ̄z κ̄x ]T
⎡ ⎤
EA EAy EAz 0 0 0
⎢ EAy EIy EJ 0 0 0 ⎥
⎢ ⎥
⎢ EAz EJ EIz 0 0 0 ⎥
D=⎢
⎢
⎥
⎥
⎢ 0 0 0 k GA 0 −kGAz ⎥
⎣ 0 0 0 0 k GA k GAy ⎦
0 0 0 −k GAz k GAy k G(Iy + Iz )
100 Y. Luo

By introducing the relations in (8) into Eq. (10), the virtual strain energy
is now expressed by displacements, rotations and their derivatives,

L
   
δV = {(EA u − EAy θ + EAz ψ ) δu
0    
−(EAy u − EIy θ + EJ ψ ) δθ
   
+(EAz u − EJ θ + EIz ψ ) δψ (13)
  
+[k GA (v − θ) − k GAz ϕ ] (δv − δθ)
  
+[k GA (w + ψ) + k GAy ϕ ] (δw + δψ)
   
+[k GAy (w + ψ) − k GAz (v − θ) + k G (Iy + Iz ) ϕ ] δϕ } dx

With integrating by parts and collecting the terms with respect to δu, δv,
δw, δϕ, δψ and δθ, the expression in ( 13) becomes

L
  
δV = {(EA u − EAy θ + EAz ψ ) δu
0   
+[k GA (v − θ ) − k GAz ϕ ] δv
  
+[k GA (w + ψ ) + k GAy ϕ ] δw
    
+[−EAy u + EIy θ − EJ ψ − k GA (v − θ) + k GAz ϕ ] δθ
    
+[EAz u − EJ θ + EIz ψ + k GA(w + ψ) + k GAy ϕ ] δψ
   
+[k GAy (w + ψ ) − k GAz (v − θ ) + k G (Iy + Iz )] δϕ} dx + δVb
(14)
2
d
where () = 2 ; δVb collects all the terms related to boundary values.
dx
Based on the principle of virtual work, the stored strain energy is equal to
the virtual work done by external forces. As it is assumed that no external
force is applied on the element, and external virtual work only contributes
to the inhomogeneous terms, therefore, the homogeneous Euler-Lagrangian
equations are obtained as

⎧   
⎪
⎪ EA u − EAy θ + EAz ψ = 0
⎪
⎪   
⎪
⎪ k GA (v − θ ) − k GAz ϕ = 0
⎨   
k GA (w + ψ ) + k GAy ϕ = 0
     (15)
⎪
⎪ −EAy u + EIy θ − EJ ψ − k GA(v − θ) + k GAz ϕ = 0
⎪
⎪     
⎪
⎪ EAz u − EJ θ + EIz ψ + k GA (w + ψ) + k GAy ϕ = 0
⎩     
k GAy (w + ψ ) − k GAz (v − θ ) + k G (Iy + Iz ) ϕ = 0

If beam cross-section is symmetrical, or the element local coordinate axes are

selected to pass through the cross-section shear center, then, Ay = 0, Az = 0,
and J = 0, the homogeneous Euler-Lagrangian equations in (15) are simplified
An efficient 3D Timoshenko beam element 101

as ⎧ 
⎪
⎪ EA u = 0
⎪
⎪  
⎪
⎪ k GA (v − θ ) = 0
⎨  
k GA (w + ψ ) = 0
  (16)
⎪ EIy θ − k GA(v − θ) = 0
⎪
⎪
⎪  
⎪
⎪ EIz ψ + k GA (w + ψ) = 0
⎩ 
k G (Iy + Iz ) ϕ = 0

3 Consistent Shape Functions and Explicit El-

ement Stiffness Matrix
From observing the homogeneous Euler-Lagrangian equations in (15) or (16),
it can be concluded that the field functions, u, v, w, θ, ψ and ϕ are dependent
to each other, as the equations are coupled. Nevertheless, in the conven-
tional Finite Element Method, these field functions are assumed completely
independent and their variations over an element are interpolated by simple
shape functions. It is demonstrated in [6, 7] that the ignorance of the inter-
dependence between field functions is the fundamental reason responsible for
various numerical deficiencies such as shear locking and membrane locking ap-
peared in finite elements. An effective way to consider the inter-dependence
of field functions is the use of general solutions to the homogeneous Euler-
Lagrangian equations to construct element shape functions. With the use of
symbolic software such as Maple, it is possible to obtain general solutions to
the differential equations in (15), but the obtained mathematical expressions
are too complex to be displayed here. Instead, as a demonstration, the sim-
plified version of the equations in (16) are solved, and the obtained general
solutions are used to construct shape functions.
The general solutions to (16) are obtained as
⎧
⎪
⎪ u(x) = c7 + c8 x
⎪
⎪ v(x) = c1 + c2 x + c3 x2 + c4 x3
⎪
⎪
⎨ w(x) = c + c x + c x2 + c x3
9 10 11 12
(17)
⎪
⎪ ϕ(x) = c 5 + c 6 x
⎪
⎪
⎪
⎪ θ(x) = (c2 − 6k EI y
) + 2 c3 x + 3 c4 x2
⎩ GA
6 EIz
ψ(x) = (−c10 + k GA ) − 2 c11 x − 3 c12 x2
where the twelve coefficients c1 , c2 , · · ·, c12 can be expressed with element
nodal unknowns, u1 , v1 , · · ·, ψ2 , by enforcing the following element ‘boundary
conditions’,
u(0) = u1 , v(0) = v1 , w(0) = w1 , ϕ(0) = ϕ1 , θ(0) = θ1 , ψ(0) = ψ1
u(L) = u2 , v(L) = v2 , w(L) = w2 , ϕ(L) = ϕ2 , θ(L) = θ2 , ψ(L) = ψ2
(18)
102 Y. Luo

After solving the algebraic equations resulted from the above conditions, the
element field functions are now approximated as
⎧
⎪
⎪ u = N1 u1 + N2 u2
⎪
⎪
⎪
⎪ v = Hv1 v1 + Hθ1 θ1 + Hv2 v2 + Hθ2 θ2
⎨
w = Hw1 v1 + Hψ1 ψ1 + Hw2 w2 + Hψ2 ψ2
(19)
⎪
⎪ ϕ = N1 ϕ1 + N2 ϕ2
⎪
⎪
⎪
⎪ θ = Gv1 v1 + Gθ1 θ1 + Gv2 v2 + Gθ2 θ2
⎩
ψ = Gw1 w1 + Gψ1 ψ1 + Gw2 w2 + Gψ2 ψ2
where the element shape functions N1 (ξ), N2 (ξ), · · ·, Gψ2 (ξ) have the following
expressions,
⎧
⎪
⎪ N1 = 1 − ξ
⎪
⎪
⎪
⎪ N2 = ξ
⎪
⎪
⎪
⎪ Hv1 = βy (2ξ 3 − 3ξ 2 + αy ξ + 1 − αy )
⎪
⎪
⎪
⎪ Hv2 = βy (−2ξ 3 + 3ξ 2 − αy ξ)
⎪
⎪
⎪
⎪ Hw1 = βz (2ξ 3 − 3ξ 2 + αz ξ + 1 − αz )
⎪
⎪ 3
⎪
⎪ Hw2 = βz (−2ξ
 + 3ξ 2 − αz ξ) 
⎪
⎪
⎪
⎪ Hθ1 = Lβy ξ 3 + ( 12 αy − 2)ξ 2 + (1 − 12 αy )ξ
⎪
⎪
⎪
⎪ Hθ2 = Lβy ξ 3 − (1 + 12 αy )ξ 2 + ( 12 αy )ξ
⎨ H = Lβ ξ 3 + ( 1 α − 2)ξ 2 + (1 − 1 α )ξ 
⎪
ψ1 z 2 z 2 z
3 1 2
Hψ2 = Lβz ξ − (1 + 2 αz )ξ + ( 2 αz )ξ1 (20)
⎪
⎪
⎪
⎪
⎪
⎪ Gv1 = 6βLy (ξ 2 − ξ)
⎪
⎪
⎪
⎪ Gv2 = 6βLy (−ξ 2 + ξ)
⎪
⎪
⎪
⎪ Gw1 = 6βLz (ξ 2 − ξ)
⎪
⎪
⎪
⎪ Gw2 = 6βLz (−ξ 2 + ξ)
⎪
⎪
⎪
⎪ Gθ1 = βy [3ξ 2 + (αy − 4)ξ + 1 − αy ]
⎪
⎪
⎪
⎪ Gθ2 = βy [3ξ 2 − (αy + 2)ξ]
⎪
⎪
⎪
⎪ Gψ = βz [3ξ 2 + (αz − 4)ξ + 1 − αz ]
⎪
⎩ G 1 = β [3ξ 2 − (α + 2)ξ]
ψ2 z z

with
x 12EIy 1 12EIz 1
ξ= , αy = , βy = , αz = , βz = (21)
L kGAL2 1 − αy kGAL2 1 − αz
In the above way, inter-dependence between element field functions is fully
considered in the construction of shape functions. The obtained shape func-
tions are thus consistent. Once element shape functions are constructed, it
is straightforward to calculate element stiffness matrix. For uniform beam
with symmetrical cross-section and linear elastic material, element stiffness
matrix can be explicitly calculated with analytical integration. The explicitly
derived expressions of element stiffness matrix for a three-dimensional 2-node
Timoshenko beam element are provided in Appendix A.
An efficient 3D Timoshenko beam element 103

4 Numerical Tests
The developed 3D 2-node Timoshenko beam element was tested with bench-
marks to examine its immunity from shear locking and its performance in
convergence. The obtained results are compared with those from the conven-
tional isoparametric Timoshenko beam elements. The results produced by the
developed Timoshenko beam element is denoted as B3D2 in the following fig-
ures. Exact and reduced integration are abbreviated as, respectively, ’E.I.’ and
’R.I.’ in the figures.
The cantilever beam shown in Fig. 3 was used in the test. The beam is
clamped at its left end and affected by a shear force at its right end. The
beam is made from isotropic, homogeneous material with Young’s modulus

11
00 1
0
0
1
00
11 0
1
00
11 0
1
00
11
00
11
00
11
00
11
00
11

Figure 3: Cantilever beam under transverse force

10 elements
1

0.9

0.8
Normalized displacement at loaded end

0.7

0.6

0.5

0.4

0.3

0.2

B3D2
0.1 2−node isoparametric beam element with R.I.
2−node isoparametric beam element with E.I.

0
0.5 1 1.5 2 2.5 3 3.5 4
Aspect ratio log ( L / h )

Figure 4: Immunity from shear locking test result

E = 1.0M P a and Poisson’s ratio ν = 0.3. The first test was designed to test
the immunity of shear locking. In the test, the beam was discretized using
ten beam elements; all other parameters and conditions are fixed except beam
104 Y. Luo

Aspect ratio log ( L / h ) = 3.9752

1

0.9

0.8

Normalized displacement at loaded end

0.7

0.6

0.5

0.4

0.3

0.2

B3D2
0.1 2−node isoparametric beam element with R.I.
2−node isoparametric beam element with E.I.

0
0 5 10 15 20 25 30 35 40 45 50
Number of elements

Figure 5: Convergence of developed 3D Timoshenko beam element

depth; The depth of the beam was gradually reduced so that the aspect ratio
of the beam is becoming larger and larger. The aspect ratio of a beam is
defined as the logarithm of the ratio between the beam length (L) and the
beam depth (h), i. e., log( Lh ). The obtained displacements at the right tip
of the beam are normalized with analytical solutions and shown in Fig. 4.
As expected, the conventional 2-node Timoshenko beam element with exact
integration suffers from shear locking; therefore, the predicted deflection at
the loaded tip becomes smaller and smaller with the beam becoming more and
more slender. Although the use of reduced integration can greatly improve the
situation, the performance of the element is still not optimal.

In the second test, the performance of the developed beam element in

convergence was examined. In this test, all the parameters and conditions in-
cluding the aspect ration of the beam were fixed, the number of beam elements
used to simulate the beam was gradually increased. The obtained displace-
ments at the loaded beam end are normalized by analytical solutions and
displayed in Fig. 5. From the results, it can be observed that the convergence
of the developed Timoshenko beam element is superior than the conventional
Timoshenko beam element with reduced integration. It is not surprising that it
can be observed from the results that using the developed Timoshenko beam
element, one element can produce an exact solution, as the element shape
functions are constructed from the analytical solutions of the homogeneous
Euler-Lagrangian equations.
An efficient 3D Timoshenko beam element 105

5 Concluding remarks
An efficient 3D 2-node Timoshenko beam element was developed. The shape
functions of the element are constructed from the general solutions to the
homogeneous Euler-Lagrangian equations. Numerical results show that the
developed Timoshenko beam element is not only free from shear locking, but
also has a superior convergence rate compared with its conventional isopara-
metric counterpart.

Appendix A
Explicit expressions of entries in the element stiffness matrix of a three-dimensional
2-node Timoshenko beam element, derived from consistent shape functions. The
other entries that are not listed here are zeros.
k1,1 = k7,7 = −k1,7 = EA
L
12kGA EIy (12EIy +kGAL2 )
k2,2 = −k2,8 = L(12EIy −kGAL2 )2
6kGA EIy (12EIy +kGAL2 )
k2,6 = k2,12 = (12EIy −kGAL2 )2
12kGA EIz (12EIz +kGAL2 )
k3,3 = k3,9 = L(12EIz −kGAL2 )2
2)
k3,5 = k3,11 = − 6kGA(12EI
EIz (12EIz +kGAL
z −kGAL )
2 2

G(Iy +Iz )
k4,4 = k7,7 = −k4,10 = L
4EIz [(kGA)2 L4 +3kGAL2 EIz +36(EIz )2 ]
k5,5 = L(12EIz −kGAL2 )2
6kGA EIz (12EIz +kGAL2 )
k5,9 = (12EIz −kGAL2 )2
2 2 4 2
k5,11 = − 2EIz [72(EIzL(12EI
) −(kGA) L −30kGAL
2 2
z −kGAL )
EIz ]

4EIy [(kGA)2 L4 +3kGAL2 EIy +36(EIy )2 ]

k6,6 = L(12EIy −kGAL2 )2
6kGA EIy (12EIy +kGAL2 )
k6,8 = − (12EIy −kGAL2 )2
2EIy [−(kGA)2 L4 −30kGAL2 EIy +72(EIy )2 ]
k6,12 = − L(12EIy −kGAL2 )2
12kGA EIy (12EIy +kGAL2 )
k8,8 = L(12EIy −kGAL2 )2
6kGA EIy (12EIy +kGAL2 )
k8,12 = − (12EIy −kGAL2 )2
12kGA EIz (12EIz +kGAL2 )
k9,9 = L(12EIz −kGAL2 )2
6kGA EIz (12EIz +kGAL2 )
k9,11 = (12EIz −kGAL2 )2
 106 Y. Luo

4EIz [(kGA)2 L4 +3kGAL2 EIz +36(EIz )2 ]

k11,11 = L(12EIz −kGAL2 )2
4EIy [(kGA)2 L4 +3kGAL2 EIy +36(EIy )2 ]
k12,12 = L(12EIy −kGAL2 )2

References
[1] R. Davis, R.D. Henshell, and G.B. Warburton. A Timoshenko beam
element. Journal of Sound and Vibration, 22:475–487, 1972.
[2] Z. Friedman and J.B. Kosmatka. Improved two-node Timoshenko beam
finite element. Computers and Structures, 47:473–481, 1993.
[3] A. W. Lees and D. L. Thomas. Unified Timoshenko beam finite element.
Journal of Sound and Vibration, 80:355–366, 1982.
[4] Y. Luo. Field consistence approach with application to the development
of finite element. In The 10-th Nordic Conference on Computational Me-
chanics, Tallinn, Estonia, 1997.
[5] Y. Luo. On Shear Locking in Finite Elements. KTH Bulletin (Licentiate
Thesis), Stockholm, 1997.
[6] Y. Luo. Explanation and elimination of shear locking and membrane
locking with field consistence approach. Comput. Methods Appl. Mech.
Engrg., 162:249–269, 1998.
[7] Y. Luo. On Some Finite Element Formulations in Structural Mechanics.
KTH Bulletin (Doctorial Thesis), Stockholm, 1998.
[8] S. Mukherjee and G. Prathap. Analysis of shear locaking in Timoshenko
beam elements using the function space approach. Communications in
Numerical Methods in Engineering, 17:385–393, 2001.
[9] B. Nour-Omid and C.C. Rankin. Finite rotation analysis and consis-
tent linearization using projectors. Comput. Methods Appl. Mech. Engrg.,
93:353–384, 1991.
[10] C. Pacoste and A. Eriksson. Beam elements in instability problems. Com-
put. Methods Appl. Mech. Engrg., 144:163–197, 1997.
[11] J.N. Reddy. On locking-free shear deformable beam finite elements. Com-
put. Methods Appl. Mech. Engrg., 149:113–132, 1997.
[12] T. Yokoyama. A reduced integration Timoshenko beam element. Journal
of Sound and Vibration, 169:411–418, 1994.
Received: April 2, 2008

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