Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 311

Locking of the Timoshenko Beam and Typical Remedies

Shear locking has been observed when computations for the Timoshenko beam are
performed with finite elements which are piecewise polynomials of low degree.
The locking of P 1 elements is easily verified on the basis of (3.45), and negative
as well as positive results can be completely provided.
We will see in §5 that the stored energy of a beam is given by

b b

t − 2
2

1
2

(θ ) 2 dx + (w − θ) 2 dx, (θ, w) : = ( 3 . 51 )

0 0

if b is the length of the beam and t is the thickness (multiplied by a correction
factor). Here, the Lame constants are abandoned since they enter only as a multi-
plicative factor in the 1-dimensional case. The rotation θ and the deflection w are
in H 0 ( 0 ,b) and in the above setting,

= w − θ. B(θ, w) : ( 3 . 52 )

Given g ∈ H 0 ( 0 ,b) , we obtain a pair of functions with B(θ, w) = 0 by defining

x) b
0

6
b 3 x(b

θ(x) : = g(x) − − g(ξ)dξ,

( 3 . 53 )

x

w(x) : = θ(ξ)dξ.

0

Hence, the kernel of B is infinite dimensional.
Now assume that the interval [0 ,b ] is divided into subintervals of length h
and that θ h ,w h ∈ M

1

0 , 0 . Note that
ξ + h ξ + h

h 3 h 2

6

(αx + β) 2 dx ≥

6 α 3

α 2 dx, =

ξ ξ

whenever α, β ∈ R . From this inequality it follows that on each subinterval of the

partition

ξ + h ξ + h

h 2
6

(w h − θ h ) 2 dx ≥ (θ h ) 2 dx.

ξ ξ

After summing over all subintervals we have

h
3 | θ h | 1 .

w h − θ h 0 ≥

( 3 . 54 )

w h −

Friedrichs' inequality and the triangle inequality yield +

θ h 0 ≤ w h − θ h 0 + c | θ h | 1 . We estimate h | θ h | 1 by (3.54) to obtain h | w h | 1 ≤
c w h − θ h

| w h | ≤ θ h 1 0

0 . Combining the last inequalities we conclude that

w h − θ h 0 ≥ ch( θ h 1 + w h 1 ).

( 3 . 55 )