Page 88 - Finite Element Modeling and Simulations with ANSYS Workbench

Beams and Frames 73

Then, on each element, we can represent the deflection of the beam (v) using shape func-
tions and the corresponding nodal values as

 v i 
 
i θ
 
vx() = Nu = [ Nx) Nx) Nx) Nx)]   (3.7)
(
(
(
(
3
4
2
1
 
v j
 j θ 
 
which is a cubic function. Note that,
N 1 + N 3 = 1
N 2 + N L + N 4 = x
3
which implies that the rigid-body motion is represented correctly by the assumed deformed
shape of the beam.
To derive the beam element stiffness matrix, we consider the curvature of the beam,
which is
2
dv = d 2 Nu = Bu (3.8)
dx 2 dx 2

where the strain–displacement matrix B is given by

d 2
B = N = Nx () Nx Nx Nx  
′′ 4 ()
′′ 3 ()
′′ 2 ()

′′ 1
dx 2 (3.9)
 6 12 x 4 6 x 6 12 x 2 6 x 
=− 2 + 3 − + 2 2 − 3 − + 2 

 L L L L L L L L L 
Strain energy stored in the beam element is
1
T
U = ∫ σ εdV
2
V
Applying the basic equations in the simple beam theory, we have

L T L
1  My  1  My  1 1
U =  ∫ ∫  −   −  dAdx = ∫ M T Mdx
2 I  E  I  2 EI
0 A 0
L
1  dv  T  dv  1 L
2
2
=  ∫ 2 2  EI  2  dx = ∫ (Bu ) T EI(Bu ) dx
2  dx   dx  2
0 0
1  L L 
= u  ∫ B T EIB dx u
T

2  
0  