The Finite Element Method for the Analysis of Non-Linear and ...

Special Considerations-Geometric Stiffness
Beam Element
In the case of a beam element with bending properties in which the deformed
shape is assumed to be a cubic function due to the rotations φ i and φ j at the
ends, additional moments M i and M j are developed. The force-displacement
relationship is given by the following equation:
⎡
⎢
⎣
F i
M i
F j
M j
⎤
⎥
⎦ =
T
30L
⎡
⎢
⎣
⎤
36 3L −36 3L
3L 4L 2 −3L −L 2
⎥
−36 −3L 36 −3L ⎦
3L −L 2 −3L 4L 2
⎡
⎢
⎣
v i
φ i
v j
φ j
⎤
⎥
⎦
or FG = KGv
The well-known elastic force deformation relationship, for a prismatic beam
without shearing deformations, is
⎡ ⎤ ⎡
⎤ ⎡ ⎤
F i
12 6L −12 6L v i
⎢ M i
⎥
⎣ F j
⎦ = EI
⎢ 6L 4L 2 −6L −2L 2
⎥ ⎢ φ i
⎥
L 3 ⎣ −12 −6L 12 −6L ⎦ ⎣ v j
⎦ or FE = KEv
M j
−6L −2L 2 −6L 4L 2 φ j
Therefore, the total forces acting on the beam element will be:
F T = F E + F G = [K E + K G]v = K T v
Institute of Structural Engineering Method of Finite Elements II 52