Page 90 - Finite Element Modeling and Simulations with ANSYS Workbench

Beams and Frames 75

q
x
i L j

qL/2 qL/2
2
2
qL /12 i j qL /12
FIGURE 3.13
Conversion of a constant distributed lateral load into nodal forces and moments.

q

L L

qL qL/2

2
qL /12
L L

FIGURE 3.14
Conversion of a constant distributed lateral load on two beam elements.

3.4.4 Stiffness Matrix for a General Beam Element
Combining the axial stiffness (from the bar element), we further arrive at the stiffness
matrix for a general 2-D beam element as

i θ j θ
u i v i u j v j
 EA EA 
 L 0 0 − L 0 0 
 
 0 12 EI 6 EI 0 − 12 EI 6 EI 
 L 3 L 2 L 3 L 2 
 6 EI 4 EI I 6 EI 2 EI 
 0 2 0 − 2  (3.12)
k =  L L L L 
 EA EA 
 − L 0 0 L 0 0 
 
 0 − 12 EI − 6 EI 0 12 EI − 6 EI 
 L 3 L 2 L 3 L 2 
 
 0 6 EI 2 2EI 0 − 6EI 4EI 
  L 2 L L 2 L  