Related Variational and Energy Principles - Mechanics of Structures: Variational and Computational Methods - page 152

FIGURE 3.9
A beam element.

and one slope—at each end of a beam. Such displacements are called degrees of freedom
(DOF). It is to be expected then that the vector v of displacements for the stiffness matrix
will contain these four DOF.
For the beam element of Fig. 3.9, the four independent displacements are θ b .

Castigliano's theorem, part II can be employed to find force-displacement relations when
each displacement is in turn given the value of unity, while the other displacements are set
to zero. From the theorem, the four end displacements are given by

w a , θ a , w b ,

U i /∂

w = ∂ V a , θ =

a a

= 0

U i /∂ U i /∂ U i /∂ M b , with U i M 2 ∂ M a , w = ∂ V b , θ = ∂ /( 2 EI ) dx . Consider first the

b b

forces and moments at the ends of the beam defined by The

moment of the beam of Fig. 3.9 in terms of M a ,V a at a section at coordinate x from the left
end would be M

w = 1 , θ = θ = w = 0 .

a a b b

=− M a − V a x . Then ∂ M /∂ M a =− 1 , ∂ M /∂ V a =− x . Use Castigliano's

w = θ =

theorem, part II for 1 , 0 , giving

a a

0 ( −

2 3

1
EI

M ∂

M 1

EI

M a V a

w = 1 = dx = M a − V a x )( − x ) dx = +

a

∂ V a 2 EI 3 EI

0

(1)

2

1
EI

M ∂

M M a V a

θ = 0 = dx = +

a

∂ M a EI 2 EI

0

These two relations can be solved for M a and V a , giving

6 EI 12 EI

=− =

M a V a (2)

2 3

By definition of the stiffness coefficients, M a = k 21 ,V a = k 11 . The equilibrium conditions

applied to the element of Fig. 3.9 give

M b =− V a − M a , b =− V a (3)

Thus,

12 EI 6 EI 6 EI

M b =− +

2 =− 2 =

k 41

3

(4)

12 EI

V b =− = k 31

3

The remaining stiffness coefficients are computed in a similar manner, i.e., by using
Castigliano's second theorem applied for w = 0 and θ = 1 , w = 1 and θ = 0 , w = 0

a a b b b

and θ = 1 , with the displacements at the opposite end set equal to zero. The resulting

b