Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 589

FIGURE 10.2
A plane frame.

The displacement vectors for each element are

w a θ a u b w b θ b ] T v 1 = [ u a

u b w b θ b u c w c θ c ] T

v 2

= [ (2)

u d w d θ d ] T
The element stiffness matrices k i in the local coordinate systems, the transformation ma-
trices T i , and the system stiffness matrix K are derived in Example 5.5. The corresponding
consistent mass matrices will be derived in this example.
Equations (10.10b and d) give expressions for the element mass matrices corresponding
to translatory [Eq. (10.10b)] and rotary [Eq. (10.10d)] inertia. Represent the sum of these
two mass matrices by m i , the total element mass matrix. For element 1:

u c w c θ c v 3 = [

1

10 − 4 = . 732 2 + 3 2 = 3 . 464 m , ρ = γ A = 7800 × 32 × = 24 . 96 kg / m ,

2

I

10 − 2

10 − 5 10 − 4

r y = / A = . 356 × / 32 × = 8 . 581 × m , (3)

60 ◦

α =