Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 586

T

T

u T u D T

For a linearly elastic solid, with σ = ED u u and δ = δ( D u u ) = δ [Chapter 2,

Eqs. (2.57) and (2.58a)], Eq. (10.1) becomes

S p δ

u T u D T ED u u dV u T u T p dS − δ W =

V δ

+

V δ

γ u dV − = 0 (10.2)

where D u and u D are operator matrices containing derivatives that operate on variables to
the right and left, respectively.
Introduce an approximate u , using the trial function approach of Chapters 4 and 6. The
interpolation function representation of the displacements u ] T = [ u vw and the second

time derivatives for the i th element are

Nv i

u =

(10.3)

N v i u =

where v i t 2 is the nodal acceleration and N contains shape functions. Then, for a

structure modeled as M elements,

= ∂

2 v i

/∂

v iT N T p dS

M

N T u D T ED u N dV v i N T N dV v i − δ W =

1 δ

+

V γ

−

V S p

i =

= 0 (10.4)

in which p is the loading, including boundary tractions. This can be written as

M

v iT k i v i m i v i p i

1 δ

( + − ) = 0 (10.5)

i =

where

m i N T N dV k i N T u D T ED u N dV

=

V γ

=

V

and

p i N T p dS =

S p

This m i is the definition of an element mass matrix . The matrix k i and the vector p i

are the

element stiffness matrix and loading vector of Chapter 6, Eqs. (6.30) and (6.36).

10.1.1 Consistent Mass Matrix

One of the most widely used mass matrices is the consistent mass matrix .Itis“consistent” in
the sense that the same shape functions are used to develop the mass matrix as are employed
for the stiffness matrix. Normally, this means that the polynomial shape functions developed
for a static response are to be used to form the acceleration (mass) for a dynamic response.
To illustrate a consistent mass matrix, consider a uniform beam. The principle of virtual
work can be expressed as (Chapter 2, Example 2.7)

u T

u D T ED u u

u T s ] 0 =

− δ =

x δ

( − ) − δ

W p dx [ 0 (10.6)