Structural Engineering Program CEE 536Structural DynamicsInstructor: Keith D. Hjelmstad

Spring 2014 Homework Problems

Homework 6

The single degree of freedom system shown at right has anonlinear restoring force r(u) and it subjected to an appliedforce f(t). The equations of motion (and initial conditions) are

Un-damped System0

( ) ( ( )) ( )(0)(0)

o

mu t r u t f tu uu v

+ ==

=

Consider the nonlinear response function to beelasto-plasticity with stiffness k and limiting forceo (symmetric in tension and compression).

Newmarks method satisfies the equation of motion at the discrete time points and approximates thevelocity and displacement with approximations to integrals. Hence, the discrete equations are

( )r um

( )u t

SDOFcorp

1 1

1 12 1

1 12

( ) 0(1 )( )

n n

n n n

n n n

ma r uv c h au b h a

+ +

+ +

+ +

+ =

= +

= +

Implement Newmarks method, including a Newton loop to solve the nonlinear equations of motion forelasto-plastic model. Explore the response of the system and the influence of nonlinearity relative to thelinear elastic response.

( )f t

2n n n

n n n n

c v h ab u hv h a

= +

= + +

1 2( ) sinf t F F t= +

Consider also the specific loading function

where F1 and F2, and are constants that describethe forcing function (i.e., a constant part and asinusoidally varying part.

( )r u

u

k kk

o

o

Slide Number 1