Page 74 - Dynamic Loading and Design of Structures

Page 53

Table 2.1 Newmark’s method in algorithmic form.

A. Initialization
1. Formation of stiffness matrix K, mass matrix M and damping matrix C
2. Initial values °U °Ü and °Ü
t
3. Assign values to time step ∆ and to parameters αand δComputation of
.
.
the following integration constants:

4. Formation of the effective stiffness matrix K, where
5. Triangularization of matrix K,

B. At each time step level
1. Computation of the effective load vector

2. Solution for displacements at time t+∆ t

t
3. Computation of accelerations and velocities at time step t +∆

induced horizontal loads applied at the storey levels. The frame is modelled as a three DOF
3
2
system and the interstorey stiffness is k=2(12EI/h ), where EI(=14.67 kN m ) is the flexural
rigidity of the columns and b is their clear height. The mass lumped at each storey is the total
2
static load pL, where L is the span, divided by the acceleration of gravity g (= 9.81 m/sec ).
We thus compute k 30.7, k =k =44.0 and M =141.0, M =132.0, M =66.0 for the stiff-nesses
3
2
1
3
2
1=
2
and masses, respectively, in units of (kN/m) and (N sec /m). Finally, we note that the structure
s own weight is included in the vertical load. The equations of dynamic equilibrium are
(2.69)