Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 145

with T as the total kinetic energy:

1

2

T = m (u u + hș) [4.14]

g

2

and V is the potential energy:

Vku

2

2

k u T k

2

[4.15]

h0 T

2

and GW is the work of non-conservative (damping) stresses:

įW = Cuįu + C u įuCșįș [4.16]

h0 0 ș

Lagrange's equations are written as follows:

dT § w · § w T · § w V · G W

[4.17]

¨ ¸ ¨ ¸ ¨ ¸

dt w q w q w q G q

© ¹ © ¹ © ¹ i i i i

Using the previous notations and taking the acceleration, speed and displacement
relationships into account, this means:

2

x = iȦx, x = Ȧ x [4.18]

When we introduce the critical damping ratios, we obtain:

iȦC
ȟ =, ȟ

iȦC iȦC

h ș

ȟ [4.19]

h ș

2k 2k 2k

h ș

- Z T [ Z
° Z T [ Z

2 2

m u uh)k 1 i umu
m u uh)k 1 i u mu

0 g

2 2

[4.20]

®
° Z T [T Z

0 h h 0 g

2 2

mh (u u h ) k (1 2i ) mh u

¯

0 T T g