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Dynamic analysis of Earthquake

The earthquake problem is solved by the means of dynamic analysis of a continuous body. At each point x and each time instant t the following differential equation is satisfied:

where:

c

-

coefficient of viscous damping

ρ

-

mass density

u

-

displacement

-

velocity

-

accelerationí

-

gradient

σ

-

stress

The stresses are provided by:

where:

Dijkl

-

material stiffness tensor

εkl

-

strain tensor

εklpl

-

plastic strain tensor

The strains are equal to the symmetric part of the displacement gradient:

where:

ui, j

-

derivative of the i-th component of the displacement in the direction of the j-axis.

Finite element discretization of the equations of motion gives the system of ordinary differential equations in the form:

where:

M

-

mass matrix

C

-

damping matrix

K

-

stiffness matrix

F(t)

-

vector of time-dependent loading

r(t)

-

vector of nodal displacements

As for time integration, the user may choose between the Newmark method and the Hilber-Hughes-Taylor Alpha method.

Further details are available in the theoretical manual on our website.

Literature:

Z. Bittnar, P. Řeřicha, Metoda konečných prvků v dynamice konstrukcí, SNTL, 1981.

T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice Hall, INC., Engelwood Clifts, New Jersey 07632, 1987.

Z. Bittanr, J. Šejnoha, Numerical methods in structural engineering, ASCE Press, 1996.

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