How to consider gravity in a dynamic analysis?

Question

This question may be a little simple. But it confuses me.....

In my case,the analysis is a time-history analysis of a structure under earthquake.

First, gravity analysis, and then, dynamic analysis with constant gravity load (stress and deformation obtained by dynamic analysis will superimpose on those obtained by gravity analysis). However, if the structure is damage, the superimposition is not rational.

the equilibrium equation is Mx''+Cx'+kdx = -ma - mg -R or Mx''+Cx'+kdx = -ma -R ??

is [do one step dynamic analysis (t->t+dt) and do one gravity analysis at the same time, and then superimpose them] rational?
if the 2. is not ture, How to consider gravity in a dynamic analysis?

dlzc · Accepted Answer

Dear lleshu...:...What I have seen done, is place the structure either on springs orhorizontal rollers, keep gravity constant, and apply the loading tothe base of the structure.  Never done such analysis myself...David A. Smith

freddy osbourne · Answer

i'm still learning fea. i gather the mass matix M already includes the gravity load, so all u have to include is the dynamic load on the right hand side of the equation. two, the static load and earthquake load is included in the coefficient of the factor load method. not only static and earthquake load, u have to include the wind load, snow load and hydrostatic pressure if there is any. i never got a chance to do any real analysis. it's just pure theory to me.

from Stanley.

freddy osbourne · Answer

sorry, the mass matrix only includes the mass of the structural members. so the equation of motion is mx''+cx'+kx=ma+mg+R. it's a good idea to includes all the input function at one time. also the damping coefficient is related to the mass matrix and the stiffness matrix. so the equation of motion can be further reduced to a new equation of motion so that Mx"+Kx=ma+mg+R. the input function can be break into Mx"+Kx=ma and Mx"+Kx=mg+R. u have to break it because in dynamic response, the solution of the equation of motion reasults in different mode shape start from 0, 1, 2, 3...etc. and u have to includes the first several mode shapes to find the maximums responds and then add the static solution to it.

from stanley.

freddy osbourne · Answer

sorry, when the damping coeffients is zero, the equation of motion is reduced to Mx"+Kx=ma and Mx"+Kx=mg+R.  however, when the damping coeffients is proportional to the mass matrix and stiffness matrix, the equation of motion can be solved using laplace transform.

freddy osbourne · Answer

when a building is damage, the moment of inertia of the structural member becomes the cracked moment of inertia of the structural members. if the building settles, then settlement becomes one of the boundary conditions of the displacement matrix.

from stanley.

How to consider gravity in a dynamic analysis?

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