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.18.3 Dynamic Analysis

Available dynamic analysis facilities include solution of the free vibration problem

(eigenproblem), response spectrum analysis and forced vibration analysis.

Solution of the Eigenproblem

The eigenproblem is solved for structure frequencies and mode shapes considering a lumped

mass matrix, with masses at all active d.o.f. included. Two solution methods are used: thesubspace iteration method for almost all problems, and the determinant search method for very

small problems.

Mass Modeling

The natural frequencies and mode shapes of a structure are the primary parameters that affect the

response of a structure under dynamic loading. The free vibration problem is solved to extractthese values. Since no external forcing function is involved, the natural frequencies and mode

shapes are direct functions of the stiffness and mass distribution in the structure. Results of the

frequency and mode shape calculations may vary significantly depending upon the massmodeling. This variation, in turn, affects the response spectrum and forced vibration analysis

results. Thus, extreme caution should be exercised in mass modeling in a dynamic analysis

problem.

In STAAD, all masses that are capable of moving should be modeled as loads applied in allpossible directions of movement. Even if the loading is known to be only in one direction there is

usually mass motion in other directions at some or all joints and these mass directions (loads inweight units) must be entered to be correct. Joint moments that are entered will be considered tobe weight moment of inertias (force-length2 units).

Please enter selfweight, joint and element loadings in global directions with the same sign as

much as possible so that the masses do not cancel each other.

Member/Element loadings may be used to generate joint translational masses. Note that

member end joint moments that are generated by the member loading (including concentratedmoments) are discarded as irrelevant to dynamics. Enter mass moments of inertia, if needed, at

the joints as joint moments.

STAAD uses a diagonal mass matrix of 6 lumped mass equations per joint. The selfweight or

uniformly loaded member is lumped 50% to each end joint without rotational mass moments ofinertia. The other element types are integrated but roughly speaking the weight is distributed

equally amongst the joints of the element.

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The members/elements of finite element theory are simple mathematical representations of

deformation meant to apply over a small region. The FEA procedures will converge if you

subdivide the elements and rerun; then subdivide the elements that have significantly changedresults and rerun; etc. until the key results are converged to the accuracy needed.

An example of a simple beam problem that needs to subdivide real members to better representthe mass distribution (and the dynamic response and the force distribution response along

members) is a simple floor beam between 2 columns will put all of the mass on the columnjoints. In this example, a vertical ground motion will not bend the beam even if there is a

concentrated force (mass) at mid span.

In addition, the dynamic results will not reflect the location of a mass within a member (i.e.

the masses are lumped at the joints). This means that the motion, of a large mass in the

middle of a member relative to the ends of the member, is not considered. This may affect

the frequencies and mode shapes. If this is important to the solution, split the member into

two. Another effect of moving the masses to the joints is that the resulting shear/moment

distribution is based as if the masses were not within the member. Note also that if one endof a member is a support, then half of the that member mass is lumped at the support and

will not move during the dynamic response.

Damping Modeling

Damping may be specified by entering values for each mode, or using a formula based on the

first two frequencies, or by using composite modal damping. Composite modal damping permits

computing the damping of a mode from the different damping ratios for different materials(steel, concrete, soil). Modes that deform mostly the steel would have steel damping ratio,

whereas modes that mostly deform the soil, would have the soil damping ratio.

Response Spectrum Analysis

This capability allows the user to analyze the structure for seismic loading. For any supplied

response spectrum (either acceleration vs. period or displacement vs. period), jointdisplacements, member forces, and support reactions may be calculated. Modal responses may

be combined using one of the square root of the sum of squares (SRSS), the complete quadratic

combination (CQC), the ASCE4-98 (ASCE), the Ten Percent (TEN) or the absolute (ABS)methods to obtain the resultant responses. Results of the response spectrum analysis may be

combined with the results of the static analysis to perform subsequent design. To account for

reversibility of seismic activity, load combinations can be created to include either the positive or

negative contribution of seismic results.