# V. Steady State - With Damping

Determine amplitude of motion of a beam due to harmonic force.

## Reference

Paz, Mario. 1985. Structural Dynamics Theory and Computation. 2nd Edition. New York, NY:Van Nostrand Reinhold. pp54-55, Example 3.5

## Problem

A 10' long, simply-supported beam simulating a damped oscillator.

• Material modulus of elasticity, E = 30,000 ksi
• Section moment of inertia, Iz = 120 in4
• Weight applied at mid-span, Fy = 3.86 kips

A steady state analysis is performed for the harmonic forcing function 7000 sin 60t with the forcing frequency = ω = 60 rad/sec = 9.5493 cyc/sec & the magnitude = 7000 lb = 7 kip applied at the midpoint of the beam. Damping is considered for this problem and the value of 0.1 is specified.

## Comparison

Steady State Amplitude (in) 0.1075 0.10749 negligible

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Steady State - With Damping.std is typically installed with the program.

``````STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 31-Mar-17
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
SET SHEAR
JOINT COORDINATES
1 0 0 0; 2 10 0 0; 3 5 0 0;
MEMBER INCIDENCES
1 1 3; 2 3 2;
START USER TABLE
TABLE 1
UNIT FEET KIP
PRISMATIC
Beam
0.001 0.005787 0.001 0.001 0.001 0.001 0 0
END
DEFINE MATERIAL START
ISOTROPIC XYZ
E 4.32e+006
END DEFINE MATERIAL
MEMBER PROPERTY
1 2 UPTABLE 1 Beam
CONSTANTS
MATERIAL XYZ ALL
SUPPORTS
1 2 PINNED
3 FY -3.86
END DEFINE REFERENCE LOADS
R1 1.0
MODAL CALCULATION REQUESTED
PERFORM STEADY STATE ANALYSIS
STEADY FORCE FREQ 9.5493 DAMP 0.1
```   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = PLANE