VM77

VM77
Transient Response to a Constant Force

Overview

Reference:

J. M. Biggs, Introduction to Structural Dynamics, McGraw-Hill Book Co., Inc., New York, NY, 1964, pg. 50, ex. E.

Analysis Type(s):

Mode-Superposition Transient Dynamic Analysis (ANTYPE = 4)

Element Type(s):

3D Elastic Beam Elements (BEAM188)

Structural Mass Elements (MASS21)

Input Listing:

vm77.dat

Test Case

A steel beam of length and geometric properties shown below, is supporting a concentrated mass, m. The beam is subjected to a dynamic load F(t) with a rise time t_r and a maximum value F₁. If the weight of the beam is considered negligible, determine the time of maximum displacement response t_max and the maximum displacement response y_max. Additionally, determine the maximum bending stress σ_bend in the beam.

Figure 108: Constant Force Problem Sketch

Material Properties

Geometric Properties

E = 30 x 10³ ksi

m = 0.0259067 kips-sec²/in

I = 800.6 in⁴

h = 18 in

= 20 ft = 240 in

F₁ = 20 kips

t_r = 0.075 sec

Analysis Assumptions and Modeling Notes

The final time of 0.1 sec allows the mass to reach its largest deflection. A static solution is done at the first load step. The integration time step (0.004 sec) is based on 1/25 of the period to allow the abrupt change in acceleration to be followed reasonably well and to produce sufficient printout for the theoretical comparison. Symmetry could have been used in the model. The time of maximum response (0.092 sec) is selected for the expansion pass calculation.

Results Comparison

		Target[1]	Mechanical APDL	Ratio
Transient	t_max , sec	0.092	0.092	1.00
Transient	y_max , in	0.331	0.335	1.01
Expansion Pass	Stress_bend , ksi	-18.6	-18.9	1.01

Based on graphical values