Response spectrum analysis of a simply supported beam

This problem verifies the Abaqus capability for response spectrum analysis by comparing the Abaqus results to an exact solution for a simple case.

This page discusses:

Problem description
Response spectra definition
Results and discussion
Input files
References
Tables
Figures

ProductsAbaqus/Standard

Problem description

The problem is a simply supported beam analyzed by Biggs (1964) and is shown in Figure 1. The beam has a rectangular cross-section of width 37 mm (1.458 in) and depth 355.6 mm (14 in). The mass density of the beam is 1.0473 × 10⁵ kg/m³ (0.0098 lb-s²/in⁴).

The finite element model is also shown in Figure 1. The response spectrum is applied in the vertical direction at both supports, and the response is determined based on the first mode of the model. Analyses are run using element types B21 and B23, with response spectra defined in the following section. Zero damping is specified for the problem. The beam section is defined as a beam section and a general beam section to test both specifications.

Response spectra definition

The response spectrum is defined as the peak response of a single degree of freedom spring-mass system excited by a given acceleration history applied to its base. Biggs (1964) defines the problem as having both supports moving vertically according to an acceleration history that ramps linearly from +g to −g (where g is the acceleration due to gravity) over a time period of 0.1 seconds and is zero after that. With this base acceleration history, the acceleration of the mass in the single degree of freedom spring-mass system is

\ddot{u} = g (1 - \cos ω t + \frac{1}{ω t_{d}} \sin ω t - \frac{t}{t_{d}}) for t \leq 2 t_{d},

\ddot{u} = - {\dot{u}}_{t = 2 t_{d}} ω \cos ω (t - 2 t_{d}) - u_{t = 2 t_{d}} ω^{2} \sin ω (t - 2 t_{d}) for t > 2 t_{d},

where $ω$ is the natural frequency and $2 t_{d}$ is the time of the ramp of the acceleration from +g to −g.

The solution of these two equations for the maximum acceleration as a function of frequency defines the response spectrum. This has been done for frequencies of 5., 6., 6.098, 7., and 8. Hz. The following table shows the resulting response spectrum:


FREQUENCY (Hz)	ACCELERATION (g's)
5.	2.0000
6.	1.6667
6.098	1.6399
7.	1.4286
8.	1.4530

Abaqus provides options for spectrum input in terms of acceleration, velocity, and displacement. The table above is expanded to these forms using the definitions that $v = \bar{a} / ω$ and $u = \bar{a} / ω$ ², where $\bar{a}$ is the peak acceleration (in m/s² or in/sec²), v is the peak velocity, and u is the peak displacement. The response spectra used in the four runs are shown in the Table 1. In the table the acceleration spectrum in m/s² (in/sec²) has been doubled and a compensating scale factor of 0.5 is used in the input.

Results and discussion

Biggs (1964) calculates the exact natural frequency of the first mode as 6.1 Hz, with a modal participation factor of 1.27324. Abaqus gives the first mode frequency as 6.098 Hz for the 10-element model using element type B23 and 6.0808 Hz for the model using element type B21. The corresponding modal participation factors are 1.2733 and 1.2628. Both of the Abaqus results are quite close to Biggs's values, with the cubic beam (B23) results giving better agreement—possibly because the linear beam, B21, allows transverse shear deformation, which adds flexibility to the model and, hence, reduces the stiffness.

Biggs also gives the values of the maximum displacement, bending moment, curvature, and bending stress at the beam midspan using SRSS summation. These values are used in Table 2 to check the Abaqus calculations (the stress, moment, and curvature values reported from the Abaqus runs are obtained by extrapolation of integration point values to the midspan node). The Abaqus results compare well for all four test cases.

Input files

responsespecbeam.inp: Displacement response spectrum problem.
responsespecbeam_velocity.inp: Velocity response spectrum.
responsespecbeam_acc.inp: g response spectrum.
responsespecbeam_absacc.inp: Absolute acceleration spectrum.

References

Biggs, J. M., Introduction to Structural Dynamics, McGraw-Hill, pp. 256–263, 1964.

Tables

Table 1. Response spectra definition.
Frequency		Acceleration			Velocity		Displacement
Hz	rad/sec	g's	m/s²	in/sec²	m/s	in/sec	m	in
5.	31.4159	2.0000	39.258	1545.60	.6248	24.5990	.0199	.7830
6.	37.6991	1.6667	32.716	1288.02	.4339	17.0830	.0115	.4531
6.098	38.3418	1.6399	32.190	1267.32	.4201	16.5382	.0110	.4316
7.	43.9823	1.4286	28.042	1104.02	.3188	12.5507	.0072	.2854
8.	50.2654	1.4530	28.521	1122.88	.2837	11.1695	.0056	.2222

Table 2. Response spectrum analysis results.
Model	Spectrum	Midspan displacement	Midspan stress	Midspan moment	Midspan curvature
		mm	MPa	N-m	rad/m
		(in)	(lb/in²)	(lb-in)	(rad/in)
Biggs		14.2	140.4	5.479 × 10³	3.778 × 10⁻³
Biggs		(.56)	(20,100)	(9.595 × 10⁵)	(9.595 × 10⁻⁵)
B23	Displ.	14.0	n/a	5.420 × 10³	3.738 × 10⁻³
B23	Displ.	(.549)	n/a	(9.493 × 10⁵)	(9.493 × 10⁻⁵)
B21	Vel.	14.0	n/a	5.282 × 10³	3.642 × 10⁻³
B21	Vel.	(.550)	n/a	(9.251 × 10⁵)	(9.251 × 10⁻⁵)
B23	g	14.0	139.3	5.420 × 10³	n/a
B23	g	(.550)	(19,937)	(9.493 × 10⁵)	n/a
B21	Acc.	14.0	135.8	5.282 × 10³	n/a
B21	Acc.	(.551)	(19,443)	(9.251 × 10⁵)	n/a
n/a in the table above means that this variable is not available in the run, because of the beam section definition used.

Figures