VM134

VM134
Plastic Bending of a Clamped I-Beam

Overview

Reference:	N. J. Hoff, The Analysis of Structures, John Wiley and Sons, Inc., New York, NY, 1956, pg. 388, article 4.5.
Analysis Type(s):	Static, Plastic Analysis (ANTYPE = 0)
Element Type(s):	3D 2 node Beam Elements ( BEAM188)
Input Listing:	vm134.dat

Test Case

A wide-flanged I-beam of length , with clamped ends, is uniformly loaded as shown. Investigate the behavior of the beam at load w₁ when yielding just begins at the ends, at load w₂, when the midpoint begins to yield, and at load w₃, when pronounced plastic yielding has occurred. The beam's cross-section is shown in Figure 187: Clamped I-Beam Problem Sketch (the cross-section is not show to scale)).

Figure 187: Clamped I-Beam Problem Sketch

Material Properties

Geometric Properties

E = 29 x 10⁶ psi

E_T = 5.8 x 10⁶ psi (C = 5)

σ_y = 38,000 psi

= 144 in

b = 19.80 in

h = 10.6 in

t_f = 0.504 in

t_w = 0.001 in

w₁ = 2190 lb/in

w₂ = 3771 lb/in

w₃ = 9039 lb/in

Analysis Assumptions and Modeling Notes

The beam cross-section is modeled as an idealized section to compare with the assumptions of the analytical solution. The loading is assumed to be applied through the centroid of the element cross-section (the neutral axis). Only half the beam is modeled, taking advantage of symmetry. Classical bilinear kinematic hardening behavior is used.

Results Comparison

		Target	Mechanical APDL	Ratio
w₁ = 2190 lb/in	Midspan Deflection , in	-0.160	-0.166	1.04
	End Moment N, in-lb	-3.784 x 10⁶	-3.784 x 10⁶	1.00
	Mid Moment M, in-lb	-1.892 x 10⁶	-1.892 x 10⁶	1.00
	End Status	At Yield	At Yield	-
	Mid Status	Elastic	Elastic	-
w₂ = 3771 lb/in	Midspan Deflection , in	-0.357	-0.368	1.03
	End Moment N, in-lb	-5.98 x 10⁶	-6.01 x 10⁶	1.01
	Mid Moment M, in-lb	-3.78 x 10⁶	-3.76 x 10⁶	1.00
	End Status	Plastic	Plastic	-
	Mid Status	At Yield	At Yield	-
w₃ = 9039 lb/in	Midspan Deflection , in	-2.09	-2.16	1.03
	End Moment N, in-lb	-1.51 x 10⁷	-1.50 x 10⁷	1.00
	Mid Moment M, in-lb	-8.36 x 10⁶	-8.36 x 10⁶	1.00
	End Status	0.0200	0.0201	1.00
	Mid Status	0.0089	0.0089	1.00

Note: δ_mid(midspan deflection) is UZ at node 10.

N (fixed-end moment and M (midspan moment) are obtained from the reaction moments MY at nodes 1 and 10 respectively. The end and mid status are determined by comparing SAXL to the yield stress (σ_y).

The total end strain is obtained by adding the quantities EPELAXL and EPPLAXL for element 1 (end l). The total mid strain is obtained by adding EPELAXL and EPPLAXL for element 9 (end J).