# SS-V: 5050 Pure Bending of a Plastic Beam

Test No. VNL06Find elastic core and maximum displacements in a cantilever beam loaded with a moment.

## Definition

Beam dimensions are b x h x L.

Where,
Dimension
Value
b
=10 mm
h
=40 mm
L
=200 mm
Material of the beam is rigid plastic with the strain-stress curve (Figure 2).
The material properties are:
Properties
Value
Yield stress
${\sigma }_{y}$ = 2.1188e+8 Pa
Yield strain
${\epsilon }_{y}$ = 1.73425e-3
Poisson's Ratio
0

The study was performed for the following moment $M$ values: 776.893 N*m and 847.518 N*m

## Reference Solution

Plastic beam bending theory assumes that there are two zones in the beam material: plastic zone at the outer layers of the beam and elastic core at the beam centerline.
The relation between applied moment $M$ and the size $H$ of the elastic core in the beam.(1)

Where, $b$ is the beam thickness.

Maximum deflection:(2)

## Results

The theory of bending of plastic beams is based on hypothesis of flat cross-sections which remain flat during deformations. In order to emulate this basic assumption as close as possible the problem was modeled as an assembly of two solids. One solid represented the beam itself with material properties defined by the curve (Figure 2). Another small solid was attached to the beam end and was set as an absolutely rigid body (Figure 4). The rigid solid was loaded with the moment $M$ and it served as a moment transfer element.
The following tables summarize the simulation results which are depicted in the figures below.
Moment M [N*m] SOL Reference, Elastic Core Size [mm] SimSolid, Approximate Elastic Core Size [mm] % Difference
776.893 10.0 12.0 20.00%
847.518 5.0 6.0 20.00%
Moment M [N*m] SOL Reference, Maximum Displacement [mm] SimSolid, Maximum Displacement [mm] % Difference
776.893 3.468 3.351 -3.37%
847.518 6.937 6.918 -0.27%
It should be mentioned that SimSolid 3D solution does not have sharp boundary between elastic and plastic zones as it is in simplified beam bending formulations. This makes difficult direct comparison; however, the elastic zone approximate sizes are in good correlation.
1 Mase, George E.,“Theory and Problems of Continuum Mechanics”, McGraw-Hill Company, New York, 1970