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)

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