# SS-V: 5030 Reactions at the Ends of Axially Loaded Plastic Bar

Test No. NVL04Find reactions at the fixed ends and maximum displacement of a bar axially loaded beyond plasticity.

## Definition

Bar dimensions are 10 x 10 x 200 mm. Distance between loaded point and left end A=50 mm. Strain-stress curve of the bar material is defined by the power law:(1)
$\sigma =K{\epsilon }^{n}$
Where,
$K$
Strength coefficient
$n$
Must be in the range [0,1]
$n$ =0
Material is perfectly plastic.
$n$ =1
Material is elastic.
The material properties are:
Properties
Value
$K$
530 MPa
$n$
0.26
Poisson's Ratio
0

The study was performed for the following load F values: 30000 N, 47000 N, 55000 N, and 60000 N. These loads cover the full range of elastic-plastic response of the bar.

## Reference Solution

One-dimensional analytical reference solution is described here.

The length of the bar does not change under the load.(2)
or,(3)
Where,
$\epsilon 1$
Tensile strain at the left span of the bar.
$\epsilon 2$
Compressive strain at the right span of the bar,
$N$
Reaction force at left end of the bar.
$R=F-N$
Reaction force at the right end of the bar.
$A$
Bar cross-section area
From this equation you can find the reaction at the left end of the bar.(4)

and at the right end.

## Results

Bar was modeled as a 3D solid with immovable ends. Axial force F could not be applied precisely at the solid bar axis, so four line spots were created at the bar sides and total load F was uniformly distributed over the spots (Figure 3).
The following table summarizes the reaction force results.
Force F [N] SOL Reference, Reaction [N] SimSolid, Reaction [N] % Difference
30000 17128 18151 5.97%
47000 26834 27146 1.16%
55000 31401 31788 1.23%
60000 34256 34591 0.98%
Typical von Mises stress distributions are shown in Figure 4 and Figure 5. The distribution has high gradients at load application lines; yet the reactions values correlate to the 1D solution because the reactions are applied far from the active force.