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  - SS-V: 5000 Coupled Analysis - Bimetallic Beam Under Thermal Load
    Test No. VNL01 Find temperature at the material interface and maximum deflection of the beam.
  - SS-V: 5010 Z-Shaped Cantilever
    Test No. VNL02 Find vertical dsiplacements of a Z-shaped cantilever beam under transverse loads of varying magnitude.
  - SS-V: 5020 Lateral Buckling of a Right Angle Frame
    Test No. VNL03 Perform analysis of buckling and post-buckling of a thin-walled frame-like structure in the shape of right angle.
  - SS-V: 5030 Reactions at the Ends of Axially Loaded Plastic Bar
    Test No. VNL04 Find reactions at the fixed ends and maximum displacement of a bar axially loaded beyond plasticity.
  - SS-V: 5040 Residual Deformations in an Axially Loaded Plastic Bar
    Test No. VNL05 Find residual deformations in a bar axially loaded beyond plasticity.
  - SS-V: 5060 Separate Beams
    Test No. VNL07 Find total or partial beam separation/slippage depending on the applied loads.
  - SS-V: 5070 Pinched Hemispherical Shell
    Test No. VNL08 Find displacements of a hemispherical shell loaded with inward and outward concentrated forces.
  - SS-V: 5080 Rigid Punch Plasticity
    Test No. VNL09 Find out reactions at the punch pressed on large block for perfectly plastic and isotropic hardening materials.
  - SS-V: 5090 Bushing Stiffness
    Test No. VNL10 Find the rotation of cylinder subjected to a moment of 10 N-m along its length.
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Frequently Asked Questions
This section provides quick responses to typical and frequently asked questions regarding

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

Test No. VNL04 Find reactions at the fixed ends and maximum displacement of a bar axially loaded beyond plasticity.

Definition

Figure 1.

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:

$σ = K ε^{n}$

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

Figure 2. Corresponded strain-stress curve

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.

$\int_{0}^{A} ε 1 d x - \int_{0}^{L - A} ε 2 d x = 0$

or,

$\int_{0}^{A} \sqrt[n]{N / (K * A)} d x - \int_{0}^{L - A} \sqrt[n]{(F - N) / (K * A)} d x = 0$

Where,

$ε 1$: Tensile strain at the left span of the bar.
$ε 2$: 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.

$N = F / (1 + {(a / b)}^{n})$

and $R = F - N$ 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).

Figure 3.

The following table summarizes the reaction force results.


Force F [N]	SOL Reference, Reaction [N]	SimSolid, Reaction [N]	% Difference
30000	17128	17522	2.3%
47000	26834	26987	0.5%
55000	31401	31660	0.8%
60000	34256	34490	0.6%