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SS-V: 5040 Residual Deformations in an Axially Loaded Plastic Bar
Test No. VNL05 Find residual deformations in a bar axially loaded beyond plasticity.

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- Non-linear
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  - SS-V: 5010 Z-Shaped Cantilever
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  - 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.
- Dynamics

SS-V: 5040 Residual Deformations in an Axially Loaded Plastic Bar

Test No. VNL05 Find residual deformations in a bar axially loaded beyond plasticity.

Definition

Figure 1.

Bar dimensions are 10 x 10 x 200 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 left end of the bar is clamped and the right end is loaded with force F.

The material properties are:

Properties: Value
$K$: 530 MPa
$n$: 0.26
Poisson's Ratio: 0

Figure 2. Corresponded strain-stress curve

Note: Elasticity modulus defined by the first two points of the strain-stress curve is E=2.67324e+10 Pa.

The study was performed for the following load F values: 20000 N, 25000 N, and 30000 N

Reference Solution

One-dimensional analytical reference solution is described here.

At strain $ε$ and stress $σ$ , the residual strain is:

ε r = ε - ε e = (\frac{σ}{K}) \frac{1}{n} - \frac{σ}{E} = (\frac{F}{(K * A)}) \frac{1}{n} - (\frac{F}{(E * A)})

Where,

$ε$: Total strain in the bar.
$ε e$: Elastic component of the total strain.
$ε r$: Residual strain.
$A$: Bar cross-section area.

Then residual displacement at the right end of the bar.

U r = \int_{0}^{L} (\sqrt[n]{\frac{F}{K * A}} - F / (E * A)) d x

Results

The bar was modeled as a 3D solid. The left end of the solid was fixed and the right end loaded with axial force (Figure 3).

Figure 3.

The following table summarizes the residual deformations results.


Force F [N]	SOL Reference, Residual Displacement [mm]	SimSolid, Residual Displacement [mm]	% Difference
20000	3.22	3.3	2.48%
25000	9.16	9.23	0.76%
30000	19.077	20.2	5.89%