LIRALAND

Test 1.10. Tension of a prismatic bar under self-weight

Updated Dec 30, 2025 | Published Oct 22, 2025

Objective

To determine the stress–strain state of a vertical prism subjected to loading due to its self-weight.

Reference

S. P. Timoshenko, J. N. Goodier. Theory of Elasticity, M., Nauka, 1975, pp. 289–291.

Problem statement

To determine the vertical displacements Z (w) at points B and C, the horizontal displacement X (u) at point D, as well as the stresses near the upper face and at mid-height of the bar.

Design model

A vertical prism subjected to its self-weight and to a load applied at the upper face, equal to the weight of the bar.

Initial geometry of analytical model

Initial geometry of analytical model

Initial geometry of FE model, variant 1

Initial geometry of FE model, variant 2

Variant 1

Variant 2


Initial geometry of FE model

Geometry

Length L = 3 m
Dimensions of cross-section a = b = 0,5 m

Material properties

Modulus of elasticity Е = 2 * 107 tf/m2
Poisson's ratio ν = 0,3
Unit weight R0 (ρg) = 7,8 tf/m3

Boundary conditions

Point A is restrained in the Z direction (Z (w_A) = 0) to prevent geometric instability.
Restraints are applied in the Y direction (v = 0) on the XZ cut plane, and in the X direction (u = 0) on the YZ cut plane.

Loads

Self-weight acting on the prism (downwards).
A uniformly distributed load (equal in total value to the unit weight of the prism) is applied to the upper face of the prism (acting upwards).

Output data

Mosaic plot of vertical displacements Z (w), m, variant 1

Mosaic plot of vertical displacements Z (w), m, variant 2

Variant 1

Variant 2


Mosaic plot of vertical displacements Z (w), m

Mosaic plot of horizontal displacements X (u), m, variant 1

Mosaic plot of horizontal displacements X (u), m, variant 2

Variant 1

Variant 2


Mosaic plot of horizontal displacements X (u), m

Vertical stress σzz (Nz), tf/m2, variant 1

Vertical stress σzz (Nz), tf/m2, variant 2

Variant 1

Variant 2


Vertical stress σzz (Nz), tf/m2

Analytical solution

σzz = ρgz
σxx = σyy = σxy = σyx = σzx = 0
u = -νρgxz/E

Comparison of calculation results

Without additional side nodes:

Point The unknown Analytical solution LIRA-FEM Error, %
Variant 1 Variant 2 Variant 1 Variant 2
В w (Δz), m -1,755*106 -1,755*106 -1,755*106 0 0
С w (Δz), m -1,74*106 -1,7584*106 -1,7447*106 1,0464 0,2694
D u (Δx), m -1,755*107 -1,7182*107 -1,7418*107 2,0969 0,7521
А σZZ, tf/m2 23,4 22,444 22,9148 4,0855 2,0735
Е σZZ, tf/m2 11,7 11,7 11,7 0 0

With additional side nodes:

Point The unknown Analytical solution LIRA-FEM Error, %
Variant 1 Variant 2 Variant 1 Variant 2
В w (Δz), m -1,755*106 -1,7547*106 -1,7549*106 0,0171 0,0057
С w (Δz), m -1,74*106 -1,7437*106 -1,7439*106 0,2122 0,2236
D u (Δx), m -1,755*107 -1,7533*107 -1,7545*107 0,0969 0,0285
А σZZ, tf/m2 23,4 22,444 22,9226 4,0855 2,0402
Е σZZ, tf/m2 11,7 11,7 11,7 0 0

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