LIRALAND

Test 1.18. Square plate framed by a bar

Updated May 19, 2026 | Published Oct 14, 2025

Objective

To verify the equilibrium of plate sections parallel to the supported edges based on shear stresses.

Reference

Anatoly V. Perelmuter, Vladimir I. Slivker. 'Design models of structures and the possibility of their analysis.' Moscow, DMK Press, 2007, pp. 238–240.

Problem statement

To determine the shear stresses τxz at the nodal points of the design model and verify that the area of the shear stress diagrams in the horizontal sections of the plate at y = 0; 2; 4; 6; 8; 10; 12; 14; 16 corresponds to the value of the horizontal reaction at point A.

Design model

A square plate is considered. Along the upper edge of the plate, a bar is placed that is absolutely rigid in bending and tension-compression. At point A (on the upper edge), the node is restrained against horizontal displacement. Along the lower edge, the plate is rigidly fixed. Two concentrated forces acting in opposite directions are applied to the plate.

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

Plate thickness h = 1,0 m
Overall plate dimension a = 16 m

Material properties

Modulus of elasticity Е = 3 * 105 kPa
Poisson's ratio ν = 0,25

Boundary conditions

Restraints in all degrees of freedom (DOF) of the FE are applied along the lower edge of the plate  (X, Z, uY).
At point A, the node is restrained against horizontal displacement (X=0).
At the vertical centreline, the nodes are restrained against displacement along the Z-axis (Z=0).

Loads

Concentrated load P = 1000 kN

Output data

Design and deformed shapes, variant 1

Design and deformed shapes, variant 2

Variant 1
Variant 2

Design and deformed models

Mosaic plot of shear stress τxz, kN/m2, variant 1

Mosaic plot of shear stress τxz, kN/m2, variant 2

Variant 1
Variant 2

Mosaic plot of shear stress τxz, kN/m2

Contour plot of shear stress τxz, kN/m2, variant 1

Contour plot of shear stress τxz, kN/m2, variant 2

Variant 1
Variant 2

Contour plot of shear stress  τxz, kN/m2

Comparison of calculation results

The unknown Point y, m Analytical solution LIRA-FEM Error, %
Var. 1 Var. 2 Var. 1 Var. 2
Q, кН 0 872,45 878,188 878,196 0,6534 0,6543
2 878,188 878,196 0,6534 0,6543
4 878,188 878,196 0,6534 0,6543
6 878,188 878,196 0,6534 0,6543
8 878,188 878,196 0,6534 0,6543
10 878,188 878,196 0,6534 0,6543
12 878,188 878,196 0,6534 0,6543
14 878,188 878,196 0,6534 0,6543
16 878,188 878,196 0,6534 0,6543
Note:
The area of shear stress diagrams in plate sections parallel to the supported edge and at a distance y from the simply supported edge is compared with the support reaction H.

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