2.1 Tapered Membrane End Load

Problem Description

NAFEMS benchmark problem for a tapered membrane subjected to an end load, including geometry configuration, thickness, loading conditions, boundary constraints, material properties, and mesh details for finite element analysis validation.

Figure 1. Description of the validation problem

Extracted Details

Field	Value
Origin	NAFEMS report LSB2
Units	M, KN
Analysis Type	Linear elastic membrane
Geometry	Tapered membrane, length = 4, height varies (2 to 1), Thickness = 0.1
Loading	Uniformly distributed horizontal load of 10 MN/m (pressure = 100 MPa) along outer edge DE
Boundary Conditions	Edge AC: zero displacement in x-direction; Point B: zero displacement in y-direction
Material Properties	Isotropic, E = 210 × 10³ MPa, ν = 0.3
Element Types	Plane stress quadrilaterals or triangles
Meshes	2 × 2 uniform mesh
Output	Direct stress σxx at point B
Target	61.3 MPa (mesh refinement)

Stress Distribution

Figure 2 shows the computed ${\sigma }_{xx}$ stress distribution for the 718 number of nodes tapered membrane. The stress increases along the loading direction as the membrane width decreases, producing the expected stress concentration near the loaded edge. The stress field varies smoothly throughout the domain, indicating a stable finite element solution.

Figure 2. Direct stress (${\sigma }_{xx}$) distribution in the tapered membrane.

Mesh Convergence Study

The stress at Point B was evaluated for progressively refined meshes using both Continuous Galerkin (CG) Interpolation degree 1 and 2 finite elements.

Continuous Galerkin Degree 1 (CG1)

Number of Nodes in mesh	${\sigma }_{xx}$ (MPa)	Error (%)
24	59.18	3.46
104	60.66	1.04
142	61.05	0.41
718	61.24	0.10
2760	61.31	0.02
11250	61.33	0.05

Continuous Galerkin Degree 2 (CG2)

Number of Nodes in mesh	${\sigma }_{xx}$ (MPa)	Error (%)
24	59.21	3.41
104	60.89	0.67
142	61.09	0.34
718	61.27	0.05
2760	61.32	0.03
11250	61.33	0.05
Figure 3 presents the convergence history for both element formulations. As the mesh is refined, the numerical solution approaches the NAFEMS reference value of 61.3 MPa. The CG2 formulation converges slightly faster than CG1; however, both formulations reach essentially the same converged solution for fine meshes.

Figure 3. Mesh convergence of ${\sigma }_{xx}$ at Point B compared with the NAFEMS reference value.

Converged Solution

For the finest mesh, the numerical solution produced:

Quantity	Value
${\sigma }_{xx}$ at Point B	61.33 MPa
Benchmark Value	61.30 MPa
Relative Error	0.05 %

The computed stress converges towards the benchmark value with mesh refinement. Continuous Galerkin Degree 2 (CG2) elements achieve the reference solution with fewer degrees of freedom, while both formulations converge to the same final value for sufficiently refined meshes.

Conclusion

The developed Linear Elastic Membrane Solver successfully reproduces the NAFEMS tapered membrane benchmark solution. The converged stress at Point B is 61.33 MPa compared to the benchmark value of 61.30 MPa, resulting in a relative error of only 0.05%. The excellent agreement verifies the correctness of the numerical implementation and provides confidence for subsequent structural and shell element analyses.