Stresses in a tensile strip with a circular hole are computed in order to obtain the stress intensity factor in a postprocessing step. The analytical solution ^{[7]} σ_{max} for the stress intensity factor is
where P is the force with the strip is pulled, D is the width of the strip, t the thickness of the strip, and r the radius of the hole.
The FE model consist of a single branch meshed with either HE8 or HE20 solid elements (mesh provided by CIRA). As the material can be isotropic or laminated the complete model has been meshed.
The width of the strip D=25.4, the thickness
t=2.616, and the radius of the circular hole
r=3.18. The strip is clamped at one end and loaded with
a 'displacement load', i.e an essential boundary condition, at the other
end, where a the DOF 1 (xdirection) is constrained to 0.013. To obtain the
total reaction force and to compute the theoretical solution a Python script
is included in the test directory (see file
theor_stress_conc.py
).
The analysis is performed with HE8.S.TL or HE20.S.TL total Lagrange formulation elements and a material with E=146.86 10^{3} and ν=0.3. Note that the material has no influence on the stress concentration factor, as long as the material is isotropic and elastic and the calculations are linear. The following files are available:
MDL file  he8.mdl (8 node solid elements
mesh).he20.mdl (20 node solid
elements mesh).

Viewer files  view.py plots stresses. compute_stress_intensity_factors.py calculates
stress intensity factors.

When executing the test example with the make he8
or make he20 shell command (see
Makefile
for details), the Python programs
theor_stress_conc.py
is invoked after running B2000++.
It extracts the reaction forces at the constrained nodes, sums them up to
get the total reaction force P, and calculates the
theoretical σ_{max} for the given
P:
Total reaction force = 614.221 for component Sxx Theoretical stress concentration factor h = 2.422 sigmanominal = 12.332 sigmamax = h*S_nominal = 29.864 sigmamax (computed) = 27.400 Error (percent): 8.251
The maximum calculated stress can also be extracted from a stress sampling plot as the one of the figure below.
With higher order elements (HE20.S.TL) the result is  as can be expected  better:
Total reaction force = 613.414 for component Syy Theoretical stress concentration factor h = 2.422 sigma_nominal = 12.315 sigma_max = h*sigmal_nominal = 29.824 sigma_max (computed) = 29.935 Error (percent): 0.370
The stress plot as illustrated below
is obtained with the make view shell command (see
Makefile
for details).
^{[7] }R.J. Roarke; Formulas for stress and strain; McGrawHill 975; ISBN 0070530319
^{[8] }Each sampling point (here: Gauss integration point) value is represented by a sphere. The size of the sphere is proportional to the magnitude of the field value, and the colour corresponds to the colour map.