Stress Concentration in an
Elastic Plate
Nathaniel Anker
MANE 4240
Professor Ernesto Gutierrez-Miravete
April 27, 2014
Abstract
Understanding the effects of stress concentrations is a crucial aspect of structural design and
engineering. It is important to analyze the elevated stresses found in these local regions as
failure is likely to occur in this area due to fatigue or other overstresses. In this report a simple
plate model with a hole in the center will be loaded under uniaxial tension will be analyzed.
Using the finite element method the effects of the hole diameter on stress concentration will be
determined and used to validate the analytical solution to this problem.
Introduction
In this report the finite element method will be used to demonstrate the effects of a stress
concentration on a plate subjected to uniaxial tension. Understanding stress concentrations is
crucial to structural design as higher stresses are present in these areas. Using the finite element
method software COMSOL a uniform plate with a hole in the center will be modeled. The plate
will be loaded in uniaxial tension and stress contour plots will be generated. To observe the
effects of stress concentrations the diameter of the hole will be modified. Additionally the finite
element method will be validated by determining the effect of altering the mesh density and
comparing the results to analytical methods.
Formulation and Solution
As stress is generally defined as force divided by area, any reduction in area will create an
increased stress in that area. Area reduction can be created by a number of sources such as
cracks, holes, or grooves. As discussed above the effect of a uniaxial tension on a plate with a
center hole will be studied. The system to be analyzed can be seen below.
Figure 1-Free Body Diagram
The plate is subjected to tension of uniform magnitude S in the x-direction. Due to the reduction
of area due to the hole the stress distribution is expected to increase locally. The Saint-Venant’s
Principle states that the stress at large distances compared to the hole radius shall have negligible
difference from the average stress. The stresses in the plate are derived to be the following:
σr =
𝑆
𝑎2
𝑆
3𝑎4 4𝑎2
(1 − 2 ) + (1 + 4 − 2 ) cos(2𝜃) 𝐸𝑞. 1
2
𝑟
2
𝑟
𝑟
σθ =
𝑆
𝑎2
𝑆
3𝑎4
(1 + 2 ) − (1 + 4 ) cos(2𝜃) 𝐸𝑞. 2
2
2
𝑟
𝑟
𝑆
2
τrθ = − (1 −
3𝑎4 2𝑎2
+ 2 ) sin(2𝜃) 𝐸𝑞. 3
𝑟4
𝑟
For large plates (i.e. b>>a), in accordance with the Saint-Venants Principle the stresses at the
outer edge of the plate are equal to the applied stress. However the stresses within the plate are
greatly affected by the shearing stresses and therefore have a different distribution. At the edge
of the hole (i.e. when r=a) the stresses are as follows:
𝑆
𝑎2
𝑆
3𝑎4 4𝑎2
σr = (1 − 2 ) + (1 + 4 − 2 ) cos(2𝜃) = 0 𝐸𝑞. 4
2
𝑎
2
𝑎
𝑎
σθ =
𝑆
𝑎2
𝑆
3𝑎4
(1 + 2 ) − (1 + 4 ) cos(2𝜃) = 𝑆 − 2𝑆𝑐𝑜𝑠(2𝜃) 𝐸𝑞. 5
2
2
𝑎
𝑎
𝑆
3𝑎4 2𝑎2
τrθ = − (1 − 4 + 2 ) sin(2𝜃) = 0 𝐸𝑞. 6
2
𝑎
𝑎
As demonstrated above the radial and shear components of stress are equal to zero at all points
on the inner edge of the hole. However the angular stress component varies depending on the
location on the edge of the hole. The maximum stresses are found to be at the edges of the hole
perpendicular to the applied load. The stress at this point (𝜃 = π/2, 3π/2) is shown to be 3S, or
three times the uniform stress applied to the end of the plate. Additionally, the stress in the
directions the load is applied ((𝜃 = 0, π) is shown to be –S and as such, is in compression. The
amplification of the applied uniform stress is called the stress concentration factor. The stress
concentration factor, Kt, is defined as the ratio of the maximum stress to the nominal stress. In
the case of the large plate with a hole in the center Kt is equal to 3.
Utilizing the finite element method via COMSOL, a model will be created to replicate the results
obtained via the analytical method above. The following geometry was created in COMSOL.
Figure 2: COMSOL Geometry
To simulate the large plate a two meter by two meter ‘cube’ was created with a thickness of 0.1
m. A hole was created in the center of the plate with a radius of 0.025 m. This hole was chosen
as it is much smaller in comparison to the size of the plate. Structural steel was chosen as the
material for the study with a Young’s Modulus of 2◦1011 Pa, a Poisson’s Ration of 0.3, and a
density of 7800 kg/m3. A prescribed displacement boundary condition was used on the left
surface of zero in all directions to simulate the plate being clamped. On the right surface a
boundary load condition of 1,000,000 Pa was applied. All other surfaces used free boundary
conditions. Utilizing the COMSOL meshing tools a normal density physics controlled mesh was
used.
Figure 3 - Finite Element Mesh
Using the above model to the numerical solution will be found via the finite element method.
The results will then be compared to the analytical model to validate the finite element method
provides an accurate solution to the problem. Additional variations of the finite element method
will be performed to optimize the solution.
Results and Discussion
Based on the geometry and mesh discussed above the following stress plot was solved for. The
fringe plot represents the stress in z-direction, i.e. the direction of applied load.
Figure 4- Normal Mesh Stress
As can be seen from the above figure the stress is fairly uniform except in way of the hole and
the boundary conditions. The additional stress is created due to the clamped boundary condition.
As expected from the analytical results, the stress concentrations at the edge of the hole have the
greatest disparity from the rest of the plate. The maximum stress at the edge of the hole is found
to be 3.22◦107 Pa. The minimum stresses found at the edge of the hole is -1.21◦107 Pa. This
gives a stress concentration of 3.22 and -1.21 at these points based on the uniform loading of 107
Pa. In comparison to the analytical results they are within 7.33% and 21% respectively. While
these results are fairly accurate it is possible to improve on them. A closer look at the finite
element mesh shows that the mesh density around the hole was poor. Utilizing a finer mesh
should allow for a more accurate solution as a greater number of algebraic equations is used to
focus on the area in question. The model is then rerun using the finer mesh density. The results
of the mesh are detailed below.
Figure 5- Finer Mesh Density
As can be seen from above the mesh density is very fine around the hole which will lead to more
precise results. The minimum element size is 0.008. These leads to maximum and minimum
stresses of 3.22 and -1.25 around the edge of the hole. As there was very small changes with the
increase in mesh density the results produced by the normal mesh density are valid. Therefore,
further analysis is required to validate the finite element model. The above analytical formulas
for stress were determined under the assumption that the size of the plate was infinite. In simple
terms the length and width of the plate is far larger than the size of the hole such that the limit of
hole diameter to plate length approaches zero. In the model above this ratio is 0.05 m to 2 m, or
2.5%. Additional models shall be analyzed changed the diameter to length ratio to investigate
the accuracy of the assumptions made by the analytical solution. As this ratio decreases it is
expected that the large plate assumption will hold true. On the other hand as the ratio increases
the local stresses shall increase. In the plot below the diameter to length ratio has been reduced
to 5 meters to 0.05m. Although the differences in mesh density have been shown to be minimal,
an extra fine mesh density will be used to accurately model the hole.
Figure 6 - Larger Plate Size Local Stress
As can be seen in the above fringe plot the peak stresses are approximately 3.19 ◦107 Pa,
corresponding to a stress concentration of 3.20. The percent error is now reduced to 6.33%. As
the ratio of plate width to hole diameter decreases the stress concentration become more local
and therefore more consistent with the analytical results. On the other hand the stress
concentration for the area of the hole in compression do not agree with the analytical results.
This occurs because the analytical solution assumes this peak stress occurs locally at one point.
However as load is applied in this direction there is deformation present. This deformation
reduces the stress in this region. Therefore, the reduced stresses seen above and in the other
finite element models is expected. These principles are further verified when the hole to plate
width ratio is increased. The results below are for a geometry with a 0.5 meter diameter hole in a
2 meter plate.
Figure 7- Large Hole Diameter Stress
Noted from the above stress distribution that with the increase in hole diameter there is an
increase in stress distribution across the plate. This is in accordance with the Saint-Venants
Principle as both the compression and tensile stresses are found in further from the hole.
Additionally, it takes a longer distance for the stress concentration to converge to the applied
uniform stress found in the rest of the plate. This resulted is expected as additional terms are
present as they no longer approach zero in Equations 1-3 when the plate width is not extremely
large in comparison to the hole diameter. The peak stresses are also higher than in this case.
This can simply be explained the definition of stress as force over area. As the load carrying
area is reduced due to the larger hole diameter the corresponding stresses increase as well.
Conclusion
In conclusion the study showed that similar results can be obtained using either finite element
analysis or analytical solution methods. The finite element method accurately predicts the tensile
stress concentration factor created by the hole. However, the compressive stress concentration is
not accurately predicted due to deformation in the plate reducing the local stresses. This result is
expected as the analytical solution makes assumptions for the model to come up with a solution.
The finite element method can further be used as a tool to predict stress for geometries that are
complicated such that an analytical solution is impractical or not readily available. Overall the
finite element method is an accurate tool for understanding the effects of stress concentrations as
was validated by changing the parameters such as mesh density as well as the hole and plate
sizes.
References
Theory of Elasticity , McGraw-Hill Book Company, 1st Ed. 1934, 2nd Ed. 1951 (with J. N.
Goodier)