Thermoelastic Determination of Individual Stresses in Vicinity of a Near-Edge Hole Beneath a Concentrated Load

Abstract

Thermoelastic data are combined with an Airy stress function to determine the individual stresses on and near the boundary of a circular hole which is located below a concentrated edge-load in a plate. Coefficients of the stress function are evaluated from the measured temperatures and the local traction-free conditions are satisfied by imposing ${\sigma_{r{\rm{r}}}} = {\tau_{r\theta }} = 0$ analytically on the edge of the hole. The latter has the advantage of reducing the number of coefficients in the stress function series. The method simultaneously smoothes the measured input data, satisfies the local boundary conditions and evaluates individual stresses on, and in the neighbourhood of, the edge of the hole. Attention is paid to how many coefficients to retain in the stress function series. Although the presence of high stress concentration factors, together with a hole-diameter-to-plate-thickness ratio of only two, result in some three-dimensional effects, these are relatively small and the agreement between the thermoelastic values, those from recorded strains and FEM-predicted surface stresses is good.

Appendices

Appendix A: Equations

From equations (1) and (2), the stresses at any point in a loaded plate such as that of Fig. 1 can be written in terms of polar coordinates r, θ (coordinate origin at distance D below the top edge of the plate) as follows [3]

$$ \begin{gathered} \,\,\,\,{\sigma_{rr}} = \left\{ \begin{gathered} \frac{{2 \cdot \left( {D - r \cdot \cos \theta } \right)}}{{\pi \cdot {{\left( {{r^2} + {D^2} - 2 \cdot D \cdot r \cdot \cos \theta } \right)}^2}}} \cdot \hfill \\\left[ \begin{gathered} - {\left( {D - r \cdot \cos \theta } \right)^2} \cdot {\cos^2}\theta - {r^2} \cdot {\sin^4}\theta \hfill \\+ r \cdot \sin \theta \cdot \left( {D - r \cdot \cos \theta } \right) \cdot \sin \left( {2 \cdot \theta } \right) \hfill \\\end{gathered} \right] \hfill \\\end{gathered} \right\} \cdot P \hfill \\+ \frac{{{b_0}}}{{{r^2}}} + 2 \cdot {c_0} + \left( { - \frac{{2 \cdot {c_1}}}{{{r^3}}} + 2 \cdot {d_1} \cdot r} \right) \cdot \cos \theta \hfill \\- \sum\limits_{n = 2,3,4...}^N {\left[ \begin{gathered} {a_n} \cdot n \cdot \left( {n - 1} \right) \cdot {r^{\left( {n - 2} \right)}} + {b_n} \cdot \left( {n + 1} \right) \cdot \left( {n - 2} \right) \cdot {r^n} \hfill \\+ {c_n} \cdot n \cdot \left( {n + 1} \right) \cdot {r^{ - \left( {n + 2} \right)}} + {d_n} \cdot \left( {n - 1} \right) \cdot \left( {n + 2} \right) \cdot {r^{ - n}} \hfill \\\end{gathered} \right] \cdot \cos \left( {n \cdot \theta } \right)} \hfill \\\end{gathered} $$

(A1)

$$ \begin{gathered} \,\,\,{\sigma_{\theta \theta }} = \left\{ \begin{gathered} \frac{{2 \cdot \left( {D - r \cdot \cos \theta } \right)}}{{\pi \cdot {{\left( {{r^2} + {D^2} - 2 \cdot D \cdot r \cdot \cos \theta } \right)}^2}}} \cdot \hfill \\\left[ { - {{\left( {D - r \cdot \cos \theta } \right)}^2} \cdot {{\sin }^2}\theta - {r^2} \cdot {{\sin }^2}\theta \cdot {{\cos }^2}\theta - r \cdot \sin \theta \cdot \left( {D - r \cdot \cos \theta } \right) \cdot \sin \left( {2 \cdot \theta } \right)} \right] \hfill \\\end{gathered} \right\} \cdot P \hfill \\- \frac{{{b_0}}}{{{r^2}}} + 2 \cdot {c_0} + \left( {\frac{{2 \cdot {c_1}}}{{{r^3}}} + 6 \cdot {d_1} \cdot r} \right) \cdot \cos \theta \hfill \\+ \sum\limits_{n = 2,3,4...}^N {\left[ \begin{gathered} {a_n} \cdot n \cdot \left( {n - 1} \right) \cdot {r^{\left( {n - 2} \right)}} + {b_n} \cdot \left( {n + 1} \right) \cdot \left( {n + 2} \right) \cdot {r^n} \hfill \\+ {c_n} \cdot n \cdot \left( {n + 1} \right) \cdot {r^{ - \left( {n + 2} \right)}} + {d_n} \cdot \left( {n - 1} \right) \cdot \left( {n - 2} \right) \cdot {r^{ - n}} \hfill \\\end{gathered} \right] \cdot \cos \left( {n \cdot \theta } \right)} \hfill \\\end{gathered} $$

(A2)

$$ \begin{gathered} \,\,\,\,{\tau_{r\theta }} = \left\{ \begin{gathered} \frac{{2 \cdot \left( {D - r \cdot \cos \theta } \right)}}{{\pi \cdot {{\left( {{r^2} + {D^2} - 2 \cdot D \cdot r \cdot \cos \theta } \right)}^2}}} \cdot \hfill \\\left[ \begin{gathered} \frac{1}{2} \cdot {\left( {D - r \cdot \cos \theta } \right)^2} \cdot \sin \left( {2 \cdot \theta } \right) - \frac{1}{2} \cdot {r^2} \cdot {\sin^2}\theta \cdot \sin \left( {2 \cdot \theta } \right) \hfill \\+ r \cdot \sin \theta \cdot \left( {D - r \cdot \cos \theta } \right) \cdot \cos \left( {2 \cdot \theta } \right) \hfill \\\end{gathered} \right] \hfill \\\end{gathered} \right\} \cdot P \hfill \\+ \left( { - \frac{{2 \cdot {c_1}}}{{{r^3}}} + 2 \cdot {d_1} \cdot r} \right) \cdot \sin \theta \hfill \\+ \sum\limits_{n = 2,3,4...}^N {\left[ \begin{gathered} {a_n} \cdot n \cdot \left( {n - 1} \right) \cdot {r^{\left( {n - 2} \right)}} + {b_n} \cdot n \cdot \left( {n + 1} \right) \cdot {r^n} \hfill \\- {c_n} \cdot n \cdot \left( {n + 1} \right) \cdot {r^{ - \left( {n + 2} \right)}} - {d_n} \cdot n \cdot \left( {n - 1} \right) \cdot {r^{ - n}} \hfill \\\end{gathered} \right] \cdot \sin \left( {n \cdot \theta } \right)} \hfill \\\end{gathered} $$

(A3)

As indicated in section 2, “Relevant Equations,” the stress function of equation (1), and hence the stresses of equations (A1) through (A3), assume symmetry about the x-axis; single-valued stresses, strains and displacements; and the loaded member can contain a hole at distance D below the top of the plate of Fig. 1. However, the shape of the hole is not yet defined. While ϕ of equation (1) contains ${a_0}$ and ${a_1}$, these coefficients disappear during the differentiations of equations (2) and hence do not occur in any of the stresses. Individual stresses could be obtained from equations (A1) through (A3) if the Airy coefficients, b ₀ , c ₀ , c ₁ , d ₁ , a _n , b _n , c _n and d _n , and P were known, and without knowledge of constitutive information (other than assuming elastic isotropy), geometry, or other external boundary conditions. For load P, and circular hole of radius R and centered at location D below the top of the plate, values of the coefficients b ₀ , c ₀ , c ₁ , d ₁ , a _n , b _n , c _n and d _n of equations (A1) and (A2) could be obtained by imposing σ _rr  = 0 of equation (A1) at r = R at discrete locations around the hole². Having determined these Airy coefficients based on σ _rr  = 0 at the edge of the hole, the hoop and shear stresses, σ _θθ, and σ _rθ , would then be available from equations (A2) and (A3). However, without even addressing how many coefficients to retain and/or at how many positions to impose this traction-free condition at r = R, it is unlikely the resulting components of stress would be reliable except at positions very close to the edge of the hole. On the other hand, combining the imposed tractions-free conditions at r = R with TSA-measured values of σ = σ _rr  + σ _θθ away from the hole enables the three individual stresses of equations (A1) through (A3) to be evaluated reliably at and in the vicinity of the hole. Since neither b _o , c ₁ , a _n , nor c _n would appear in the isopachic stress σ = σ _rr  + σ _θθ based on equations (A1) and (A2), these Airy coefficients cannot be determined solely from experimentally-determined values of the isopachic stress, σ, based on these expressions, thereby necessitating combining some local collocation or other information with measured data.

As noted previously, equations (A1) through (A3) contain coefficients that do not appear in the isopachic stress σ = σ _rr + σ _θθ based on equations (A1) and (A2), thereby precluding determining the individual stresses of equations (A1) through (A3) from equations (A1) and (A2) and TSA alone. However, imposing the traction-free conditions analytically at r = R for all θ (this also defines the shape of the hole) results in identical coefficients existing in the isopachic stress expression (σ = σ _rr + σ _θθ) and individual stresses of equations (4) through (16) [3]. In addition to now being able to evaluate the remaining Airy coefficients from only TSA data, incorporating these local boundary conditions at the hole analytically enables one to acquire accurate stresses with fewer coefficients (imposing such boundary conditions analytically typically reduces the number of independent Airy coefficients by about 50% compared with doing so non-analytically/discretely) and consequently potentially needing fewer measured input data [3].

Appendix B: Three-Dimensional FEM

Acknowledging the two planes of symmetry of the plate of Fig. 1, a quarter symmetry 3-D ANSYS model was used. To prevent rigid body motion, nodes on the bottom surface of this plate were constrained to have zero displacements in the vertical x-direction. Post-processing checks confirmed that reaction forces at all these nodes were in the positive x-direction. The applied load P of Fig. 1 was simulated by constraining those nodes on the top surface of the plate which lay in the symmetry plane containing the axis of the hole to have a small fixed displacement (0.1 mm) in the negative x-direction. The reaction force acting at these nodes was determined in a post-processing step and used to scale the resulting stresses to the desired applied force (889.6 N = 200 pounds), and then those stresses were normalized using the uniform stress of 1.05 MPa (152.38 psi). This approach avoids the need to assume a force distribution on the top surface of the plate and gives a good approximation of the experimental application of load using a roller.

Three-dimensional mesh density was varied by controlling the number of element divisions around the circumference of the hole. Element sizes in the remainder of the model were linked to this parameter so that a change in mesh density around the hole propagated in a controlled way throughout the model. Models with 10, 20 and 40 element divisions around one quarter of the hole circumference were analyzed, see Fig. 14. Linear hexahedral brick elements (ANSYS type SOLID45) were used. Some degenerate (prism) elements were generated in the meshing process, but these were all well away from the regions of interest, the latter being above and around the hole boundary where approximately equilateral elements were ensured by using mapped meshing.

Fig. 14

Quarter models of the plate for number of elements NEL = (a) 10, (b) 20 and (c) 40, respectively

The stability of the finite element solution as a function of mesh density was assessed by plotting the stresses on the boundary of the hole as a function of mesh density, see Figs. 15 and 16. Even for the coarsest mesh, the hoop stress at the plate surface on the hole boundary at the side of the hole, θ = 0 of Figs. 15 and 16, is 97.5% of the value for the finest mesh. This suggests that the difference between the solution for the finest mesh and the “true” solution is small. All subsequent results were generated using this finest mesh of Fig. 14(c), NEL (number of elements/mesh density) = 40. Component stresses on the external surface at a radius r = 1.17 R are plotted in Figs. 17 and 18, whereas Figs. 19, 20 and 21 illustrate these stresses at r = 1.17 R also as a function of angular position, θ, but through the plate thickness (center of plate at left in these figures).

Fig. 15

Three-dimensional ANSYS normalized hoop stress, σ _{θθ /} σ _o , at the hole boundary as a function of θ on the external surface of the plate for the various mesh densities of Fig. 14

Fig. 16

Three-dimensional ANSYS normalized stresses on the external surface of the plate at θ = 0 at the hole boundary for various mesh densities of Fig. 14

Fig. 17

Three-dimensional ANSYS normalized circumferential stress, σ _{θθ /} σ _o , at r = 1.17R on the external surface of the plate for a mesh density of NEL = 40

Fig. 18

Three-dimensional ANSYS normalized shear stress τ _rθ/ σ _o , at r = 1.17R on external surface of the plate for a mesh density of NEL = 40

Fig. 19

Three-dimensional ANSYS normalized circumferential stress, σ _θθ/ σ _o , at r = 1.17R through the plate thickness and at various angular positions θ for a mesh density of NEL = 40

Fig. 20

Three-dimensional ANSYS normalized radial stress, σ _rr/ σ _o , at r = 1.17R through the plate thickness and at various angular positions θ for a mesh density of NEL = 40

Fig. 21

Three-dimensional ANSYS normalized shear stress, τ _{rθ /} σ _o, at r = 1.17R through the plate thickness and at various angular positions θ for a mesh density of NEL = 40