# Photoelastic analysis of a thick plate with an elliptical hole subjected to simple out-of-plane bending

## Abstract

A three-dimensional photoelastic analysis using the stress freezing and slicing techniques was employed to study the stress distribution and the stress-concentration factors around an elliptical hole in a plate of finite thickness. The plate was subjected to simple out-of-plane bending. A special bending device was designed to produce uniform bending moment at the two opposite free edges of the plate. Six plates with various elliptical holes were studied. The stress variation across the plate thickness at the periphery of the elliptical hole was also investigated. The experimental results were correlated with the existing theoretical solutions.

## Keywords

Mechanical Engineer Fluid Dynamics Stress Distribution Plate Thickness Thick Plate## Nomenclature

- \({\left. {\begin{array}{*{20}c} {Q_{xx} } \\ {Q_{yy} } \\ \end{array} } \right\}}\)
shear forces per unit length in

*X*and*Y*directions respectively*D*plate rigidity

*a*semi axis of ellipse in the

*X*direction*b*semi axis of ellipse in the

*Y*direction*f*_{o}stress-optical coefficient (kPa/fringe/mm)

*h*total plate thickness

*K*_{b}stress-concentration factor

*M*_{o}applied bending moment (N.m/m)

*t*thickness of calibration specimen

*t′*thickness of slice

- α=(
*b/h*) ratio of axis of ellipse in the

*Y*direction to plate thickness- β=(
*b/a*) ratio of the semi axes of ellipse

- δ=(
*a/h*) ratio of the semi axis of ellipse in the

*X*direction to plate thickness- η
angle measured from

*X*axis- λ
*z/h*- ν
Poisson’s ratio

- \(\xi _o\)
boundary of elliptical hole

- ϱ
radius of curvature

- \(\sigma _{\eta (max)}\)
tangential stress at η=π/2 (kPa)

*A*_{o},*A*_{n}constants

*B*_{o},*B*_{n}constants

*G*_{n},*H*_{n}constants

- \(\left. {\begin{array}{*{20}c} {F_e k_n } \\ {G_e k_n } \\ \end{array} } \right\}\)
modified Mathieu function of the 2nd kind

*R*\(\frac{{Gek'_2 }}{{Gek_2 }}\)

- \(M_\eta (\xi _o )\)
bending moment at the edge of the ellipse for any angle η

*B*\(\frac{{M_{max} (\eta = \tfrac{\pi }{2})}}{{M_o }}\)

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## References

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