Kirsch solution: stress concentration around a circular hole

With θ measured from the loading direction, maxima occur at 90° and 270°, on the sides of the hole. There cos(2θ)=−1 and the boundary equation gives σθθ=3σ∞. Along the loading axis cos(0)=1 and hoop stress is −σ∞. This angle check prevents confusing the load axis with the maximum tensile stress direction. Assume an infinite isotropic linear-elastic plate, circular hole and uniaxial remote stress; finite geometries differ. Application: Estimate the first-order elastic stress peak around a hole or opening. The numerical example At remote tension 80 MPa, maximum hoop stress at the sides of the hole is 3×80=240 MPa. At θ=0°, the boundary equation gives 80(1−2)=−80 MPa; at θ=90°, it gives 80(1−2 cos 180°)=240 MPa. This checks both the sign pattern and factor 3 anchors the idea in a concrete measurement case.

The value belongs to an analytical limiting solution for an infinite plate with an ideal circular hole under uniaxial remote stress. Finite width, nearby edges, multiple openings, plasticity or different notch shapes change the stress field. Use geometry-specific or code/numerical factors for a real component; Kt alone does not predict fatigue life. Assume an infinite isotropic linear-elastic plate, circular hole and uniaxial remote stress; finite geometries differ. The corresponding units are Stresses in Pa; Kt dimensionless; radius and coordinates in m. At remote tension 80 MPa, maximum hoop stress at the sides of the hole is 3×80=240 MPa. At θ=0°, the boundary equation gives 80(1−2)=−80 MPa; at θ=90°, it gives 80(1−2 cos 180°)=240 MPa. This checks both the sign pattern and factor 3. This checks that the reference quantities are compatible.

At a traction-free hole boundary, stress normal to the surface vanishes; hoop stress acts tangentially along the edge. It varies with θ and can be tensile or compressive. The Kirsch solution also describes other stress components in the field. Here Kt summarizes only the maximum tangential boundary stress relative to remote stress. Assume an infinite isotropic linear-elastic plate, circular hole and uniaxial remote stress; finite geometries differ. Scope: The classical Kirsch solution assumes a circular hole in an infinite, homogeneous, isotropic, linear-elastic plate under uniaxial remote stress and plane stress. Angle θ is measured from the loading direction; the boundary maximum occurs at 90° and 270° and equals three times the remote stress. Finite width, nearby holes, plasticity and real notch geometry change Kt; a peak stress alone is not a fatigue assessment. A practical application is Estimate the first-order elastic stress peak around a hole or opening.

Insert σ∞=80 MPa. At θ=90°, cos(180°)=−1, so σθθ=80[1−2(−1)]=240 MPa. At θ=0°, cos(0)=1 and σθθ=80(1−2)=−80 MPa. Opposite signs show the local stress pattern; peak-to-remote ratio is 240/80=3. Assume an infinite isotropic linear-elastic plate, circular hole and uniaxial remote stress; finite geometries differ. Check the calculation with this case: At remote tension 80 MPa, maximum hoop stress at the sides of the hole is 3×80=240 MPa. At θ=0°, the boundary equation gives 80(1−2)=−80 MPa; at θ=90°, it gives 80(1−2 cos 180°)=240 MPa. This checks both the sign pattern and factor 3. Keep the model limits and units in view.

For an initial elastic screen, hoop stress at the hole edge shows the local peak. A strength assessment must consider the full stress state, material properties, load combinations, safety approach and possible plasticity or fatigue. Local yielding can occur near a notch despite an elastic peak calculation. Use the solution for understanding and preliminary estimates, not as sole component approval. Assume an infinite isotropic linear-elastic plate, circular hole and uniaxial remote stress; finite geometries differ. The relation describes The Kirsch solution gives elastic hoop stress at the boundary of a circular hole in an infinite plate under uniaxial remote tension. This helps distinguish its target quantity from neighbouring quantities.