Tools › Structures › Stress Concentrations

Stress Concentrations

18 tools in the Structures area.

Round Hole Uniaxial Stress

Stresses around a round hole in an infinite plate under uni-axial stress, from the Kirsch (1898) elasticity solution: radial, tangential and shear stress at a point given by its distance from the hole center and its angle from the load axis, and the stress concentration at that point.

1 worked example reproduced, listed on the tool page.

Round Hole General Load

Stresses around a round hole in an infinite plate under any uniform in-plane far-field state given as stress resultants N_xx, N_yy, N_xy and the plate thickness t (far-field stresses N/t). The Kirsch solution is linear, so the general field is the superposition of the uni-axial solutions along the far-field principal directions: the stress components, principal stresses and maximum shear at a user input point, the hoop-stress extremes on the hole edge (3S_1 - S_2 and 3S_2 - S_1) with K_t referred to the maximum far-field principal stress S_1, and the stress field around the hole.

3 worked examples reproduced, listed on the tool page.

Round Hole Finite Strip

Stresses around a central round hole in a long strip of finite width W under an axial load P, from Howland's (1930) series solution carried to convergence: the stress components, principal stresses and maximum shear at a user input point, the gross and net stress concentration factors K_tg = σ_max / σ_g and K_tn = σ_max / σ_n (σ_g = P / (W t), σ_n = P / ((W − D) t)), the stresses at the hole edge and at the strip edge on the net section, and the stress field.

2 worked examples reproduced, listed on the tool page.

Pin-Loaded Hole

Stresses around a round hole in an infinite plate loaded by a pin: the pin presses on the half of the hole facing the load with a frictionless cosine pressure p = p_0 cos θ, and the load P is reacted far away by the plate (Bickley 1928; the far field is the Kelvin point-force field, decaying as 1/r). An optional uniform bypass far-field state N_xx, N_yy, N_xy is superposed by Kirsch's solution. Outputs: the bearing stress P/(D t) and peak pressure, the stress components, principal stresses and maximum shear at a user input point, the hoop stress at the sides, crown and back of the hole with K referred to the bearing stress, and the stress field around the hole.

1 worked example reproduced, listed on the tool page.

Pressurized Hole

Stresses around a round hole in an infinite plate that carries a uniform internal pressure (fluid pressure, or the contact pressure of an interference-fit pin or bushing) together with any uniform in-plane far-field state given as N_xx, N_yy, N_xy and the thickness t. The Lamé pressure field (σ_rr = −p a²/r², σ_θθ = +p a²/r²) superposes on the Kirsch field: stresses at a user input point, hoop-stress extremes on the hole edge, and the stress field around the hole. The interference-fit pressure of a rigid pin can be computed from the radial interference.

1 worked example reproduced, listed on the tool page.

Hole in Plate Bending

Moments and surface stresses around a round hole in a thin plate under uniform remote bending and twisting moments M_xx, M_yy, M_xy per unit width (Kirchhoff plate theory, Goodier 1936). Closed-form moment field, the hole-edge extremes with the bending stress concentration factor K = (5 + 3ν)/(3 + ν) for uni-axial bending, surface stresses 6M/t², and the field plotted around the hole.

1 worked example reproduced, listed on the tool page.

Rotating Disk With Hole

Stresses in a thin uniform disk with a central hole spinning at constant speed (Lamé-type closed form): radial and hoop stress at any radius, the peak hoop stress at the hole edge, the peak radial stress, the radial growth, and the stress concentration of the hole relative to a solid disk (which tends to 2 for a small hole).

1 worked example reproduced, listed on the tool page.

Elliptic Hole Uniaxial Stress

Stresses around an elliptic hole in an infinite plate under uni-axial stress, from the Inglis (1913) / Kolosov-Muskhelishvili elasticity solution: the stress components, principal stresses and maximum shear at a point given by its distance from the hole center and its angle from the load axis, the peak stress on the hole edge (1 + 2a/b at the ends of the axis normal to the load for the aligned ellipse, or found on the edge for an inclined one), and the stress field around the hole.

1 worked example reproduced, listed on the tool page.

Elliptic Hole General Load

Stresses around an elliptic hole, aligned with the plate axes or inclined, in an infinite plate under any uniform in-plane far-field state given as stress resultants N_xx, N_yy, N_xy and the plate thickness t (far-field stresses N/t). The Inglis / Kolosov-Muskhelishvili solution is linear, so the general field is the superposition of the uni-axial solutions along the far-field principal directions: stress components, principal stresses and maximum shear at a user input point, the hoop-stress extremes on the hole edge with K_t referred to the maximum far-field principal stress S_1, and the stress field around the hole.

1 worked example reproduced, listed on the tool page.

Rounded Square Hole

Stresses around a square hole with rounded corners in an infinite plate under any uniform in-plane far-field state given as stress resultants N_xx, N_yy, N_xy and the plate thickness t. The hole is Savin's one-term conformal map of a circle, z = R(ζ − m/ζ³), fitted to the hole width and corner radius; the Kolosov-Muskhelishvili potentials for it are closed-form and are superposed over the far-field principal directions: stress components, principal stresses and maximum shear at a user input point, the hoop-stress extremes on the hole edge with K_t referred to the maximum far-field principal stress S_1, and the stress field around the hole.

3 worked examples reproduced, listed on the tool page.

Two Holes

Stresses around two interacting round holes of diameters D_1 and D_2, centers s apart on the x axis, in an infinite plate under any uniform in-plane far-field state given as stress resultants N_xx, N_yy, N_xy and the plate thickness t (far-field stresses N/t). The field is the superposition of the two single-hole Laurent expansions carried to convergence (the problem of Ling 1948 and Haddon 1967): the stress components, principal stresses and maximum shear at a user input point about hole 1, the hoop-stress extremes on each hole with K_t referred to the maximum far-field principal stress S_1, the ligament stresses, and the stress field around the holes.

4 worked examples reproduced, listed on the tool page.

Row of Holes

Stresses around an infinite row of equal round holes in an infinite plate under any uniform in-plane far-field state given as stress resultants N_xx, N_yy, N_xy and the plate thickness t (far-field stresses N/t). The holes lie along x at a pitch s. The field is the periodic Muskhelishvili solution (Howland 1935, Schulz 1941, Meijers 1967) carried to convergence: the stress components, principal stresses and maximum shear at a user input point about the central hole, the hoop-stress extremes on the hole edge with K_t referred to the maximum far-field principal stress S_1, the net-section factor for the load across the row, and the stress field around the holes.

3 worked examples reproduced, listed on the tool page.

Drawn Cutout

Stresses around one or more cutouts of any shape drawn on the screen, in an infinite plate under a uniform far-field state given as the resultants N_xx, N_yy, N_xy and the thickness t. The plate is solved by the boundary element method (Kelvin's fundamental solution, constant elements on the cutout edges, the far field superposed), so no formula for the particular shape is needed: slots, rounded squares, D-holes, keyholes, clusters of holes. Sharp corners are rounded with the entered corner radius, because a sharp re-entrant corner has an infinite elastic stress. The tool reports the hoop-stress extremes on every cutout edge with their positions, K_t referred to the maximum far-field principal stress, the stress field as fringe plots, and the hoop stress around each edge.

4 worked examples reproduced, listed on the tool page.

Semicircular Edge Notch

Stresses around a semicircular notch of radius ρ on the straight free edge of a semi-infinite plate under uniform tension σ = N_yy / t parallel to the edge (Maunsell 1936, Ling 1947: K_t = 3.065 at the notch root). The field is the Muskhelishvili series with the straight edge traction-free exactly, carried to convergence: the stress components, principal stresses and maximum shear at a user input point, the hoop stress along the notch with K_t at the root, the stresses on the root line and along the free edge, and the stress field around the notch.

1 worked example reproduced, listed on the tool page.

Neuber Hyperbolic Notch

Stress concentration factors for a flat bar with symmetric hyperbolic notches or a round bar with a hyperbolic circumferential groove, from Neuber's exact deep-notch solutions (Kerbspannungslehre) in tension, bending and torsion, with the shallow-notch solution and Neuber's interpolation when the notch depth is given; nominal and peak stresses at the notch root for the entered loads.

2 worked examples reproduced, listed on the tool page.

Circular Inclusion

Stresses in and around a circular elastic inclusion (a plug, bushing, fiber or second-phase particle) perfectly bonded in an infinite plate under any uniform in-plane far-field state given as stress resultants N_xx, N_yy, N_xy and the plate thickness t. Closed-form Kolosov-Muskhelishvili solution (Goodier 1933): the stress inside the inclusion is uniform, the matrix field has the Kirsch form; the rigid inclusion and the empty hole are the limits. Interface stresses, stress concentration referred to the maximum far-field principal stress S_1, and the stress field around the inclusion.

2 worked examples reproduced, listed on the tool page.

Elliptic Inclusion

Stresses in and around an elliptic elastic inclusion (a plug, fiber cross-section or second-phase particle), aligned with the plate axes or inclined, perfectly bonded in an infinite plate under any uniform in-plane far-field state given as N_xx, N_yy, N_xy and the thickness t. Kolosov-Muskhelishvili solution on the conformal map of the ellipse (Hardiman 1954): the stress inside the inclusion is uniform (Eshelby); the matrix potentials are Laurent series solved to machine precision. Rigid inclusion and empty hole (Inglis) are the limits.

1 worked example reproduced, listed on the tool page.

Spherical Cavity

Stress concentration at a spherical cavity (a pore, void or casting defect) in an infinite elastic solid under remote uniaxial tension, from the closed-form solution of Southwell and Gough (1926) and Goodier (1933): the peak stress on the equator, the compressive stress at the poles, their dependence on Poisson's ratio, and the decay of the axial stress along the equatorial plane; compared with the plate hole (K_t = 3).

1 worked example reproduced, listed on the tool page.

Other Structures sections

Materials · Composites · Frames and Trusses · Beams · Loads and Dynamics · Joints · Stress and Strain · Stability · Plates and Shells · Thin-Walled Structures · Contact · Fatigue · Damage Tolerance