How to Calculate Max Shear Stress

Written by brad painting
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How to Calculate Max Shear Stress
Scissors create shear stress in paper. (to cut a bit of sky image by Stasys Eidiejus from

Shear stress is the distribution of shear force over an area. Any force that acts on a plane in a parallel direction is a shear force. If you hold a pencil in two tightly clenched fists and try to slide them past each other, you apply shear force along the pencil's cross-section. When an object fails under shear stress, the pieces will slide past each other, rather than pulling apart or crunching together. Under a force with uniform direction, shear stress will be greatest at the cross section's midpoint and weakest at its edges.

Skill level:
Moderately Challenging

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  1. 1

    Calculate the shear force acting on the cross-section. Do this by drawing the object and schematically cutting it along the cross section you want to analyse. If the object is static, the sum of forces acting in any direction will be zero. If there are two downward forces of 22.7 Kilogram acting on a beam to the left of a certain point, there must be a single upward force of 45.4 Kilogram acting within the beam at that cross section, because 100 - 50 - 50 = 0.

  2. 2

    Identify the cross-section's neutral axis--the point where maximum shear stress will occur. For symmetrical shapes such as a rectangle or circle, the neutral axis simply runs through the midpoint between the top and bottom of the cross section. For more complex shapes, use the formula Atyc = A1y1 + A2y2 + A3y3.... , where A1, A2 and A3 are the areas of sub-shapes that make up the cross section, and y1, y2 and y3 are the distances to the centroids of those subshapes. The variable At represents the total area, and yc represents the distance to the neutral axis. Solve this formula for yc.

  3. 3

    Calculate the moment of area. This equals the area on either side of the neutral axis multiplied by the distance to its centroid. The moment of area is for calculation purposes and does not represent a concrete property.

  4. 4

    Determine the moment of inertia about the cross section's neutral axis. The moment of inertia is a geometry-based calculation usually used to find a beam's resistance to bending. You can find the moment of inertia of most cross-sections in a table of previously calculated values.

  5. 5

    Apply the following formula: shears stress = QV / Ib, where Q is moment of area, V is shear force, I is moment of inertia, and b is the object's thickness at the point of examination.

Tips and warnings

  • The formula stress = Q*V / I*b is only for shear force applied in a uniform direction. Shafts subjected to torsion experience shear stress, but it wraps around the shaft in circles and is greatest at its outer edge. For shafts subjected to torsion, use the formula stress = T*r / Ip, where T is torsion, r is the distance of the analysed point from the shaft's centre, and Ip is the polar moment of inertia.

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