Shear Stress Calculator

This shear stress calculator calculates the horizontal shear stress due to transverse loads and the shear stress owed to torsion applied on a circular shaft.

The shear stress owed to transverse forces is critical in the design of thin-walled members. For non-tin-walled members, like a circular shaft, the most important will probably be the axial stress caused by the bending moment, and you'll likely need to use our section modulus calculator. Our stress calculator can also be helpful if you're dealing with axial stress. If that's not the case, use this calculator!

The shear stress equation to use will depend on whether we apply a transverse load to a beam or a torsion couple to a circular shaft. In the next section, you can look at the formulas for shear stress under these two conditions.

Shear stress is not only present in solid bodies; it's also an essential quantity in the study of liquids and gases. For this reason, we created a FAQ section dedicated to answering questions like how to calculate the shear stress of a fluid or how to calculate its viscosity from shear stress and shear rate.

Stress is defined as force per unit area. In a beam section subjected to shear force, the average shear stress in that section is:

This is just an average, as the shear stress across a beam subjected to transverse shear varies widely.

You can find the exact shear stress at any distance from the centroidal axis with the following equation:

where:

• $\tau$ – Shear stress at a specific distance from the neutral axis;
• $V$ – Internal shear force at the section of interest in the beam;
• $Q$ – First moment, about the neutral axis, of the cross-sectional area that lies above the point where we'll calculate the stress;
• $t$ – Section width at the point where we'll calculate the shear stress; and
• $I$ – Second moment of area of the entire beam cross-section, about the neutral axis (you can find it with our moment of inertia calculator).

The previous equation also gives us an average value of the stress, as it doesn't only vary with vertical distance from the neutral axis but also across the width of the beam. Fortunately, this variation is usually minor, and we can neglect it.

To calculate the shear stress from a torque applied to a circular shaft, use the torsional shear stress formula:

where:

• $\tau$ – Shear stress at the point of interest;
• $\rho$ – Radial distance from the shaft center to the point where we want to calculate the stress;
• $T$ – Torque applied to the shaft; and
• $J$ – Polar moment of inertia of inertia of the cross-section.
  • For a solid shaft of radius c, $J = π c^4/2$.
  • For a hollow shaft of inner radius $c_1$ and outer radius $c_2$, $J = π(c_2^4-c_1^4)/2$.

The maximum shear stress for torsional loads occurs at the outer surface of the circular shaft, where $\rho = c$ or $\rho = c_2$. In other words, the maximum shear stress is given by:

where:

• $\tau_{max}$ — Maximum shear stress; and
• $c$ — Shaft radius.

You must also consider this when using these formulas:

• The shear stress units are psi or pascals.
• This formula only applies to homogenous materials with a linear elastic behavior.

As mentioned before, the maximum shear stress of a circular beam under torsion is the shear stress at its outermost part. But if you're interested in how to calculate the maximum shear stress for a beam under transverse shear $V$, it's not that easy.

Fortunately, our calculator can do it for you using the following formulas for different beams under transverse shear $V$:

In an I-beam (wide-flange beam), the $\tau_{min}$ value (the stress value at the joint between the flange and web) is a critical quantity, as sometimes the joint in that zone is created through welding or glue, and we need to calculate how much stress those elements will withstand.

Please keep this in mind:

• When we apply a transverse load to a beam, transverse and longitudinal shear stresses arise. Due to the complementary property of shear, these longitudinal and transversal stresses have the same magnitude. Therefore, the horizontal shear stress calculation only requires knowing the transverse shear stress value, as they are both the same.

• $Q$ is a tricky quantity. Mathematically, it is the area above the point of interest multiplied by the distance between the centroid of that area to the neutral axis. For example, for a cross-section with cam geometry, if you wanted to calculate the shear stress at a distance $y'$ from the neutral axis, the corresponding $Q$ would be $Q = \bar y' \bar A$.

• From the previous image, we can also note what the $t$ term means.

The shear stress units are pascal (Pa) in the International System of Units and pound per square inch (psi) in the United States customary units system. We can also describe this quantity in bar units.

The polar moment of inertia of a circle of 10 cm radius is 15708 cm⁴. For the United States customary units system, its polar moment of inertia is 377.4 in⁴.

To calculate the shear stress 𝜏ᵧₓ of a fluid:

1. Determine whether it is a Newtonian or non-Newtonian fluid.
2. For a Newtonian fluid, multiply the viscosity μ by the shear rate du/dy: 𝜏ᵧₓ = μ(du/dy).
3. If it's non-Newtonian, multiply the apparent viscosity η by the shear rate du/dy:
   𝜏ᵧₓ = η(du/dy).

η and μ are analog terms, with the difference that η varies with du/dy.

To calculate viscosity μ from shear stress 𝜏ᵧₓ and shear rate du/dy, divide the shear stress by the shear strain:

μ = 𝜏ᵧₓ/(du/dy)

To calculate shear stress from the flow rate in a pipe, use the formula 𝜏ᵣₓ = -(4μrQ)/(πR⁴), where:

• 𝜏ᵣₓ – Shear stress (in the flow direction) at the point of interest;
• μ – Absolute (or dynamic) viscosity of the fluid;
• r – Radial distance from the pipe center to the point of interest;
• Q – Flow rate; and
• R – Pipe radius.

That's how you calculate shear stress from flow rate! Take into account that this formula applies to Newtonian fluids under fully developed laminar flow.

The maximum shear stress of a screw body subjected to torsion can be expressed as:

𝜏ₘₐₓ = 16T / (πdᵣ³)

where:

• 𝜏ₘₐₓ – Maximum nominal shear stress;
• T – Torque applied to the screw; and
• dᵣ – Minor diameter of the screw.