Shear Stress Calculator

Find the maximum transverse shear stress in a rectangular beam cross-section, with the parabolic stress distribution plotted across the height.

🔻 Shear Stress Calculator
kN
mm
mm
Max shear stress (τmax)
—
Cross-section area
—
Step-by-step working

🔻 What is Shear Stress?

Transverse shear stress is the internal stress that develops within a beam's cross-section as it resists the shear force created by applied loads. Unlike bending stress, which is zero at the neutral axis and peaks at the extreme fibers, shear stress follows the opposite pattern: it is zero at the top and bottom extreme fibers and reaches its maximum at the neutral axis, following a parabolic distribution across a rectangular section's height.

Structural and mechanical engineers check shear stress alongside bending stress whenever sizing a beam, especially for short, deep beams where shear can govern the design instead of bending. Common applications include checking beam-column connections, deep transfer beams, timber joists near supports, and any short-span member where the load path is dominated by shear rather than flexure. The general formula tau = VQ/(Ib) applies to any cross-section, but for a solid rectangle it simplifies to the convenient shortcut tau_max = 1.5V/A.

A common point of confusion is assuming shear stress is distributed evenly across a cross-section, like V/A. It is not: the true distribution is parabolic, and for a rectangle the true peak value at the neutral axis is exactly 1.5 times that naive average. Ignoring this factor of 1.5 can under-predict the true peak stress by a third.

This calculator computes tau_max = 1.5V/A directly from the shear force and rectangular section dimensions, and plots the full parabolic shear stress distribution across the section height, with the peak stress at the neutral axis marked.

📐 Formula

τmax = 1.5V / A
τmax = maximum shear stress, at the neutral axis (MPa)
V = shear force at the section (converted internally to N)
A = cross-sectional area = b × h (mm²)
Full distribution: τ(y) = τmax × [1 − (2y/h)²], where y is measured from the neutral axis
Example: V = 20 kN, b = 150 mm, h = 300 mm → τmax ≈ 0.6667 MPa.

📖 How to Use This Calculator

Steps

1
Enter the shear force. Type V, the shear force at the section being checked, in kilonewtons.
2
Enter the section width and height. Type b and h, the rectangular cross-section's width and height, in millimeters.
3
Read the maximum shear stress. See tau_max at the neutral axis, with the parabolic distribution plotted across the section height.

💡 Example Calculations

Example 1 — Timber Beam Near a Support

V = 20 kN, b = 150 mm, h = 300 mm

1
A = b × h = 150 × 300 = 45,000 mm²
2
τmax = 1.5V / A = (1.5 × 20,000 N) / 45,000
3
τmax = 0.6667 MPa (at the neutral axis)
τmax = 0.6667 MPa
Try this example →

Example 2 — Steel Beam-Column Connection

V = 50 kN, b = 200 mm, h = 400 mm

1
A = b × h = 200 × 400 = 80,000 mm²
2
τmax = 1.5V / A = (1.5 × 50,000 N) / 80,000
3
τmax = 0.9375 MPa (at the neutral axis)
τmax = 0.9375 MPa
Try this example →

Example 3 — Short-Span Transfer Beam

V = 10 kN, b = 100 mm, h = 250 mm

1
A = b × h = 100 × 250 = 25,000 mm²
2
τmax = 1.5V / A = (1.5 × 10,000 N) / 25,000
3
τmax = 0.6000 MPa (at the neutral axis)
τmax = 0.6000 MPa
Try this example →

❓ Frequently Asked Questions

What is transverse shear stress in a beam?+
Transverse shear stress is the internal stress that develops within a beam's cross-section as it resists the shear force from applied loads. Unlike bending stress, which peaks at the extreme fiber, shear stress is zero at the top and bottom edges and reaches its maximum at the neutral axis.
What is the formula for maximum shear stress in a rectangular beam?+
tau_max = 1.5V/A, where V is the shear force and A is the cross-sectional area (A = b x h for a rectangle). This is a simplified shortcut of the general shear formula tau = VQ/(Ib), valid specifically for a solid rectangular section.
Why is the maximum shear stress 1.5 times the average shear stress?+
The average shear stress is simply V/A, but shear stress is not distributed uniformly, it follows a parabolic curve across the height, zero at the top and bottom edges and peaking at the neutral axis. For a rectangular section, that parabolic peak works out to exactly 1.5 times the average value.
Where is shear stress highest in a beam cross-section?+
Shear stress is highest at the neutral axis (the horizontal centerline) of a rectangular cross-section and decreases parabolically toward zero at the top and bottom extreme fibers, the opposite pattern from bending stress.
Does this formula work for I-beams or circular sections?+
No, tau_max = 1.5V/A applies specifically to a solid rectangular cross-section. I-beams have a different, more complex shear distribution (most of the shear is carried by the web), and circular sections use tau_max = 4V/(3A). Each shape requires its own version of the general formula tau = VQ/(Ib).
What units does this calculator use?+
Shear force is entered in kilonewtons (kN), width and height in millimeters (mm), and the resulting maximum shear stress is shown in megapascals (MPa).
How does shear stress differ from bending stress?+
Bending stress (sigma = Mc/I) is a normal stress acting perpendicular to the cross-section, caused by the bending moment, and peaks at the extreme fiber. Shear stress (tau = VQ/Ib) acts parallel to the cross-section, caused by the shear force, and peaks at the neutral axis. Both are checked separately in a complete beam design.
What is Q in the general shear stress formula?+
Q is the first moment of area of the portion of the cross-section above (or below) the point where stress is being calculated, taken about the neutral axis. For a rectangle at the neutral axis, Q works out such that the general formula tau = VQ/(Ib) simplifies exactly to the 1.5V/A shortcut used here.
Why does shear stress matter for short, deep beams?+
Shear stress becomes proportionally more significant relative to bending stress in short, deep beams (low span-to-depth ratio), such as beam-column connections or deep transfer beams, where it can govern the design instead of bending stress, which usually dominates for long, slender beams.
What is a typical allowable shear stress for structural steel?+
Allowable shear stress is typically taken as a fraction of the material's allowable normal stress, commonly around 0.4 times the yield stress under many design codes, but always check the governing code (such as AISC for steel or ACI for concrete) for the exact allowable value and any additional shear reinforcement requirements.
Tags: Shear Stress Calculator Transverse Shear Stress Rectangular Beam Shear Vq Over Ib Parabolic Shear Distribution Calculator Formula Free Online

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What is transverse shear stress in a beam?

Transverse shear stress is the internal stress that develops within a beam's cross-section as it resists the shear force from applied loads. Unlike bending stress, which peaks at the extreme fiber, shear stress is zero at the top and bottom edges and reaches its maximum at the neutral axis.

What is the formula for maximum shear stress in a rectangular beam?

tau_max = 1.5V/A, where V is the shear force and A is the cross-sectional area (A = b x h for a rectangle). This is a simplified shortcut of the general shear formula tau = VQ/(Ib), valid specifically for a solid rectangular section.

Why is the maximum shear stress 1.5 times the average shear stress?

The average shear stress is simply V/A, but shear stress is not distributed uniformly, it follows a parabolic curve across the height, zero at the top and bottom edges and peaking at the neutral axis. For a rectangular section, that parabolic peak works out to exactly 1.5 times the average value.

Where is shear stress highest in a beam cross-section?

Shear stress is highest at the neutral axis (the horizontal centerline) of a rectangular cross-section and decreases parabolically toward zero at the top and bottom extreme fibers, the opposite pattern from bending stress.

Does this formula work for I-beams or circular sections?

No, tau_max = 1.5V/A applies specifically to a solid rectangular cross-section. I-beams have a different, more complex shear distribution (most of the shear is carried by the web), and circular sections use tau_max = 4V/(3A). Each shape requires its own version of the general formula tau = VQ/(Ib).

What units does this calculator use?

Shear force is entered in kilonewtons (kN), width and height in millimeters (mm), and the resulting maximum shear stress is shown in megapascals (MPa).

How does shear stress differ from bending stress?

Bending stress (sigma = Mc/I) is a normal stress acting perpendicular to the cross-section, caused by the bending moment, and peaks at the extreme fiber. Shear stress (tau = VQ/Ib) acts parallel to the cross-section, caused by the shear force, and peaks at the neutral axis. Both are checked separately in a complete beam design.

What is Q in the general shear stress formula?

Q is the first moment of area of the portion of the cross-section above (or below) the point where stress is being calculated, taken about the neutral axis. For a rectangle at the neutral axis, Q works out such that the general formula tau = VQ/(Ib) simplifies exactly to the 1.5V/A shortcut used here.

Why does shear stress matter for short, deep beams?

Shear stress becomes proportionally more significant relative to bending stress in short, deep beams (low span-to-depth ratio), such as beam-column connections or deep transfer beams, where it can govern the design instead of bending stress, which usually dominates for long, slender beams.

What is a typical allowable shear stress for structural steel?

Allowable shear stress is typically taken as a fraction of the material's allowable normal stress, commonly around 0.4 times the yield stress under many design codes, but always check the governing code (such as AISC for steel or ACI for concrete) for the exact allowable value and any additional shear reinforcement requirements.

📌 Quick Tips

💡For a rectangular cross-section, the general shear formula tau = VQ/(Ib) simplifies to the shortcut tau_max = 1.5V/A, exactly 1.5 times the average shear stress V/A, both are used here internally.
💡Shear stress is zero at the top and bottom extreme fibers and reaches its maximum at the neutral axis, the opposite pattern from bending stress, which peaks at the extreme fibers and is zero at the neutral axis.
💡The 1.5V/A shortcut applies only to a solid rectangular section. Circular, I-beam, and other cross-sections have different shear distribution shapes and different maximum-to-average stress ratios.
💡Compare the computed maximum shear stress against the material's allowable shear stress (often taken as roughly 0.4 to 0.6 times the allowable normal stress for common design codes) to check whether a section is adequate.
💡A complete beam design checks bending stress at the extreme fiber and shear stress at the neutral axis separately, since they peak at different locations within the same cross-section.