Calculator · Structural Mechanics
Transverse Shear Stress Calculator
Shear stress across a beam section, using the shear force the beam engine already computed. Q and t describe the level in the section you are cutting at, so the same beam gives a different τ at the neutral axis than at a flange junction.
Beam
Scroll the schematic sideways to see the whole beam; every result is also listed in the tables below.
Beam and loads
The beam runs from x = 0 to x = L.
Supports
Point loads
Point moments
Uniform distributed loads
Downward loads are negative (q < 0, Fy < 0); an applied couple is positive counter-clockwise. Internally, M > 0 is sagging, v > 0 is upward and θ > 0 is counter-clockwise.
Section and material (optional)
Leave a field empty to leave it unspecified: reactions, shear and bending moment are reported without any section data. A value that is zero or negative is not unspecified — it is reported as an invalid entry. Bending stress additionally needs I and an extreme-fibre distance c; slope and deflection need both E and I.
Section level and station
Measured from the left end of the beam.
Q of the area beyond the level you are cutting at, about the neutral axis.
Width of the section at that level — the web thickness for an I-beam.
Results
| Shear force V(x) | 5 kN |
|---|---|
| Transverse shear stress τ | 1.5 MPa |
| Largest shear force on the beam | 5 kN at 0 m |
| τ at that station, same Q and t | 1.5 MPa |
Assumptions
- · Statically determinate planar beam; reactions come from the shared equilibrium solver.
- · Small displacements, linear elasticity, plane sections remain plane (Euler-Bernoulli); shear deformation is neglected.
- · Uniform E and I along the beam when slope and deflection are reported.
Open in the Lab
The Lab shows the shear diagram this stress is taken from.
Technical reference
τ = V Q / (I t)
Q = ∫ y dA over the area beyond the cut level
rectangle: τ_max = 3V / (2A) at the neutral axis
- V
- internal shear force at the station[N]
- Q
- first moment of the area beyond the level, about the neutral axis[m³]
- I
- second moment of area of the whole section[m⁴]
- t
- section width at the level considered[m]
- τ
- average transverse shear stress across that width[Pa]
τ is the average across the width t, not a point value. The formula is accurate for narrow rectangles and for the web of a thin-walled section, and less so across a wide flange.
Q is zero at the extreme fibres and largest at the neutral axis, so a rectangular section carries no shear stress at top and bottom and its maximum at mid-height.
Prismatic linear-elastic beam, bending about a principal centroidal axis, away from concentrated loads and supports.