# Beam Shear Stress

on . Posted in Classical Mechanics

Beam shear stress, abbreviated as $$\tau$$ (Greek symbol tau), is the horizontal shear stress of a beam.  It refers to the internal stress that occurs within a beam when it is subjected to transverse or shear loads.  When a beam is loaded in such a way that forces are applied parallel to its cross-sectional plane, shear stresses develop along the beam's cross-section.

Shear stress is highest at the neutral axis of the beam, which is an imaginary line that divides the beam into equal top and bottom portions.  The shear stress distribution across the cross-section of a beam is typically triangular, with zero stress at the neutral axis and increasing toward the top and bottom surfaces.  The magnitude of shear stress in a beam is directly proportional to the applied shear force and inversely proportional to the area over which the shear force is distributed.

The first moment of area, Q, represents the product of the area and the perpendicular distance from the neutral axis to the centroid of the area above or below the neutral axis.  It quantifies the distribution of the shear force within the beam's cross-section.  Shear stress is an important consideration in the design of beams and other structural elements.  Engineers analyze shear stress to ensure that beams are designed to withstand the applied shear forces without experiencing failure or excessive deformation.

### Beam Shear Stress formula

$$\tau \;=\; V \; Q \;/\; I \; t$$     (Beam Shear Stress)

$$V \;=\; \tau \; I \; t \;/\; Q$$

$$Q \;=\; \tau \; I \; t \;/\; V$$

$$I \;=\; V \; Q \;/\; \tau \; t$$

$$t \;=\; V \; Q \;/\; \tau \; I$$

Symbol English Metric
$$\tau$$ (Greek symbol tau) = Shear Stress $$lbf\;/\;in^2$$ $$Pa$$
$$V$$ = Total Shear Force at the Point of Location $$lbf\;/\;in^2$$ $$Pa$$
$$Q$$ = First Moment of Area $$in^4$$ $$mm^4$$
$$I$$ = Moment of Inertia of Entire Area Cross-section $$in^4$$ $$mm^4$$
$$t$$ = Material Thickness Perpendicular to Shear $$in$$ $$mm$$