Weld Bending Stress⚠ unverified

Understanding the Parameters

• Bending moment $M$ — for an eccentric load $F$ at eccentricity $e$, $M = Fe$. It is the agent that bends the group; the whole calculation exists to convert it into an edge stress.

• Distance $y$ — measured from the group's neutral (centroidal) axis. Stress is largest at the outermost weld fibre, so the design value uses $y = y_{\max}$; that outer point is where cracking initiates.

• Second moment $I$ — the geometric stiffness of the weld pattern in bending, from Weld Group Moment Of Inertia (or the exact line/throat sum). A larger $I$ — a taller group — lowers the stress for the same moment.

• Section modulus link — $\sigma_{\max} = M/S$ with $S = I/y_{\max}$ the section modulus (Section Modulus); the two forms are the same statement.

Derivation (Approaching a Proof)

Adopt the Euler–Bernoulli assumption that plane sections remain plane, so the longitudinal strain varies linearly with distance $y$ from the neutral axis: $\varepsilon = y/\rho$, with $\rho$ the radius of curvature. For linear-elastic weld metal, $\sigma = E\varepsilon = E y/\rho$.

Equilibrium requires the internal stresses to reproduce the applied moment:

$$M = \int y\,\sigma\,dA = \frac{E}{\rho}\int y^2\,dA = \frac{E}{\rho}\,I.$$

Eliminating the curvature $E/\rho = M/I$ and substituting back into $\sigma = (E/\rho)\,y$ gives the flexure formula for the weld group:

$$\sigma = \frac{M\,y}{I}.$$

The weld group borrows this beam result by treating its throat area as the bending section; the neutral axis passes through the group centroid, and $I$ is computed for the pattern (line method or the bounding-rectangle approximation of Weld Group Moment Of Inertia).

Dimensional check. $\dfrac{M\,y}{I} = \dfrac{(\text{N}\cdot\text{m})\cdot\text{m}}{\text{m}^4} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$ — a stress, as required.

History and Development

The flexure formula $\sigma = My/I$ is the cornerstone of beam theory, developed from Galileo's cantilever problem through the elastic curve of Bernoulli and Euler to Navier's general statement (1820s). Its application to weld groups — treat the weld as a thin section, find its centroid and $I$, evaluate the stress at the outer fibre — is the bending half of Blodgett's line method (Design of Welded Structures, 1966). Combined with direct shear, it sizes the eccentrically loaded brackets, seat angles, and base plates that fill structural steelwork.

Related Concepts: Weld Group Moment Of Inertia, Section Modulus, Eccentric Load Angle, Eccentric Weld Load, Fillet Weld Size, Weld Throat Stress

Notes: Euler–Bernoulli flexure applied to the weld pattern; neutral axis through the centroid. Use $y = y_{\max}$ (outer fibre) for the design stress. $\sigma = M/S$ with $S = I/y_{\max}$ is the equivalent section-modulus form. Combine vectorially with direct shear for the resultant.