After determining the maximum internal shear forces ($V$) and bending moments ($M$) acting on a beam, the next step is calculating the resulting stresses that develop within the beam's physical cross-section. This is the crucial bridge between structural analysis (finding the abstract internal forces) and structural design (sizing the physical member to safely resist those forces without yielding or rupturing).

The stress caused by an internal bending moment is called flexural stress or bending stress. It is a normal stress (acting perpendicular to the cross-section), just like axial tension or compression, but unlike a uniform axial load, bending stress varies linearly across the depth of the beam.

The general flexure formula calculates the normal bending stress ($\sigma$) at any vertical distance $y$ from the neutral axis:

To design a safe beam, we must find the maximum flexural stress ($\sigma_{max}$). Since stress increases linearly with distance from the NA, the maximum stress occurs at the extreme outer fibers. We denote the distance from the NA to this outermost fiber as $c$:

Where:

The flexure formula $\sigma = \frac{My}{I}$ is derived based on several critical assumptions, known collectively as the Euler-Bernoulli beam theory:

The flexure formula ($\sigma = \frac{Mc}{I}$) introduced earlier is highly useful, but it relies on two very strict geometric assumptions:

When these conditions are violated, we must apply advanced stress analysis techniques.

If a beam cross-section is completely asymmetrical (like an unequal leg angle, a Z-section, or a channel section loaded outside its shear center), or if the applied load does not pass through an axis of symmetry, the beam will undergo unsymmetrical bending.

In pure symmetrical bending, the neutral axis is perfectly horizontal and the beam simply deflects downwards. However, in unsymmetrical bending, the member simultaneously bends downwards and twists laterally (sideways). The true neutral axis becomes inclined at an angle to the horizontal cross-section.

To determine the true bending stresses at any point ($y, z$) on an unsymmetrical section under moments $M_y$ and $M_z$, engineers must use the generalized, expanded flexure formula involving the product of inertia ($I_{yz}$):

Many architectural members are constructed by permanently bonding two completely different materials together to leverage their individual strengths (e.g., a timber joist reinforced with a steel flitch plate, or a concrete slab poured over steel decking). These are composite beams.

Because the two materials have different Moduli of Elasticity ($E_1$ and $E_2$), they do not behave uniformly under load. A steel plate will carry significantly more stress than an identically sized piece of wood because steel is much stiffer.

To apply the standard flexure formula to a composite beam, we must mathematically convert the entire cross-section into a single, homogeneous, imaginary material. This is done using a modular ratio ($n$):

We then multiply the width (but not depth) of the stronger material by $n$ to create an equivalent "transformed" cross-section entirely made of the weaker material. We can then calculate the transformed section properties ($I$, $y_{bar}$) and apply $\sigma = \frac{Mc}{I}$ to find the true stresses.

To simplify the iterative process of beam design, the geometric properties of a cross-section (Moment of Inertia $I$ and extreme fiber distance $c$) are often combined into a single ratio defined as the Section Modulus ($S$).

The Section Modulus directly quantifies the bending strength of a specific geometric shape, independent of the material it is made from.

Substituting $S$ back into the flexure formula, the maximum flexural stress equation simplifies significantly to:

While bending moments cause normal stress, shear forces ($V$) cause shear stress ($\tau$).

When a beam bends, the horizontal layers of fibers tend to slide past one another. Think of bending a deck of cards; the cards slide horizontally against each other. In a solid beam, the material's cohesive strength resists this sliding, creating internal horizontal (longitudinal) shear stress.

The general shear formula for beams is:

Where:

Many structural members are not solid pieces of material but are "built-up" from smaller components (like an I-beam constructed from three steel plates welded together, or a box beam made of glued plywood boards).

To ensure these separate pieces act together as a single solid beam, we need to design the fasteners (nails, bolts, screws, or welds) connecting them. We use the concept of shear flow ($q$), which represents the horizontal shear force acting per unit length along the longitudinal axis of the beam.

Once $q$ is known (in $N/mm$ or $lb/in$), an engineer can determine how many nails or welds are required per inch to safely transfer that shear force between the built-up components.

The primary objective in designing structural beams is to select a cross-section that can safely carry the applied loads without exceeding the allowable flexural (bending) stress ($\sigma_{allow}$) or the allowable shear stress ($\tau_{allow}$). Because bending typically governs the design of beams (except for very short, heavily loaded spans), the standard design procedure starts with flexure and is subsequently checked for shear.