This module focuses entirely on the internal forces (shear and bending moments) generated within structural beams, their strict sign conventions, and how to construct Shear and Moment Diagrams (V and M diagrams). While previous modules calculated external support reactions, this module finally cuts the beam open to see what is happening inside the material itself.

When an external load (like a heavy concrete slab) rests on a steel beam, the external support reactions (the columns) push back up to keep the entire beam in static equilibrium.

However, the material between the load and the supports must transfer these forces. To do so, internal stresses are generated inside the steel. If we make an imaginary mathematical "cut" completely through the beam at any point, we expose these internal forces. To keep the newly cut Free Body Diagram in equilibrium, three internal forces must exist on the cut face:

While shear ($V$) and bending moments ($M$) are dominant in horizontal beams supporting vertical gravity loads, structural members can also experience an internal Normal Force ($N$).

The normal force acts strictly perpendicular to the cross-section (along the longitudinal axis) and is responsible for axial tension or compression. It becomes significant in inclined beams, columns, and frames subjected to lateral wind loads or thermal expansion.

Establishing a strict mathematical sign convention for internal forces is critical for accurately drawing diagrammatic curves.

Because external loads are distributed differently along a beam (e.g., point loads vs. uniform loads), the internal Shear ($V$) and Moment ($M$) change drastically depending on exactly where you cut the beam.

Why are they important? Architects and structural engineers don't just guess where a beam might fail. They look precisely at the $V$ and $M$ diagrams to mathematically locate the absolute maximum values ($V_{max}$ and $M_{max}$). The beam's physical size, material strength, and internal reinforcement must be designed specifically to safely resist these maximum peaks, which usually dictate the entire structural design.

Drawing V and M diagrams point-by-point using repetitive free-body equations is incredibly tedious. Instead, we use powerful mathematical relationships based on differential calculus. Let $w$ be the intensity of a distributed load acting downward, $V$ be the internal shear, and $M$ be the internal bending moment.

The mathematical slope (rate of change) of the Shear Diagram exactly equals the negative intensity of the distributed Load at that specific point.

Therefore, by integrating, the change in Shear ($\Delta V$) between any two points on the beam is exactly equal to the negative mathematical area under the external Load Diagram between those exact same two points.

The mathematical slope of the Bending Moment Diagram exactly equals the value of the Shear Diagram at that specific point.

Therefore, by integrating, the change in Bending Moment ($\Delta M$) between any two points is exactly equal to the total mathematical area under the Shear Diagram between those points.

While calculus handles distributed loads, concentrated external forces create sudden discontinuities in the diagrams:

Drawing shear and moment diagrams by hand using the method of sections requires writing a new set of equations every time the load changes (e.g., passing a point load). For a complex beam with many loads, this results in a piecewise nightmare of equations.

Advanced engineering analysis utilizes Singularity Functions (often called Macaulay's Method). These special bracketed mathematical functions $\langle x - a \rangle^n$ "turn on" only when the variable $x$ exceeds the load's starting position $a$. This allows an engineer to write one single, continuous, integrable mathematical equation that defines the shear and bending moment for the entire length of a complex architectural beam, making computer programming of beam analysis possible.

For complex loading conditions, the Principle of Superposition allows engineers to calculate shear and moment diagrams for individual simple loads separately, and algebraically sum them together to find the final combined diagram.

To ensure consistent mathematical results across complex structural analyses, all engineers adhere strictly to a standard sign convention for internal forces on a Free Body Diagram (FBD) of a cut beam section:

The entire process of constructing V and M diagrams visually is strictly based on three differential calculus relationships ($w(x)$ is the distributed load intensity):