Mastering Shear Force and Bending Moment Diagrams for Beams

Overview

Shear force and bending moment diagrams are essential tools for analyzing beams under loads in mechanical and civil engineering. This video explains what shear forces and bending moments are, how to represent them with two resultants, and how to build accurate diagrams along a beam using free body diagrams and equilibrium.

Key Concepts

We cover reaction forces at supports, static determinacy, and the sign conventions for forces and moments. The presenter demonstrates how to determine internal forces by cutting the beam at successive locations and ensuring equilibrium on each segment. The video also introduces relationships between loads, shear, and moment, including how the derivative of shear with respect to x equals minus the distributed load, and the derivative of bending moment with respect to x equals the shear.

Introduction and core idea

The video presents a focused, practical approach to mastering shear force and bending moment diagrams for beams. When beams are loaded, internal forces develop to maintain equilibrium. These internal forces have vertical components, called sheer forces, and axial components, called normal forces. The cross section of a beam can be represented by two resultants: a shear force and a bending moment. The diagrams visualize how these internal forces vary along the beam as loads are applied and as supports restrain movement.

Free body diagrams and equilibrium

There are three main steps to find the internal forces along the beam. First, draw a free body diagram of the beam showing applied loads and reactions at supports. Second, use equilibrium to determine the unknown reactions and reaction moments. If all reaction loads can be found using the three static equilibrium equations, the beam is statically determinate. If not, the beam is statically indeterminate and different methods are needed; however, the video focuses on determinate cases.

Third, with the reactions known, move a cut along the beam from left to right and compute the shear force and bending moment on each segment. The sign convention used is that downward applied loads are positive, and for shear, the left side of the cut uses downward positive, while the right side uses upward positive. Positive bending moments are those that put the lower beam in tension, corresponding to sagging as positive, and hogging is negative.

Sign conventions and example problems

The video walks through several examples. A simple beam with pin and roller supports subjected to two concentrated forces shows how to compute reactions via force and moment balance. After finding RA and RB, the shear diagram jumps to match the reaction forces, while the bending moment diagram is a straight line segment between loads, increasing linearly with distance from the reactions. A more complex case examines a distributed load combined with a concentrated force, illustrating the relationship between the areas under the shear diagram and the changes in the bending moment diagram. The presenter uses the derivative relationships to verify the diagrams and demonstrates a quick sense check by differentiating the bending moment equation to recover the shear and the distributed load to recover the area under the shear diagram.

Key relationships and checks

The video emphasizes two fundamental relationships: the slope of the shear diagram equals minus the distributed load, and the slope of the bending moment diagram equals the shear. Integrating the first relation shows that the change in shear between two points equals the area under the loading diagram between those points. Integrating the second relation shows that the change in bending moment equals the area under the shear diagram. These relationships help ensure the diagrams are consistent with the loads and support conditions.

Advanced example and visualization

An extension considers a cantilever with a concentrated moment and a distributed load. The moment diagram experiences a jump at the location of the concentrated moment and a curved segment under the distributed load due to the quadratic bending moment equation. The video notes that the deformation of the beam can be inferred from the bending moment: positive moment causes sagging, negative moment hogging, and a zero moment corresponds to a straight segment.

Practical takeaways

Although displacement cannot be read directly from the diagrams, the bending moment information helps predict the beam’s qualitative shape under load. The diagrams provide a powerful sense-check tool and a structured method to analyze many beam configurations, from simple simply supported spans to cantilevers with complex loading.