Beam Deflection Analysis: Five Common Methods Explained

In this video the deflection of beams under load is examined from first principles. The presenter defines the elastic curve, the axis x along the undeformed beam, and y the vertical displacement, noting that small deflections lead to a simple slope dy/dx approximation. Five common methods for predicting y are then introduced: the double integration method, Macaulay brackets and singularity functions, the superposition principle, the moment area method, and Castigliano's energy theorem. For each method the speaker shows how to obtain the deflection or rotation at points along the beam, highlights boundary conditions at fixed or supported ends, and explains when linear theory is valid. The video also discusses when to split the beam into sections, how to apply continuity, and how to leverage standard deflection formulas from reference texts.

Introduction and Core Concepts

The discussion begins by establishing the framework for beam deflection in the elastic, small deflection regime. The axis x runs along the undeformed beam while y denotes the vertical deflection. The slope of the elastic curve is approximated by the derivative dy/dx, and the angle of rotation theta is treated as this slope under the small angle assumption. Engineers use the moment curvature relation to connect bending moments M to the curvature of the beam through the flexural rigidity EI. The objective is to determine how y varies along the beam length, given loads and support conditions. Boundary conditions at supports determine how y and its slope behave, which drives the calculation of deflections.

Double Integration Method for Cantilevers

The double integration method starts from the differential equation that relates curvature to bending moment and then integrates twice with respect to x to obtain the deflection y. With EI treated as constant, the moment M is inserted into the equation, and two constants of integration are introduced. Boundary conditions at the fixed support, where the slope and deflection are zero, fix these constants. A cantilever with a load at the end yields a linearly varying moment M = minus P times x, which leads to a straightforward integration. The method can be extended to more complex layouts by splitting the beam into sections where the moment expression changes. Continuity of displacement and slope at section boundaries ensures a consistent deflection curve.

Singularity Functions and Macaulay Brackets

To avoid splitting the beam into multiple regions, Macaulay brackets are introduced as a way to encode piecewise bending moment expressions in a single formula. Macaulay brackets effectively switch terms on and off as we move along the beam, allowing a single M(x) to describe the entire length even when loads or supports change. This approach also handles distributed loads by representing the load and its center of action with appropriate bracket terms. The result is a more compact, yet still exact, description of the bending moment along the full length that can be integrated with the same double integration method.

Superposition in Beam Deflection

Superposition leverages the linearity of small deflections to combine the effects of different loads. If EI is constant and the deflections respond linearly to each load, the total deflection is the sum of the deflections caused by each load independently. Reference tables of deflection formulas for common configurations can be used as building blocks, and the overall response to complex loading is obtained by summing the contributions from each load. This approach is powerful because it extends beyond simple one load cases to more intricate loading patterns through the principle of additivity.

Moment Area Method

The moment area method focuses on the area under the bending moment diagram. The first theorem states that the change in angle between two points on the elastic curve equals the area under the bending moment diagram between those points divided by EI. The second theorem relates the tangential deviation to the first moment of the area of the bending moment diagram and the distance to the centroid. This method is particularly convenient when a point has zero slope, since the tangential deviation directly yields the deflection or angle at that location. Cantilever problems are a common setting for applying the moment area method.

Castigliano's Theorem and Energy Methods

Castigliano's theorem uses strain energy to determine deflections. For a beam bent elastically, the strain energy U is the integral along the beam of M squared over 2EI. The deflection at a point along the line of action of a load equals the partial derivative of U with respect to that load. The derivative can be moved inside the integral, simplifying the calculation. A key feature is that a fictitious load can be introduced to determine deflections at locations without a real load, enabling computation of displacements or angles by differentiating with respect to the fictitious force and then evaluating at zero. Castigliano is powerful because it applies to a wide range of problems beyond straightforward bending cases, provided the small deflection and elastic assumptions hold.

Choosing a Method and Calculus Foundations

All of these methods rely on calculus and the assumptions of elastic behavior and small deflections. The choice of method depends on the geometry, loading pattern, and boundary conditions. In practice, engineers combine these tools with standard deflection tables and the superposition principle to tackle real world structures efficiently.