Elastic Curve from the Load Distribution

JoVE Core
Mechanical Engineering
Zum Anzeigen dieser Inhalte ist ein JoVE-Abonnement erforderlich.  Melden Sie sich an oder starten Sie Ihre kostenlose Testversion.
JoVE Core Mechanical Engineering
Elastic Curve from the Load Distribution

Nächstes Video25.4: Deflection of a Beam

When a beam carries a distributed load, the shear force and bending moment at any point on the beam can be expressed in a differential form. A third-order linear differential equation is created by further differentiating the expression and assuming the flexural rigidity to be constant. Differentiating once again, results in a fourth-order linear differential equation. This equation essentially governs the shape or elastic curve the beam will take when supporting a distributed load. By multiplying the fourth-order linear differential equation by the flexural rigidity constant and integrating it four times, an expression for the curve can be obtained. The constants in the equation defining the beam's curve are dictated by end conditions, such as tilt or deflection from supports and whether the shear force and bending moment are null at a cantilever beam's free end or only the bending moment is zero at both ends of a supported beam. This technique accurately calculates curves for cantilevered or supported beams with distributed loads. However, overhanging beams' support reactions cause shear irregularities, requiring distinct functions for the full-length curve definition.

Elastic Curve from the Load Distribution

The structural behavior of beams under distributed loads is critical for engineering analysis, which focuses on predicting how beams bend and react under such conditions. Different types of beams (e.g., cantilever, supported, or overhanging) behave differently under distributed load conditions.

For all beams, the analysis of the beam's reaction to distributed loads begins by understanding the relationship between a beam's load and the resulting shear forces and bending moments. Initially, this relationship is expressed as a differential equation.

Further differentiation of this expression, assuming constant beam flexibility, leads to a more complex differential equation that describes the beam's deflection curve or how it will bend under the load.

This complex equation is integrated multiple times to determine the actual shape of this curve. Each integration step introduces a constant that must be defined by the beam's boundary conditions, such as how the beam is supported or connected at its ends.

Boundary conditions vary based on how a beam is supported. For instance, a cantilever beam, fixed at one end and free at the other, will have different constraints regarding deflection and force at each end. Conversely, a supported beam will have conditions focused primarily at the points of support. Particularly challenging is the analysis of overhanging beams, where parts of the beam extend beyond its supports. These segments experience unique forces and bending moments, requiring distinct calculations to describe the beam's behavior along its entire length accurately. This detailed understanding ensures that structures are safe and functional.