A cantilever beam of length L is embedded at its right end,...

A cantilever beam of length $L$ is embedded at its right end, and a horizontal tensile force of $P$ pounds is applied to its free left end. When the origin is taken at its free end, as shown, the deflection $y(x)$ of the beam can be shown to satisfy the differential equation $$ E I y^{\prime \prime}=P y-w(x) \frac{x}{2} $$ Find the deflection of the cantilever beam if $w(x)=w_{0} x$, $0<x<L$, and $y(0)=0, y^{\prime}(L)=0$

A cantilever beam of length $L$ is embedded at its right end, and a horizontal tensile force of $P$ pounds is applied to its free left end. When the origin is taken at its free end, as shown, the deflection $y(x)$ of the beam can be shown to satisfy the differential equation
$$
E I y^{\prime \prime}=P y-w(x) \frac{x}{2}
$$
Find the deflection of the cantilever beam if $w(x)=w_{0} x$, $0<x<L$, and $y(0)=0, y^{\prime}(L)=0$

Key Concepts

Distributed Loads

Distributed loads are forces spread over a portion or the entire length of the beam, and they can vary along the beam's length. Recognizing how these loads influence the bending moment and integrating them appropriately into the differential equation is key to accurately predicting the beam's deflection.

Cantilever Beam Analysis

A cantilever beam is fixed at one end and free at the other, which imposes specific boundary conditions on the problem. Understanding these boundary conditions, such as zero deflection and slope at the free end or at the built-in support, is crucial for correctly solving the differential equation governing deflection.

Euler-Bernoulli Beam Theory

This theory provides the foundational relationship between bending moments, deflection, and curvature in beams. By assuming that plane sections remain plane and that deflections are small, it leads to a differential equation linking the bending moment to the beam deflection, which is essential for analyzing how beams respond to various loads.

Beam Deflection Differential Equation

This concept involves setting up a second order differential equation that relates beam deflection to the applied loads and the beam's stiffness (represented by EI). The differential equation is central to determining the deflection profile of a beam under distributed and concentrated loads.

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