When a compressive instead of a tensile force is applied at...

When a compressive instead of a tensile force is applied at the free end of the beam in Problem 7 , the differential equation of the deflection is
$$
E I y^{\prime \prime}=-P y-w(x) \frac{x}{2}
$$
Solve this equation if $w(x)=w_{0} x, 0<x<L$, and $y(0)=0$ $y^{\prime}(L)=0 .$

Key Concepts

Forcing Functions and Load Distribution

The presence of a forcing function, representing a distributed load, introduces a non-homogeneous term in the differential equation. Understanding how the shape and variation of the load (for example, a linearly varying load with distance) affect the beam's response is an essential aspect of solving the equation. The solution typically involves finding a particular solution that satisfies the non-homogeneous part, in addition to the homogeneous solution.

Boundary Conditions in Structural Analysis

Boundary conditions are crucial for a unique solution to the differential equation that describes the beam's deflection. They reflect the physical constraints imposed at the supports or free ends of the beam, such as zero deflection or zero slope. These conditions allow the determination of integration constants in the general solution of the beam’s differential equation.

Differential Equation Formulation

In structural mechanics, the deflection of a beam under load is governed by a differential equation that balances internal resistive forces (like bending moments) with external forces (such as distributed loads and axial loads). The inclusion of terms like the compressive or tensile force multiplied by the displacement reflects the axial load effects on the beam’s stability and deflection behavior.

Euler-Bernoulli Beam Theory

This theory forms the basis for analyzing the bending of beams. It relates the bending moment along the beam to its curvature using the flexural rigidity (usually denoted as EI, where E is the elastic modulus and I is the moment of inertia). The theory allows deriving differential equations that describe the deflection profile of a beam under various loading conditions.

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