The ends of a beam of length L rest on two supports as shown...

The ends of a beam of length $L$ rest on two supports as shown in FIGURE 5.1 .6 . With a uniform load on the beam, its shape (or elastic curve) is determined from $$ E I y^{\prime \prime}=\frac{1}{2} q L x-\frac{1}{2} q x^{2} $$ where $E, I,$ and $q$ are constants. Find $y=f(x)$ if $f(0)=0$ and $f^{\prime}(L / 2)=0 .$

The ends of a beam of length $L$ rest on two supports as shown in FIGURE 5.1 .6 . With a uniform load on the beam, its shape (or elastic curve) is determined from
$$
E I y^{\prime \prime}=\frac{1}{2} q L x-\frac{1}{2} q x^{2}
$$
where $E, I,$ and $q$ are constants. Find $y=f(x)$ if $f(0)=0$ and $f^{\prime}(L / 2)=0 .$

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Key Concepts

Elastic Curve

The elastic curve represents the deflected shape of a beam under loading. It is determined by the differential equation relating bending moments to the curvature of the beam, and describes how the beam deforms along its length when subjected to loads.

Euler-Bernoulli Beam Theory

This theory relates the bending moment of a beam to its curvature and is given by the equation EI * y'' = M(x), where E is the modulus of elasticity, I is the moment of inertia, and M(x) is the bending moment due to applied loads. It forms the basis for analyzing beam deflections and stresses in structural mechanics.

Uniform Load

A uniform load means the force is distributed evenly along the entire length of the beam. In beam theory, this simplifies the expression for the bending moment, leading to a quadratic or parabolic form in the governing differential equation, which then needs to be integrated to find the deflected shape.

Integration with Boundary Conditions

Solving for the deflection of the beam involves integrating the second-order differential equation derived from Euler-Bernoulli beam theory. The integration yields constants that are determined by applying specific boundary conditions, such as known deflections or slopes at certain points along the beam.

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