Question

A simply supported uniform beam of length \(L\), bending stiffness \(E I\) and mass per unit length \(m\) is subjected to the uniform distributed load \(p(x, t)=q_{0} \sin \Omega t\). Determine the bending moment at the center of the span if the behavior of the beam is predicted using Timoshenko beam theory for a structure where \(r_{G} / L=0.1\) and \(E / k G\) \(=5\).

Short answer

Answer: The bending moment at the center of the span can be calculated as: \[M\left(\frac{L}{2}\right)= {-q_0 EI \sum_{n=1}^{\infty} \frac{(1 - (-1)^{n})}{(n\pi)^{2}} \left(\frac{n \pi}{L}\right)^{2} \sin\left(\frac{n\pi}{2}\right) } \] Compute and evaluate the sum to obtain the bending moment M(L/2) at the center of the span.

Step-by-step solution

Write down the equation for Timoshenko beam theory

The governing equation for Timoshenko beam theory can be written as: \[ \frac{\partial^{2}w}{\partial t^2}+\frac{ρ_{G}G A}{E I w} \frac{\partial^{4}w}{\partial x^{4}}-\frac{ρ_{G}G A}{E I w} \frac{\partial^{2}}{\partial x^{2}}\left[\frac{\partial^{2}w}{\partial t^{2}}\right]=q_{0} \sin \Omega t \] where w is the transverse displacement, ρG is the mass per unit length, G is the shear modulus, A is the cross-sectional area, E is the Young's modulus, and I is the moment of inertia.

Apply the boundary conditions for the simply supported beam

For a simply supported beam, the boundary conditions are: 1. At x = 0 and x = L, w = 0 (simply supported on both ends) 2. At x = 0 and x = L, M = 0 (simply supported on both ends) To apply these boundary conditions to the Timoshenko beam equation, we will express the displacement w in terms of a Fourier series and apply the boundary conditions: \[ w(x, t) = \sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi x}{L}\right)}{n \pi} \] \[\sum_{n}^{\infty} \frac{A_n \sin \left(\frac{n \pi x}{L}\right)}{(n \pi)(r_G / L)} \left(\frac{E}{k G}\right) = q_{0} \sin \Omega t \]

Solve for the response function

To solve for the response function An, which represents the Fourier coefficients in the displacement series, we will integrate both sides of the equation over the domain 0 < x < L and simplify the expression: \[\int_{0}^{L} \left(\sum_{n}^{\infty} \frac{A_n \sin \left(\frac{n \pi x}{L}\right)}{(n \pi)(r_G / L)} \left(\frac{E}{k G}\right) - q_{0} \sin \Omega t \right) dx = 0 \] \[ \sum_{n=1}^{\infty} \left[ \frac{2A_n \sin \left(\frac{n \pi L}{L}\right)}{(n \pi)^{2}} - \frac{q_{0} (\sin\left(\frac{n \pi L}{L}\right))}{n^{2} \pi^{2}} \right] = 0 \] \[ A_n = \frac{q_{0} (1 - (-1)^{n})}{n^{2} \pi^{2}} \]

Calculate the bending moment at the center of the span

To calculate the bending moment M at the center of the span x = L/2, we need to calculate the second derivative of the displacement with respect to x and substitute the expression for An: \[ M(x) = E I \frac{\partial^{2}w}{\partial x^2} \] \[ M\left(\frac{L}{2}\right) = E I \sum_{n=1}^{\infty} A_n \left(-\frac{n \pi}{L}\right)^{2} \sin \left(\frac{n \pi}{2}\right) \] \[M\left(\frac{L}{2}\right)= {-q_0 EI \sum_{n=1}^{\infty} \frac{(1 - (-1)^{n})}{(n\pi)^{2}} \left(\frac{n \pi}{L}\right)^{2} \sin\left(\frac{n\pi}{2}\right) } \] Compute and evaluate the sum, and the bending moment M(L/2) at the center of the span can be obtained.

Key concepts

Bending Moment Calculation

The calculation of the bending moment is vital in understanding how beams respond to loads. In essence, the bending moment at a particular point on a beam reflects the internal forces trying to bend the beam at that point.

For the given problem, the bending moment calculation is performed by determining the second spatial derivative of the displacement of the beam, which is denoted by the function \( w(x, t) \). The Timoshenko beam theory provides a framework to consider both bending and shear deformations to achieve more accurate results than traditional Euler-Bernoulli beam theory.

The bending moment, particularly at the center of the span of a simply supported beam, informs us of how the material will react to external loads and forces. It is given by:
- The formula \( M(x) = E I \frac{\partial^{2}w}{\partial x^2} \) where \( E \) and \( I \) are the Young's modulus and moment of inertia, respectively.

Inserting the displacement equation derived from the Fourier series solution into this formula, we amplify accuracy by taking into account distributed loads represented by the term \( q_0 \sin \Omega t \). The correct computation and interpretation of this moment help in designing beams that are structurally sound and long-lasting.

Simply Supported Beam

A simply supported beam is one of the most straightforward support structures used in engineering. It is supported at its endpoints but is free to deflect vertically. This type of support does not restrict rotation, making it ideal for understanding basic beam behavior.

In the analysis, the key characteristics of a simply supported beam are:

• The supports do not apply moments at the endpoints since the beam can freely rotate.
• Boundary conditions typically include zero displacement at the supports, i.e., \( w(0, t) = 0 \) and \( w(L, t) = 0 \), indicating that the ends are pinned but able to rotate.
• The absence of moment on the ends, \( M(0) = 0 \) and \( M(L) = 0 \), emphasizes this rotational freedom.

When modeled using Timoshenko beam theory, the simply supported beam allows us to consider shear deformations, accounting for dynamic effects like vibrations due to distributed loads.

These conditions simplify the mathematical modeling, but still, the calculation of moments and deflections must consider all acting loads to ensure a safe and efficient design process.

Distributed Load Analysis

Distributed load analysis is an essential concept for evaluating how uniformly applied forces affect structures like beams. Instead of point loads, distributed loads apply force along the length of the beam, leading to a more complex stress distribution.

In the problem provided, the beam experiences a uniform distributed load described by the function \( p(x, t) = q_0 \sin \Omega t \). This sinusoidal expression indicates that the load varies with time, imposing dynamic effects on the beam.

Key points about distributed load analysis within the context of Timoshenko beam theory include:

• Consideration of both shear and bending deformations. This makes Timoshenko theory more apt for real-world applications involving high loads or short beams.
• The load function formulates an oscillating load, introducing temporal dependencies which must be captured in the solution.
• Application of boundary conditions for simply supported beams helps in accurately determining the structural response under load.

By analyzing the distributed load using such advanced methods, we obtain a precise understanding of stress, deflection, and moment distribution along the beam. This ensures the design can withstand the applied loads without failure.