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 Rayleigh Beam Theory with \(r_{G} / L=0.1\).

Short answer

Question: Determine the expression for the bending moment at the center of a simply supported uniform beam subjected to a uniform distributed load that varies with time, using Rayleigh Beam Theory. Answer: The bending moment at the center of the span as a function of time is given by: $$ M\left(\frac{L}{2}, t\right) =q_{0} E I \sum_{n=1}^{\infty} \frac{n^{3}\pi}{L^{2}} \left[\frac{1-\cos\left(n \pi (0.1)\right)}{1-\cos\left(n \pi\right)}\right] \cos \frac{n \pi}{2}\left[\frac{L}{n\pi}(\cos(n\pi)-1)\right]\left[-\frac{1}{\Omega}+ \frac{1}{\Omega}\cos(\Omega t)\right] $$

Step-by-step solution

Write down Rayleigh Beam Theory expression for deflection

Rayleigh beam theory provides an approximate expression for the deflection, \(w(x,t)\), of a beam as: $$ w(x,t) = \sum_{n=1}^{\infty} \frac{1}{n \pi} \left[ \frac{1-\cos(n \pi r_{G})}{1-\cos(n \pi)} \right] \sin \frac{n \pi x}{L} \int_{0}^{t} \int_{0}^{L} p(x',t')\sin\frac{n\pi x'}{L}\,dx'\,dt' $$

Substitute the given values and solve for deflection

Now, we need to substitute the given values of \(p(x, t)=q_{0} \sin \Omega t\) and \(r_{G}/L = 0.1\) into the Rayleigh Beam Theory equation for deflection, and solve for \(w(x,t)\). $$ w(x,t) = \sum_{n=1}^{\infty} \frac{1}{n \pi} \left[ \frac{1-\cos(n \pi (0.1))}{1-\cos(n \pi)} \right] \sin \frac{n \pi x}{L} \int_{0}^{t} \int_{0}^{L} (q_{0}\sin\Omega t')\sin\frac{n\pi x'}{L}\,dx'\,dt' $$ Now, let's integrate the expression with respect to \(x'\) and \(t'\): $$ w(x,t) = \sum_{n=1}^{\infty} \frac{1}{n \pi} \left[ \frac{1-\cos(n \pi (0.1))}{1-\cos(n \pi)} \right] \sin \frac{n \pi x}{L} (q_{0}\int_{0}^{t} \sin\Omega t'\,dt'\int_{0}^{L} \sin\frac{n\pi x'}{L}\,dx') $$ Compute the \(x'\) integral: $$ \int_{0}^{L} \sin\frac{n\pi x'}{L}\,dx' = \frac{L}{n\pi}(\cos(n\pi)-1) $$ Compute the \(t'\) integral: $$ \int_{0}^{t} \sin\Omega t'\,dt' = -\frac{1}{\Omega}+ \frac{1}{\Omega}\cos(\Omega t) $$ Put these two integrals back into the expression for \(w(x,t)\): $$ w(x,t) = -q_{0} \sum_{n=1}^{\infty} \frac{1}{(n \pi)^{2}} \left[\frac{1-\cos\left(n \pi (0.1)\right)}{1-\cos\left(n \pi\right)}\right] \sin \frac{n \pi x}{L}\left[\frac{L}{n\pi}(\cos(n\pi)-1)\right]\left[-\frac{1}{\Omega}+ \frac{1}{\Omega}\cos(\Omega t)\right] $$

Calculate bending moment using deflection

Now that we have the expression for deflection, \(w(x,t)\). We can determine the bending moment \(M(x,t)\) at the center of the span by using the following relationship between deflection and bending moment: $$ M(x, t) = -E I \frac{d^2 w(x,t)}{dx^2} $$ To calculate the second derivative of the deflection with respect to x, we need to differentiate the deflection expression \(w(x,t)\) twice: $$ \frac{d^2 w(x,t)}{dx^2} = -q_{0} \sum_{n=1}^{\infty} \frac{n^{3}\pi}{L^{2}} \left[\frac{1-\cos\left(n \pi (0.1)\right)}{1-\cos\left(n \pi\right)}\right] \cos \frac{n \pi x}{L}\left[\frac{L}{n\pi}(\cos(n\pi)-1)\right]\left[-\frac{1}{\Omega}+ \frac{1}{\Omega}\cos(\Omega t)\right] $$ Now substitute the expression for the second derivative into the expression for bending moment and use x = L/2 for the center of the span: $$ M\left(\frac{L}{2}, t\right) = E I \left. \frac{d^2 w(x,t)}{dx^2}\right|_{x=\frac{L}{2}} =q_{0} E I \sum_{n=1}^{\infty} \frac{n^{3}\pi}{L^{2}} \left[\frac{1-\cos\left(n \pi (0.1)\right)}{1-\cos\left(n \pi\right)}\right] \cos \frac{n \pi}{2}\left[\frac{L}{n\pi}(\cos(n\pi)-1)\right]\left[-\frac{1}{\Omega}+ \frac{1}{\Omega}\cos(\Omega t)\right] $$ This expression represents the bending moment at the center of the span as a function of time.

Key concepts

Bending Moment

The bending moment is a key concept in beam design and mechanics. It refers to the internal force that induces bending in a beam whenever external loads are present. Bending moments can vary across the length of the beam, and their calculation is crucial to determine how beams will respond to loads.

For a simply supported beam, the bending moment is typically greatest at the mid-span if the load is uniformly distributed. This happens because the beam bends most at this point. In our specific example using Rayleigh Beam Theory, the bending moment calculation at the center of the span involves differentiating the deflection equation twice with respect to the beam's length.

• The bending moment formula is determined by the beam's material and cross-sectional properties, represented by the bending stiffness, EI.
• The calculation involves integrating over the beam's length and time to account for the dynamic load's effect.
• Finally, evaluating the bending moment equation at the central point \(x = \frac{L}{2}\), provides the bending moment at the center of the span.

Deflection Calculation

Deflection in beams is an important aspect of structural engineering because it defines how much a beam will bend under a given load. Understanding deflection ensures that the beam remains functional and safe.

Using Rayleigh Beam Theory, we approximate deflection by expanding it into a series. This series involves calculating integrals over the beam's length and time, capturing the distributed load's effects.
Solving for deflection involves:

• Setting up the general deflection expression, which accounts for residual vibrating effects sometimes absent in simpler models.
• Substituting known expressions like load function and given ratios into the deflection equation.
• Performing integral operations first with respect to the spatial variable then with respect to time.

Simply Supported Beam

A simply supported beam is a basic structural element widely used in engineering and architecture. It rests on supports at both ends but experiences free rotation and horizontal displacement between them.

The boundary conditions of a simply supported beam are:

• Zero deflection at both supports, allowing the beam to pivot easily.
• Equal and opposite reactions at the supports if there's an external load to maintain equilibrium.

These conditions simplify solving for deflection and bending moments. The understanding of its fundamental behavior is critical when using sophisticated theories like Rayleigh Beam Theory, which verifies how predisposed the beam is to flex under long-term cyclical loads.

Vibration Analysis

Vibration analysis in beams is crucial when dynamic loads, like time-varying or cyclical loads, are applied. These loads can induce oscillations that affect stability and performance.

Rayleigh Beam Theory incorporates these elements by including factors such as mass per unit length and frequency of applied dynamic loads. Here’s why this specific analysis method is useful:

• Accounts for inertial effects, thus improving the accuracy for how beams behave under varying load conditions.
• Evaluates how natural frequency and external oscillation frequencies may cause resonance, which increases deflection.
• Considers the beam’s geometry, material properties, and boundary conditions collectively to provide a more comprehensive solution.

Importantly, vibration analysis helps predict structural performance under fluctuating forces, thus preventing fatigue or potential failure over time.