Problem 20 - Concepts and Applications of Finite Element Analysis

Question

A uniformly loaded beam of constant flexural stiffness \(E I\) is simply supported at its ends \(x=0\) and \(x=L\). In parts (a) and (b), determine the deflection and bending moment predicted at \(x=L / 2\) by a Rayleigh-Ritz solution that has a single d.o.f. Compare exact and approximate results. (a) Use a single-d.o.f. algebraic expression-that is, a d.o.f. \(a_{1}\) times a function that contains \(x\) and \(x^{2}\). (b) Use one term of a sine series. (c) Why should you anticipate that part (b) will be better than part (a) if part (a) is the simplest admissible function?

Step-by-Step Solution

Answer

The mid-point deflection and bending moment of a beam can be calculated using Rayleigh-Ritz method with different degrees-of-freedom models. While an algebraic expression involving polynomial of \(x\) can approximate these quantities, a function involving sine series tends to give more accurate results due to the periodic nature of the deflection of the beam.

Step 1: Using a single d.o.f. algebraic expression

Firstly, assume a function of deflection in terms of \(x\) and \(x^{2}\) multiplied with a degree-of-freedom \(a_{1}\). \[v(x) = a_{1}(x - x^{2})\] Using this function, derive the expression for bending moment \(M(x)\) by solving the equation \(M = E.I.(d^2v/dx^2)\). Substitute the value of \(x = L/2\) in the derived equation to calculate bending moment at the midpoint.

Step 2: Using one term of a sine series

Next, assume the function of deflection as a term of a sine series multiplied with a degree-of-freedom \(a_{2}\). \[v(x) = a_{2}.sin(πx/L)\] Repeat the same steps of deriving the bending moment equation, and substituting the value of \(x = L/2\) to calculate the bending moment.

Step 3: Comparing results and giving reasons

Compare the results of the two models. In general, the sine series solution will be more accurate than the polynomial solution. This is because the sine series solution takes into consideration the periodic nature of the beam - due to points of support at the ends thereby allowing the beam to move up and down periodically like a wave which is naturally captured by the sine function but not by a simple polynomial function of \(x\).

Key Concepts

Rayleigh-Ritz Method

The Rayleigh-Ritz method is a powerful technique used in engineering and physics to approximate solutions to complex problems. This method is particularly useful in structural analysis for calculating deflections or stress in beams. The basic idea is to assume a trial function that satisfies the essential boundary conditions of the beam. This function usually contains arbitrary parameters, known as degrees of freedom (d.o.f.), which can be adjusted to minimize the error in the approximation.
The trial function should be chosen such that it resembles the actual physical behavior of the system. The accuracy of the Rayleigh-Ritz solution depends significantly on the choice of the trial function. By using one or more terms with parameters, the Rayleigh-Ritz method simplifies complex differential equations into simple algebraic equations.
In the given exercise, two trial functions are used: one polynomial and one sine series. Comparing solutions from different functions helps reveal the most efficient method for predicting deflection and bending moments accurately.

Bending Moment

Understanding the bending moment is essential when analyzing beams in structural engineering. The bending moment at any section of the beam is the result of internal forces that create bending without causing acceleration. It measures the intensity of the force needed to make the beam bend at that section.
The bending moment is mathematically defined as the derivative of the shear force along the beam. For a simply supported beam, it can be calculated by the relation: \(M = EI \frac{d^2v}{dx^2} \), where \(EI\) is the flexural stiffness and \(v(x)\) is the deflection function.
In our example, calculating the bending moment at the midpoint \(x = L/2\) involves first defining the deflection function, then finding its second derivative, and finally substituting the specific \(x\)-value. This calculation differs for different assumed functions, such as the polynomial and sine series introduced in the exercise.

Beam Deflection

Beam deflection refers to the displacement of the beam under loading. It’s a critical consideration in construction as excessive deflection can lead to structural failure. Understanding how much a beam will deflect under certain conditions helps in designing safer structures.
It is essential to accurately define the deflection curve, \(v(x)\), which describes how the beam bends. One method to define \(v(x)\) is by assuming formulations based on algebraic expressions or trigonometric functions. For example: - Polynomial of the form \( v(x) = a_1(x - x^2) \) - Sine function of the form \( v(x) = a_2 \sin(\pi x/L) \)
Both methods aim to model the real bending shape of the beam under a given load and support condition. Using these deflection functions, bending moments at key points such as the midpoint can be calculated, providing insight into how the beam behaves under load.

Flexural Stiffness

Flexural stiffness is a vital property in material science and structural engineering that quantifies a beam’s resistance to bending under load. It is denoted by the product \(EI\), where \(E\) is the modulus of elasticity of the material and \(I\) is the moment of inertia of the beam's cross-section.
Higher flexural stiffness indicates a stronger resistance to bending, meaning the beam can support more load without deflecting excessively. This concept helps engineers design beams that are strong yet economical.
In problems involving beam deflection and bending, flexural stiffness plays a crucial role in determining how much the beam will bend under load. When calculating the bending moment or deflection equations as illustrated in the exercise, \(EI\) acts as a constant that impacts the beam’s overall response to loading.