Problem 18 - An Introduction to the Mechanics of Solids in SI Units

Question

The singularity functions introduced in Sec. \(3.6\) are useful for writing the bending moment at a section \(x\) based on a free body which extends to the lefi end of the beam. Investigate the family of reversed singularity functions $$ g_{n}(x)=\langle a-x\rangle^{n} $$ and show that they are useful for writing the bending moment at a section \(x\) based on a free body which extends to the right end of the beam. Show that when \(n \geqq 0\) $$ \int_{x}^{\infty}\langle a-x\rangle^{n} d x=\frac{\langle a-x\rangle^{n+1}}{n+1} $$ and $$ \int\langle a-x\rangle^{n} d x=-\frac{\langle a-x\rangle^{n+1}}{n+1}+c $$

Step-by-Step Solution

Answer

The reversed singularity functions \( g_{n}(x) = \langle a-x \rangle^n \) do indeed allow the calculation of bending moment extending to the right end of a beam. The result of the integrals \( \int_{x}^{\infty} \langle a-x \rangle^n dx = \frac{\langle a-x \rangle^{n+1}}{n+1} \) for \( n \ge 0 \), and \( \int \langle a-x \rangle^n dx = -\frac{\langle a-x \rangle^{n+1}}{n+1} + c \).

Step 1: Understand the Reversed Singularity Function

The reversed singularity function \( g_{n}(x) \) defined as \( \langle a-x \rangle^n \) is a piece-wise function where the function equals to 0 when \( x \) is greater than \( a \), and to \( (a-x)^n \) when \( x \) is less than or equal to \( a \). This demonstrates how the function is designed to handle a segment of a beam extending to the right.

Step 2: Performing the First Integral

To calculate \( \int_{x}^{\infty} \langle a-x \rangle^n dx \), split the integral into two based on the possible values of \( x \). When \( x \ge a \), the function is zero and its integral from \( x \) to \( \infty \) is also zero. When \( x < a \), the integral is \( \int_{x}^{a} (a-x)^n dx \) which can be computed with the power rule for integration resulting in \( \frac{(a-x)^{n+1}}{n+1} \). Thus, \( \int_{x}^{\infty} \langle a-x \rangle^n dx = \frac{\langle a-x \rangle^{n+1}}{n+1} \) for \( n \ge 0 \).

Step 3: Performing the Second Integral

To compute the indefinite integral \( \int \langle a-x \rangle^n dx \), again split the integral into two based on the possible values of \( x \). The integral is 0 for \( x > a \). For \( x \leq a \), the integral is \( -\int (a-x)^n dx \), which gives \( -\frac{(a-x)^{n+1}}{n+1} + c \). Hence, \( \int \langle a-x \rangle^n dx = -\frac{\langle a-x \rangle^{n+1}}{n+1} + c \).

Key Concepts

Beam Bending Moment

When you hear 'beam bending moment', think of it as a measure of the force causing a beam to bend. It's critical in the design and analysis of structures, especially in civil engineering. How do we calculate this bending moment? Imagine you have a beam supported at some points, and various forces are applied to it. The bending moment at any point along the beam is the result of these forces acting upon the segment of the beam. The bending moment depends on the magnitude of the applied force, the distance from the point of interest to where the force is applied (the moment arm), and the distribution of forces and supports along the beam.

To calculate bending moments, we need to consider the internal forces in the beam, which essentially 'resist' the applied loads—this is where integration plays a crucial role. By integrating the load distribution along the length of the beam, we can determine the bending moment at any section. To provide an extra layer of understanding, consider a seesaw with two children of different weights sitting on opposite ends. The bending moment would be the 'tipping effect' caused by these weights. If you were out there handling the seesaw, your hands would feel this moment. In structural engineering, we use math instead of hands to quantify that 'feeling' and ensure safety and stability.

Reversed Singularity Function

The reversed singularity function might sound intimidating, but it's actually a simple and intuitive concept once broken down. Think of it as a mathematical tool that lets us describe a certain condition in our problem - specifically, we're dealing with loads or forces acting on a beam in directions you may not be used to. The 'reverse' part means that the function is reflecting a scenario where we are interested in a free body that extends towards the right end of the beam.

Just like the regular singularity function predominantly used for leftward extents, the reversed singularity function vanishes, meaning it equals zero, after a certain point—this point is defined by the location 'a'. The key takeaway is that these functions help us to neatly describe how loads affect bending moments without having to break them into multiple cases. It's like having a precise vocabulary for expressing exactly when and where things happen along the beam. The reason why you reverse the singularity function is similar to why you might use a mirror to look at something from a different angle—it's about perspective and context.

Integration in Mechanics of Solids

Integration is like the Swiss Army knife of calculus, particularly when you dive into mechanics of solids. Picture this: you have an object, and you need to figure out the total impact of varying conditions (like forces) along its body. Integration gives us superpowers to add up all these tiny influences to get a clear picture of what's happening overall.

In the context of beams and bending moments, integration allows us to accumulate the effects of distributed loads to find out the bending moment at any point. It's like adding slices of load influence along the beam's length to understand the total 'twist' at a point. When learning mechanics of solids, getting comfortable with integration is crucial. Think about trying to know every grain of sugar in a jar by individually counting—impractical, right? Integration helps us 'count' these influences in a practical, efficient manner, giving you the sweet (pun intended) total influence of forces or moments. By mastering integration in the context of mechanics, students can navigate complex problems with more ease and bring precision to their understanding of solid mechanics.