Problem 43 - An Introduction to the Mechanics of Solids in SI Units

Question

A straight slender member along the \(x\) axis is subjected to only point loads along its length which are all parallel to the \(y z\) plane. Show that the diagrams for \(M_{b y}\) and \(M_{b z}\) must be made up of a number of straight- line segments connected at the points of loading, and the maximum of either component must always occur at a loading point. Now consider the resultant bending moment $$ M_{b}=\sqrt{M_{b y}^{2}+M_{b z}^{2}} $$ Show that if $$ \begin{aligned} &M_{b y}=A x+B \\ &M_{h z}=C x+D \end{aligned} $$ where \(A, B, C\), and \(D\) are arbitrary constants in a certain segment of the member, then $$ \frac{d^{2} M b}{d x^{2}} \geqq 0 $$ and therefore that the resultant moment curve is either straight or concave outward.

Step-by-Step Solution

Answer

The bending moment diagrams for \(M_{b y}\) and \(M_{b z}\) consist of straight-line segments connected at points of loading, with maximum values occurring at these points. For the resultant bending moment \(M_{b}\) in a certain segment, if \(M_{b y}=Ax+B\) and \(M_{b z}=Cx+D\), then the second derivative of \(M_{b}\) with respect to x is zero or positive. This indicates that the resultant moment curve is either straight or concave upwards.

Step 1: Demonstrate \(M_{b y}\) and \(M_{b z}\)

For a beam subject to point loads, the bending moments \(M_{b y}\) and \(M_{b z}\) are determined by the loads and their distances from a particular point along the x-axis. Because each point load contributes a linearly varying moment between the points of loading, the moment diagrams for \(M_{b y}\) and \(M_{b z}\) will consist of straight-line segments connected at the points of loading. The maximum moment for either component will occur at a loading point, as that is where the bending effect of that load is greatest.

Step 2: Derive \(M_{b}\) using \(M_{b y}\) and \(M_{b z}\)

The resultant moment \(M_{b}\) is the square root of the sum of squares of \(M_{b y}\) and \(M_{b z}\). Therefore, we get \(M_{b}=\sqrt{M_{b y}^{2}+M_{b z}^{2}}\). This gives us the magnitude of the resultant moment for any point along the beam.

Step 3: Translate Linear Equations to \(M_{b y}\) and \(M_{b z}\)

For a certain segment of the beam, we are given that \(M_{b y}=Ax+B\) and \(M_{b z}=Cx+D\). Here, A, B, C and D are constants, and x is the distance from the origin.

Step 4: Calculate the Second Derivative of \(M_{b}\)

Now we need to substitute the expressions of \(M_{b y}\) and \(M_{b z}\) in \(M_{b}\), differentiate it twice with respect to x, and show that the second derivative of \(M_{b}\) is either 0 or positive.

Step 5: Conclude about the Shape of the Moment Curve

Because \( \frac{d^{2} M_{b}}{d x^{2}} \geqq 0 \) , the curve of the resultant bending moment \(M_{b}\) has either no curvature (when second derivative is zero; a straight line) or is concave upwards (when second derivative is positive).

Key Concepts

Beam Bending Analysis

Beam bending analysis is a fundamental part of structural mechanics and serves to predict the behavior of beams under various load conditions. Beams are structural members designed to carry and distribute loads to supports, and understanding how they bend due to these loads is crucial for ensuring safety and reliability.

When dealing with straight, slender beams subjected to transverse point loads, we can use the principles of statics to create mathematical models that represent the bending moments at different points along the beam's length. The bending moments, typically denoted as \(M_{by}\) and \(M_{bz}\), represent the internal forces induced by the applied loads, causing the beam to bend about its respective axes.

These bending moment values change along the length of the beam and create what we call moment diagrams. These diagrams are pivotal in structural engineering because they visually represent the magnitude of bending at any given point, enabling engineers to identify the points of maximum stress and evaluate the beam's capacity to withstand these moments without failure.

Moment Diagrams

Moment diagrams are graphical representations that show how bending moments vary along the length of a beam. For a beam under point loads, as described in the exercise, the bending moment at any point is a function of the distance from the loads and the magnitude of these loads. Since the loads are applied at distinct points, the bending moments change linearly between these points. This results in a moment diagram made up of straight-line segments, each connecting at the points where the loads are applied.

The moments \(M_{by}\) and \(M_{bz}\) at any point in the beam can be described using equations of straight lines in segments between the loads. The lines show a clear relationship: the moment increases linearly from one point load to the next. At the points where loads are applied, there is typically a sudden change in the slope of the moment diagram, corresponding to the abrupt change in bending moment caused by the load.

Maximum moments usually occur at these load points, as they are where the lever arm - the perpendicular distance from the line of action of the load to the point of interest - is greatest. Understanding how to construct and interpret these diagrams is essential for evaluating the behavior of beams and ensuring safe design in structural systems.

Structural Mechanics

Structural mechanics is the science of analyzing and predicting the behavior of structures under various forces and moments. It provides a foundational framework that engineers use to ensure the safety and performance of buildings, bridges, and other structures.

In the context of beam bending, structural mechanics allows us to derive equations that predict the internal stresses and deformations a beam experiences under loading conditions. By applying calculus to the fundamental equations provided in this exercise, engineers can determine not just the magnitude of the bending moments, but also the curvature of the beam at any point along its length.

The second derivative of the bending moment equation with respect to the distance, \(\frac{d^{2} M_b}{d x^{2}}\), tells us about the beam's curvature. When this derivative is non-negative, it implies that the curve is flat or concave upwards, corresponding to a segment of a beam under a uniform bending moment or one that is getting stiffer, respectively. Such insights from structural mechanics are vital for safe and efficient structural design, as they allow engineers to identify potential failure points and ensure that structures can sustain the loads they are subjected to throughout their service life.