Problem 12 - An Introduction to the Mechanics of Solids in SI Units

Question

Sketch shear-force and bending-moment diagrams for the cantilever beam which carries a concentrated force \(P\) and a distributed load of intensity \(w_{o}\) force per unit length.

Step-by-Step Solution

Answer

The shear-force and bending-moment diagrams visualize the internal forces acting on a cantilever beam subject to a concentrated force and a distributed load. The shear-force diagram starts with a value equal to the reaction at the support, and decreases linearly due to the distributed load until it drops suddenly at the point of the concentrated force. The bending-moment diagram starts from zero at the free end of the cantilever beam and increases linearly until the point of the concentrated force where it jumps up suddenly.

Step 1: Calculate the reactions at the support.

Since the beam is a cantilever, there will be a single support at one end (let's say the right end). This support will have to resist the vertical load and a moment. This force is the sum of the concentrated load P and the total distributed load, which is \(w_{o}\) times the total length of the beam (say L). The total downward force is thus \(P + w_{o}L\). This support must then push upwards with the equal force to resist it, so the reaction force at the support is \(P + w_{o}L\). The moment is the sum of the moment caused by the concentrated load (which is \(P \times L\)) and the moment caused by the distributed load (which is \(w_{o}L \times \frac{L}{2}\)). Thus, the total moment at the support is \(P \times L + w_{o}L \times \frac{L}{2}\).

Step 2: Draw the shear force diagram

Shear force is a measure of the internal force acting perpendicular to the plane of the beam. The shear force immediately to the right of the support is equal to the reaction at the support. As we move leftwards, the distributed load decreases the shear force linearly, and therefore the shear force diagram is a straight line sloping downwards. Once we reach the concentration force, shear force drops suddenly by \(P\), thus the diagram has a vertical line at this point.

Step 3: Draw the bending moment diagram

The bending moment diagram is drawn by deriving the area under the shear force diagram. At the far right end of the beam, the bending moment is zero, since that's the free end of the cantilever. As we move leftwards, it increases linearly because of the distributed load, the diagram will be a curve upwards. When we reach the concentrated load, the bending moment jumps suddenly by the force times the distance to the free end of the beam, so, there will be a sudden upward bump in the diagram at that point. The peak value of the bending-moment diagram is at the right end of the beam, since that's where it's supported.

Key Concepts

Understanding Cantilever Beams

A cantilever beam is a structural element that is anchored at only one end while the other end is free. This type of beam is commonly used in building structures like balconies, overhanging roofs, and bridges. The anchored end of the beam must be strong enough to support the load applied to the free end.
This unique structure allows cantilever beams to bear loads without requiring a support or column at the other end, making them convenient for when an open space is required underneath.
Cantilever beams experience two main forms of stress: bending and shear. Bending occurs due to external forces acting on the free end, while shear stress is due to internal forces acting across the cross-section of the beam. Understanding these forces and stresses is essential when analyzing cantilever beams.

Exploring Concentrated Loads

A concentrated load is a force that is applied at a specific point on a structure.
In a cantilever beam, this load can cause significant bending moments and shear forces, especially if the force is applied close to the free end. This leads to a jump in the shear force and bending moment diagrams at the point of application.
Consider the concentrated force labeled as \(P\). It produces a moment calculated as \(P \times L\), where \(L\) is the distance from the point of force application to the anchored support. This concentrated moment will be represented as a sudden drop in the shear force diagram and a sudden rise in the bending moment diagram at that location.

Examining Distributed Loads

Distributed loads are forces spread out along the length of a beam. Their intensity varies with position and is often measured in force per unit length, commonly denoted by \(w_0\).
In the context of a cantilever beam, a distributed load can be thought of as a consistent force acting across the beam's entire length, such as the weight of the beam itself or other evenly spread weight sources.
The effect of a distributed load on shear force and bending moment diagrams is linear. The shear force decreases linearly as you move along the beam with the distributed load subtracted from the initial reaction force. The bending moment due to a distributed load forms a parabolic curve, starting from zero at the free end and reaching a maximum at the supported end. The total moment due to a distributed load can be determined using the formula: \(w_0L \times \frac{L}{2}\).

Understanding Support Reactions

Support reactions are the forces exerted by the support to hold the beam in place. For cantilever beams, support reactions include both a vertical force and a moment since the beam is only fixed at one end.
The vertical force, also known as the reaction force, is equal to the sum of the applied loads—consisting of both the concentrated and distributed loads (calculated as \(P + w_{o}L\)). This balances the downward forces acting on the beam.
The reaction moment is a result of both the point load's moment and the distributed load's moment. This is calculated using the expression \(P \times L + w_{o}L \times \frac{L}{2}\). These calculations ensure that the beam remains in equilibrium, resisting both bending and shear stresses at the support.