Problem 52 - An Introduction to the Mechanics of Solids in SI Units

Question

The beam \(A D\) is built-in at both ends. The magnitude of its fully plastic bending moment is \(M_{L}\). Find the limiting value of the load \(P\) on the swing seat which corresponds to plastic collapse, as a function of the dimension \(a\), where \(0

Step-by-Step Solution

Answer

The limit of the load \(P\) is a function of the fully plastic bending moment \(M_{L}\), the distance \(a\), and the length of the beam \(L\). This function varies based on whether \(a\) is less than, equal to, or greater than the value of \(L/2\).

Step 1: Assigning Symbols to Variables

Let's denote the bending moments at A, D, and E as \(M_{A}\), \(M_{D}\), and \(M_{E}\) respectively. Similarly, the reactions at A and E are represented as \(R_{A}\) and \(R_{E}\). We also know the fully plastic bending moment as \(M_{L}\). Furthermore, \(L\) is the total length of the beam (from A to E), while \(a\) is the distance from A to the load P.

Step 2: Calculating Bending Moments

Firstly, the bending moment at point D can be calculated by using the relation \(R_{A} \cdot a = M_{D}\). This result can then be used to calculate the bending moments at points A and E using the equations \(M_{A} = M_{D} + P \cdot (\frac{L}{2} - a)\) and \(M_{E} = M_{D} + P \cdot (\frac{L}{2} + a)\) respectively.

Step 3: Determining the Limit of Load P

The limit of the load \(P\) is when the bending moments \(M_{A}\), \(M_{D}\), and \(M_{E}\) reach the fully plastic moment \(M_{L}\). This gives three equations: \(M_{A} = M_{L}\), \(M_{D} = M_{L}\), \(M_{E} = M_{L}\). Solving these three equations simultaneously gives the limit of the load \(P\).

Step 4: Combining Equations

Combining our moment equations with these limit equations, we can solve for \(P\) in different cases based on the value of \(a\) (whether \(a\) is less than, equal to, or greater than \(L/2\)). In each case, \(P\) will be a function of \(M_{L}\), \(a\) and \(L\).

Key Concepts

Fully Plastic Bending Moment

The concept of the fully plastic bending moment is crucial in understanding the limits of material bending. Specifically, it is the moment when a section of the beam yields fully across its cross-section. This is the maximum moment that the beam can withstand before undergoing large, permanent deformations.
In engineering terms, this is the point at which all the material in the cross-section has reached its yield stress, leading to a plastic hinge. It represents the ultimate load-carrying capacity of the section. It's helpful to think of it as the beam's breaking point, allowing engineers to know how far they can push a structure when designing it.
When designing beams, knowing the fully plastic bending moment can prevent structural failure and ensure safety. Understanding this concept is essential when analyzing structures for potential collapse as it governs the permissible moment threshold.

Bending Moment Calculation

Bending moment calculation is a fundamental part of structural analysis. It involves determining the internal moment needed to keep a beam from bending under external forces.
For a built-in beam like the one in the exercise, the bending moment varies along the length of the beam, influenced by the position and magnitude of loads. The step-by-step solution involves calculating bending moments at different points, like at A, D, and E.

• At point D, the moment is calculated using the reaction at A and the distance to the load.
• At points A and E, additional factors like the load's position relative to the center influence the moment calculations.

Bending moment calculations account for varying conditions that impact how the beam, or any structural element, responds to load. Correct calculations ensure that structures can withstand applied forces without collapsing.

Built-in Beams

Built-in beams are beams that are fixed or clamped at both ends. This type of support ensures no rotation can occur at the ends, creating a unique distribution of internal forces.
They are known for being more rigid and having higher load capacities compared to other types of support, which makes them suitable for many structural applications. However, this fixed condition also implies that built-in beams need careful analysis, as they are subject to complex interactions between moments and shear forces.
This exercise focuses on a built-in beam setup where understanding load placement, like distance "a", is critical to evaluating the plastic collapse load. Engineers prioritize such analyses to prevent failure by accurately predicting how loads will affect a built-in beam's structural integrity.

Load Limit Evaluation

Load limit evaluation determines the maximum permissible load before a structure reaches failure, ensuring safety and stability. It involves calculating the highest load that will not exceed the fully plastic bending moment at any point of the beam.
In the exercise, the load limit is evaluated under the condition where moments at points A, D, and E all reach the fully plastic level. This involves:

• Establishing equations based on the moments equating to the fully plastic value.
• Solving these equations to find the load value "P" that corresponds to plastic collapse.

This approach helps identify the point of maximum stress and ensures that a beam can perform its intended function without risk of sudden failure. Evaluating the load limit is fundamental in designing components that are both safe and efficient.