Problem 30 - An Introduction to the Mechanics of Solids in SI Units

Question

The beams \(A B\) and \(C D\) are each of length \(L\) and both have the same bending stiffness \(E I .\) The beam \(C D\) is simply supported, while \(A B\) is built-in at \(A\). Before the load \(P\) is applied the two beams make light contact at their midpoints. Find the deflection under the load \(P\).

Step-by-Step Solution

Answer

The deflection of the beam \(CD\) under load \(P_{CD}\) is \[v_{CD} = - \frac{{3 P L^3}}{{64 E I}}\] and the deflection of the beam \(AB\) under load \(P_{AB}\) is \[v_{AB} = - \frac{{P L^3}}{{64 E I}}\]. The total deflection under the load \(P\) is zero as the beams make light contact at their midpoints before applying the load \(P\).

Step 1: Calculate the Deflection of Beam \(CD\)

Since \(CD\) is a simply supported beam with a mid-span point load \(P\), it has a well-established deflection equation: \(v_{CD} = - \frac{{P L^3}}{{48 E I}}\). This is the maximum deflection occurring at the center.

Step 2: Calculate the Deflection of Beam \(AB\)

For the built-in beam \(AB\), the deflection also occurs at the mid-span under the load \(P\). The equation for this is \(v_{AB} = -\frac{{P L^3}}{{192 E I}}\)

Step 3: Determine the Amount of Load on Each Beam

The total deflection at the midspan where beams \(AB\) and \(CD\) make light contact before applying the load \(P\), must be zero after the load application. This means the algebraic sum of individual deflections should be zero, \(v_{CD} + v_{AB} = 0\). Replacing \(v_{CD}\) and \(v_{AB}\) with their respective deflection equations, the load shared by beams \(AB\) and \(CD\) can be calculated as, \(P_{AB} = 3P_{CD}.\)

Step 4: Find Deflection Under the Load \(P\)

Summing the loads on both beams equals to the total applied load, \(P_{AB} + P_{CD} = P\). Solving the simultaneous equations obtained in Step 3 and Step 4 yields the values of \(P_{AB}\) and \(P_{CD}\). Now, substitute \(P_{AB}\) and \(P_{CD}\) into their deflection equation calculated in Step 1 and Step 2 separately, we get the deflection under the load \(P\).

Key Concepts

Bending Stiffness

Bending stiffness is a key concept in understanding how beams behave under loads. It is represented by the product of the modulus of elasticity ( E ) and the moment of inertia ( I ) of the beam's cross-section. The stiffness influences how much a beam will bend when a load is placed on it. In formula terms, it is expressed as EI .

• Modulus of Elasticity ( E ): This is a measure of a material's ability to withstand changes in length when under lengthwise tension or compression. A higher modulus means the material is stiffer.
• Moment of Inertia ( I ): This describes how the area of a beam's cross-section is distributed about a certain axis, affecting its resistance to bending.

Together, these factors determine a beam's ability to resist bending under a load. Beams with high bending stiffness do not deflect much, whereas those with lower stiffness will bend more for the same amount of applied force.

Simply Supported Beam

The simply supported beam is one of the most fundamental types of beam supports. This setup allows the beam to freely rotate and deflect vertically but not horizontally. It is supported at its ends by pivots, which offer only vertical reactions.

• Characteristics: End supports may consist of a pin and a roller, allowing rotation and translational movements at these points, but they do not resist bending moments.
• Application: It is often used to model bridges or structures that need this kind of flexibility.
• Deflection: The deflection formula for a simply supported beam with a point load applied at mid-span is \(v = - \frac{{P L^3}}{{48 EI}}\).

This type of beam setup allows for a clear understanding of how loads affect overall bending and deflection.

Built-In Beam

A built-in beam is quite different from a simply supported beam. Also known as a fixed beam, it is rigidly connected at its ends, preventing rotations and translational movements at these points.

• Support Condition: This type of support keeps both ends of the beam completely immobile.
• Resulting Effects: The built-in nature of the beam causes it to sustain moment and shear, reducing the overall deflection.
• Deflection Formula: For a point load applied at the center, the deflection for a built-in beam is given by \(v = - \frac{{P L^3}}{{192 EI}}\).

The properties of a built-in beam make it well-suited for conditions where minimal movement is desirable, and it provides structural stability under different load scenarios.

Point Load

A point load is a load that is applied at a specific location on a structure, rather than distributed along its length. In beam theory, understanding point loads is crucial because they have significant effects on deflection and bending moments.

• Definition: Unlike distributed loads, which stretch over an area or length, a point load acts at a precise point, often idealized as being concentrated over an infinitely small area.
• Impact on Beams: Point loads cause maximum bending moments and result in concentrated deflections at their point of application, especially in simply supported and built-in beams.
• Calculation: By using specific deflection equations for simply supported and built-in beams, engineers can predict precisely how a point load will affect the beam's structural integrity and performance.

Point loads are extensively analyzed in structural engineering to ensure that beams can support specific loads without failing.