2.1. Example 1

Suppose we have a straight beam which is clamped at one end and is subjected to a transverse load P at its other end as shown in Figure 6.

In this scenario, we have a straight beam that is supported in such a way that one end is fixed (clamped) while the other end is free. A transverse load, denoted as P, is applied at the free end. This setup is important in structural mechanics as it allows us to analyze how the beam will react to the load, specifically how much it will deflect.

Imagine a diving board at a swimming pool. One end of the board is fixed to the pool deck (like the clamped end of the beam), while the other end is free for divers to jump and put weight on it. The diving board bends downward due to the weight of the diver, similar to how the beam bends under the force P.

Let us obtain the deflection of the beam using the Euler-Bernoulli Theory (EBT). We first need to find the bending moment profile (i.e., bending moment M as a function of X). Let us cut a section in the beam at a distance X from the clamped end.

To determine how the beam bends under the applied load, we need to calculate the bending moment at any point along the length of the beam. This can be achieved by analyzing a section of the beam taken at a distance X from the clamped end. The bending moment at this section will depend on the distance to where the force is applied and is calculated based on equilibrium principles.

Consider the same diving board. If you were to measure how much the board bends at the middle as more people jump off the end, you'd be examining a section of the board. The bending moment at that point is like measuring how much force is pulling down on that segment due to the weight of the diver and how the board reacts to it.

Moment balance for the right part of the beam about the centroid of its left-end cross-section results in −M(X) + P(L−X) = 0 ⇒ M(X) = P(L−X).

Using moment balance, we derive the equation for the bending moment M at a distance X. This shows the relationship between the applied load P and the moment M at a section of the beam. The positive and negative signs indicate the direction of the forces acting on the beam, allowing us to formulate this as an equation we can solve.

In terms of the diving board, if we think about the force exerted by a diver and how far they are from the clamped end, we can picture how much 'twisting' or bending the board experiences. The further away from the clamped end a diver is, the more bend there will be. This moment balance is essential in understanding how much the board will sag.

Plugging this expression in equation (8), we get a second-order linear differential equation.

After determining the bending moment, we substitute this expression into the governing equation of beam theory (equation 8). This results in a second-order linear differential equation that describes the relationship between deflection and the position along the beam, enabling us to calculate how much the beam will bend under the load.

Think of this as creating a blueprint for a new diving board after analyzing how it bends. By formulating this equation, we're essentially mapping out the deflective behavior of the board - just like an engineer would design a board that minimizes sag.

If there were additional unknown parameters in the equation, we would have required more boundary conditions to find them. At the clamped end (X=0), the centerline cannot deflect and the cross-section cannot rotate either. Thus, we get the following two boundary conditions: ...

Boundary conditions are necessary to solve our differential equation effectively. In this case, we set conditions based on the physical situation: at the fixed end of the beam, there is no deflection or rotation. These boundary conditions are needed to uniquely determine the constants that will appear in the solution of our differential equation.

Returning to our diving board analogy, think of these as safety measures: if a board were to have no limits on how far it could bend or rotate at the fixed end, we wouldn't be able to determine its safe usage. The constraints provided by these boundary conditions help us ensure we understand the board’s performance under varying loads.

The tip deflection can be obtained by substituting X=L in the above equation which yields. ... Thus, larger the force, larger is the tip deflection. Also, larger is the bending stiffness EI, smaller is the deflection.

Once we solve the equation, we are particularly interested in the deflection at the free end of the beam. By substituting the length of the beam in our derived equation, we find out how much the beam tips downward due to the load. The results indicate that increased force results in greater deflection, while greater stiffness of the beam reduces the tip deflection.

In our diving board example, this would mean if a heavier diver jumps from further up, the board will deflect more than if a lighter diver jumped. Conversely, if a board is made stronger (more rigid), it will bend less under the same conditions. This is vital for safety and performance.

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