the deflection of a beam under a load is directly proportional to the load applied.

The statement you provided is a basic principle in structural engineering and physics. It's based on Hooke's Law, which states that the force needed to extend or compress a spring by some distance is proportional to that distance.

Here's a step-by-step explanation:

1. A beam, when subjected to a load, will bend or deflect. This is a natural response of the material to the applied force.

2. The amount of deflection depends on several factors: the load applied, the length of the beam, the material properties (like Young's modulus), and the moment of inertia of the beam's cross-sectional area.

3. The direct proportionality between load and deflection means that if you double the load, the deflection will also double, assuming all other factors remain constant. Similarly, if you halve the load, the deflection will also halve.

4. This relationship is often used in engineering to design beams that can safely carry specific loads without excessive deflection, which could lead to failure or other problems.

5. However, it's important to note that this direct proportionality only holds up to a certain point. If the load becomes too great, the beam may yield or fail, and the relationship between load and deflection will no longer be linear.