Question

I am teaching students at a high school about bending of a beam (the beam is clamped at one end and then an applied force acts downwards close to the end of the beam to bend it).

I thought that it did not require further explanation that the deflection of the beam due to the force at the end is inversely proportional to the stiffness of the beam, proportional to the size of the applied force, and that it is dependent on the distance from the fixed point where the beam is clamped.

However, I have been asked to provide justification for these statements and not to appeal to physical intuition which is not obvious to students who are not seasoned physicists. What type of justification would be enough to use these statements at an elementary level? How could I explain that these statements are true?

Answer 1

Hooke's law states that deformation is proportional to the applied force, with stiffness inversely affecting deformation. A stiffer beam is less prone to bending and the point of application for the force also contributes to the deflection extent.

Step-by-step explanation:

To justify the statements about beam bending at an elementary level, you can refer to Hooke's law, which states F = -kx. This law establishes that the deformation x of a material (including the bend of a beam) is proportional to the applied force F. The constant k represents the stiffness of the material and is measured in newtons per meter (N/m).

A larger k means the material is stiffer and less prone to deformation; thus, the deflection is inversely proportional to the stiffness of the beam. It also expresses that the deflection of a beam is proportional to the force applied: increased force leads to a larger deformation if all other factors remain unchanged. A beam's length also affects its deflection; the further the applied force is from the clamping point, typically the greater the deflection, as there is more beam to bend.

Additionally, for small deformations, this relationship is linear, which implies the deformation is directly proportional for small changes in length, sideways bending, and volume. This concept reinforces the proportional relationship between an applied force and the resulting deformation. In the case of beam bending, applying these principles helps explain why a beam deflects more with greater force, less with higher stiffness, and varies with the point of force application.