For a simply supported beam with a length L and a point load P at a distance a f

For a simply supported beam with a length L and a point load P at a distance a from the left support, what is the reaction at the left support?

For a simply supported beam with a length L and a point load P at a distance a from the left support, what is the reaction at the left support?

Step 1: Understand the problem. You have a beam that is supported at both ends (left and right) and has a load (P) applied at a certain point (a) from the left end.
Step 2: Identify the total length of the beam, which is L.
Step 3: Recognize that the beam is in static equilibrium, meaning the sum of forces and moments (torques) acting on it must be zero.
Step 4: The reaction forces at the supports are denoted as R_A (left support) and R_B (right support).
Step 5: Write the equation for the sum of vertical forces: R_A + R_B = P.
Step 6: Write the moment equation about the left support (point A) to find R_A. The moment caused by the load P is P * a, and the moment caused by R_B is R_B * L.
Step 7: Set the moment equation: P * a = R_B * L.
Step 8: Solve for R_B: R_B = P * a / L.
Step 9: Substitute R_B back into the vertical forces equation: R_A + (P * a / L) = P.
Step 10: Rearrange the equation to solve for R_A: R_A = P - (P * a / L).
Step 11: Factor out P: R_A = P(1 - a/L).
Step 12: Simplify the equation: R_A = P(L - a) / L.

Static Equilibrium – Understanding how forces and moments balance in a simply supported beam.
Reaction Forces – Calculating the support reactions based on applied loads and their positions.
Moment Calculation – Using the moment about a point to find unknown forces in a static system.