The maximum bending moment in a simply supported beam length L loaded by a concentrated load W at midpoint is given by
WL/4
Determining Maximum Bending Moment for a Simply Supported Beam
This explanation focuses on finding the maximum bending moment experienced by a simply supported beam when subjected to a single concentrated load placed exactly at its center.
Understanding the Problem Setup
We consider a beam that is supported at both ends (simply supported) and has a total length denoted by L. A concentrated force, represented as W, is applied exactly at the midpoint of the beam's length (at distance L/2 from either support).
Calculating Support Reactions
Due to the symmetrical loading (the load W is at the center), the total load W is distributed equally between the two supports. Let the reactions at the supports be RA and RB.
Using the principle of equilibrium (sum of vertical forces = 0):
$$ R_A + R_B = W $$
Since the load is at the midpoint, the reactions are equal:
$$ R_A = R_B $$
Substituting this into the equilibrium equation:
$$ R_A + R_A = W \implies 2R_A = W \implies R_A = \frac{W}{2} $$
Therefore, the reaction at each support is W/2.
Calculating Bending Moment
The bending moment at any section of the beam is the algebraic sum of the moments of the forces to the left (or right) of that section. Let's consider a section at a distance x from the left support (where 0 ≤ x ≤ L/2).
The bending moment M(x) at this section is caused by the reaction force RA acting at a distance x:
$$ M(x) = R_A \times x $$
Substituting the value of RA:
$$ M(x) = \frac{W}{2} \times x $$
This formula gives the bending moment for any point between the left support and the center of the beam.
Finding the Maximum Bending Moment
The bending moment varies linearly from 0 at the left support (x=0) to its maximum value at the center of the beam (x=L/2). To find the maximum bending moment (Mmax), we substitute x = L/2 into the moment equation:
$$ M_{max} = M\left(\frac{L}{2}\right) = \frac{W}{2} \times \frac{L}{2} $$
$$ M_{max} = \frac{WL}{4} $$
Conclusion
The maximum bending moment in a simply supported beam of length L, carrying a concentrated load W at its midpoint, occurs at the center and is equal to WL/4.
Important Questions from Bending Moment
Slope and deflection of a cantilever beam carrying a moment M at the free end is given by:
Which of the following beams is likely to have the point of contraflexure?
The point of contraflexure is the point at which ___________ changes its sign.
The maximum bending moment of the center of laminated spring of span L due to load W is given by-
If a simply supported beam is loaded with point load W at the centre then what is the ratio of bending moment at the support to the bending moment at the centre?
