Short Answer:

The maximum deflection in a simply supported beam with a central point load occurs exactly at the midpoint of the beam. It depends on the magnitude of the load, the span length, the material properties, and the moment of inertia of the beam section.

The mathematical expression for the maximum deflection () in a simply supported beam carrying a central point load  is given by:

where  is the length of the beam,  is the modulus of elasticity, and  is the moment of inertia.

Detailed Explanation :

Maximum Deflection in a Simply Supported Beam with Point Load

When a simply supported beam carries a point load (W) at its center, it bends due to the bending moment developed inside it. This bending causes deflection, which is the vertical displacement of the beam from its original straight position. The deflection depends on the load intensity, beam length, material stiffness, and beam cross-section.

The deflection is not uniform throughout the beam. It is zero at the supports (since the beam ends are held on supports and cannot move vertically) and maximum at the midpoint, where the bending moment is greatest. The goal of deflection analysis is to determine this maximum deflection value to ensure that the beam does not bend excessively under the applied load.

Derivation of Maximum Deflection Formula

Consider a simply supported beam of length  subjected to a central point load .

Reactions at Supports:
Since the load is symmetrically placed at the center, both supports share the load equally:

Bending Moment at a Distance  from the Left Support:
For ,

Basic Bending Equation:

Integrating once to find slope:

Integrating again to find deflection:

Applying Boundary Conditions:

At , :

At , slope () = 0 (because the deflection is maximum and the tangent is horizontal):

Substitute  and  back into the equation of deflection:

The maximum deflection occurs at :

 

Hence, the maximum deflection is:

The negative sign indicates the direction of deflection (downward).

Physical Meaning of the Formula

The equation

shows that:

• Deflection is directly proportional to load (W):
  Doubling the load doubles the deflection.
• Deflection increases rapidly with beam length (L³):
  Small increases in beam length cause large increases in deflection.
• Deflection decreases with higher stiffness (EI):
  Increasing modulus of elasticity (E) or moment of inertia (I) makes the beam stiffer and reduces bending.

Thus, beams made of high-strength materials or with larger cross-sectional areas will show less deflection under the same load.

Units of Maximum Deflection

In the SI system:

•  → Newton (N)
•  → Meter (m)
•  → N/m²
•  → m⁴
  Thus, deflection () has units of meters (m).

Example Calculation

Given:
A simply supported beam of length  carries a central load of .

Solution:

 

Hence, the maximum deflection at the midpoint is 0.26 mm.

Factors Affecting Maximum Deflection

1. Load Magnitude (W):
   Higher loads cause greater deflection.
2. Beam Length (L):
   Deflection increases rapidly with the cube of the beam length ().
3. Material Property (E):
   A stiffer material with a higher modulus of elasticity reduces deflection.
4. Cross-Sectional Shape (I):
   Beams with higher moments of inertia (like I-beams) bend less.
5. Support Conditions:
   Fixed beams show less deflection than simply supported beams for the same loading conditions.

Importance of Calculating Maximum Deflection

• Ensures structural safety: Excessive deflection can lead to cracks or even failure.
• Improves serviceability: Controls vibrations and maintains comfort in buildings and bridges.
• Ensures proper function: In machinery, deflection limits are crucial for maintaining alignment and precision.
• Helps in economical design: Prevents overdesign by balancing strength and stiffness.

Conclusion

The maximum deflection in a simply supported beam with a central point load occurs at the midpoint of the span and is given by:

It depends on the load applied, beam length, material stiffness, and moment of inertia. By understanding and controlling deflection, engineers ensure that structures remain safe, functional, and durable under various loading conditions.