Short Answer:
The moment of resistance is the internal moment developed in a beam or structural member that balances the external bending moment applied to it. When a beam bends due to an external load, internal stresses are produced, and these stresses create a resisting moment known as the moment of resistance.
In simple words, the moment of resistance is the beam’s internal capability to oppose bending caused by external forces. It ensures equilibrium and prevents structural failure. The moment of resistance acts opposite to the external bending moment and maintains the beam in a stable condition.
Detailed Explanation :
Moment of Resistance
When a beam or structural member is subjected to external loads, it experiences bending moments that tend to bend the beam. To resist this bending, internal stresses develop within the material — compressive stresses in the upper portion and tensile stresses in the lower portion. These stresses together create an internal resisting moment, known as the moment of resistance (Mₙ).
The moment of resistance is defined as:
“The internal moment developed in a beam section due to bending stresses, which is equal in magnitude but opposite in direction to the external bending moment acting on the beam.”
In equilibrium,
Hence, the moment of resistance ensures that the beam remains stable and does not fail or deform excessively under the applied loads.
Formation of Moment of Resistance
When a beam is loaded, it bends and forms a curvature. Due to bending:
- The upper fibers (above the neutral axis) are compressed and try to shorten.
- The lower fibers (below the neutral axis) are stretched and try to elongate.
- The neutral axis separates the compression and tension zones and experiences zero stress.
These tensile and compressive stresses are distributed linearly along the beam’s cross-section. The resultant compressive and tensile forces act at certain distances from the neutral axis, forming a couple. This couple produces the internal moment of resistance, which balances the external bending moment.
Thus, the moment of resistance arises due to the combined effect of compression and tension in the beam fibers.
Mathematical Expression for Moment of Resistance
From the bending equation,
Where,
- = External bending moment
- = Moment of inertia of the beam cross-section about the neutral axis
- = Bending stress at distance from neutral axis
- = Modulus of elasticity
- = Radius of curvature
At the limiting condition, the maximum stress in the beam section reaches the permissible bending stress . The moment of resistance of the section can then be expressed as:
Since is the section modulus (Z),
This equation shows that the moment of resistance depends on:
- Material strength (through )
- Geometric strength of the cross-section (through )
Thus, the greater the material’s allowable stress or the section’s modulus, the higher the moment of resistance and the greater the beam’s ability to carry loads safely.
Distribution of Stresses and Moment of Resistance
- Compression Zone:
- The upper part of the beam experiences compressive stresses.
- The resultant compressive force acts at the centroid of this zone above the neutral axis.
- Tension Zone:
- The lower part of the beam experiences tensile stresses.
- The resultant tensile force acts at the centroid of the tension zone below the neutral axis.
The tensile and compressive forces are equal in magnitude and opposite in direction. Their line of action is separated by a perpendicular distance, forming a couple. This couple’s moment equals the moment of resistance.
Hence,
This internal moment resists the bending action of external loads and maintains equilibrium in the beam.
Factors Affecting Moment of Resistance
- Material Strength:
The higher the permissible stress of the material (), the greater the moment of resistance. For example, steel beams have a much higher moment of resistance than wooden beams. - Shape of Cross-section:
Cross-sections with a larger section modulus (Z) provide higher resistance to bending. I-sections are widely used because most material is concentrated away from the neutral axis, increasing . - Size of Beam:
Increasing the beam’s depth or moment of inertia (I) increases the moment of resistance significantly. - Type of Loading:
Uniformly distributed loads create constant bending moments, while point loads cause localized maximum moments affecting the section’s required resistance. - Support Conditions:
Fixed beams can develop higher moments of resistance compared to simply supported beams due to restraint at the supports.
Importance of Moment of Resistance
- Structural Safety:
It helps ensure that the internal strength of a beam can safely counter the applied external bending moments. - Beam Design:
Moment of resistance is used to calculate the required beam dimensions for given loads and material properties. - Material Optimization:
Engineers use the concept to design economical structures by choosing sections with higher strength-to-weight ratios. - Failure Prevention:
Helps prevent structural failure due to bending by keeping stresses within allowable limits. - Performance Prediction:
The moment of resistance predicts how much load a beam can carry before yielding or bending excessively.
Example of Moment of Resistance
For a rectangular beam of width and depth :
- Moment of inertia,
- Distance to outer fiber,
Then section modulus,
Moment of resistance,
Hence, a beam’s resistance increases with both its depth and material strength.
Conclusion
In conclusion, the moment of resistance is the internal moment developed in a beam or structural member to counteract the external bending moment caused by applied loads. It ensures that the beam remains in equilibrium and resists bending safely. The moment of resistance depends on the material’s strength () and the cross-section’s geometry (). Beams with higher section modulus and stronger materials have greater bending resistance, making this concept fundamental in the design and analysis of all structural elements.