Section Modulus calculation and optimization
Section Modulus:
Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design includes area for tension & shear, radius of gyration for compression, moment of inertia & polar moment of inertia for stiffness.
Types of section modulus:
- Elastic section modulus
- Plastic section modulus
Elastic section modulus(S):
For general design, the elastic section modulus is used, applying upto the yield point for most metals and other common materials.
The elastic section modulus is defined as S=I/y, where I is the second moment of area or area moment of inertia and y is the distance from the neutral axis to any given fibre. It is also often used to determine the yield moment such that My=S.σ where σ is the yield strength of material.
Plastic section modulus(Z):
The plastic section modulus is used for materials where elastic yielding is acceptable & plastic behaviour is assumed to be an acceptable limit. Designs generally strive to ultimately remain below the plastic limit to avoid permanent deformations, often comparing the plastic capacity against amplified forces or stresses.
The plastic section modulus depends on the location of the Plastic Neutral Axis(PNA). The PNA is defined as the axis that splits the cross-section such that the compression force from the area in compression equals the tension force from the area in tension. So, for section with constant yielding stress, the area above & below the PNA will be equal, but for composite sections, this is not necessarily the case.
The plastic section modulus is not the first moment of area. Both relate to the calculation of the centroid, but plastic section modulus is the sum of all areas on both sides of PNA and multiplied with the distances from the centroid of the corresponding areas to the centroid of cross-sections, while the first moment of area is calculated based on either side of the considering point of the cross-section and it is different along the cross-section and depends on the point of consideration.
The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, Fy, by Mp=Fy.Z. Plastic section modulus and elastic section modulus are related by a shape factor which can be denoted by k, used for an indication of capacity beyond elastic limit of material. This could be shown mathematically with the formula: k = Z/S