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 include area for tension and shear, radius of gyration for compression, and second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

formula S = I/y

S = Section Modulus

I = Second Moment Of Area MOI (Min)

y = distance between the neutral axis and the extreme end of the object

Elastic section modulus

For general design, the elastic section modulus is used, applicable up to 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, not to be confused with moment of inertia) and y is the distance from the neutral axis to any given fibre. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre, as seen in the table below. It is also often used to determine the yield moment (My) such that My = S ⋅ σy, where σy is the yield strength of the material.

Plastic section modulus

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior 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 sections with constant, and equal compressive and tensile yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarily the case.

The plastic section modulus is the sum of the areas of the cross section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA:

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 (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross section, 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.

Section Modulus calculation and optimization

1. Hood Design as per master section

• take intersection curve & perform section inertia Analysis

Second Moment Of Area MOI (Min) : 2.743622458*10^5 = 274362.2458 mm^4

Y = L/2 (distance between the neutral axis and the extreme end of the object)

Y = 879.4753/2 mm

y = 439.73765 mm

Section Modulus, S = Second Moment Of Area MOI (Min) / Distance between the neutral axis and the extreme end of the

S1 = 274362.2458/439.73765 mm^3

S1 = 623.92257 mm^3

2. Hood Design Improved

Second Moment Of Area MOI (Min) : 2.755360853*10^5 = 275536.0853 mm^4

Y = L/2 (distance between the neutral axis and the extreme end of the object)

Y = 879.4753/2 mm

y = 439.73765 mm

Section Modulus, S = Second Moment Of Area MOI (Min) / Distance between the neutral axis and the extreme end of the

S2 = 275536.0853/439.73765 mm^3

S2 = 626.59198 mm^3

Percentage increase in section modulus = ((S2-S1)/S1)*100

= ((626.59198-623.92257)/623.92257)*100

= 0.42 %

Conclusion

With Improved Hood Design Section Modulus Improve by 0.42 %