Section Modulus calculation and optimization

Aim

To calculate section modulus of hood and optimize the design to have improved section modulus.

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 inertia 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 modulus and the plastic section modulus. The section module of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

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 = l/y, where l is the second moment of area 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 (M_y = S ⋅ σ_y, where σ_y  is the yield strength of the material.

                       Section modulus equations

Plastic section modulus

The plastic section modulus is used for materials where plastic 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 constants, and equal compressive and tensile yielding stress, the area above and below the PNA will be equal, but for composite section, 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 multiplied by the distance from the local centroids of the two area to the PNA:

z_{p =} A_{cyc +  ^Atyt}

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 the question, Fy, by Mp=F*z. Plastic section modulus and elastic section modulus are related by a shape factor which can be denoted by ‘k’, used for indication of  capacity beyond elastic limit of material. This could be shown mathematically with the formula:-

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Shape factor for a rectangular section is 1.5

So here we are calculating elastic section modulus.

 Case 1:  Calculating section modulus for the hood design:

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Maximum moment of inertia = 8.934777263e+07[mm^4]

Minimum moment of inertia = 2.935860007e+05[mm^4]

Total hood distance = 876.80mm

          Distance between the end of the hood and neutral axis(y)=438.40mm

  Section modulus S = I/Y

                                 = 2.935860007*10^4/438.40

                                 = 29358.60007/438.40

                                 = 66.96mm^3 

Case 2: Modified hood cross section:

The original cross section curve is modified by offsetting the lower half of the cross section loop 0.5mm downwards. This simulates the increase of the depth of emboss within the drawing operation by 0.5mm. 

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Maximum moment of inertia = 8.919411477e+07[mm^4]

Minimum moment of inertia = 2.974248583e+05[mm^4]

Total hood distance = 876.80mm

          Distance between the end of the hood and neutral axis(y)=438.40mm

  Section modulus S = I/Y

                                 = 2.974248583e *10^4/438.40

                                 = 29742.48583/438.40

                                 = 67.84mm^3 

Conclusion:  In this study, we take the minimum of inertia moment to analyze the minimum limit of hood deformation. Thus the section modulus of the front hood was calculated and compared with the new design.

From the result we found out that increasing the cross-section area would give a better section modulus and helps in increasing the strength of the hood.