Section Modulus calculation and optimization

                             SECTION MODULUS CALCULATION AND OPTIMIZATION

OBJECTIVE:

The objective of the project is to find the section modulus of the designed front hood and optimizing it to get an improved section modulus compared to the previous section.

INTRODUCTION:

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 the design include an area for tension and shear, the radius of gyration for compression, and the moment of inertia and polar moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape. There are two types of section moduli,

Elastic section modulus(S).

Plastic section modulus(Z).

The elastic section modulus is defined as,

                                                                 S = I / y

Were,

• S is the Section Modulus
• I is the second moment of area (or area moment of inertia, not to be confused with the moment of inertia)
• y is the distance from the neutral axis to any given fiber. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fiber.

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.

The neutral axis is an axis in the cross-section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic, and is not curved before a bend occurs, then the neutral axis is at the geometric centroid. All fibers on one side of the neutral axis are in a state of tension, while those on the opposite side are in compression.

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 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: Zp = Acyc + 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 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 the elastic limit of the material. This could be shown mathematically with the formula.

                                                                    K = (Z/S)

The shape factor for a rectangular section is 1.5.

SECTION MODULUS CALCULATION:

For finding better results I have analyzed two sections and considered it as case 1 and case 2 and again the two sections were optimized and analyzed and then compared to get better results.

CASE 1(FRONTAL CRASH):

For case 1 the below section was considered parallel to x-coordinate for calculating the section modulus.

BEFORE OPTIMIZATION

By using the Section Inertia Analysis in Siemens NX, the below-shown results were achieved.

Moment of Inertia (MAX) = 4.881146244x 10^6 [mm^4]

Moment of Inertia (MIN) = 1.336007965x 10^4 [mm^4]

Section Modulus, S = Moment of Inertia (MOI) / Distance between the neutral axis and the extreme end of the object (y)

For MOI max,

                      S = (4.881146244x 10^6 [mm^4]) ÷ (440.36 mm)

                      S = 11084.445099 mm^3

For MOI min,

                      S = (1.336007965 x 10^4 [mm^4]) ÷ (440.36 mm)

                      S = 30.3389945 mm^3

AFTER OPTIMIZATION:

The hood inner panel width was increased to 0.50 mm than the original one and achieved the below-shown results in Siemens NX,

 

Moment of Inertia (MAX) = 4.872688410 x 10^6 [mm^4]

Moment of Inertia (MIN) = 1.359210692 x 10^4 [mm^4]

Section Modulus, S = Moment of Inertia (MOI) / Distance between the neutral axis and the extreme end of the object (y)

For MOI max,

                     S = (4.872688410 x 10^6 [mm^4]) ÷ (440.36 mm)

                     S = 11065.238464 mm^3

For MOI min,

                     S = (1.359210692 x 10^4 [mm^4]) ÷ (440.36 mm)

                     S = 30.8658981 mm^3

CASE 2(SIDE CRASH):

For case 1 the below section was considered parallel to x-coordinate for calculating the section modulus.

BEFORE OPTIMIZATION:

By using the Section Inertia Analysis in Siemens NX, the below-shown results were achieved.

Moment of Inertia (MAX) = 8.350598964 x 10^5 [mm^4]

Moment of Inertia (MIN) = 4.914498247 x 10^3 [mm^4]

Section Modulus, S = Moment of Inertia (MOI) / Distance between the neutral axis and the extreme end of the object (y)

For MOI max,

                      S = (8.350598964 x 10^5 [mm^4]) ÷ (244.14 mm)

                      S = 3420.41409 mm^3

For MOI min,

                      S = (4.914498247 x 10^3 [mm^4]) ÷ (244.14mm)

                      S = 20.129836 mm^3

AFTER OPTIMIZATION:

The hood inner panel width was increased to 0.50 mm than the original one and achieved the below-shown results in Siemens NX,

Moment of Inertia (MAX) = 8.328039517 x 10^5 [mm^4]

Moment of Inertia (MIN) = 5.106158437 x 10^3 [mm^4]

Section Modulus, S = Moment of Inertia (MOI) / Distance between the neutral axis and the extreme end of the object (y)

For MOI max,

                     S = (8.328039517 x 10^5 [mm^4]) ÷ (244.14 mm)

                     S = 3411.17371 mm^3

For MOI min,

                     S = (5.106158437 x 10^3 [mm^4]) ÷ (244.14 mm)

                     S = 20.914878 mm^3

COMPARING THE CASE 1 AND 2 TO FIND THE BETTER RESULT:

COMPARISON TABLE
CASE1(FRONTAL CRASH)	CASE2(SIDE CRASH)
BEFORE OPTIMIZATION
Smax = 11084.445099 mm^3	Smax = 3420.41409 mm^3
Smin = 30.3389945 mm^3	Smin = 20.129836 mm^3
AFTER OPTIMIZATION
Smax = 11065.238464 mm^3	Smax = 3411.17371 mm^3
Smin = 30.8658981 mm^3	Smin = 20.914878 mm^3

From the comparison table we have achieved the section modulus result and for the maximum moment of inertia the section modulus value decreases after optimization but for the minimum moment of inertia the value increases. 

• But the minimum limit of deformation of the hood needs to be analyzed so we’ll go with the minimum moment of inertia.
• After increasing the sectional area to 0.5 mm from the original the Section Modulus value increased which shows that the load-carrying capacity and resistance of the hood will also be increased simultaneously.

CONCLUSION:

Thus, the section modulus of the Front Hood of a car was calculated and compared with the optimized hood and found a better result.