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.

There are two types of section moduli

• The Elastic Section Modulus
• The Plastic Section Modulus

THE ELASTIC SECTION MODULUS

The elastic section modulus is used for general design applicable up to the yield point for most metals and other common materials.It can be defined as

S = I / y

where I is the second moment of area

y is the distance between the neutral axis of any given fibre

It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre

It is also often used to determine the yield moment (M_y) such that M_y = S ⋅ σ_y where σ_y is the yield strength of the material

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle			Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		,	NA indicates neutral axis
doubly symmetric I-section (minor axis)			NA indicates neutral axis
Circle			Solid arrow represents neutral axis
Circular hollow section			Solid arrow represents neutral axis
Rectangular hollow section			NA indicates neutral axis
Diamond			NA indicates neutral axis
C-channel			NA indicates neutral axis

THE 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.

MODULE 1

To find the strength of hood panel

The section modulus of hood panel is shown below

MOI (Max) = 4.803235538e+06[mm^4]

MOI (Min) = 1.687370765e+04[mm^4]

second moment of area I = 1.687370765e+04[mm^4]

The distance between the neutral axis of given fibre Y

Total distance Y = 874.0861/2

Y = 437.04305 mm

S = I / y

S = 1.687370765e+04 mm^4 / 437.04305 mm

S = 0.01964737 MM^3

MODULE 2

To improve strength of material changing the dimensions of hood.

The hood inner panel’s curve is offset by 0.5mm outwards

Optimized hood panel is shown below

MOI (Max) = 4.808738735e+06 [mm^4]

MOI (Min) = 1.712247090e+04 [mm^4]

second moment of area I =1.712247090e+04 [mm^4]

Total distance Y = 875.24 /2

Y = 437.62 mm

S = I / y

S = 1.712247090e+04 [mm^4] / 437.62 mm

S = 0.019785 mm^3

Finally the strength of hood panel is increased compared to module 1.