Section Modulus calculation and optimization

• Elastic Section Modulus
  • Elastic section modulus is calculated when we are to use the material behavior till the yield point. For general design purposes, we use elastic section modulus.
  • Generally, elastic section modulus is denoted by Zₑ.
    So,
    Zₑ = I/ymax
    Where,
    I = Moment of inertia
    ymax = Distance from NA to extreme layer
  • ELASTIC SECTION MODULUS FOR HOLLOW RECTANGULAR SECTION
    • As we know the moment of inertia of the hollow rectangular section,
      
      I = (BD³/12) - (bd³/12)
      ymax = D/2
      After putting these values,
      Zₑ = I/ymax
      (BD³/12) - (bd³/12)/(D/2)
      Zₑ = (BD³ - bd³)/6D
• Plastic Section Modulus
  • The plastic section modulus is used for materials where the elastic yield is acceptable and the plastic behavior is considered to be in an acceptable range.
    It depends on the location of the plastic neutral axis. The PNA is the axis that bisects the cross-section such that the compressive force from the region in compression is equal to the tension force from the region under tension. So, for sections with a constant yield stress, the area above and below the PNA will be equal, but for mixed sections, this is not necessarily the case.
    Mathematically,
    The plastic section modulus is the sum of the areas of cross-sections on each side of the plastic neutral axis that may or may not be equal and multiplied by the distance from the local centroid of the two regions to the plastic neutral axis.
    
    Zₚ = (Ac × yc) + (Aₜ + yₜ)
    Where,
    Ac = Area of cross-section under compression.
    Aₜ = Area of cross-section under tension.
    yc = Distance from the plastic neutral axis to the center of gravity of the areas under compression.
    yₜ = Distance from the plastic neutral axis to the center of gravity of areas under tension.
    Since on the neutral axis, the area under compression is equal to the area under tension.
    So,
    Ac = Aₜ
    And,
    yc = yₜ

Section Modulus calculation for a car hood

Since,

We know section modulus Z = I/ymax. So while designing we check for the minimum strength for safety. To get minimum strength we must minimize Section Modulus (Z).

So, Z(Min)=I(Min)/Y(max). We need the minimum value of the moment of inertia (I) and the Maximum distance of the extreme layer from the neutral axis.

1. First, we create an intersection of the hood's outer surface with the YC-ZC plane.
2. We then create a datum plane at the middle and normal to the above intersection line and create an intersection with the whole assembly to get the cross-section.
3. We create a sketch and project the cross-section and trim the inside edges and create splines to create a closed curve.
4. We calculate the MOI using the Sectional Inertia Analysis. So Minimum MOI = 2.9534 x 10^5 mm^4.
5. To calculate Ymax, we create 2 points at the edges and measure between them. So, Ymax = 881.2351/2 = 440.61755 mm.
6. Initial Minimum Section Modulus (Z1) = min. MOI/ max. Y = 2.9534 x 10^5 mm^4/440.61755 mm = 670.2865 mm^3.

Section Modulus Optimization

1. We modify the Sketch and Calculate the above calculations again.
2. Calculation of Final Section Modulus.
3. Final Minimum Section Modulus (Z2) = min. MOI/ max. Y = 2.9374 x 10^5 mm^4/440.61755 mm = 666.655 mm^3.

Percentage change in the section modulus

Percentage Decrease = {(Starting Value−Final Value)/|Starting Value|}×100

% decrease in section modulus = {(Z1-Z2)/Z1}x100 = {(670.2865 mm^3 - 666.655 mm^3)/670.2865 mm^3} x 100 = 0.541783 % decrease