Section Modulus calculation and optimization

WHAT IS SECTION MODULUS



It is termed as the ratio of MOMENT OF INERTIA and distance from N.A (Neutral axis) to the extreme OBJECT. Also it is the measure of strength of given member. The STRESS in the outermost section of beam is computed with the help of section modulus. It is indicated by S.

Section modulus is indicated as follows:

S = Section Modulus

I = Moment of Inertia

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

TYPES OF SECTION MODULUS

Elastic section modulus

For general design, the elastic section modulus is used, applying 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 MOMONENT OF INERTIA (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 (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yeild strngth of the material.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle		{\displaystyle S={\cfrac {bh^{2}}{6}}}	Solid arrow represents neutral axis
doubly symmetric I-SECTION (major axis)		{\displaystyle Sx={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}} {\displaystyle Sx={\tfrac {Ix}{y}}}, with {\displaystyle y={\cfrac {H}{2}}}	NA indicates neutral axis
doubly symmetric I-SECTION(minor axis)		{\displaystyle Sy={\cfrac {B^{2}(H-h)}{6}}+{\cfrac {(B-b)^{3}h}{6B}}}	NA indicates neutral axis
Circle		{\displaystyle S={\cfrac {\pi d^{3}}{32}}}	Solid arrow represents neutral axis
Circular hollow section		{\displaystyle S={\cfrac {\pi \left(r_{2}^{4}-r_{1}^{4}\right)}{4r_{2}}}={\cfrac {\pi (d_{2}^{4}-d_{1}^{4})}{32d_{2}}}}	Solid arrow represents neutral axis
Rectangular hollow section		{\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}	NA indicates neytral axis
Diamond		{\displaystyle S={\cfrac {BH^{2}}{24}}}	NA indicates neutral axis
C-channel		{\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}	NA indicates neutral axis

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:

{\displaystyle Z_{P}=A_{C}y_{C}+A_{T}y_{T}}

CASE 1 : SECTION MODULUS OF HOOD WITHOUT ANY CHANGES

CALCULATION OF SECTION MODULUS FOR 1ST CASE



Formula S=I/Y


Section Modulus=Moment of Inertia / distance between the neutral axis and the extreme end of the object

Section Modulus= 4.567185143*10^3 / 241.051

= 4,567.185143 / 241.051

= 18.946915 mm^3

CASE 2 : OUTER HOOD THICKNES CHANGED TO 0.75 TO 1.25














CALCULATION OF SECTION MODULUS FOR 2ST CASE

Formula S=I/Y


Section Modulus=Moment of Inertia / distance between the neutral axis and the extreme end of the object

Section Modulus= 4.566733772 * 10^3 / 241.051

= 4,566.733772 / 241.051

= 18.945093 mm^3

CASE 3 : INCRESED THE GAP BETWEEN OUTER PANEL AND INNER PANEL FROM 3.02 MM TO 5.250 MM