Section Modulus calculation and optimization

Bending:- when a beam is loaded under pure moment M. The beam will bend in a circular arc. as shown below 

In sagging upper layer will experience compression while the lower layer will experience tension as shown above. The distribution of internal force in bending is linear i.e `sigma=ky`.The distribution of internal forces in sagging and hogging is shown below 

The nature of internal forces of the beam is normal stress(`sigma`)

The longitudinal surface of the beam at which Axial strain and hence Axial Stress is zero is known as Neutral Surface or Neutral layer of the beam 

Neutral Axis :- Neutral Axis of a cross section is the line at which the Neutral Surface cut the cross section .

On Applying Hooke's law and equilibrium equation i.e 

By equilibrium 

`sigma=M/I*y`

where `sigma`is stress experienced by a beam 

M is a Bending Moment 

I is the inertia 

and y is a distance from the neutral axis 

And by the help of Hooke's law `sigma=E(1/R)*y`  

For Maximum value 

`sigma`max=`M/I*ymax`

Beam Strength 

Maximum Bending Moment (limit of load capacity ) can be resisted by the beam without failure also known by the name of Moment of Resistance 

For safe case 

`sigma_max le sigma_p`

where `sigma_p`is the permissible stress so

`M/I*y le sigma_p`

`M le sigma_p*(I/y` `sigma_p`is a function of material and  `I/y`is a function of section which is called Section Modulus `

`M_(max)=sigma_p*Z`

where Z is a section modulus 

More the value of Section modulus More will be the bending strength 

Now consider a case of bending strength of circular section and square section of same material and same cross section 

On comparing Strength of circular section and square section

`(M_(max))_(square)=1.182*(M_(max))_(circ)`

We have to design for a min strength for factor of safety .

For car hood of current design section inertia can be seen by the help of analysis on nx using section inertia as shown below 

SECTION-1

the detail report will be uploaded as pdf on attachment 

The section modulus can be calculated by the help of below formula 

`Z_(max)=I/y`

for designing we have to choose minimum MOI for factor of safety 

for `y``=881.1/2=440.55`

`(Z_(max))_(1)=I/y=(2.892*10^5 )/441.55=654.97mm^3`

SECTION-2

Now we are going to increase emboss height instead of increasing in model we are going to do offset current curve to 0.5 mm and with the help of section inertia analysis we can find the section modulus 

the detail report will be uploaded as pdf on attachment 

for `y``=881.1/2=440.55`

`(Z_(max))_(2)=I/y=(2.93*10^5 )/441.55=663.57mm^3`

On Comparision above section (1) and section (2) 

`(Z_(max))_1<(Z_(max))_2`

Hence we can say that we can improve strength by offseting or by increasing emboss height