Section Modulus calculation and optimization

Aim 

To calculate the section modulus of the previously designed hood for analyzing its strength and also optimizing the design to see the difference in the bending strength of the hood. Higher the section modulus of a structure, the more the resistive it becomes to bending.

Section modulus 

Section modolus 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 area  and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

S=I/Y

Here,

S = Section modulus

I= Moment of Inertia (Unit mm 4)

Y= Distance from Neutral axis to the extreme end (unit mm)

The Unit of a “section modulus” is (mm3)

 

 

 Changing the position can show the difference in bending which is because of the centroid distance from extreme fiber end.

  

In Layman's terms, in this position there is more material from axis to the specific direction, to resist the load which makes less bend amd improves the strength.

Elastic section modulus

For general design, the elastic section modulus is used, applicable 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 second moment of area (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 yield strength  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 S_{x}={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}} {\displaystyle S_{x}={\tfrac {I_{x}}{y}}}, with {\displaystyle y={\cfrac {H}{2}}}	NA indicates neutral axis
doubly symmetric I-section  (minor axis)		{\displaystyle S_{y}={\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 neutral 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, 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:

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

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.

Description	Figure	Equation	Comment
Rectangular section		{\displaystyle Z_{P}={\cfrac {bh^{2}}{4}}}	{\displaystyle A_{C}=A_{T}={\cfrac {bh}{2}}} , {\displaystyle y_{C}=y_{T}={\cfrac {h}{4}}}
Rectangular hollow section		{\displaystyle Z_{P}={\cfrac {bh^{2}}{4}}-(b-2t)\left({\cfrac {h}{2}}-t\right)^{2}}	where: b = width, h = height, t = wall thickness
For the two flanges of an I-beam with the web excluded		{\displaystyle Z_{P}=b_{1}t_{1}y_{1}+b_{2}t_{2}y_{2}\,}	where: {\displaystyle b_{1},b_{2}} =width, {\displaystyle t_{1},t_{2}}=thickness, {\displaystyle y_{1},y_{2}} are the distances from the neutral axis to the centroids of the flanges respectively.
For an I Beam including the web		{\displaystyle Z_{P}=bt_{f}(d-t_{f})+0.25t_{w}(d-2t_{f})^{2}}
For an I Beam (weak axis)		{\displaystyle Z_{P}=(b^{2}t_{f})/2+0.25t_{w}^{2}(d-2t_{f})}	d = full height of the I beam
Solid Circle		{\displaystyle Z_{P}={\cfrac {d^{3}}{6}}}
Circular hollow section		{\displaystyle Z_{P}={\cfrac {d_{2}^{3}-d_{1}^{3}}{6}}}

The plastic section modulus is used to calculate the plastic moment, M_p, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, F_y, by M_p = F_y ⋅ 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 elastic limit of material. This could be shown mathematically with the formula :-

{\displaystyle k={\cfrac {Z}{S}}}

Shape factor for a rectangular section is 1.5.

SECTION MODULUS OF AN HOOD

• Section modulus is a direct measure of strength of any  sectional area of an object about an reference axis(neutral axis).
• Strength is nothing but, capable of supporting greater loads, so higher the strength it has a high capable of supporting  greater loads.
• This strength can measured directly from Section modulus formula, and defined by below formula.
• Section modulus represented with a symbol of 'S'.

                                                                                        S=I/Y

                                                                                       where 'I'moment of inertia

                                                                                                 'Y' is the distance from the reference line(neutral axis) to one of the extreme end of an object.

• Moment of inertia is nothing but offering resistance against of moment, bending or twisting, i.e if we applied any external load on the body, how much resistance its offering to tend to deform.
• So, higher the value of moment of inertia , have higher resistance to deform under certain load, hence we can say that moment of inertia directly proportional to the strength of the material , and strength of the material can be calculated with the help of section modulus.
• If we vary the  neutral axis towards positive or negative side the overall strength of the body will decrease , because when nuetral axis line lies on the middle of the section only have higher strength at particular point, so,  here 'Y'  value was inversely proportional to section modulus.

Thats why formula represented as above.

By using the same principle i going to calculate the strength of an hood panel.

Here the moment of inertia was calculated by using the NX-software obtained values are

I_MAX:-8.485481X10⁵mm⁴

I_min:-3.18672x10³mm⁴

Total length of 'Y' value was 488.8252mm

So,at neutral axis we need 'Y' value:-Y/2=488.8252/2=244.412mm

1)Section modulus(S):-I/Y=8.485481X10⁵mm⁴/244.412mm

                                                =3.471X10³mm³(Maximum strength).

2) Section modulus(S):-I/Y=3.18672x10³mm⁴/244.412mm

                                                =0.013x10³mm³(minimum strength).

So the strength of an hood varies from 0.013x10³mm³to 3.471X10³mm³

Now, one more trail is that, here the inner panel curve offsetting outside by 1mmi.e increasing in cross sectional area,the results shown below.

I obtained the values of

I_MAX:-8.4462X10⁵mm⁴

I_min:-3.61202x10³mm⁴

Total length of 'Y' value was 488.8252mm

So,at neutral axis we need 'Y' value:-Y/2=488.8252/2=244.412mm

1)Section modulus(S):-I/Y=8.4462X10⁵mm⁴/244.412mm

                                                =3.455X10³mm³(Maximum strength).

2) Section modulus(S):-I/Y=3.61202x10³mm⁴/244.412mm

                                                =0.014x10³mm³(minimum strength).

So the strength of an hood varies from 0.014x10³mm³to 3.455X10³mm³.

• So, here when comparing both the hoods there is changes in both MAX and MIN strength values,what i came to know that when i increased sectional area by 1mm the value of minimum strength was increased from 0.013x10³mm³ to 0.014x10³mm³,Hence with increase in cross sectional area results in increase in strength.
• But, when we seen the value of maximum strength its slightly decreases from 3.471X10³mm³ to 3.455X10³mm³,i think this due to design of hood, here what we want to know that when increase in cross sectional area there will be increase in minimum strength or maximum strength or increase in both.
• So Newton's first law also explains that ,if higher the mass , the force required to move the object is also high due to offering more resistance in order to move the object, This resistance we called it as a MOMENT OF INERTIA.
• In case of hood also same principle will work, i.e when increase in cross sectional area by 1mm the minimum load carrying capacity or strength was increased by approximately 8% percentage what we observed from the above discussion.
• The strength of the material also depends upon the composition of material ,thickness and cross sectional area of an object.

CONCLUSION:-

The strength of the material is directly proportional to the moment of inertia, hence increase in sectional area there is have increase in overall mass of an object ,hence have increase in strength.