Tuesday, 25 April 2017

WHAT IS SECTION MODULUS OF BEAMS?

WHAT IS SECTION MODULUS OF BEAMS?

We were discussing basic concept of bending stress and derivation for beam bending equation in our previous session. We have also discussed assumptions made in the theory of simple bending and formula for bending stress or flexural formula for beams during our last session.

Now we are going ahead to start new topic i.e. Section modulus of beams in the strength of material with the help of this post.

Let us go ahead step by step for easy understanding, however if there is any issue we can discuss it in comment box which is provided below this post. So let us come to the main subject i.e. Section modulus.

Strength of any section indicates the moment of resistance offered by the section. Let us recall the concept of moment of resistance which is defined as the total moment of the forces, due to bending stresses, about the neutral axis of the section.
 
Where,
Z: Section modulus of the section
M: Moment of resistance
σ = Bending stress

Now we will define here the section modulus of the beam

Section modulus of a beam will be defined as the ratio of area moment of inertia of the beam about the neutral axis or centroidal axis of the beam, subjected to bending, to the distance of the outermost layer of the beam from its neutral axis or centroidal axis.

If we consider the moment of resistance offered by a section for a given value of bending stress, we can easily say that moment of resistance will be directionally proportional to the section modulus of the section.

Therefore, strength of any section will be dependent over the section modulus and we can say that if a beam has higher value for section modulus then beam will have more capacity to bear the bending moment for a given value of bending stress or beam will be stronger and hence section modulus of the beam will indicate the strength of the section.

Unit of section modulus

Unit of section modulus = Unit of area moment of inertia / Unit of distance
Unit of section modulus = m3

Section modulus of a beam having rectangular cross-section

Let us consider a beam, as displayed in following figure, with rectangular cross-section. Let us consider that width of the rectangular cross-section is B and depth or height of the rectangular cross-section is D.

IXX = Area moment of inertia of the rectangular section about the XX axis
 
Recall the derivation and concept of area moment of inertia of the rectangular section and we will have
IXX = BD3/12
y = distance of the outermost layer of the beam from its neutral axis or centroidal axis
y = D/2

Section modulus of the rectangular section about XX axis could be secured as mentioned here
ZXX = BD2/6

Similarly, we can find out here the value of section modulus of the rectangular section about YY axis
ZYY = DB2/6

Section modulus of a beam having circular cross-section

Let us consider a beam, as displayed in following figure, with circular cross-section. Let us consider that diameter of the circular cross-section is D.
IXX = Area moment of inertia of the circular cross-section about the XX axis
IYY = Area moment of inertia of the circular cross-section about the YY axis
 
Recall the derivation and concept of area moment of inertia of the circular cross-section and we will have
IXX = IYY = ПD4/64
y = distance of the outermost layer of the section from its neutral axis or centroidal axis
y = D/2

Section modulus of the circular cross-section about XX axis and YY axis could be secured as mentioned here
ZXX = ZYY = ПD3/32

Similarly we can secure the value of section modulus for various cross-sections as we have discussed above.

We will discuss another topic i.e. in the category of strength of material in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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