Monday, 24 April 2017

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

Also read

Continue Reading

Sunday, 23 April 2017

MOMENT OF RESISTANCE OF BEAM SECTION

We were discussing basic concept of bending stress 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. moment of resistance of a beam 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.

Moment of resistance

 
Let us consider one structural member such as beam with rectangular cross section, we can select any type of cross section for beam but we have considered here that following beam has rectangular cross section as displayed in following figure.
As we have discussed that when a beam will be subjected with a pure bending, as displayed in above figure, layers above the neutral axis will be subjected with compressive stresses and layers below the neutral axis will be subjected with tensile stresses.

Therefore, there will be force acting on the layers of the beams due to these stresses and hence there will be moment of these forces about the neutral axis too.

Total moment of these forces about the neutral axis for a section will be termed as moment of resistance of that section.

As we have already assumed that we are working here with a beam having rectangular cross-section and let us consider the cross-section of the beam as displayed here in following figure.
Let us assume one strip of thickness dy and area dA at a distance y from the neutral axis as displayed in above figure.

Let us determine the force acting on the layer due to bending stress and we will have following equation.
dF = σ x dA

 
Let us determine the moment of this layer about the neutral axis, dM as mentioned here
dM = dF x y
dM = σ x dA x y
Recall here the concept of bending stress, which is mentioned below, and use the value of bending stress (σ) in above equation and we will have equation for bending moment of the layer about the neutral axis.
σ = (E/R) x y
dM = (E/R) x y x dA x y
dM = (E/R) x y2 dA

Total moment of the forces on the section of the beam around the neutral axis, also termed as moment of resistance, could be secured by integrating the above equation and we will have
dM = (E/R) x y2 dA
Where M will be termed as moment of resistance

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

Reference:

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

Also read

Continue Reading

Saturday, 22 April 2017

BENDING STRESS ANALYSIS FOR SYMMETRICAL AND UNSYMMETRICAL CROSS-SECTIONS

We were discussing basic concept of bending stress in our previous session. We have also discussed assumptions made in the theory of simple bending and expression for bending stress in pure bending during our last session.

Now we are going ahead to start new topic i.e. bending stress analysis for symmetrical and unsymmetrical cross-sections with the help of this post. 

First of all we must have to understand here the meaning of symmetrical sections and unsymmetrical sections.

 
In case of symmetrical sections, neutral axis will pass through the geometrical center of the section.
Cross-sections such as circular cross-section hollow circular cross-section, square cross-section, hollow square cross-section, rectangular cross-section, hollow rectangular cross-section and I cross-section are the best examples of symmetrical sections.

In case of unsymmetrical sections, neutral axis will not pass through the geometrical center of the section. Cross-sections such as T cross-section and L cross-section are the best examples of unsymmetrical sections.

Bending stresses in symmetrical sections

Neutral axis of symmetrical sections such as for circular section will lie at a distance d/2 from the outermost layer of the section. Where d will be diameter of the circular cross-section.

As we have discussed the formula for bending stress in pure bending of beams, we have concluded that stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis.

Let us see the formula for bending stress in pure bending of beams
If we consider the value of stress at neutral axis, we can easily say that it will be zero because value of y will be zero here.

 
Stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis; hence maximum stress will occur at the outermost layer of the section.

If we consider the case of simply supported beam, we must note it here that due to bending action, top portion of the beam will be in compression whereas bottom portion of the beam will be in tension.

If we will draw the diagram for stress distribution, we will have following figure showing the stress distribution for symmetrical sections.

Bending stresses in unsymmetrical sections

In case of unsymmetrical sections, neutral axis will not pass through the geometrical centre of the section and therefore value of y, which is the distance of the layer from the neutral axis, for outermost layers i.e. for topmost layer and bottom layer of the section will not be same.

In order to calculate the bending stress for unsymmetrical sections, we must have to find the value of centre of gravity of the given unsymmetrical section. 

As we know that neutral axis will pass through the center of gravity of the section and hence after determining the center of gravity of the section, we can have value of y for topmost layer and bottom layer of the section.

 
In order to calculate the bending stress for unsymmetrical sections, we will use the bigger value of y.
If we will draw the diagram for stress distribution for unsymmetrical section such as for T section, we will have following figure showing the stress distribution for T sections.
We will discuss another topic i.e. bending equation or flexural formula for beam subjected to a simple bending in the category of strength of material in our next post.

Reference:

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

Also read

Continue Reading

Friday, 21 April 2017

FORMULA FOR BENDING STRESS IN A BEAM

We were discussing basic concept of bending stress 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. expression for bending stress in pure bending of beam 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.

Let us consider one structural member such as beam with rectangular cross section, we can select any type of cross section for beam but we have considered here that following beam has rectangular cross section. 

Bending stress

 
Let us assume that following beam PQ is horizontal and supported at its two extreme ends i.e. at end P and at end Q, therefore we can say that we have considered here the condition of simply supported beam.
Once load W will be applied over the simply supported horizontal beam PQ as displayed above, beam PQ will be bending in the form of a curve and we have tried to show the condition of bending of beam PQ due to load W in the above figure.

Now let us consider one small portion of the beam PQ, which is subjected to a simple bending, as displayed here in following figure. Let us consider two sections AB and CD as shown in following figure.
Now we have following information from the above figure.

AB and CD: Two vertical sections in a portion of the considered beam
N.A: Neutral axis which is displayed in above figure
EF: Layer at neutral axis
dx = Length of the beam between sections AB and CD

Let us consider one layer GH at a distance y below the neutral layer EF. We can see here that length of the neutral layer and length of the layer GH will be equal and it will be dx.

Original length of the neutral layer EF = Original length of the layer GH = dx

Now we will analyze here the condition of assumed portion of the beam and section of the beam after bending action and we have displayed here in following figure. 
As we can see here that portion of the beam will be bent in the form of a curve due to bending action and hence we will have following information from above figure.

Section AB and CD will be now section A'B' and C'D'

Similarly, layer GH will be now G'H' and we can see here that length of layer GH will be increased now and it will be now G'H'

Neutral layer EF will be now E'F', but as we have discussed during studying of the various  assumptions made in theory of simple bending, length of the neutral layer EF will not be changed.

Length of neutral layer EF = E'F' = dx

A'B' and C'D' are meeting with each other at center O as displayed in above figure
Radius of neutral layer E'F' is R as displayed in above figure
Angle made by A'B' and C'D' at center O is θ as displayed in above figure
Distance of the layer G'H' from neutral layer E'F' is y as displayed in above figure

Length of the neutral layer E'F' = R x θ

 
Original length of the layer GH = Length of the neutral layer EF = Length of the neutral layer E'F' = R x θ

Length of the layer G'H' = (R + y) x θ

As we have discussed above that length of the layer GH will be increased due to bending action of the beam and therefore we can write here the following equation to secure the value of change in length of the layer GH due to bending action of the beam.

Change in length of the layer GH = Length of the layer G'H'- original length of the layer GH
Change in length of the layer GH = (R + y) x θ - R x θ
Change in length of the layer GH = y x θ

Strain in the length of the layer GH = Change in length of the layer GH/ Original length of the layer GH
Strain in the length of the layer GH = y x θ/ R x θ
Strain in the length of the layer GH = y/R

As we can see here that strain will be directionally proportional to the distance y i.e. distance of the layer from neutral layer or neutral axis and therefore as we will go towards bottom side layer of the beam or towards top side layer of the beam, there will be more strain in the layer of the beam.

At neutral axis, value of y will be zero and hence there will be no strain in the layer of the beam at neutral axis.

Let us recall the concept of Hook’s Law
According to Hook’s Law, within elastic limit, stress applied over an elastic material will be directionally proportional to the strain produced due to external loading and mathematically we can write above law as mentioned here.

Stress = E. Strain
Strain = Stress /E
Strain = σ/E
Where E is the Young’s Modulus of elasticity of the material

 
Let us consider the above equation and putting the value of strain secure above, we will have following equation as mentioned here.
σ/E = y/R
σ= (y/R) x E
Therefore, bending stress on the layer will be given by following formula as displayed here
We can conclude from above equation that stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis.

We will discuss another topic i.e. derivation of flexure formula or bending equation for pure bending in the category of strength of material in our next post.

Reference:

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

Also read


Continue Reading

DERIVATION OF BEAM BENDING EQUATION

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

Now we are going ahead to start new topic i.e. derivation of flexure formula or bending equation for pure bending 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.

Let us consider one structural member such as beam with rectangular cross section, we can select any type of cross section for beam but we have considered here that following beam has rectangular cross section. 

 
First of all we will find here the expression for bending stress in a layer of the beam subjected to pure bending and aftre that we will understand the concept of moment of resistance and once we will have these two information, we can easily secure the bending equation or flexure formula for beams. 

So let us first find out the expression for bending stress acting on a layer of the beam subjected to pure bending.

Bending stress

Let us assume that following beam PQ is horizontal and supported at its two extreme ends i.e. at end P and at end Q, therefore we can say that we have considered here the condition of simply supported beam.
Once load W will be applied over the simply supported horizontal beam PQ as displayed above, beam PQ will be bending in the form of a curve and we have tried to show the condition of bending of beam PQ due to load W in the above figure.

Now let us consider one small portion of the beam PQ, which is subjected to a simple bending, as displayed here in following figure. Let us consider two sections AB and CD as shown in following figure.
Now we have following information from the above figure.

AB and CD: Two vertical sections in a portion of the considered beam
N.A: Neutral axis which is displayed in above figure
EF: Layer at neutral axis
dx = Length of the beam between sections AB and CD

Let us consider one layer GH at a distance y below the neutral layer EF. We can see here that length of the neutral layer and length of the layer GH will be equal and it will be dx.
Original length of the neutral layer EF = Original length of the layer GH = dx

Now we will analyze here the condition of assumed portion of the beam and section of the beam after bending action and we have displayed here in following figure. 
As we can see here that portion of the beam will be bent in the form of a curve due to bending action and hence we will have following information from above figure.

Section AB and CD will be now section A'B' and C'D'

Similarly, layer GH will be now G'H' and we can see here that length of layer GH will be increased now and it will be now G'H'

Neutral layer EF will be now E'F', but as we have discussed during studying of the various 
assumptions made in theory of simple bending, length of the neutral layer EF will not be changed.
Length of neutral layer EF = E'F' = dx

A'B' and C'D' are meeting with each other at center O as displayed in above figure

 
Radius of neutral layer E'F' is R as displayed in above figure

Angle made by A'B' and C'D' at center O is θ as displayed in above figure

Distance of the layer G'H' from neutral layer E'F' is y as displayed in above figure

Length of the neutral layer E'F' = R x θ

Original length of the layer GH = Length of the neutral layer EF = Length of the neutral layer E'F' = R x θ

Length of the layer G'H' = (R + y) x θ

As we have discussed above that length of the layer GH will be increased due to bending action of the beam and therefore we can write here the following equation to secure the value of change in length of the layer GH due to bending action of the beam.

Change in length of the layer GH = Length of the layer G'H'- original length of the layer GH
Change in length of the layer GH = (R + y) x θ - R x θ
Change in length of the layer GH = y x θ

Strain in the length of the layer GH = Change in length of the layer GH/ Original length of the layer GH
Strain in the length of the layer GH = y x θ/ R x θ
Strain in the length of the layer GH = y/R

As we can see here that strain will be directionally proportional to the distance y i.e. distance of the layer from neutral layer or neutral axis and therefore as we will go towards bottom side layer of the beam or towards top side layer of the beam, there will be more strain in the layer of the beam.

At neutral axis, value of y will be zero and hence there will be no strain in the layer of the beam at neutral axis.

Let us recall the concept of Hook’s Law

According to Hook’s Law, within elastic limit, stress applied over an elastic material will be directionally proportional to the strain produced due to external loading and mathematically we can write above law as mentioned here.

Stress = E. Strain
Strain = Stress /E
Strain = σ/E
Where E is the Young’s Modulus of elasticity of the material

Let us consider the above equation and putting the value of strain secure above, we will have following equation as mentioned here.

σ/E = y/R
σ= (y/R) x E
Therefore, bending stress on the layer will be given by following formula as displayed here
We can conclude from above equation that stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis.

 

Moment of resistance

As we have discussed that when a beam will be subjected with a pure bending, layers above the neutral axis will be subjected with compressive stresses and layers below the neutral axis will be subjected with tensile stresses.

Therefore, there will be force acting on the layers of the beams due to these stresses and hence there will be moment of these forces about the neutral axis too.

Total moment of these forces about the neutral axis for a section will be termed as moment of resistance of that section.

As we have already assumed that we are working here with a beam having rectangular cross-section and let us consider the cross-section of the beam as displayed here in following figure.
Let us assume one strip of thickness dy and area dA at a distance y from the neutral axis as displayed in above figure.

Let us determine the force acting on the layer due to bending stress and we will have following equation
dF = σ x dA

Let us determine the moment of this layer about the neutral axis, dM as mentioned here
dM = dF x y
dM = σ x dA x y
dM = (E/R) x y x dA x y
dM = (E/R) x y2 dA

Total moment of the forces on the section of the beam around the neutral axis, also termed as moment of resistance, could be secured by integrating the above equation and we will have
dM = (E/R) x y2 dA
Let us consider the above equation of moment of resistance and equation that we have secured for bending stress in case of bending action; we will have following equation which is termed as bending equation or flexural formula of bending equation.
We will discuss another topic in the category of strength of material in our next post.

Reference:

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

Also read

Continue Reading

Archivo del blog

Copyright © ENGINEERING MADE EASY | Powered by Blogger | Designed by Dapinder