Hi,
Shear Centre is new topic introduced into the syllabus of the 3rd semester of the b.tech. in Civil engineering by HPTU. So this was a topic was new to me, and I think it might be new to someone like me. So after studying it from the online video lectures offered by Prof. S.K. Maiti from IIT Bombay(nptel lectrures). I thought to form some online notes in writing by myself. So here is what I have prepared till now.
Introduction:
Shear Center is a point on the cross section of a beam at which when shear force is applied there is zero twisting moment in the cross section of the beam. If the load(Shear force) does not pass through the shear center then, the beam gets a twisting moment, so in order to avoid the twisting moment in the beams one must know the location of the shear center of a given cross section.
Consider a channel section subjected a shear force S. The shear stress near to the center of Gravity will be maximum and then its magnitude will decrease towards the ends of the web, similarly it will be maximum near to C.G. in the flange and will decrease when moving away from the C.G.
To find out the total internal shear forces acting along the flange and the web we have to write use the equation of the shear stress at a point in the beam and then multiply it to a small element of the beam and then integrate it to the whole length of the element.
Shear stress at a point is given by = S.Q/ Iz.b = S.A.y/ Iz.b
where S = Applied shear force
A = Area under above the point under consideration
y = Distance of the centroid of the area under consideration from the centroid of the whole cross section
Iz = moment of inertia of the cross section about the z-axis.
b= width of the cross section
Nos if you want to find the the value of V in the web, multiply the equation of the shear stress with the a small element of the area and then integrate it for the whole depth of the web.
Similarly for the flanges.
Once the V and F and are known, you have to calculate the amount of internal twisting moment about the center of the gravity of the section which in this case will be = 2F*d/2 + V*z2.
This internal moment has to be balanced by some external twisting moment. So external shear force S has to be applied in such a way that it create the equal twisting moment, but in opposite direction.
So S must be applied at some eccentricity from the centre of gravity such that
S*z1 = 2F*d/2+ V.z2
Here z1 tells us about the location of the shear center. So in same manner you can find out the internal twisting moment for any given section and then you will find out the location of the shear center.
I hope the article was of some help to you. Please leave a comment if you feel that there are some changes to be made.
Thank you!
Sanjay.
Shear Centre is new topic introduced into the syllabus of the 3rd semester of the b.tech. in Civil engineering by HPTU. So this was a topic was new to me, and I think it might be new to someone like me. So after studying it from the online video lectures offered by Prof. S.K. Maiti from IIT Bombay(nptel lectrures). I thought to form some online notes in writing by myself. So here is what I have prepared till now.
Introduction:
Shear Center is a point on the cross section of a beam at which when shear force is applied there is zero twisting moment in the cross section of the beam. If the load(Shear force) does not pass through the shear center then, the beam gets a twisting moment, so in order to avoid the twisting moment in the beams one must know the location of the shear center of a given cross section.
To find out the total internal shear forces acting along the flange and the web we have to write use the equation of the shear stress at a point in the beam and then multiply it to a small element of the beam and then integrate it to the whole length of the element.
Shear stress at a point is given by = S.Q/ Iz.b = S.A.y/ Iz.b
where S = Applied shear force
A = Area under above the point under consideration
y = Distance of the centroid of the area under consideration from the centroid of the whole cross section
Iz = moment of inertia of the cross section about the z-axis.
Nos if you want to find the the value of V in the web, multiply the equation of the shear stress with the a small element of the area and then integrate it for the whole depth of the web.
Similarly for the flanges.
Once the V and F and are known, you have to calculate the amount of internal twisting moment about the center of the gravity of the section which in this case will be = 2F*d/2 + V*z2.
This internal moment has to be balanced by some external twisting moment. So external shear force S has to be applied in such a way that it create the equal twisting moment, but in opposite direction.
So S must be applied at some eccentricity from the centre of gravity such that
S*z1 = 2F*d/2+ V.z2
Here z1 tells us about the location of the shear center. So in same manner you can find out the internal twisting moment for any given section and then you will find out the location of the shear center.
I hope the article was of some help to you. Please leave a comment if you feel that there are some changes to be made.
Thank you!
Sanjay.
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