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There are some cases where the beams are not symmetrical. In these cases, one can find a point where the moment due to shear flow is zero and you can avoid the twisting by applying a transverse load to that point. This point is called as shear center. Hence, only bending will be seen on the beam. It can also be called as center of twisting.
The stress formula is :
This formula can only be used if the load is applied to shear center on the beam. Also, as the beam deflects downwards, the deflection equations can be applied to the beam as well.
In order to calculate a shear center a beam cross section will be considered. One can find the cross section geometry and the shear force applied on the figure below.
The rectangular area of the sections has to be multiplied by its distance to the neutral axis. Neutral axis here is the symmetry axis at x direction. The distances between the upper and lower sections and the neutral axis is 5, between the flange and the neutral axis is 2.5 for two separated parts of the flange. (Considering the separated parts middle points). This comes from the formula of:
One can see the calculation of area with this formula y*A on the figure below:
Depending on the shear flow which can be seen at the right corner of the figure, the distribution is considered and the equation has been calculated as following:
At the intersection points of the web and the flange, the flow is considered to be the same amount. Another assumption is the moment has to be zero at the neutral center. Considering that the moment caused by shear force and shear flow, the formula of the equilibrium can be written as:
Developing the formula of moment equation equals to zero, one get:
On the left part; shear force multiplied by the shear center distance to neutral center. On the right part of the formula; shear flow multiplied by lever arm. In order to apply momentum with all shear flow distribution, one needs to calculate the integral which can be summarized as following:
The sign of bending moment can be found by checking the shear flow directions. To find the area of the shear flow integral and then multiply by its (web and flange) distance to the neutral center:
Area of two triangle: (1/2*15*4)*2 //15 equals to the max magnitude for rectangular
Distance of the flanges to the neutral center: 5
Area of the web: (15*10) //15 equals to the max magnitude for triangle
Distance of the web to the neutral center: 1.081
Area of the parabolic section: (2/3*10*6.25)
Distance of the web to the neutral center: 1.081 //Because parabola is on web as well
Finally what we get is:
After the simplification, one needs to calculate the Inertia of the beam to find the shear center:
Ix=192.7cm^4
Finally the shear center is calculated as:
You can follow our other articles to find out more about aircraft structural calculations, beam equations, structural analysis and mechanics of materials.
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