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Whenever a beam is subjected to transverse loading, or bending moment, it will deflect to a certain amount depending upon the rigidity of the beam and, the applied moment. Double integration method and Macaulay's methods are used to find out the slope and deflection at any section of a given statically determinate beam.
Consider a beam shown in the figure which is subjected to a pure bending moment of 'M'. This is assumed that the beam has a circular shape when get deflected. Consider the section z-z, let M is the moment at the section. It has been studied in the flexure theory(Theory of simple bending)
M/I = f/y = E/R
or
1/R =M/EI -----(1)
Consider section at z-z, it can be well figured out that
1/R = -(d2Y/ dx2)----(2)
So from one and 2,
M/EI = -(d2Y/ dx2) ---(3)
equation(3) is known as the differential equation of the elastic curve of deflection of beam.
If we integrate the eqn(3) once we get dy/dx i.e. slope at any point x distant, and if we further integrate this once more, we get 'y', which is deflection at any point x distant.
After integrating the differential equation of elastic curve of deflection of beam, we get the deflection at the required point.
This must be noted that we have to put the equation of the Mx in the equation(3) and then we have to integrate the equation from o to x(required point). It may happen that the equation of the moment may vary from section to section of the beam, so you have to write different equation of moment for different sections, when the loading/shear force varies.
Macaulay's method is similar to double integration method, only modification is that, there is only one equation written for whole of the span for the bending moment, instead of writing different moment equation for different sections of the beam whenever shear force gets varied.
Reference: Strength of materials by R.K.Rajput
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- Introduction:
Whenever a beam is subjected to transverse loading, or bending moment, it will deflect to a certain amount depending upon the rigidity of the beam and, the applied moment. Double integration method and Macaulay's methods are used to find out the slope and deflection at any section of a given statically determinate beam.
- Moment Curvature Relationship and Differential equation of elastic curve:
Consider a beam shown in the figure which is subjected to a pure bending moment of 'M'. This is assumed that the beam has a circular shape when get deflected. Consider the section z-z, let M is the moment at the section. It has been studied in the flexure theory(Theory of simple bending)
or
1/R =M/EI -----(1)
Consider section at z-z, it can be well figured out that
1/R = -(d2Y/ dx2)----(2)
So from one and 2,
M/EI = -(d2Y/ dx2) ---(3)
equation(3) is known as the differential equation of the elastic curve of deflection of beam.
- Double integration Method:
If we integrate the eqn(3) once we get dy/dx i.e. slope at any point x distant, and if we further integrate this once more, we get 'y', which is deflection at any point x distant.
After integrating the differential equation of elastic curve of deflection of beam, we get the deflection at the required point.
This must be noted that we have to put the equation of the Mx in the equation(3) and then we have to integrate the equation from o to x(required point). It may happen that the equation of the moment may vary from section to section of the beam, so you have to write different equation of moment for different sections, when the loading/shear force varies.
- Macaulay's Method:
Macaulay's method is similar to double integration method, only modification is that, there is only one equation written for whole of the span for the bending moment, instead of writing different moment equation for different sections of the beam whenever shear force gets varied.
Reference: Strength of materials by R.K.Rajput
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