Slope of a Beam: Slope of a beam is the angle between deflected beam to the actual beam at the same point.
Deflection of Beam: Deflection is defined as the vertical displacement of a point on a loaded beam. There are many methods to find out the slope and deflection at a section in a loaded beam.
The maximum deflection occurs where slope is zero. The position of the maximum deflection is found out by equating the slope equation zero. Then the value of x is substituted in the deflection equation to calculate the maximum deflection
Double Integration Method: This is most suitable when concentrated or udl over entire length is acting on the beam.The double integration method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve.
Integrating one time
Integrating again
Where,
M = Bending moment
I = Moment of inertia of the beam section
y/v = Deflection of the beam
E = Modulus of elasticity of beam material.
Area Moment Method:Another method of determining the slopes and deflections in beams is the area-moment method, which involves the area of the moment diagram.The moment-area method is a semigraphical procedure that utilizes the properties of the area under the bending moment diagram. It is the quickest way to compute the deflection at a specific location if the bending moment diagram has a simple shape.
Method of Superposition: The method of superposition, in which the applied loading is represented as a series of simple loads for which deflection formulas are available. Then the desired deflection is computed by adding the contributions of the component loads(principle of superposition).
Mostly direct formula are used in questions,hence it is advised to look for the beam deflection formula which are directly asked from this topic rather than going for long derivations.
Beam Deflection Formula:
Cantilever Beams:
