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A cantilever beam of span l, carrying point load W at free end, slope at free end will be
3EIWl3тАЛ
2EIWl2тАЛ
6EIwl3тАЛ
8EIwl4тАЛ
2EIWl2тАЛ
For a cantilever beam of length l carrying a concentrated point load W at its free end, the slope ╬╕ at the free end is determined by integrating the bending moment equation or using the area-moment method. The slope is the result of the accumulation of curvature due to the bending moment induced by the load.
For a cantilever beam of length l carrying a concentrated point load W at its free end, the slope ╬╕ at the free end is determined by integrating the bending moment equation or using the area-moment method. The slope is the result of the accumulation of curvature due to the bending moment induced by the load.
╬╕=2EIWl2тАЛ тАФ Slope at the free end of a cantilever beam with point load
╬┤=3EIWl3тАЛ тАФ Deflection at the free end of a cantilever beam with point load
The bending moment MxтАЛ at a section x from the free end is WтЛЕx. Using the governing differential equation EIdx2d2yтАЛ=MxтАЛ, the slope equation is derived by integrating once with respect to x. Since the slope at the fixed end (x=l) is zero, the integration constant is determined, yielding the maximum slope at x=0 as 2EIWl2тАЛ.
The slope is maximum at the free end and zero at the fixed support.
The deflection is maximum at the free end.
The result is derived using Macaulay's method or the Area-Moment Method.
The units of slope are radians (dimensionless).
Standardized formula for quick calculation
Applicable for linear elastic analysis
Assumes material follows Hooke's Law
Neglects shear deformation (Euler-Bernoulli beam theory assumption)
Structural analysis of cantilever overhangs
Calculation of deflection in machine tool spindles
Structural health monitoring of beams
Option A represents the deflection at the free end, not the slope.
Option C is for a cantilever beam carrying a Uniformly Distributed Load (UDL) of intensity w, which is 6EIwl3тАЛ for slope.
Option D is for the deflection of a cantilever beam with UDL, which is 8EIwl4тАЛ.
B is correct тАФ The slope at the free end of a cantilever beam carrying a point load W at its tip is 2EIWl2тАЛ.
Always verify if the load is a point load or UDL and whether the question asks for 'Slope' (quadratic in l) or 'Deflection' (cubic/quartic in l).
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