CANTILEVERED BEAM WITH CONCENTRATED FORCE AT INTERMEDIATE POINT

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SI/Metric Units
INPUT   DATA ♥ EXAMPLE Of Input/Output

Title  

Length, L ft

    

Loadpoint distance, a ft
Modulus of elasticity, E   106lb/in2 
Area moment of inertia, I   in4
Force applied, F lb

     Reset

OUTPUT   VARIABLES   &   GRAPHS

Variables   Values   Units
 ♦  Maximum Shear force, Vmax lb Graphs:
 Shear force Vs Distance  
 Bending moment Vs Distance  
 Deflection Vs Distance  
 ♦  Maximum Bending moment, Mmax   ft.lb  
 ♦  Maximum Deflection, Dmax in
 ♦  Distance of point of Dmax ft
 ♦  Deflection at loadpoint in
 ♦  Reaction force, R1 lb
 ♦  Reaction force, R2 lb
 ♦  Slope angle, θ1 °
 ♦  Slope angle, θ2 °
THEORY  &   FORMULAE

Bending Of A Straight Elastic Prismatic Beam

Consider a cantilevered bar, having a concentrated force acting vertically at any intermediate point along its length (including the tip of the free end). The following equations describe the distribution of shear force, bending moment and deformation:

    

where
     F = applied force at any intermediate point
     L = length of beam
     a = location of load point from left end of beam
     x = distance from left end of beam
     E = modulus of elasticity of beam material
     I = area moment of inertia of cross-sectional area about axis through centroid
     V = shear force
     M = bending moment
     D = deflection
     R1 = vertical reaction at left end
     R2 = vertical reaction at right end
     θ1 = angle of slope at left end
     θ2 = angle of slope at right end

The delection at load point is given by D=[F(L-a)3/3EI]. The maximum deflection is:
Dmax=[F(L-a)2(2L+a)/6EI] occuring at the free end.

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point

BIBLIOGRAPHY