SIMPLY-SUPPORTED BEAM WITH SINGLE OVERHANG AND A CONCENTRATED FORCE AT FREE END

SI/Metric Units

US Customary Units
INPUT   DATA ♥ EXAMPLE Of Input/Output

Title  

Distance between supports, L m

    

Length of overhang, a m
Modulus of elasticity, E   109N/m2 
Area moment of inertia, I   cm4
Force applied, F N

     Reset

OUTPUT   VARIABLES   &   GRAPHS

Variables   Values   Units
 ♦  Maximum Shear force, Vmax N Graphs:
 Shear force Vs Distance  
 Bending moment Vs Distance  
 Deflection Vs Distance  
 ♦  Maximum Bending moment, Mmax   N.m  
 ♦  Maximum Deflection, Dmax cm
 ♦  Distance of point of Dmax m
 ♦  Max upward Deflection, Du cm
 ♦  Location of Du m
 ♦  Reaction force, R1 N
 ♦  Reaction force, R2 N
 ♦  Slope angle, θ1 °
 ♦  Slope angle, θ2 °
THEORY  &   FORMULAE

Bending Of A Straight Elastic Prismatic Beam

Consider a simply-supported slender bar, having a single overhang and a concentrated force acting vertically and located at the free end of the beam. The following equations describe the distribution of shear force, bending moment and deformation:

    

where
     F = applied force at the tip of the overhang
     L = distance between supports
     a = length of the overhang
     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 support
     R2 = vertical reaction at right support
     θ1 = angle of slope at left support
     θ2 = angle of slope at right support

The maximum upward deflection between the support points is given by D=[(FaL2)/9√ 3EI] and occurs at x = L/√ 3. The maximum downward deflection (which is also the max overall deflection) is: Dmax=[(Fa2/3EI)*(L+a)] occuring at the tip x=L+a.

Tips

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

BIBLIOGRAPHY