SIMPLY-SUPPORTED BEAM WITH CONCENTRATED COUPLE AT INTERMEDIATE POINT

US Customary Units

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
Couple applied, C ft.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 simply-supported bar, having a concentrated couple acting counter-clockwise at any intermediate point along its length. The following equations describe the distribution of shear force, bending moment and deformation:

    

where
     C = applied couple (moment) about any intermediate point, +ve anticlockwise
     L = length of beam or distance between supports
     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 support
     R2 = vertical reaction at right support
     θ1 = angle of slope at left support
     θ2 = angle of slope at right support

The delection at load point is given by D=[(Ca(L-a)(2a-L)/3EIL]. If a < L/2, this deflection will be above the axis of the beam; if a=L/2 the deflection is zero, and if a > L/2, the deflection will be below the beam axis.

Tips

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

BIBLIOGRAPHY