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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

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

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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
♦  Deflection at loadpoint		cm
♦  Reaction force, R1		N
♦  Reaction force, R2		N
♦  Slope angle, θ1		°
♦  Slope angle, θ2		°

Bending Of A Straight Elastic Prismatic Beam

Consider a simply-supported bar, having a concentrated force acting vertically at any intermediate point (including the mid-point) along its length. 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 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=[(Fa²(L-a)²/3EIL]. The maximum deflection eg. for the case where x < a, is:
D_max=[(F(L-a)/3EIL)*(a(2L-a)/3)^3/2] occuring at x=√[a(2L-a)/3].

Tips

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

BIBLIOGRAPHY

• Brown TH; Marks' Calculations for Machine Design; McGraw Hill, New York, 2005, Chapter 2.
• Hicks TG; Mechanical Engineering Formulas, Pocket Guide; McGraw Hill, New York, 2003, Section 3.
• Young WC & Budynas RG; Roark's Formulas for Stress and Strain; 7th Edition, McGraw Hill, New York, 2002, Chapter 8.
• Gieck K & Geick R; Engineering Formulas, 7th Edition, McGraw Hill, New York, 1997, Section P.