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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 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:
D_max=[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

• 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.