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Strength of Materials
(GATE Aerospace and GATE Mechanical) By
Mr Dinesh Kumar
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Shear Forces and Bending Moments
A slender structural member is usually classified according to the types of loads, they support. For example, a beam is a type of structural member subjected to transverse loads, that is, forces or moments having their vectors perpendicular to the axis of the bar.
Type of supports :-
o Fixed support
o Hinge support or pinned support
→ u=v = 0
→ 휃 = 0
→ Force ≠0
→ Moment ≠0
→ Shear stress ≠0
→ Bending stress ≠0
→ u=v = 0
→ 휃 ≠ 0
→ Force ≠0
→ Moment =0
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o Roller support
Types of Beams:-
o Simply supported Beam
→ v = 0
→ u ≠ 0
→ 휃 ≠ 0
→ Force ≠0
→ Moment =0
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o Cantilever beam
o Beam with overhang
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o Propped Cantilever beam
o o Fixed beam
o Continuous beam
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Type of load o Concentrated load
o Uniformly distributed load (UDL)
o Uniformly varying load (VDL)
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o Concentrated moment
]
Sign convention of shear force and bending moment
Consider the sign conventions for shear forces and bending moments.
According to above figure, a positive shear force acts clockwise against the material and a negative shear force acts counterclockwise against the material. Also, a positive bending (sagging) moment compresses the upper part of the beam and a negative bending (hogging) moment compresses the lower part.
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Relationship between loads, shear force and bending moments Consider a small section of beam of length dx as shown in the figure, it is subjected to a udl of intensity of q(x) as shown. Free body dia for shear forces and bending moment as shown.
o Force equilibrium In Vertical direction ∑F = 0 q(x)dx – V + (V+dV) = 0 q(x)dx + dV = 0 q(x) = - 풅푽
풅풙
o Moment equilibrium
Taking Moment equilibrium about z axis passing through CG of the section
∑M = 0
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Vdx + M + (V+ dV) dx – M – dM = 0 V푑푥+ Vdx + dV dx – dM = 0 (here 푑푣푑푥 is very small) Vdx – dM = 0 V = 풅푴
풅풙
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Shear- force and bending moment diagrams o Cantilever beam
Cantilever with an end point lod
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Cantilever with UDL
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Cantilever with point load at mid-span and UDL
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Cantilever with moment at mid-span and UDL
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Simply supported beam o Simply supported beam with point load
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o Simply supported beam with UDL
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Problems
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