Strength of Materials (GATE Aerospace and GATE ......o Cantilever beam Cantilever with an end point...
Transcript of Strength of Materials (GATE Aerospace and GATE ......o Cantilever beam Cantilever with an end point...
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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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