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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
‘•‹†‡” ƒ •ƒŽŽ •–”‹’ ‘ˆ ™‹†–Š Ǯ †Ʌǯ ƒ† Ž‡‰–ŠȀ ‹” —ˆ‡”‡ ‡ ǮʹɎ” α ʹɎ •‹ɅǯǤ Š‡ ƒ”‡ƒ ‘ˆ –Š‡
•–”‹’ ‹• ‰‹˜‡ „›ǣ ȏʹɎ •‹Ʌ ȗ †ɅȐǤ

‡– —• ‹–‡‰”ƒ–‡ –Š‡ ƒ”‡ƒ ‘ˆ –Š‡ ‡Ž‡‡–ƒŽ •–”‹’ ˆ”‘ ǮͲ –‘ ɅͳǯǤ

θ 1 θ 1
2
2
2π Rsinθ ∗ Rdθ = 2πR sinθ dθ = 2πR 1 − cosθ - - - - (1.14)
1
0 0
“—ƒ–‹‘ ȋͳǤͳͶȌ ‹• –Šƒ– ‘ˆ –Š‡ •—”ˆƒ ‡ ƒ”‡ƒ ‘ˆ –Š‡ •’Š‡”‹ ƒŽ ƒ’Ǥ

‡– —• ˜‡”‹ˆ› –Š‡ ƒ —”ƒ › ‘ˆ ‡“—ƒ–‹‘ ȋͳǤͳͶȌǤ ˆ ™‡ •—„•–‹–—–‡ ˆ‘” ǮɅͳ α ͻͲ ǯǡ ™‡ ‰‡– ǮʹɎ ǯ ™Š‹ Š ‹•
ʹ
–Š‡ •—”ˆƒ ‡ ƒ”‡ƒ ‘ˆ –Š‡ Š‡‹•’Š‡”‡Ǥ ˆ ™‡ •—„•–‹–—–‡ ˆ‘” ǮɅͳ α ͳͺͲǯǡ ™‡ ‰‡– ǮͶɎ ǯ ™Š‹ Š ‹• –Š‡ ƒ”‡ƒ ‘ˆ
ʹ
–Š‡ •’Š‡”‡Ǥ
Š‡ ˜‘Ž—‡ ‘ˆ –Š‡ •’Š‡”‡ ‹• ‘’—–‡† ƒ• ˆ‘ŽŽ‘™•ǣ

ȋ ‘Ž—‡ ‘ˆ –Š‡ ‡Ž‡‡–ƒŽ •–”‹’ α ȏƒ”‡ƒ ‘ˆ –Š‡ ‡Ž‡‡–ƒŽ •–”‹’ ȗ Š‡‹‰Š– ȋ†ŠȌȐȌ

2
V strip = π Rsinθ ∗ Rdθ sinθ physically, this the volume of the plate
Š‡ ˜‘Ž—‡ ‘ˆ –Š‡ •’Š‡”‹ ƒŽ ƒ’ ‹• ‰‹˜‡ „›ǣ

θ 1 θ 1 θ 1
π Rsinθ ∗ Rdθ sinθ = πR sin θ dθ = πR sin θ 1 − cos θ dθ - - - - (1.15)
2
3
3
3
2
0 0 0
‘ ˆƒ ‹Ž‹–ƒ–‡ ‹–‡‰”ƒ–‹‘ǡ Ž‡– —• ƒ‡ –Š‡ ˆ‘ŽŽ‘™‹‰ •—„•–‹–—–‹‘Ǥ
dt
Assume cosθ = t , differentiating, we get = − sinθ or dt = − sin θ dθ
dθ

Ž•‘ ™Š‡ (θ = θ ), t = cosθ , and, when θ = 0 , t = 1 .
1
1
Š‡”‡ˆ‘”‡ǡ ‡“—ƒ–‹‘ ȋͳǤͳͷȌ ƒ „‡ ”‡™”‹––‡ ƒ•ǣ

cos θ 1 cos θ 1
t 3 cos θ 1
2
3
3
2
3
πR − 1 − t dt = πR t − 1 dt = πR − t cos θ 1
3 1
1 1 1
—„•–‹–—–‹‰ –Š‡ —’’‡” ƒ† Ž‘™‡” Ž‹‹–•ǡ ™‡ ‰‡–ǣ
3
3
cos θ 1 1 cos θ 1 2
3
3
πR − − cosθ + 1 = πR + − cosθ - - - - (1.16)
1
1
3 3 3 3
QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 17