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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics

′
2 2
1 f x P x Px 3
′
Equation . is of the form: f x = = 0 or f x = 0 , where f x = −
2 f x 4 2
2
d f x P x 3Px 2 P
′
Therefore, we have: f x = = − = 0 , by where, we get: x =
dx 2 2 3
P
P − P
3
Substituting the value of x in the expression . , we get: y = =
2 3
‹ ‡ ȋšȌ ‹• ‡“—ƒŽ –‘ ȋ›Ȍǡ –Š‡ –”‹ƒ‰Ž‡ ‹• ‡“—‹Žƒ–‡”ƒŽ ˆ‘” ‘’–‹— ƒ”‡ƒǤ

Š‡ ‘’–‹— ƒ”‡ƒ ȋ Ȍ ‹• ‰‘– „› •—„•–‹–—–‹‰ –Š‡ ˜ƒŽ—‡ ‘ˆ ȋšȌ ‹ ‡“—ƒ–‹‘ ȋͳǤͷȌǣ

P 2 P 3
2
1 P P 1 P 4 P 4 P 2 18 P 2
3
3
Area A = 2 4 − 2 = − 54 = 2 36 ∗ 54 = 12 3
2 36

P 2
Therefore, we have the optimum area of the triangle to be:
12 3

– ƒ ƒŽ•‘ „‡ •Š‘™ –Šƒ– –Š‡ ƒ”‡ƒ ȋ Ȍ ‹• –Š‡ ƒš‹— ˜ƒŽ—‡ǡ ƒ• ˆ‘ŽŽ‘™•ǣ

2
2
d f x P x 3Px 2 d f x P 2
′
f x = = − , therefore, = − 3Px
dx 2 2 dx 2 2
—„•–‹–—–‹‰ –Š‡ ˜ƒŽ—‡ ‘ˆ ȋšȌ ‹ –Š‡ ƒ„‘˜‡ ‡š’”‡••‹‘ǡ ™‡ ‰‡–ǣ

2
2
d f x P 2 P P 2 d y
= − 3P = − , which means < 0 , indicating maximum value.
dx 2 2 3 2 dx 2

‘–‡ǣ ™Š‡‡˜‡” –Š‡ •— ‘ˆ –™‘ ˜ƒ”‹ƒ„Ž‡• ‹• ϐ‹š‡†ǡ –Š‡ǡ –Š‡‹” ’”‘†— – ™‘—Ž† „‡ ƒš‹— ‘Ž›
™Š‡ –Š‡› ƒ”‡ ‡“—ƒŽǤ

šƒ’Ž‡ǣ ‹ˆ –Š‡ •— ‘ˆ –™‘ —„‡”• ‹• ʹͷǡ –Š‡ ƒš‹— ˜ƒŽ—‡ ‘ˆ –Š‡‹” ’”‘†— – ‹• ‰‹˜‡ „›ǣ
ȋͳʹǤͷȗͳʹǤͷȌǡ ™Š‹ Š ‹• ͳͷ͸ǤʹͷǤ

• ƒ ’”ƒ –‹ ƒŽ ƒ•‡ǡ ™Š‡ –Š‡ •— ‘ˆ –Š”‡‡ •‹†‡• ‘ˆ ƒ —„‘‹† ‹• ‰‹˜‡ǡ –Š‡ ˜‘Ž—‡ ‘—Ž† „‡
ƒš‹‹œ‡† ™Š‡ –Š‡ •‹†‡• ƒ”‡ ‡“—ƒŽ –‘ ‡ƒ Š ‘–Š‡”ǡ –Šƒ– ‹•ǡ ‹– „‡ ‘‡• ƒ —„‡Ǥ

Š‹• Ž‘‰‹ ‘—Ž† „‡ ‡š–‡†‡† –‘ ƒ› —„‡” ‘ˆ ˜ƒ”‹ƒ„Ž‡•Ǥ

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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