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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
ͳǤͶǤͶ –‡‰”ƒ–‹‘

–‡‰”ƒ–‹‘ ‹• ƒ –‡ Š‹“—‡ –Šƒ– Š‡Ž’• “—ƒ–‹ˆ› –Š‡ Šƒ‰‡ ‘ˆ ƒ ˆ— –‹‘ ‘˜‡” ƒ ‹–‡”˜ƒŽ ‘ˆ –‹‡ ‘”
•’ƒ ‡Ǥ  ‡‰‹‡‡”‹‰ǡ –Š‹• ‹• ’”‹ƒ”‹Ž› —•‡† –‘ ϐ‹† ƒ”‡ƒ• ƒ† ˜‘Ž—‡• ƒ† •‘Ž˜‡ †‹ˆˆ‡”‡–‹ƒŽ
‡“—ƒ–‹‘•Ǥ

–‡‰”ƒ–‹‘ ‘ˆ ƒ› ˆ— –‹‘ ‹• —†‡”•–‘‘† ƒ• ˆ‘ŽŽ‘™•ǣ

[Fig 1.13: function f(x) between limits 'a' and 'b']

Š‡ ‹–‡‰”ƒ–‹‘ „‡–™‡‡ Ž‹‹–• ȋš α ƒȌ ƒ† ȋš α „Ȍ ‘ˆ –Š‡ ˆ— –‹‘ ˆȋšȌǡ ™Š‹ Š ‹• ƒ —”˜‡ ƒ• •Š‘™ ‹
ȏ ‹‰ ͳǤͳ͵Ȑ ‹• –Š‡ ƒ”‡ƒ —†‡” –Š‡ —”˜‡Ǥ Š‡ ˆ—†ƒ‡–ƒŽ ƒ’’”‘ƒ Š –‘ ϐ‹†‹‰ ƒ”‡ƒ —†‡” •— Š ƒ
‰‡‡”ƒŽ —”˜‡ ‹• ƒ• ˆ‘ŽŽ‘™•ǣ

‘•‹†‡” –Š‡ ƒ”‡ƒ —†‡” –Š‡ —”˜‡ „‡–™‡‡ –Š‡ Ž‹‹–• ȋš α ƒȌ ƒ† ȋš α „ȌǤ

Š‹• ƒ”‡ƒ ‹• †‹˜‹†‡† ‹–‘ •‡˜‡”ƒŽ ”‡ –ƒ‰—Žƒ” •–”‹’• ‘ˆ ™‹†–Š ȋ†šȌǤ – ƒ „‡ ”‡ƒ•‘‡† –Šƒ–ǡ ƒ• ȋ†šȌ
„‡ ‘‡• •ƒŽŽ‡” ƒ† •ƒŽŽ‡”ǡ –Š‡ •–”‹’ –‡†• –‘™ƒ”†• ƒ ’‡”ˆ‡ – ”‡ –ƒ‰Ž‡Ǥ ‡ ‡ǡ –Š‡ ƒ”‡ƒ ‹•
ƒ’–—”‡† „‡––‡”Ǥ

Š‡ ƒ”‡ƒ —†‡” –Š‡ —”˜‡ ƒ „‡ ƒ–Š‡ƒ–‹ ƒŽŽ› ˆ‘—†ǡ ƒ• –Š‡ •— ‘ˆ ƒŽŽ –Š‡ ƒ”‡ƒ• ‘ˆ ƒŽŽ –Š‡
”‡ –ƒ‰—Žƒ” •–”‹’•ǡ ™Š‹ Š ‹• ‰‹˜‡ „›ǣ

x=b
Total area under the curve = f x dx
x=a
b
The above expression is symbolically given to be: f x dx
a

Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ ǮˆȋšȌǯ ƒ† Ǯšǯ ‘—Ž† „‡ ‘ˆ ƒ› ƒ–—”‡ǡ ‹ˆ ‹– ‹• ‘–‹—‘—•Ǥ Š‡ ‹–‡‰”ƒŽ ‘ˆ
ǮˆȋšȌǯ ‹• ƒ ƒ–‹-†‡”‹˜ƒ–‹˜‡ǡ ‡ƒ‹‰ǡ †‹ˆˆ‡”‡–‹ƒ–‹‰ –Š‡ ‹–‡‰”ƒŽ ™‘—Ž† Ž‡ƒ† —• –‘ –Š‡ ˆ— –‹‘ ‹–•‡ŽˆǤ
Š‹• ƒ „‡ ƒ–Š‡ƒ–‹ ƒŽŽ› ”‡’”‡•‡–‡† ƒ• ˆ‘ŽŽ‘™•ǣ

d
f x dx = f x
dx

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