Foundations — Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E

3.6.3 · D1 · Physics › Spacecraft Structures & Systems Engineering › Stress and strain — σ = F - A, ε = ΔL - L, Young's modulus E

Is page pe assume kiya gaya hai ki tumhe kuch nahi pata. Parent note Stress and Strain mein jo bhi letters, slashes, aur units hain — woh sab yahan groundup se build kiye jayenge. Top se bottom tak padho — har idea agli cheez ki neenv hai.

0. Woh picture jis par hum baar baar aate hain

Is topic mein sab kuch ek hi experiment hai: ek bar ko dono siron se kheeenchna. Abhi woh image fix kar lo.

Us bar ki ek length hai (left se right kitni lambi hai), ek cross-section hai (woh flat face jo dikhti hai agar tum use kaat do), aur woh ek force feel karti hai (khichaav). Yeh teeno simple cheezein hamare teeno symbols ban jaati hain. Chalte hain ek ek ko samajhte hain.

1. Force — symbol $F$

Picture: figure mein bar ke dono siron ko khenchte hue do arrows. Ek newton roughly ek chote seb ka weight hota hai jo tumhare haath mein rakhha ho — gravity jo pull deti hai usse.

Topic ko iske kya zaroorat hai: strut ka poora maqsad hi load carry karna hai. Load ek force hai. Lekin — aur parent note ki shuruaati warning yahi hai — force akela nahi bata sakta ki kuch tutega ya nahi. $1000\text{N}$ ka pull ek baal tod deta hai aur ek girder pe kuch nahi karta. Hume force ko kitne material ne share kiya hai usse compare karna hoga. Yahi agla symbol hai.

2. Area — symbol $A$

Picture: socho bar ko knife se kaata aur kati hui face dekhi — ek chota sa rectangle ya circle. Uski size hi $A$ hai.

"Cross-sectional" kyun, sirf "area" kyun nahi? Kyunki bar ki kai faces hoti hain (lambe sides, sir). Sirf woh face jo pull ki taraf dekh rahi ho matter karti hai — woh surface jise internal force cross karni padti hai. Figure dekho: moti bar ki cut face badi hai, patli bar ki choti, chahe pull same hi ho.

3. Length aur change in length — $L$ aur $\Delta L$

Picture: bar $L$ lambi thi; khichaav ke baad woh $L+\Delta L$ lambi ho gayi. Ant mein jo extra takda hai woh $\Delta L$ hai.

$\Delta$ ko zor se padhna: "delta L" ka matlab hai "L ka woh thoda sa hissa jo badla." Yeh $\Delta$ multiplied by $L$ nahi hai — $\Delta$ koi number nahi, ek instruction hai: difference lo.

Topic ko iske kya zaroorat hai: $1\text{mm}$ stretchna ek $2\text{mm}$ sample ke liye bahut badi baat hai aur $2\text{m}$ beam ke liye kuch nahi. Toh raw stretch $\Delta L$ utna hi misleading hai jitna raw force $F$. Hume ise original $L$ se compare karna hoga — woh ratio hi strain hai, section 6.

4. Ratios, aur divide karna hi poora trick kyun hai

Is topic ke dono key moves divisions hain:

• crowding $=\frac{F}{A}$ (force compared to area)
• proportion of stretch $=\frac{\Delta L}{L}$ (stretch compared to length)

Slash $/$ ka matlab sirf "divided by" hai. $\text{N}/{\text{m}}^2$ padha jaata hai "newtons divided by square metres." Dhyan rakho — units exactly numbers ki tarah divide hote hain.

5. Stress — symbol $\sigma$

$\sigma =\frac{F}{A}$

Picture: cut face pe spread hue chote internal-force arrows ki density. Choti face mein bahut saare arrows = zyada $\sigma$.

Units padhna: $1\text{Pa}$ ek square metre pe ek newton hai — bahut choti bheed. Real metals millions of pascals pe jeete hain, isliye hum megapascal use karte hain: $1\text{MPa}=10^6\text{Pa}$. Chota superscript $10^6$ matlab "1 ke baad 6 zeros," yaani ek million.

6. Strain — symbol $\epsilon$

$\epsilon =\frac{\Delta L}{L}$

Picture: $\Delta L$ ka takda puri bar $L$ ke fraction ke roop mein measure kiya gaya. Aadha bar zyada lamba? $\epsilon =0.5$. Ek hazarwa hissa zyada? $\epsilon =0.001$.

Units kyun nahi? Metres divided by metres hai — units cancel ho jaate hain, ek pure number bachta hai (percentage ki tarah). Kyunki real strains bahut chote hote hain, hum aksar microstrain kehte hain: $1\mu \epsilon =10^{-6}$, yaani ek million mein se ek hissa. $\mu$ ("mu") ka matlab "millionth" hai.

7. Proportional to — symbol $\propto$

Picture: $\sigma$ upar plot karo, $\epsilon$ across — chote pulls ke liye points ek seedhi line pe girte hain jo origin se shuru hoti hai. Woh seedhi-line wala region hi hai jahan hamare saare formulas kaam karte hain. (Poora curved graph tumhe Stress-Strain Curve mein milega.)

"Proportional" aur sirf "related" mein fark kyun? Kyunki "related" koi bhi tedhi-medhi connection ho sakti hai. Proportional special, clean, straight-line relationship hai — aur sirf seedhi line ka ek constant slope hota hai. Woh slope hi haara aakhiri symbol hai.

8. Young's modulus — symbol $E$

Picture: section 7 ki seedhi line ki steepness. Khadi line = bada $E$ = material bade stress ke liye mushkil se strain karta hai = stiff. Dheemi line = chota $E$ = aasaani se khinchta hai.

Yeh aakhiri wala — "engineer's stretch formula" — poora topic hai jo paanch symbols mein pack hai, jo tum ab samajhte ho. Yeh proportional law hi Hooke's Law hai jo ek material pe apply kiya gaya hai.

9. Greek letters zor se padhna (taaki notation kabhi daraye nahi)

Foundations topic ko kaise feed karte hain

Har arrow ek division ya slope hai jo tumne upar khud build kiya. Bahar se kuch nahi aaya.

Yeh foundations aage kahan jaate hain

• Seedhi line Stress-Strain Curve mein poora curve ban jaati hai.
• Proportional law khud Hooke's Law hai.
• Ek taraf kheencho, doosri taraf patla ho jaata hai — Poisson's Ratio.
• Heat bhi $\Delta L$ cause karti hai — Thermal Stress.
• Real struts in forces ko route karte hain — Spacecraft Load Paths and Struts.
• Ceiling ke neeche safely rehna — Factor of Safety aur Yield Strength and Plastic Deformation.

Equipment checklist

Khud test karo — daayaa side cover karo.