Foundations — Sandwich structures — face sheets, core

3.6.17 · D1 · Physics › Spacecraft Structures & Systems Engineering › Sandwich structures — face sheets, core

Yeh page assume karta hai ki aapne parent note ki koi bhi notation pehle nahi dekhi. Hum har symbol ko ek picture se build karte hain pehle ki woh kabhi use ho. Agar koi term doosre vault note se link karti hai, toh woh note aur gehri jaati hai — lekin aapko uss line ko follow karne ke liye uski zaroorat nahi jo aap abhi padh rahe hain.

1. Length, thickness, width, depth — shape ke words

Kisi bhi physics se pehle, hume words chahiye pieces kitne bade hain ke liye. Ek sandwich panel ek flat slab hai, isliye iske teen sizes hain.

Fig 1 padhna. Upar aur neeche ki dono magenta bars face sheets hain, har ek ki height $t_f$ hai — notice karo ki yeh kitni patli hain. Unke beech ka hatched pink band core hai, jiska height $c$ hai. Orange double arrow daayein taraf poore stack $h$ ko measure karta hai; violet arrow neeche span $L$ ko measure karta hai; aur baayein taraf ka chhota navy arrow yaad dilata hai ki panel page ke andar bhi apni width $b$ tak extend karti hai. Woh paanch arrows dimag mein fix karo — is page ka har formula unmen se ek squared, multiplied, ya compared hai.

Kyunki dono skins core ke baahir rehti hain, depths add up ho jaati hain:

Topic ko yeh kyun chahiye. Sandwich ka har advantage ek story hai $h$ ke bade hone ki jabki $t_f$ chhoti rehti hai. Poora subject comparison $t_f\ll h$ mein jeeta hai (padho: "$t_f$, $h$ se bahut chhota hai"). Symbol $\ll$ "less than" nahi hai — iska matlab hai "itna kam ki hum use bade cheez ke saath ignore karne ki ijazat rakhte hain."

2. Force, aur force phela hua: $F$, $w$, $p$

Yeh simple multiplication se connect hote hain:

Topic ko yeh kyun chahiye. Launch ke dauran panel ko ek finger se push nahi kiya jaata; yeh apne weight times rocket ki acceleration se har jagah dabi jaati hai. Hum us even pressure $p$ ko ek line load $w$ mein convert karte hain taaki panel ko ek beam ki tarah treat kar sakein — sabse simple object jiska bending hum predict karna jaante hain. (Dekho Beam Bending Theory.)

3. The bending picture: neutral axis, stress, aur $y$

Ek beam ko bend karo aur kuch sundar hota hai: top surface squeeze hoti hai (compression) aur bottom stretch hoti hai (tension). Beech mein kahin ek line hai jo na tension mein hai na compression mein — neutral axis.

Fig 2 padhna. Beech se guzarti violet dashed line neutral axis hai, jahan $y=0$ hai. Daayein taraf ki magenta sloped line stress hai: yeh vertical se door jhukti hai, matlab stress beech se jitni door jaao utni zyada badhti hai. Axis ke upar orange shading compression mark karti hai (top fibres squeezed); neeche magenta shading tension mark karti hai (bottom fibres stretched). Navy $y$ arrow woh distance dikhata hai jo hum measure karte hain. Key visual yeh hai ki stress line neutral axis par exactly zero touch karti hai aur surfaces par sabse tezi se fan out hoti hai.

$\sigma =My/I$ kahan se aata hai

Law ko poora nigal lene ki bajaye, dekho yeh khud do ideas se kaise banta hai jo tum pehle se jaante ho.

Idea 1 — strain $y$ ke saath straight-line grow karta hai. Jab beam gently arc mein bend hoti hai, flat cross-sections flat rehte hain aur sirf tilt karte hain. Height $y$ par ek fibre isliye $y$ ke proportion mein stretch hoti hai: neutral axis se door fibres sabse zyada stretch hoti hain, $y=0$ par fibres bilkul nahi. Stretch-per-length (the strain) ko $\epsilon =\kappa y$ kaho, jahan $\kappa$ (Greek "kappa") curvature hai — beam kitni sharply arc mein hai. Yeh ek arc ki pure geometry hai; phir se Fig 2 mein fanning stress line dekho, jo actually yeh straight-line strain hai.

Idea 2 — stress is stiffness times strain. Elastic material ke liye, stress = $E$ (defined in §5) times strain, isliye $\sigma =E\kappa y$. Stress isliye $y$ mein bhi ek straight line hai.

Ab moment $M$ sirf un saari chhoti stresses ka total turning effect hai neutral axis ke baare mein: height $y$ par area $\mathrm{d}A$ ki har thin slice force $\sigma \mathrm{d}A$ carry karti hai aur ek lever arm $y$, isliye

$M=\int y\sigma \mathrm{d}A=\int y(E\kappa y)\mathrm{d}A=E\kappa \int y^2\mathrm{d}A.$

Woh bacha hua integral $\int y^2\mathrm{d}A$ exactly woh "distance-squared weighting" hai jo hum §4 mein milte hain — yeh $I$ hai. Toh $M=E\kappa I$ hai, jo rearrange hota hai $E\kappa =M/I$. $\sigma =E\kappa y$ mein wapas substitute karne par law milta hai:

4. Second moment of area $I$ — "material kitna spread out hai?"

Upar ki derivation se nikla integral $\int y^2\mathrm{d}A$ ka ek naam aur ek picture hai. Yeh is topic ki sabse important quantity hai, isliye hum ise slowly build karte hain.

Rectangle formula aur uska $1/12$ kahan se aata hai

Width $b$ aur kuch thickness ${\tau }_{\text{rect}}$ ka ek solid rectangle lo (ek generic thickness — "whatever height that rectangle happens to have," nahi face thickness $t_f$). Iska neutral axis beech se guzarta hai, isliye $y$, $-{\tau }_{\text{rect}}/2$ se $+{\tau }_{\text{rect}}/2$ tak jaata hai. Height $y$ par ek horizontal strip ka area $\mathrm{d}A=b\mathrm{d}y$ hai, isliye:

$I_{\text{rect}}={\int }_{-\tau /2}^{+\tau /2}{y}^2(b\mathrm{d}y)=b{\left[\frac{y^3}{3}\right]}_{-\tau /2}^{+\tau /2}=\frac{b}{3}\left(\frac{{\tau }^3}{8}+\frac{{\tau }^3}{8}\right)=\frac{b{\tau }^3}{12}.$

Jab hum ise ek face sheet par apply karte hain toh ${\tau }_{\text{rect}}=t_f$ substitute karte hain; jab hum ise poori solid plate par apply karte hain toh plate ki apni thickness substitute karte hain. Letter ek placeholder hai, koi nayi physical quantity nahi.

Parallel-axis theorem — aur tum $A{d}^2$ add kyun karte ho

Hum bahut kam kabhi material ko neutral axis par rakhte hain. Ek sandwich skin uss se door hoti hai. Ise handle karne ke liye hum ek shape ko axis se door shift karte hain:

Fig 3 padhna. Violet dashed line wapas neutral axis hai. Magenta bar usse oopar hai, ek akela face sheet, jiska apna centre navy dot se mark kiya gaya hai. Orange double arrow us dot se neutral axis tak ki doori $d$ hai — ek real skin ke liye yeh lagbhag $h/2$ hai.

Tum $A{d}^2$ add kyun karte ho. Shape ka apna centre height $d$ par rakho. Ek patch jo shape ke apne centre se measured height $y^{\prime }$ par baitha hai, woh ab neutral axis se $y=d+y^{\prime }$ height par baitha hai. Use definition mein daalo:

$I=\int (d+y^{\prime }{)}^2\mathrm{d}A=\int (d^2+2d{y}^{\prime }+y^{\prime 2})\mathrm{d}A=d^2\int \mathrm{d}A+2d\int y^{\prime }\mathrm{d}A+\int y^{\prime 2}\mathrm{d}A.$

Teen pieces padho: pehla total area $A$ times $d^2$ hai; middle integral $\int y^{\prime }\mathrm{d}A$ zero hai kyunki $y^{\prime }$ shape ke apne balance point (centroid) se measure kiya gaya hai; aakhri sirf shape ka apna $I_{\text{own}}$ hai. Toh cross-term vanish ho jaata hai aur tum ek clean sum ke saath bache ho:

Ek sandwich skin ke liye, $d\approx h/2$, isliye iska contribution $A{d}^2=b{t}_f(h/2{)}^2$ hai. Kyunki $t_f\ll h$, yeh outboard term skin ke apne tiny $b{t}_f^3/12$ ko utterly dwarf karta hai — exactly isliye parent note chhota term throw away karta hai.

5. Stiffness properties: $E$, $G$, aur stiffness $EI$

§3 ki derivation ne pehle hi $E$ borrow kar liya tha; yahan uski poori picture hai.

Subscripts batate hain ki ek symbol kaunse part ka hai. Do families seedha rakho:

• Material properties faces vs. core ki: $E_f$ (faces) aur $E_c,G_c$ (core).
• Geometry faces vs. core ki: $t_f$ (face thickness) aur $c$ (core thickness). Yeh lengths hain, material properties nahi — wohi $f$/$c$ subscript sirf batata hai kaunsa part measure kiya gaya hai.

Do alag moduli kyun? Kyunki dono kaam alag hain. Faces bending se stretch aur squeeze hoti hain → hume $E_f$ ki parwah hai. Core panel shear karne par sideways skew hota hai → hume $G_c$ ki parwah hai. $E$ wahan use karna jahan tumhe $G$ chahiye (ya vice versa) ek category error hai jiske baare mein topic warn karta hai.

Stiffness se sag tak: deflection $\delta$

Bada $EI$ matlab kam droop. Ek simply-supported beam (har end par ek support par rakha hua) ke liye uniform line load $w$ ke neeche, mid-span par peak sag hai:

Fig 4 padhna. Navy line unloaded straight beam hai. Magenta curve wohi beam hai orange downward arrows (line load $w$) ke neeche sag karti hui. Violet double arrow mid-span par $\delta$ mark karta hai. Caption yaad dilata hai ki $EI$ double karne par woh arrow half ho jaata hai — yeh sandwich ka poora promise hai, ek picture mein bana hua.

6. Density $\rho$, yield stress ${\sigma }_y$, aur "specific" idea

Space mein, mass hi paisa hai — har kilogram launch karna bahut mehnga padta hai. Isliye engineers "kitna stiff?" nahi poochte balki "per kilogram kitna stiff?" poochte hain.

Topic ko yeh kyun chahiye. Ek sandwich ek solid plate se per kilogram $5$–$10\times$ zyada stiff ho sakti hai, bhale hi solid plate per cubic metre zyada stiff ho. Winning number hamesha "per kilogram" wala hota hai, isliye $E/\rho$, ${\sigma }_y/\rho$ aur $\rho$ har jagah aate hain.

7. Shear, core stress, aur section modulus: $V$, $\tau$, $S$

Do last symbol families loop close karti hain.

$I$ ko $c_{\text{edge}}$ se divide karna "yeh kitna moment survive kar sakta hai" kyun answer karta hai. §3 se bending law rearrange karo: $\sigma =My/I$. Material pehli baar apni limit stress ${\sigma }_y$ most stressed point par reach karti hai, jo outermost fibre hai jahan $y=c_{\text{edge}}$ sabse badi hai. Wahan $\sigma ={\sigma }_y$ set karo aur moment ke liye solve karo: $M_{\max }={\sigma }_y\cdot \frac{I}{c_{\text{edge}}}={\sigma }_{y}S.$ Toh $S=I/c_{\text{edge}}$ exactly woh geometry factor hai jo ek material limit ${\sigma }_y$ ko ek survivable moment mein convert karta hai — bada $S$, yield se pehle zyada moment. Isliye yeh $I$ aur $c_{\text{edge}}$ ko ek number mein bundle karta hai.

Topic ko yeh kyun chahiye. Bending stiffness $EI$ batata hai panel kitna sag karta hai ($\delta$); core shear ${\tau }_c$ batata hai light filling tear hogi ya nahi; aur $S$ batata hai faces kaun sa moment survive kar sakti hain. Ek real design teeno pass karna chahiye — ek stiff panel jo weak core ke saath hai woh bhi fail hogi. (Shear failure aur modal effects Buckling and Instability aur Vibration and Modal Analysis se connect hote hain.)

Foundations kaise ek number mein chain hote hain

Neeche ka map causal hai, table of contents nahi: har arrow ek operation hai ("integrate", "multiply", "divide") jo usse pehle wali quantity ko uske baad wali quantity mein transform karta hai, do numbers par khatam hota hai jo ek designer actually report karta hai — sag $\delta$ aur core margin.

Ek path end to end follow karo: arc geometry straight-line strain deta hai, times $E$ stress deta hai, lever arms sum karna ($\int y^2\mathrm{d}A$) $I$ deta hai, times $E$ se $EI$ milta hai, aur sag formula $EI$ ko droop $\delta$ mein badal deta hai. Do side branches $V$ ko core-stress check mein aur ${\sigma }_y$ ko survivable moment mein turn karte hain. Teeno single verdict ko feed karte hain: pass ya fail.

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