Structural design process — load cases, FOS (factor of safety): Foundations

3.6.2 · D1 · Physics › Spacecraft Structures & Systems Engineering › Structural design process — load cases, FOS (factor of safet

Ye Structural design process — load cases, FOS (factor of safety) ka toolbox page hai. Hum assume karte hain tumne kuch nahi dekha. Hum ek push se start karte hain.

1. Force — ek push ya pull, size aur direction ke saath

Ye picture: ek arrow draw karo. Arrow ki length size $F$ hai; jis taraf point karta hai wo direction hai. Poora arrow milake $F$ hai.

Topic ko iske kyun zaroorat hai: spacecraft par har load — rocket ka upar dhakka dena, vibration se bracket hilna, tank ka bahar push karna — ultimately forces ka ek collection hai. Agar hum arrow draw nahi kar sakte, toh hum uske against design nahi kar sakte.

2. Mass, gravity, aur number " $g$ "

Gravity se weight force ka size $F = m g$ hai. Ye pehli force hai jo object hamesha feel karta hai.

Figure — Structural design process — load cases, FOS (factor of safety)

Figure s01 — Alt text: a green mass block with two downward arrows. A short lavender arrow labelled "1g = m·9.81" shows normal gravity; a long coral arrow labelled "9g = m·9.81·9" shows the launch acceleration as nine times longer. The picture defines a "g-load" as a multiple of gravity's pull.

3. Forces add karna — general vectors, aur perpendicular shortcut

Kabhi kabhi do forces ek saath alag alag directions mein ek hi part pe act karti hain. Forces arrows hain ( $F$ ), isliye ye waise add hoti hain jaise arrows add hote hain: tip-to-tail, sirf unke sizes add karke nahi.

Size formula kahan se aati hai? (ek picture, koi magic nahi.) $F_{1}$ ko zameen ke saath flat rakho aur $F_{2}$ ko angle $θ$ par uski tip pe khada karo. $F_{2}$ ki tip ko seedha ground line par giraa do: ye $F_{2}$ ko ek flat part jiska length $F_{2} cos θ$ hai ( $F_{1}$ ke along) aur ek upright part jiska length $F_{2} sin θ$ hai ( $F_{1}$ ke right angles par) mein split karta hai. Ab resultant ek right triangle ka lamba slanted arrow hai jiska horizontal leg $F_{1} + F_{2} cos θ$ aur vertical leg $F_{2} sin θ$ hai. Us right triangle par plain Pythagoras apply karo:

$F^{2} = (F_{1} + F_{2} cos θ)^{2} + (F_{2} sin θ)^{2} .$

Multiply out karo aur identity $cos^{2} θ + sin^{2} θ = 1$ use karo: $F_{2}^{2} cos^{2} θ$ aur $F_{2}^{2} sin^{2} θ$ ek single $F_{2}^{2}$ mein collapse ho jaate hain, chhod ke

$F^{2} = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} cos θ .$

Toh "law of cosines" kuch nahi hai bas Pythagoras jo tilted arrow ko flat aur upright parts mein split karne ke baad apply kiya gaya hai — figure dekho.

Figure s02 — Alt text: tip-to-tail construction. A lavender arrow $F_{1}$ lies flat; a mint arrow $F_{2}$ stands on its tip at angle $θ$ and is split by a dashed line into a flat leg $F_{2} cos θ$ and an upright leg $F_{2} sin θ$ . The coral resultant $F$ is the hypotenuse of the right triangle with legs $F_{1} + F_{2} cos θ$ and $F_{2} sin θ$ — the geometric source of the law of cosines.

Ab teen special cases padhte hain:

Same line, same way ( $θ = 0^{\circ}$ , so $cos θ = 1$ ): $F = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} = (F_{1} + F_{2})^{2} = F_{1} + F_{2}$ — bas add kar do.
Opposite ways ( $θ = 18 0^{\circ}$ , $cos θ = - 1$ ): $F = (F_{1} - F_{2})^{2} = ∣ F_{1} - F_{2} ∣$ — subtract kar do.
Right angles ( $θ = 9 0^{\circ}$ , $cos θ = 0$ ): middle term vanish ho jaata hai aur hume clean RSS rule milta hai neeche.

Topic ko iske kyun zaroorat hai: axial ( $9 g$ ) aur lateral ( $4 g$ ) launch loads simultaneously aur right angles par hote hain, isliye RSS unke liye exact hai. Unhe plain numbers ke roop mein add karna reality se bada load fake karta — aur mass space mein dushman hai. Quasi-static Loads and Launch Environment dekho.

Figure s03 — Alt text: a right triangle of forces. The lavender horizontal leg is the axial force $F_{a} = 4415$ N, the mint vertical leg is the lateral force $F_{l} = 1962$ N, and the coral hypotenuse is the resultant $F = F_{a}^{2} + F_{l}^{2} = 4832$ N. Shows why perpendicular loads combine by Pythagoras, giving less than the scalar sum.

4. Stress — ek force ko area par spread karna (aur uska sign)

Sirf force se ye nahi pata chalta ki part toot jaayega ya nahi. 20 kN pull ek patli wire ko snap kar deti hai lekin ek moti bar ko barely stretch karti hai. Jo matter karta hai wo hai force kitni concentrated hai.

Ye picture: imagine karo wahi pull ek moti rope se aur ek patli thread se carry ki ja rahi hai. Same force $F$ , lekin thread ka small $A$ hai, isliye $σ = F / A$ bahut bada hai — ye snap ho jaata hai. Stress force concentrated hai.

Figure s04 — Alt text: two bars side by side. Left, a lavender bar pulled by two outward coral arrows labelled "σ > 0 (tension)". Right, a butter-yellow bar squeezed by two inward slate arrows labelled "σ < 0 (compression)". Defines the sign convention of stress by picture: pulling is positive, squashing is negative.

5. Strength — yield aur ultimate, ek material ki do limits

Material tabhi tak fight karta hai ek point tak. Do important points hain, ek nahi.

Ye picture: ek metal bar ko stretch karo aur ek stress-vs-stretch graph dekho. Ye ek straight line ki tarah upar jaata hai (springy), phir yield par bend ho jaata hai, upar climb karta rehta hai, aur finally ek peak pe pahunchta hai — ultimate — jahan ye snap ho jaata hai. Do alag heights, do alag failures. Full detail Stress and Strain — Yield vs Ultimate Strength mein hai.

Figure s05 — Alt text: a stress-vs-strain curve in lavender. It rises straight, bends over at a coral dot marked "yield σ_y = 300 MPa (bends permanently)", climbs further, and peaks at a mint dot marked "ultimate σ_ult (ruptures)". Shows the two distinct failure heights of a metal.

6. Force vs stress — ek letter "S", aur convert kaise karte hain

Parent note design quantities ke liye symbol $S$ use karta hai — $S_{allow}$ , $S_{limit}$ , $S_{design}$ . Ye crucial rule hai jo tumhe trouble se door rakhta hai.

7. Limit load — worst push jo hum expect karte hain

Ye picture: Lego-bridge story ki taraf wapas — ye toy car ka weight kitna hai uska tumhara honest guess hai. Real, lekin ek guess.

Topic ko iske kyun zaroorat hai: ye wo number hai jise hum safety factor se multiply karenge. Har cheez downstream is ek "worst expected" value par build hoti hai.

8. 1 se bada ek ratio — factor of safety, aur hum demand kyun multiply karte hain

Ratio kyun, added amount nahi? Ek fixed added margin (say "+50 N") scale jaane bina kuch nahi kehta — 50 N ek wire ke liye huge hai, beam ke liye trivial. Ek ratio automatically load ke saath scale hota hai, isliye ek number ( $1.25$ ) har jagah kaam karta hai.

9. Margin of safety — "kitna spare hai?" number

Ab hamare paas supply ( $S_{allow}$ ) aur inflated demand ( $F O S \cdot S_{limit}$ ) dono same basis aur units par hain. Final question: kya supply kaafi hai, aur kitna bacha hua hai?

Foundations topic ko kaise feed karte hain

Equipment checklist

Bold

F

ka matlab kya hai versus plain

F

Bold

F

poora force arrow hai (size aur direction); plain

F = ∣ F ∣

sirf uska size hai, ek positive number.

9 g

" jo mass

m

par act kar raha hai use real force mein kaise convert karte hain?

F = m g \times 9

, with

g = 9.81 m/s^{2}

Kisi bhi angle par do forces add karne ka general rule?

Tip-to-tail add karo; resultant size

F = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} cos θ

hai, jo bas Pythagoras hai tilted arrow ko flat aur upright parts mein split karne ke baad.

Right angles par do forces — unhe kaise combine karte hain?

RSS:

F = F_{a}^{2} + F_{l}^{2}

(

cos 9 0^{\circ}

term zero hai), jahan

F_{a}

axial size hai aur

F_{l}

lateral size hai.

Stress kya hai aur uska unit kya hai?

Force per area,

σ = F / A

; pascals mein measure hoti hai, usually MPa

= N/mm^{2}

σ

ke sign ka matlab kya hai?

Positive = tension (pull apart ho raha hai); negative = compression (squash ho raha hai).

Crushing ke alawa, ek compressed part aur kaise fail ho sakta hai?

Buckling se — ek slender column sideways bow hota hai aur material ki compressive strength se well below stress par collapse hota hai; ye ek shape failure hai, material ki nahi.

Yield aur ultimate strength mein difference?

Yield = wo stress jahan permanently bend ho jaata hai; ultimate = zyada stress jahan rupture ho jaata hai.

Kya tum directly ek force ko ek stress se compare kar sakte ho?

Nahi — dono sides same basis aur unit ki honi chahiye; pehle

σ = F / A

(ya

F = σ A

) se convert karo.

Limit load kya hai?

Real mission mein expected worst load (demand ka best estimate), force ke roop mein ya stress ke roop mein jo ye produce karta hai.

FOS kya multiply karta hai, aur kyun?

Applied/limit load (demand), kyunki hum loads ke baare mein uncertain hain — hum push inflate karte hain, metal ko nahi.

Margin of safety likho aur uski pass condition.

M S = \frac{S _{allow}}{F O S \cdot S _{limit}} - 1

; passes when

M S \geq 0

Margin formula mein "

- 1

" kyun?

Taaki inflated demand ke exactly barabar supply

M S = 0

de — zero-spare pass/fail line.