FEM software — NASTRAN, ABAQUS (concepts and use): Foundations

3.6.20 · D1 · Physics › Spacecraft Structures & Systems Engineering › FEM software — NASTRAN, ABAQUS (concepts and use)

Parent note (FEM software — NASTRAN, ABAQUS (concepts and use)) mein jo kuch bhi hai woh kuch muthi bhar ideas par bana hai. Ye page har symbol aur word ko absolute zero se define karta hai, ek aisi order mein jahan har ek sirf pehle wale par depend karta hai.

1. Displacement — "ek point kitna move hua?"

Socho ki ek metal bracket par ek dot draw karo, phir bracket par push karo. Dot ek naye jagah slip kar jaata hai. Woh choti arrow jahan dot tha wahan se jahan woh ab hai tak ko displacement kehte hain.

Figure — FEM software — NASTRAN, ABAQUS (concepts and use)

Picture: figure mein red arrow — tail purani position par, head nayi jagah par.
Is topic ko iske liye kyun chahiye: displacement woh cheez hai jiske liye hum solve karte hain. Baaki sab kuch (stress, bracket crack hoga ya nahi) compute hota hai iss baat se ki points kaise move hue. Jab parent $u$ likhta hai $Ku = F$ mein, yahi hai.

Bold letter $u$ ka matlab hai ki yeh ek single number nahi balki numbers ki list hai — har direction ke liye ek number jisme point move kar sakta hai. Yeh seedha agli idea par le jaata hai.

2. Degrees of Freedom (DOF) — "kitne tareekon se move kar sakta hai?"

3D space mein ek single point teen tareekon se slide kar sakta hai: left–right ( $x$ ), forward–back ( $y$ ), up–down ( $z$ ). Move karne ka har ek independent tarika ek degree of freedom hai.

Symbol check

Bold

u

sare nodes ke saare DOF displacements ka ek column hai, ek ke upar ek stack kiya hua.

3. Nodes aur Elements — "smooth cheez ko pieces mein kaato"

Ek smooth continuous shape ko computer mein store nahi kar sakte. Isliye FEM structure ko ek grid se cover karta hai.

Picture: baayein smooth L-bracket, wahi bracket daayein red quad elements se covered. Mesh smooth shape ko approximate karta hai.
Is topic ko iske liye kyun chahiye: hum sirf ek simple shape ke liye physics equations likhna jaante hain. Isliye hum simple shapes solve karte hain aur jawab chipkate hain. Is chipkaane ko assembly kehte hain: element stiffness tables $k_{e}$ ko ek global $K$ mein stack kiya jaata hai shared nodes par contributions add karke (Section 7 dikhata hai $k_{e}$ kahan se aata hai). Yahi literally Finite Element mein "finite" hai — infinitely smooth material ki jagah finite pieces.

Thin walls ke liye quads kyun?

Ek thin bracket basically ek surface hai, isliye flat 4-node shell elements (CQUAD4) solid bricks se bahut kam DOF mein bending capture karte hain.

4. Force $F$ — "push"

Picture: ek node par push karta arrow; iski length force ka size hai.
Bold kyun? Wahi wajah $u$ ki tarah — har DOF ke liye ek force number, ek column mein stack kiya hua.

5. Stiffness $K$ — "push karna kitna mushkil hai?"

Ek soft spring push karo toh woh bahut zyada move karta hai; ek stiff spring push karo toh woh muskil se hilta hai. "Movement ki per unit force chahiye" ka ratio stiffness hai.

Picture: seedhi red line $F = k u$ . Iski slope $k$ hai. Steeper line = stiffer.
Number nahi, matrix kyun? Ek real structure mein bahut DOF hote hain, aur node 1 push karne se node 2 bhi move hota hai (woh connected hain). Isliye hume ek table $K$ chahiye jahan entry $K_{ij}$ kehti hai "DOF $j$ ke unit move se DOF $i$ par force." Woh table stiffness matrix hai, aur $F = k u$ ka many-spring version exactly $Ku = F$ hai.

6. Boundary conditions — "ise pin karo warna woh float kar jaayega"

Space mein freely floating structure ke paas push karne ke liye kuch nahi: ise koi force do aur poori cheez drift karti hai, isliye koi unique displacement exist nahi karta. Mathematically, raw $K$ singular hai (iska determinant zero hai — tum ise invert nahi kar sakte, aur $Ku = F$ ke infinitely many solutions hain).

Free-floating model unique

u

kyun nahi deta?

Kuch bhi pin nahi hone par, rigid-body drift koi bhi constant motion free mein add kar deta hai, isliye

K

singular hai aur solutions non-unique hain.

7. Stress $σ$ , Strain $ϵ$ , aur matrices $N, B, D$

Displacement batata hai ki points kaise move karte hain. Lekin yeh jaanne ke liye ki metal crack hoga ya nahi, tumhe jaanna hai ki iiske andar kya ho raha hai.

Ab hume inside quantities (strain, stress) ko nodal displacements se connect karna hai jo hum actually solve karte hain. Do matrices yeh kaam karte hain, aur use se pehle dono define hone chahiye.

Parent ke virtual-work integral mein $σ : δ ϵ$ mein colon kyun? Stress aur strain 2D tables (tensors) hain, aur " $:$ " ka matlab hai "matching entries multiply karo aur add karo" — yeh energy per volume produce karta hai.

Parent jo chain use karta hai woh hai: nodes move karo → strain milti hai → stress milta hai → margin check karo. $u_{e} B ϵ D σ$

8. Young's Modulus $E$ aur Density $ρ$ — "material facts"

$E$ aur $ρ$ dono kyun? $E$ (via $D$ ) stiffness $K$ banata hai; $ρ$ mass $M$ banata hai (agla). Stiffness bending resist karta hai; mass acceleration resist karta hai. Vibration dono ke beech ki ladaai hai.

9. Mass $M$ , Frequency $ω$ , aur Mode Shape $ϕ$ — "kaise vibrate karta hai"

Structure ko pluck karo aur woh special rates par wobble karta hai. FEM unhe dhundhta hai.

Formula Modal Eigenvalue Problem

$(K - ω^{2} M) ϕ = 0$ Yeh shape kyun? Ek free vibration ek wobble hai $u (t) = ϕ cos (ω t)$ . Structure ke liye Newton's law hai $M \ddot{u} = - Ku$ — padhte hain "mass × acceleration equals minus stiffness × displacement," minus isliye kyunki elastic force hamesha resting position ki taraf push back karta hai (ek restoring force). Wobble daalte hain, $\ddot{u} = - ω^{2} ϕ cos (ω t)$ , $¨$ ko $- ω^{2}$ mein badalta hai; do minus signs cancel hote hain aur upar box mein equation tak rearrange karte hain. Sirf special $ω$ kyun kaam karte hain: equation kehti hai ki matrix $(K - ω^{2} M)$ times ek nonzero vector $ϕ$ zero deta hai. Ek matrix nonzero vector ko zero mein squeeze kar sakta hai sirf agar woh singular ho, yaani $det (K - ω^{2} M) = 0.$ Yeh determinant condition $ω^{2}$ mein ek polynomial hai; iski roots discrete natural frequencies hain. Kisi bhi dusre $ω$ par matrix invertible hai aur sirf $ϕ = 0$ solution hai (koi vibration nahi).

Parent ke $f_{1} \approx 850$ Hz ke liye kyun matter karta hai: agar launch shaking mein 850 Hz par energy hai, bracket resonate karta hai aur toot sakta hai. Yeh Vibration and Modal Analysis aur Launch Vehicle Loads ka bridge hai.

Parent ke

f_{1}

ko convert karo

ω_{1} = 2 π f_{1} = 2 π (850) \approx 5341

rad/s.

Mode shapes sirf certain frequencies par kyun exist karte hain?

Kyunki

ϕ \neq = 0

(K - ω^{2} M)

singular chahiye, yaani

det (K - ω^{2} M) = 0

, jo sirf specific

ω

satisfy karte hain.

10. Linear vs Nonlinear — " $K$ fixed hai ya move kar raha hai?"

Nonlinearity ke teen flavours hain jinka parent use karta hai, aur har ek sirf " $K$ ab constant nahi hai" hai alag physical reason ke liye:

Material nonlinearity: ek certain stress se aage material proportionally spring back karna band kar deta hai (yielding/plasticity, composite damage), isliye load badhne par iski stiffness drop hoti hai.
Geometric nonlinearity: jab parts bahut zyada bend ya rotate karte hain (deployable booms), geometry khud itna change ho jaata hai ki stiffness current shape par depend karti hai.
Contact nonlinearity: do surfaces touch ya separate hoti hain (bolted joints), aur stiffness us instant jump karti hai jab woh contact mein aati hain.

Nonlinear mein iterate kyun?

Kyunki

F_{int}

(isliye

K

) answer

u

par depend karta hai, tum ek shot mein solve nahi kar sakte — guess karo, imbalance

R = F_{ext} - F_{int}

measure karo, correct karo, aur repeat karo jab tak

R \approx 0

11. Factor of Safety & Margin — "humne kitna over-build kiya?"

Factors of Safety aur Stress Analysis and Margins se connect hota hai — $u$ , $σ$ jaanne ke baad decision stage.

Ye foundations topic ko kaise feed karte hain

Equipment checklist

Daayein side cover karo aur reveal karne se pehle jawab do.

Arrow

u

physically kya represent karta hai?

Ek structure ka point rest se kitna door aur kis direction mein move hua.

Translation aur rotation ke saath ek shell-corner node mein kitne DOF hote hain?

6 (teen translations, teen rotations).

Node aur element mein kya difference hai?

Node ek labelled corner point hai; element ek simple shape hai jiske corners nodes hain.

F = k u

mein line ka slope kya hai?

Stiffness

k

K

ek single number ki jagah matrix kyun hai?

Kyunki ek DOF move karna connected DOF ko affect karta hai, isliye hume cross-couplings ki table

K_{ij}

chahiye.

K

ki do mathematical properties bolo aur kyun matter karti hain.

Symmetric (

K_{ij} = K_{j i}

) aur, constrained hone ke baad, positive definite — ek unique solution aur fast Cholesky-type solves deta hai.

Ku = F

solve karne se pehle boundary conditions kyun apply karni chahiye?

Pinned DOF ke bina,

K

singular hai (rigid-body drift) aur koi unique solution nahi hai.

Strain kya hai, aur iski units kya hain?

Fractional stretch (length change ÷ original length); dimensionless.

Poisson's ratio

ν

kya hai, aur ek typical metal value?

Sideways contraction strain ÷ lengthwise stretch strain; aluminium ke liye lagbhag 0.33.

Shape functions

N

kya karte hain?

Corner-node displacements blend karte hain har interior point par displacement dene ke liye (

u = N u_{e}

u_{e}

kya hai?

Sirf ek element ke nodal DOF ka chota vector, global

u

ka ek slice.

Strain–displacement matrix

B

kya karta hai?

Element nodal displacements ko strain mein badalta hai shape functions differentiate karke (

ϵ = B u_{e}

D

banane wale do material facts kya hain, aur

M

kya banata hai?

E

aur

ν

D

banate hain (isliye

K

); density

ρ

M

banata hai.

ω

ko ordinary frequency

f

mein convert karo.

f = ω / (2 π)

Nonzero mode shapes possible karne ki condition kya hai?

det (K - ω^{2} M) = 0

— matrix singular hona chahiye.

Nonlinearity ke teen flavours bolo.

Material (yielding/damage), geometric (large deformation), aur contact (surfaces touching/separating).

F_{ext}

F_{int}

, aur

R

mein distinguish karo.

F_{ext}

= applied load (

= F

);

F_{int}

= elastic push-back;

R = F_{ext} - F_{int}

= imbalance, zero tak drive kiya jaata hai.

Newton–Raphson derivative

K_{T}

kyun use karta hai?

Curved force line ka slope har jagah badalta hai; derivative agla step lene ke liye local slope deta hai.

FoS ke terms mein Margin of Safety likho.

MoS = FoS - 1 = allowable / applied - 1

1.0 ka failure index kya matlab hai?

Material exactly apne breaking point par hai.