Finite element method — nodes, elements, stiffness matrix: Foundations

3.6.18 · D1 · Physics › Spacecraft Structures & Systems Engineering › Finite element method — nodes, elements, stiffness matrix

Is page par kuch bhi assume nahi kiya gaya hai. Isse pehle ki tum parent topic padh sako, tumhe har ek symbol dekh ke turant uske peeche ka picture samajhna aana chahiye. Isliye hum har ek symbol build karte hain — "displacement" kya hota hai, yahan se lekar ek poori matrix equation ka matlab kya hota hai, tak.

0. Sabse pehla picture: ek spring

FEM ki har cheez ek physical fact par tiki hai jo tum rubber band se pehle se jaante ho: zyada kheencho, zyada stretch hogi, aur dono proportional hain.

Figure — Finite element method — nodes, elements, stiffness matrix

Relationship is page ki sabse important equation hai:

FEM ko kyun parwah hai? Kyunki har element ko ek spring ki tarah treat kiya jaata hai, aur poora method sirf yahi equation baar baar repeat aur stack karna hai. Is picture ko pakde rakho — baaki sab bookkeeping hai.

1. Ek node — woh point jis par hum nazar rakhte hain

Picture: ek seedhi bar imagine karo. Hum har atom ki motion record nahi kar sakte (infinitely many hain). Isliye hum kuch dots — nodes — lagate hain aur sirf poochhte hain "yeh dot kitna door gaya?" Dots ke beech mein hum baad mein guess karenge.

Topic ko kyun chahiye: nodes ek infinite problem ("har point kahaan gaya?") ko numbers ki ek finite list mein badal dete hain ("yeh 3 dots kahaan gaye?"). Forces nodes par apply hoti hain, supports nodes par fix hote hain, aur jo unknowns hum solve karte hain woh nodes par rehte hain.

2. Degrees of freedom (DOF) — wo tarike jisme ek point move kar sakta hai

Picture: flat 2D mein, ek dot left-right slide kar sakta hai ( $u_{x}$ ) aur up-down ( $u_{y}$ ).

Ek node ke kitne DOF hain yeh element type par depend karta hai — yeh ek common trap hai, isliye hum clearly batate hain:

Topic ko kyun chahiye: problem ka total size (number of nodes) × (DOF per node) hai. Jo simple 1D bar hum baad mein derive karenge woh sirf 1 DOF per node use karta hai (sirf axial slide). Isliye parent topic kehta hai ki global matrix 1D bar model ke liye $N \times N$ hai lekin 3D beam/shell model ke liye $6 N \times 6 N$ hai.

3. Strain $ϵ$ — stretch per unit length

Hume "material kitna stretched hai?" yeh kehne ka ek aisa tarika chahiye jo bar ki length par depend na kare. 1 cm ki bar ke liye 1 mm stretch bahut bada hai aur 1 m ki bar ke liye trivial. Toh hum divide karte hain.

Picture: ek bar par 1 metre door do dots painted hain. Kheencho, aur woh 1.01 m door ho jaate hain. Original 1 m par extra 0.01 m se $ϵ = 0.01$ milta hai.

Topic ko kyun chahiye: material behaviour strain par depend karta hai, raw stretch par nahi. Yeh woh quantity hai jisme materials ke liye Hooke's law likha jaata hai.

(Stress aur strain par deep dive ke liye, dekho 3.6.1-Stress-and-strain-in-structural-members.)

4. Stress $σ$ — force spread over area

Picture: apne thumb se press karo versus same force se needle se press karo — needle dard karta hai kyunki force ek tiny area mein squeeze hoti hai. Chhota area, bada stress.

$F$ — force (newtons), §0 ka push/pull.
$A$ — cross-sectional area (square metres): woh face kitna bada hai jis par force push karti hai. Ek bar ke liye, yeh woh area hai jo tum seedha kaat ke dekhte.

Topic ko kyun chahiye: ek real material ke andar internal forces stress se describe hoti hain, aur jo final answer engineers chahte hain ("kya yeh tutega?") woh ek stress ko material ki limit se compare karna hai.

5. Young's modulus $E$ — material ki apni stiffness

Picture: steel aur rubber. Dono ko same fraction se stretch karo (same $ϵ$ ); steel enormous stress se fight karta hai (bada $E$ ), rubber barely resist karta hai (tiny $E$ ).

6. Length $L$ aur position $x$ — element ke along measure karna

$L$ — ek element ki length (metres): woh chhota piece kitna lamba hai.
$x$ — element ke along ek position (metres): ek ruler reading node 1 par $x = 0$ se node 2 par $x = L$ tak.

Topic ko kyun chahiye: hum chahte hain ki displacement ek function ho jahan tum ho, likha $u (x)$ — "position $x$ par displacement." Woh notation ( $u$ of $x$ ) matlab hai "ek location daalo, woh spot kitna door gaya wapas milega."

7. Nodal displacements $d_{1}, d_{2}$ — woh numbers jo hum actually solve karte hain

Kuch bhi blend karne se pehle, hume har node ke displacement ke naam chahiye.

Picture: node picture (§1) mein, har red dot shifted end hoti hai; $d_{1}$ pehle dot ke liye arrow-length hai, $d_{2}$ doosre ke liye. Yeh woh unknowns hain jinhein find karne ke liye poori FEM machine exist karti hai. Baaki sab kuch — field $u (x)$ , strain, stress — $d_{1}$ aur $d_{2}$ jaanne ke baad unse compute hota hai.

8. Shape functions $N_{1}, N_{2}$ — nodes ke beech smart blending

Hume sirf nodal displacements $d_{1}, d_{2}$ pata hain. Beech wale points ka kya? Hum blend karte hain — aur shape functions blending weights hain.

Displacement kahin bhi do nodal values ka weighted blend hai: $u (x) = N_{1} (x) d_{1} + N_{2} (x) d_{2}$

$L$ length ki 2-node bar ke liye, woh do straight-line weights jo apne node par $1$ aur doosre par $0$ hain: $N_{1} (x) = 1 - \frac{x}{L}, N_{2} (x) = \frac{x}{L}$

Picture padhho: $x = 0$ par (node 1), $N_{1} = 1$ aur $N_{2} = 0$ — node 1 ka poora say hai. $x = L$ par (node 2), yeh flip ho jaata hai. Aadhe raaste par, dono aadha aadha contribute karte hain.

Topic ko kyun chahiye: shape functions woh "interpolation magic" hai jo node numbers ki ek mutthi ( $d_{1}, d_{2}$ ) ko ek continuous displacement field mein badal deta hai — finite data se real, smooth structure tak bridge.

9. Derivative $\frac{d u}{d x}$ — woh tool jo strain read karta hai

Yahan ek naya mathematical tool aata hai, aur hume bolna chahiye kyun yeh tool aur koi nahi.

Hume strain chahiye, aur strain hai stretch per length (§3). Ek aisi field ke liye jo position ke saath vary karti hai, "stretch per length" ek point par exactly woh hai ki $x$ ke saath step karne par $u$ kitni tezi se change hoti hai — displacement graph ki slope. Woh tool jo ek function ki slope measure karta hai woh derivative hai.

Kyun yeh tool: koi aur operation "change per unit length" nahi deta. Yeh strain ki literal definition hai. Toh jis moment hum displacement ko ek smooth function likhte hain, derivative hum par automatically force ho jaata hai strain extract karne ke natural tarike ke roop mein.

Kyunki hamara $u (x)$ ek seedhi line hai (linear shape functions), iska slope constant hai: $ϵ = \frac{d u}{d x} = \frac{d _{2} - d _{1}}{L}$

Ise padhho: total stretch $(d_{2} - d_{1})$ divided by length $L$ — exactly §3 ki strain definition, ab automatically recover ho gayi. Sign check: agar $d_{2} > d_{1}$ toh door wala node zyada bahar chala gaya, bar lamba ho gaya, toh $ϵ > 0$ (tension) — §0 ke consistent. Constant slope matlab har element ke andar constant strain, isliye FEM ko ek curving stress follow karne ke liye bahut saare chhote elements chahiye.

(Derivatives aur computers inka aur injaisi equations kaise handle karte hain, yeh 2.4.8-Numerical-methods-for-differential-equations ka subject hai.)

10. Vectors, matrices, aur $Kd = F$

Jab bahut saare nodes hote hain toh bahut saare displacements aur bahut saari forces hoti hain. Hum inhe lists mein stack karte hain aur stiffnesses ko ek grid mein organise karte hain.

Topic ko kyun chahiye: ek spacecraft mein hazaaron nodes hote hain. Hazaaron alag spring equations likhna hopeless hai; ek matrix equation sabko hold karta hai aur computer use ek hi baar mein solve kar leta hai.

11. Boundary conditions, loads & moments — pinning aur pushing

Ek load ek force ya moment hai jo hum apply karte hain, ek node par rakha jaata hai (§0 ka orange arrow, ab ek real structure par).
Ek boundary condition (BC) ek DOF fix karta hai: "yeh node move nahi kar sakta." Yeh us displacement ko zero set kar deta hai aur use unknowns se hata deta hai.

Picture: ek bolt jo ek bracket ko spacecraft wall se pakde hua hai — us node ka displacement zero par nail ho jaata hai. Jab tak enough BCs nahi hain jo saari rigid drifting rok sakein, $K$ singular rehti hai (§10) aur kuch solve nahi ho sakta.

Prerequisite map

Legend: Spring = Hooke spring $F = k d$ · Stiffness = $k$ · Stress = $σ$ · Strain = $ϵ$ · YoungE = Young's modulus $E$ · MatLaw = material law $σ = E ϵ$ · Nodal-d = nodal displacements $d_{i}$ · Shape-N = shape functions · K-elem = element stiffness matrix · Master = $Kd = F$ .

Equipment checklist

Right side cover karo aur khud test karo. Agar koi line tumhe atkaaye, uska section dubara padho.

F = k d

plain words mein kya kehta hai?

Force equals stiffness times displacement — zyada kheencho, ek stiffer spring zyada stretch hogi, aur dono same sign share karte hain.

Tension vs compression ke liye hum kaunsa sign convention use karte hain?

Positive = tension (stretch / chosen positive direction mein bahar kheenchna); negative = compression (squash).

Node kya hota hai?

Structure mein ek marked point jahan hum motion track karte hain aur forces/supports apply karte hain.

Ek node ke kitne DOF hote hain, aur yeh element type par kaise depend karta hai?

Solid elements: sirf translations (3D mein 3). Beam/shell elements: translations plus rotations (3D mein 6). Ek 1D bar node: 1 (axial slide).

Strain

ϵ

define karo aur uske units batao.

Change in length over original length; dimensionless; positive matlab stretched.

Stress

σ

define karo aur uske units batao.

Force over area; pascals (

Pa

); positive matlab tensile.

Young's modulus

E

kya measure karta hai, aur

σ = E ϵ

ke neeche kaunsa assumption hai?

Material ki stiffness (stress per unit strain); sirf small-strain, linear-elastic (reversible, non-yielding) conditions ke under valid.

Nodal displacement

d_{i}

kya hai?

Node

i

par displacement value — har node per DOF ek unknown number, woh cheez jise FEM solve karta hai.

Hum ek 2-node element ke liye linear (straight-line) shape functions kyun choose karte hain?

Do known nodal values exactly ek seedhi line fix karte hain; yeh unique simplest interpolation hai aur ek end-loaded bar ke liye exact hai.

Hum yahan derivative

d u / d x

kyun use karte hain?

Kyunki strain displacement-mein-change per length hai, aur derivative exactly

u

x

ke saath slope (rate of change) hai.

Matrix entry

K_{ij}

ka kya matlab hai?

Woh force jo DOF

i

par appear hoti hai jab DOF

j

ek unit se displace hota hai.

Moment

M

kya hai, uski unit aur sign kya hai?

Ek twisting load jo ek node ko rotate karta hai; newton-metres (

N\cdotm

); counter-clockwise positive.

Boundary conditions se pehle

K

singular kyun hoti hai?

Structure bina kisi strain ke rigidly drift ya spin kar sakta hai, infinitely many solutions deta hai jab tak DOFs fix na ho jaayein.