FEM for structures — assembling global stiffness: Foundations

3.6.19 · D1 · Physics › Spacecraft Structures & Systems Engineering › FEM for structures — assembling global stiffness

Parent note Assembling Global Stiffness padhne se pehle, tumhe usme har ek squiggle bina rukke padhna aana chahiye. Yeh page har symbol ko bilkul zero se introduce karta hai: uska matlab plain words mein, uski picture, aur kyun yeh topic uske bina kaam nahi kar sakta. Upar se neeche padho — har idea upar wale par lean karta hai.

1. Ek structure springs ka network hai

Figure — FEM for structures — assembling global stiffness

Figure dekho. Left mein, ek single spring: right end ko $u$ (metres mein) kheeencho, aur woh $F$ (newtons mein) force se kheench ke wapas aata hai. Graph par straight line kehti hai $F = k u$ . Slope $k$ stiffness hai. Right mein, teen springs end to end join hain — yeh ek finite element mesh ka seed hai.

Yahi poori idea hai Element Stiffness Matrices aur Finite Element Method Overview ke peechhe: ek continuous solid ko inn chhoti stiffnesses ke network se replace karo.

2. Displacement $u$ aur Degrees of Freedom (DOF)

Lekin space mein ek point ek se zyada tarike se move kar sakta hai. Yahi ek degree of freedom capture karta hai.

Figure teeno cases ko side by side dikhata hai. Chhote arrows DOFs hain — har arrow ek number hai jise solver ko find karna hai.

Symbol $u_{x}$ ka matlab hai ::: $x$ -direction mein ek node ka displacement.
Symbol $θ_{z}$ ka matlab hai ::: $z$ -axis ke baare mein ek node ka rotation.

3. Vectors aur matrices — bookkeeping

Hamare paas bahut saare DOFs hain, isliye hum unke numbers ko lists aur grids mein stack karte hain.

Matrix × vector kaise kaam karta hai (tumhe yeh padhna aana chahiye): $K$ ki row $i$ ko $u$ se entry-by-entry multiply karke sum karo toh output ki entry $i$ milti hai:

$F_{i} = \sum_{j} K_{ij} u_{j}$

$K_{ij}$ plain words mein ::: DOF $j$ ke unit displacement se DOF $i$ par aane wala force.

4. Local vs global — do coordinate worlds

Har element (har chhota spring/bar) ko apni axis ke saath describe karna sabse aasaan hai. Lekin poori structure ek shared frame mein rehti hai. Do vocabularies:

Local frame se global frame mein translate karne ka kaam Coordinate Transformations in FEM handle karta hai; $k^{(e)}$ ke andar chhote numbers $E$ , $A$ , $L$ Element Stiffness Matrices se aate hain.

5. Material symbols: $E$ , $A$ , $L$ , aur $\frac{E A}{L}$

Yeh teen letters set karte hain ki ek single bar kitna stiff hai.

Figure dikhata hai ki $\frac{E A}{L}$ kaise respond karta hai: mota $\Rightarrow$ steeper line (stiffer), lamba $\Rightarrow$ flatter line (softer).

6. Connectivity — kaunsa element kaunse node ko touch karta hai

Figure mein, element 1 global nodes 1 aur 2 ko connect karta hai; element 2 nodes 2 aur 3 ko connect karta hai. Node 2 shared hai. Yahi sharing woh wajah hai jiske liye hum contributions add karte hain — aur kyun parent ke worked example mein $K_{22}$ doubled niklata hai.

$L^{(e)}$ ka role ::: element $e$ ke global DOFs select karna (0s aur 1s ki table).

7. Scatter-add aur properties words

Teen final vocabulary items jinka parent use karta hai:

"Sparse" ka matlab ::: almost har matrix entry zero hai.
Ek rigid-body mode ::: ek aisi motion jo poori structure ko bina kisi element ko stretch kiye move kare (zero strain energy).

Prerequisite map

Har box ek cheez hai jo is page ne define ki; arrows dikhate hain kya kya feed karta hai. Sabse neeche wala box parent topic hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.

Stiffness $k$ ka matlab ::: force returned per unit stretch, units N/m.
DOF ka matlab ::: ek independent tarika jisme ek node move (ya rotate) kar sakta hai.
Ek free 3D node ke kitne DOFs hote hain ::: 6 (teen translations, teen rotations).
$u$ aur $F$ mein kya hota hai ::: mesh mein har DOF ke liye stacked displacements aur stacked forces.
$Ku = F$ kya kehta hai ::: poori structure ke liye $F = k u$ ka multi-spring version.
$K_{ij}$ mein subscript ka matlab ::: row $i$ , column $j$ — DOF $j$ ke unit displacement se DOF $i$ par force.
$\sum$ symbol ka matlab ::: listed terms ko add up karo.
$k^{(e)}$ mein $(e)$ ka matlab ::: "element $e$ se belong karta hai", label hai power nahi.
Local aur global coordinates mein fark ::: local = element ki apni axis ke saath; global = poori structure ka ek shared frame.
$E$ , $A$ , $L$ kya represent karte hain ::: Young's modulus, cross-sectional area, length.
Single bar ki stiffness ki value ::: $E A / L$ .
Connectivity kya batata hai ::: har element kaunse global nodes ko join karta hai.
Scatter-add kya karta hai ::: har element ki chhoti entries ko sahi global DOF slots par global matrix mein add karta hai.
Kyun $K_{22}$ doubled niklata hai ::: node 2 do elements se shared hai, isliye dono contribute karte hain aur hum sum karte hain.
"Sparse" ka matlab ::: almost har entry zero hai.
Positive semi-definite physically matlab ::: strain energy kabhi negative nahi; zero sirf rigid-body motion ke liye.