matlab project assembling the global stiffness matrix

CE 890Introduction to Matlab

Matlab Project

Assembly of Global Stiffness Matrix

Submitted toProfessor Dr. Phanikumar S.

Submitted ByAqeel Ahmed

ManthaCivil & Environmental Engineering

PID 36846644

Introduction

In finite element method (& structural analysis approach), a structure

is modeled as an assembly of elements or components with various

forms of connection between them. A continuous discrete system is

modeled with a finite number of elements interconnected at finite

number of nodes. The behaviour of individual elements is

characterised by the element's stiffness or flexibility relation, which

altogether leads to the system's stiffness or flexibility relation. To

establish the element's stiffness or flexibility relation, further leading to

the global stiffness/flexibility matrix, MATLAB programming can be

effectively used.

In this project paper, stiffness matrix has been obtained using different

approaches for spring elements and then extended to bar and beam

elements. A general code has also been included that is capable of

reading from any text file the connectivity matrix and compute the

global stiffness matrix. Also, the knowledge of cells in Matlab has been

included in the codes which necessarily eased the work. All codes have

been developed for a defined problem in hand and results compared to

solutions for verification.

Assembling the Global Stiffness Matrix for Spring Elements

To develop the stiffness matrix, we take an example of two springs

connected together and a force P equal to 15 kN is applied to it. The

spring constants are k1 = 100 kN and k2 = 200 kN. The layout is as

follows:

Figure 1: Spring System for Two Elements

Solution

1. Approach to Solution

a. Step 1 . It involves discretization of the problem. The

domain consists of two springs/elements and connected at

nodes.

b. Step 2 . Elements need to have connectivity as follows:

c. Step 3

stiffness

matrix for

each

spring, we

have the stiffness’s of each spring. ( k1 = 100 kN, k2 =

200 kN). Calling the function SpringElementStiffness will

give us the 2x2 stiffness matrix for each spring. The details

are:

MATLAB Code

function y = SpringElementStiffness(k)

Element

Number

Node i Node j

1 1 2

2 2 3

% This Function claculates the element stiffness matrix for springs with spring stiffness as k. It returns 2x2 stiffness matirx

y = [k -k; -k k];

MATLAB Output

>> k1 = SpringElementStiffness(100)

k1 = 100 -100

-100 100


k2 = 200 -200

-200 200

d. Step 4 (Assembling the Global Stiffness Matrix for the

System). The system has three nodes; therefore the global

stiffness matrix will be 3x3 matrix. To obtain the K matrix,

first we setup the zero matrix of size 3x3 and then call the

Matlab function “SpringAssemble” to obtain the matrix.

The details are:

MATLAB Code

function y = SpringAssemble(K,k,i,j)

% This function will assemble the element stiffness matrix k at node i(left node) and j (right hand node) into global stiffness matrix K

K(i,i)=K(i,i)+k(1,1);K(i,j)=K(i,j)+k(1,2);K(j,i)=K(j,i)+k(2,1);K(j,j)=K(j,j)+k(2,2);

y = K;

MATLAB Output

>> K = zeros(3,3)

K =

0 0 0

0 0 0

0 0 0

>> K = SpringAssemble(K,k1,1,2)

K =

100 -100 0

-100 100 0

0 0 0


K =

100 -100 0

-100 300 -200

0 -200 200

The same approach is tested for a six spring system having different

connectivity of nodes. The details are:

Figure 2: Six-element Spring System

Solution

1. Step 1. The domain consists of six elements and five nodes.

The connectivity will be:

Element Number Node i Node j

1 1 3

2 3 4

3 3 5

4 3 5

5 3 4

6 4 2

2. Step 2. Each element has 2x2 stiffness matrix and since

there are five nodes, therefore, K (global) size will be 5x5. Each

element stiffness matrix will be obtained by plugging in the ‘k’ (spring

constant in kN) of respective spring. The out put is as follows:

>> k1= SpringElementStiffness(100)

k1 =

100 -100

-100 100


k2 =

200 -200

-200 200


k3 =

300 -300

-300 300


k4 =

400 -400

-400 400


k5 =

500 -500

-500 500


k6 =

600 -600

-600 600

>> K = zeros(5,5)

K =

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0


K =

100 0 -100 0 0

0 0 0 0 0

-100 0 100 0 0

0 0 0 0 0

0 0 0 0 0


K =

100 0 -100 0 0

0 0 0 0 0

-100 0 300 -200 0

0 0 -200 200 0

0 0 0 0 0


K =

100 0 -100 0 0

0 0 0 0 0

-100 0 600 -200 -300

0 0 -200 200 0

0 0 -300 0 300


K =

100 0 -100 0 0

0 0 0 0 0

-100 0 1000 -200 -700

0 0 -200 200 0

0 0 -700 0 700


K =

100 0 -100 0 0

0 0 0 0 0

-100 0 1500 -700 -700

0 0 -700 700 0

0 0 -700 0 700


K =

100 0 -100 0 0

0 600 0 -600 0

-100 0 1500 -700 -700

0 -600 -700 1300 0

0 0 -700 0 700

Stiffness Matrix for Bar Element

Dealing with bar elements involves 2 degree of freedom (dof) per node

(similar to springs). The problem in hand is to obtain the global

stiffness matrix of 4 bars connected with node connectivity as shown:

Figure 3: Bar Elements with Node Numbering

Solution

Approach - 1

The connectivity will be read through a text file and used in the main

program to obtain the global stiffness matrix. For the problem EA/L is

assumed to be constant. The connectivity is read from the text file

(Node1.txt) and can be varied for any number of elements. The code is

as follows:

MATLAB Code

clc, clear all elcon = load('Node1.txt') % To read the file regarding the connectivity of the elements [row, col] = size(elcon) % Arranging the data in matrix form % Creating the Stiffness Matrix of Zeros Stiffness = zeros(row + 1) % The size of K(global) is one plus the number of elements %**********************************************% Defining the Element Stiffness matrix% ********************************************* a = [1 -1; -1 1] % Assuming EA/L is constant % *********************************************% Assembly of Stiffness Matrix%**********************************************

for i=1:row m = elcon(i,2); n = elcon(i,3); Stiffness(m,m) = Stiffness(m,m) + a(1,1); Stiffness(m,n) = Stiffness(m,n) + a(1,2); Stiffness(n,m) = Stiffness(n,m) + a(2,1); Stiffness(n,n) = Stiffness(n,n) + a(2,2);endStiffness

MATLAB Output

elcon =

1 1 2

2 2 3

3 3 4

4 4 5

row =

4

col =

3

Stiffness =

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

a =

1 -1

-1 1

Stiffness =

1 -1 0 0 0

-1 2 -1 0 0

0 -1 2 -1 0

0 0 -1 2 -1

0 0 0 -1 1

Approach – 2

The same problem has been addressed by writing the code in a very

generalized form. This code requires the input of number of elements

and length (L) and computes the global stiffness matrix.

MATLAB Code

clc,clear all %**********************************************% Input Data%********************************************** numel = 4 % The number of elements numnodes = numel + 1 % Total number of nodesneq = numnodes connection = [1:numel; 2:numel+1]' % Take care of any number of elements % Location of nodesL = 1x = [0:numel]'/numel*L K = zeros(neq,neq) % The Assembly of the Global Stiffness Matrix for nel = 1:numel n1 = connection(nel,1); n2 = connection(nel,2); x1 = x(n1); x2 = x(n2); ke = [1 -1;-1 1]; % Assembly of element matrix into Global K Matrix K([n1,n2],[n1,n2])=K([n1,n2],[n1,n2])+ke;end K

MATLAB Output

numel =

4

numnodes =

5

neq =

5

connection =

1 2

2 3

3 4

4 5

L =

1

x =

0

0.2500

0.5000

0.7500

1.0000

K =

1 -1 0 0 0

-1 2 -1 0 0

0 -1 2 -1 0

0 0 -1 2 -1

0 0 0 -1 1

Approach 3

Another approach to obtain the stiffness matrix is using the cell array.

The same has been done using following MATLAB Code

MATLAB Code

clc, clear all a = [1 -1;-1 1] % Input the connectivity of the nodes of elementsb1 = [1 2]b2 = [2 3]b3 = [3 4]b4 = [4 5] % Assigning the connectivity to cellb = {b1,b2,b3,b4} K = zeros(5,5) for i = 1:4 for m = 1:2 for n = 1:2 K(b{i}(1,m),b{i}(1,n))=K(b{i}(1,m),b{i}(1,n)) + a(m,n) end endend K

Stiffness Matrix for Beams

The methodology can be developed for the beam elements using 2

degree of freedom per node. The element stiffness matrix will become

4x4 and accordingly the global stiffness matrix dimensions will change.

Consider a beam discretized into 3 elements (4 nodes per element) as

shown below:

1 2 3 4

1

2

3

4

5

6

7

8

1 2 3 4

1

2

3

4

5

6

7

8

Figure 4: Beam dicretized (4 nodes)

The global stiffness matrix will be 8x8. The MATLAB code to assemble

it using arbitrary element stiffness matrix (4x4) is as follows:

MATLAB Code

clc, clear all numel = 3nnodes = numel+1 dof = {[1 2 3 4],[3 4 5 6],[5 6 7 8]} K = zeros(nnodes*2) k = {rand(4), rand(4), rand(4)} % Assembling the Global Stiffness Matrix for i = 1:numel for m = 1:4 for n = 1:4 K(dof{i}(1,m),dof{i}(1,n))= K(dof{i}(1,m),dof{i}(1,n))+k{i}(m,n) end endend K

Conclusion

The global stiffness matrix can be assembled using different

techniques as described above. The approach to address the problem

has been improved as understanding of the MATLAB functions and

code writing improved. These can be easily extended to account for

the matrix multiplication to get nodal degree of freedom and nodal

forces/reactions as required.

matlab project assembling the global stiffness matrix

Documents

element stiffness matrix

spring stiffness

elements stiffness

2x2 stiffness matrix

connectivity matrix

3x3 matrix

systems stiffness

node j