Tutorial 17 – Directed Graphs
Problem Definition
It is intended to represent a two dimensional truss in matrix notation and
develop functions to calculate the following:
1. function to calculate the length of a member given its Id
2. function to calculate the projections along x- and y-axes and angle
made by a member with the x-axis
3. function to list the start and end nodes of a specified member
4. function to calculate the total length of all members
5. function to list the members meeting at a node
6. function to calculate the number of members meeting at each joint
Data Definition
Nodes of a truss have x and y coordinates. Interconnectivity of members is
expressed in terms of a connectivity matrix. The connectivity matrix represents
the numbers of the nodes connected by a given member.
Node Coordinates
Node x y
1 0.0 0.0
2 3.0 0.0
3 6.0 0.0
4 1.5 4.0
5 4.5 4.0
Member Connectivity
Member Node i Node j
1 1 2
2 2 3
3 4 5
4 1 4
5 2 4
6 2 5
Tutorial 17 – Directed Graphs | 47
1
2 3
4 5
1 2
3
4
5 6 7
3 m 3 m
4 m
7 3 5
This data can be input into two variables, xy and conn, as follows:
-->xy = [0 0; 3 0; 6 0; 1.5 4; 4.5 4]
-->conn = [1 2; 2 3; 4 5; 1 4; 2 4; 2 5; 3 5]
Before attempting to write the functions, let us try out the commands
interactively and verify our understanding of the steps. To display the graph of
the truss, we must first define the graph data structure with the help of the
make_graph() function and then use show_graph() function.
-->tail = conn(:,1)'; head = conn(:,2)';
-->[nodes, dummy] = size(xy);
-->g = make_graph('Truss', 1, nodes, tail, head);
-->g.nodes.graphics.x = xy(:,1)'*100;
-->g.nodes.graphics.y = xy(:,2)'*100;
-->show_graph(g);
The above commands produce the figure shown above. Let us now write
the functions to complete the various assigned tasks.
Function rows(x) returns the number of rows in a matrix x.
function [r] = rows(x)
[r,dummy] = size(x);
endfunction
Function get_nodes(id, conn) returns the numbers of the nodes n1 and
n2 at the ends of member with number id.
function [n1, n2] = get_nodes(id, conn)
n1 = conn(id, 1); // Column 1 of connectivity matrix stores start
node
n2 = conn(id, 2); // Column 2 of connectivity matrix stores end
node
endfunction
Function len(id, xy, conn) returns the length L of a member.
function [L] = len(id, xy, conn)
[n1, n2] = get_nodes(id, conn);
p1 = xy(n1,:); p2 = xy(n2,:); // Coordinates of nodes n1 and n2
dx = p2 - p1; // Difference of coordinates of n1 and n2
L =sqrt(sum(dx .^2)); // Formula for length from analytical
geometry
endfunction
Tutorial 17 – Directed Graphs | 48
Function projections(id, xy, conn) returns the x and y projections dx
and dy of a member as well as the angle theta made by the member with the xaxis.
function [dx, dy, theta] = projections(id, xy, conn)
[n1, n2]=get_nodes(id, conn);
p1 = xy(n1,:); p2 = xy(n2,:);
difference = p2 - p1;
dx = difference(1); // x-projection is in column 1 of diff
dy = difference(2); // y-projection is in column 2 of diff
if dx == 0 then
theta = %pi/2;
else
theta = atan(dy/dx);
end
endfunction
Function total_length(xy, conn) returns the total length of all members
in a truss.
function [tot_L]=total_length(xy, conn)
members = rows(conn);
for id=1:members
L(id)=len(id,xy,conn);
end
// disp(L);
tot_L = sum(L);
endfunction
Function members_at_node(nodeid, conn) returns the numbers of
members meeting at a node.
function [memlst] = members_at_node(nodeid, conn)
memlst = find( (conn(:,1)==nodeid) | (conn(:,2)==nodeid) );
endfunction
Function num_members_at_node(nodeid, conn) returns the number of
members meeting at node.
function [n] = num_members_at_node(nodeid, conn)
memlst = members_at_node(nodeid, conn);
n = length(memlst);
endfunction
Function make_truss(name, xy, conn) prepares the data structure to
represent a truss.
function [g] = make_truss(name, xy, conn, scale)
t = conn(:,1)'; // tail node numbers
h = conn(:,2)'; // head node numbers
g = make_graph(name, 1, rows(xy), t, h);
g.nodes.graphics.x = xy(:,1)' * scale;
g.nodes.graphics.y = xy(:,2)' * scale;
endfunction
Tutorial 17 – Directed Graphs | 49
Using the data previously entered, we can test the above functions as
shown below:
-->g = make_truss('Truss', xy, conn, 100);
-->show_graph(g)
The other functions can also be tested as follows:
-->tot_len = total_length(xy, conn); disp(tot_len)
26.088007
-->for i = 1:rows(xy)
--> n = num_members_at_node(i, conn);
--> mem_lst = members_at_node(i, conn);
--> disp([n mem_list])
-->end
2. 1. 4.
4. 1. 2. 5. 6.
2. 2 7.
3. 3 4. 5.
3. 6. 7.
We can also test the length and projection functions in a single for loop:
-->for i = 1:rows(conn)
--> L = len(i, xy, conn);
--> [dx, dy, theta] = projections(i, xy, conn);
--> disp([dx dy theta L])
-->end
3. 0. 0. 3.
3. 0. 0. 3.
3. 0. 0. 3.
1.5 4. 1.2120257 4.2720019
-1.5 4. -1.2120257 4.2720019
1.5 4. 1.2120257 4.2720019
-1.5 4. -1.2120257 4.2720019
The node numbers and member numbers can be displayed on the graph
using the View Options from the main menu of the graphics window. Graph
settings can be modified from the Graph Settings from the main menu.
Graph settings that can be changed are node diameter, edge width, border node
width, edge colour index etc.
T
