Linear System of Equations.

[justastupidperson]

This is my first post here so im sorry if i do anything wrong...
Okay here goes. i have no idea how to answer this. im very sorry for troubling you guys and im very grateful to anyone who would answer this... thanks. )

The instruction is to set up a matrix then solve the linear equation using Gauss-Jordan Elimination Method and Gaussian Elimination.
i only need help in setting up the matrix or getting the equations for the matrix. i try to implement Gauss-Jordan and Gaussian elimination on my own... this is kind of a long one. thank you!!!

Here's the problem:

The Republic United Steel Truss (RUST) division has already began manufacturing steel beams
for a customer Monolithic Engineering Steel Structures (MESS). They have delivered some
bundles of the beams to MESS, which were used in the construction of the Unduly Gigantic
Hanger (UGH).


MESS is suing RUST's management because several UGHs shown signs of severe fatigue. The
complaint claims that the beams produced by RUST are at faults. RUST maintains that the beams
were made to proper specification and believes that the signs seen were due to the fact that MESS
constructed UGHs beyond the written specification.


According to the complaints filed by MESS, the UGH is constructed with 27 beams connected at 11
joints. MESS has specified that each beam is required to withstand extension forces of up to 20 tons
and compressive forces of up to 17 tons without fatigue failure.


The specifications for the UGH are contained in the following tables. The first table contains the
coordinates of the joints. For example, joint 8 is at point (7.20, 0.00, 5.54) relative to the origin,
which is joint 1.

Joint	x	y	z
1	0.00	0.00	0.00
2	6.50	2.15	0.00
3	6.50	-2.15	0.00
4	-1.00	0.00	5.54
5	4.10	1.63	5.54
6	4.10	-1.63	5.54
7	7.20	3.26	5.54
8	7.20	0.00	5.54
9	7.20	-3.26	5.54
10	0.50	0.00	7.83
11	9.86	0.00	6.54

Figure 1. Coordinates of the joints


The second table below contains the connection information, which in this case is given by a
number of beams, together with the two joints they connect, e.g. Beam 3 connects joint 1 and 6:

Beam	Joint1	Joint2
1	1	4
2	1	5
3	1	6
4	2	5
5	2	7
6	2	8
7	3	8
8	3	9
9	3	6
10	4	5
11	4	6
12	5	6
13	5	7
14	7	8

​

Beam	Joint1	Joint2
15	5	8
16	6	8
17	8	9
18	6	9
19	4	10
20	5	10
21	7	10
22	8	10
23	9	10
24	6	10
25	10	11
26	7	11
27	9	11

To find the forces on each of the beams, use the equations expressing the idea that the forces on the
beams at each joint must balance the applied load, The equations can be written out as follows. The
equation for the x component of the force at joint j is:

(1) Σ f_m cos(j;k;x) = -load_j,x


where the sum is over all beams m which connect to joint j and where k is the joint which is
connected to j and m. The quantity fm is the force on the beam m, which you are to determine and
cos(j;k;x) is the cosine of the angle between the x direction and the beam connecting joints j and k.
The direction cosine is computed as:

(2) cos (j;k;x)=(x(k)−x(j)) /dist(j,k)


where dist(j,k) is the distance between joints j and k.
The equations for the y and z components of the force are similar to (1) and (2) with the x replaced
by y and z. These equations give 33 equations (3 equations for each 11 joints or for the 27 forces on
the beam, fm). Six equations are eliminated from the system to account for the UGH being secured
to the ground. For joint 1, remove the equations for the y and z and for both joints 2 and 3 remove
the x and z equations. Thus, we obtain 27 equations for the 27 unknowns. The load applied to the
joint 11 has components (0.0, 1.0, -10.0).


Solve this linear system of equations to determine the forces on UGH given the load on joint 11. Set
up the matrix from the data and call the appropriate routines to obtain the forces. The forces are
compression forces if they are negative and extension forces if positive. Check if the forces on the
4 beams exceed those specified by MESS. Hint: cos(j;k;x) = -cos(k;j;x)

 

[tkhunny]

That does seem to be an excellent problem. What have you done to solve it.

Note: If you say, "I have no idea," I will become aggitated and tell you that you are in the wrong class or you haven't been paying attention.

 

[wjm11]

Some thoughts:

By the nature of the problem statement, which gives no beam I values and speaks only in terms of tension and compression loads (as opposed to bending moment loads), we must assume that all joints are "pinned" as opposed to "fixed." That means that they carry no bending moments and the joints do not react or transmit any bending moments. All beams are in simple tension or compression (or carry no load).

Look at the forces applied at the top of the structure. Attempt to determine the necessary reactions at the three base points of the structure (joints 1, 2, and 3). If that can be accomplished, one might be able to next determine the loads in the beams connected to those base points.

This approach allows us to break the problem down into smaller "bites". We might not need to solve all 27 equations at once. We might be able to solve a few at a time and feed those results into the next sets of equations corresponding to the various joints.

Just one approach...