Question

During a load test on one of the prototype frame structure at CGC Engineering, a set of three linear equations of forces has been developed. The forces In three members of this structure are F1, 1-2 & F3 are in kN. The forces are related by the following three linear equations:

2F1 - F2 + 3F3 = 2

F1 + 3F2 - F3 = 11

2F1 - 2F2 + 5F3 = 3

Convert the above linear equations into 3x3 matrix array, determine the three unknown forces F 1, F2 & F3 using the required analytical matrix methods below and validate the obtained analytical matrix solutions for forces F1, F2 & F3 using appropriate computer software of your choice.

a) Ascertain the values of F1, F2 & F3 using determinants technique

b) Use Inverse Matrix method to compare and check your results from the previous method.

Answer 1

To determine the unknown forces F1, F2, and F3 using matrix methods, we can convert the linear equations into a 3x3 matrix, apply determinants to find the values, and then validate the solutions using the inverse matrix method.

Step-by-step explanation:

To convert the given linear equations into a 3x3 matrix array, we can rewrite them as:

2F1 - F2 + 3F3 = 0

2F1 + 3F2 - F3 = 11

2F1 - 2F2 + 5F3 = 3

We can represent these equations in matrix form as:

[2 -1 3][F1] = [0]

[2 3 -1][F2] = [11]

[2 -2 5][F3] = [3]

To determine the values of F1, F2, and F3 using determinants, we can use Cramer's rule. The determinant of the coefficient matrix is non-zero, so we can solve for the unknown forces:

F1 = determinant ([0 -1 3])/determinant ([2 -1 3])

F2 = determinant ([2 11 -1])/determinant ([2 -1 3])

F3 = determinant ([2 -1 0])/determinant ([2 -1 3])

To validate the solutions obtained using determinants, we can use the inverse matrix method. We can find the inverse of the coefficient matrix and multiply it by the constant matrix to obtain the unknown forces.