Problem Statement:

For the truss in Figure below, a horizontal load of P = fg00 (N) is applied in the x-direction at node 2. Please note that the truss dimensions b and c are taken from your student number. You can use Excel or MATLAB to do matrix operations. If you do so, then insert snapshots of Excel tabs or MATLAB scripts.

ANSWER STEP BY STEP 

Code for indexing and reassemble the global stiffness matrix.

import numpy as np

Given values

E = 200 * 109 # Modulus of Elasticity in Pascals A = 160 * 10-6 # Area in square meters P = 9700 # Load in Newtons b = 2 # Length in meters c = 9 # Length in meters

Calculating the length of the diagonal member (4-2)

Stiffness coefficients for the members

Direction cosines for the diagonal member

Stiffness matrix for member 1-2 (horizontal)

Stiffness matrix for member 2-4 (vertical)

K24 = k24 * np.array([[0, 0, 0, 0], [0, 1, 0, -1], [0, 0, 0, 0], [0, -1, 0, 1]])

Stiffness matrix for member 3-4 (horizontal)

K34 = k34 * np.array([[1, 0, -1, 0], [0, 0, 0, 0], [-1, 0, 1, 0], [0, 0, 0, 0]])

Stiffness matrix for member 1-3 (vertical)

Stiffness matrix for member 4-2 (diagonal)

Assemble the global stiffness matrix K

Assigning the member stiffness matrices to the global stiffness matrix

K_global

ANSWER

1. Nodal Displacements Q: I'll proceed to solve for the nodal displacements using the elimination approach.
2. Reactions R: After finding the displacements, I will compute the reactions.

To resolve this, I would need to check the structure's boundary conditions and possibly reconfigure the problem setup to ensure the stiffness matrix is non-singular. This could involve adding constraints or double-checking the degrees of freedom that have been fixed.

To investigate and resolve the singular matrix issue, I'll need to re-examine the problem, particularly focusing on the degrees of freedom and constraints. Here are the steps I'll take:

Let me go through these steps and attempt to resolve the issue.

The matrix still appears to be singular even after reducing the system to only the free degrees of freedom. This suggests that the truss configuration might have a mechanism (a situation where parts of the structure can move without any resistance), leading to a rank deficiency in the stiffness matrix.

Potential Solutions: