: Find reactions at node 2.

Once we have found the values for the nodal displacements and rotations, we can substitute these values back into our FEM equation for the entire structure, (as we have done below, being sure to use consistent units throughout our problem, vertical displacements being in inches and rotations being in radians), the unknown nodal reaction force F₂ and moment M₂ can now be found from the following equation:

We will find F₂ by multiplying the terms in row 3 of the stiffness matrix by the column of known nodal displacements and rotations. The moment M₂ is found by multiplying the terms in row 4 of the stiffness matrix by the column of known nodal displacements and rotations. The terms in rows 1, 2, 5 and 6 will be multiplied by the column of known nodal displacements and rotations, as a check, where the resultant forces and moments should match the applied (known) forces and moments at nodes 1 and 3. After matrix multiplication, our FEM equation becomes:

The solution for F₂ and M₂ is shown below, with units, along with the forces and moments from nodes 1 and 3 which match our known applied force and moment at those nodes.

As with our solution for the vertical displacements and rotations, we must check the signs of our answers to see if they make physical sense. The positive sign for the reaction force, F₂ indicates a force in the upward (positive y) direction, which is in the opposite direction as the applied forces, as is necessary in order to satisfy equilibrium. The negative sign for the moment, M₂, in the above equation indicates a moment in the clockwise direction, which is in the proper direction to satisfy equilibrium. We can see from the free body diagram below that equilibrium can be met with the positive (upward) reaction force and a negative (clockwise) moment.