# Unable to obtain closed system of equations using method of joints

I am currently working on some code to solve 2-dimensional truss and frame problems, and I have come into an issue with one of the problems. I need to come up with a system of linear equations that I can feed to a linear solver in order to solve for the support reactions.

In the image above, the filled joints represent fixed joints (A and D) and the empty ones represent hinged joints (B and C). To solve the problem, I divided it into five different sections, shown in the following image:

Sections 1, 2, and 3 all lead to three equations each (sum of forces in x and y directions, plus sum of moments). Sections 4 and 5 only have two equations each (sum of forces in x and y directions). I have only 13 equations, while I have 15 unknowns ($$A_x$$, $$A_y$$, $$B_y$$, $$B_{1x}$$, $$B_{1y}$$, $$B_{2x}$$, $$B_{2y}$$, $$B_{3x}$$, $$B_{3y}$$, $$C_{1x}$$, $$C_{1y}$$, $$C_{2x}$$, $$C_{2y}$$, $$C_{3x}$$, and $$C_{3y}$$). The entire system is solvable, so I must be missing something here (or I have extra variables that I should not have), but I have not been able to figure this out. Any help would be greatly appreciated.

## External Reactions

Solving for the support reactions of an externally determinate structure like this one does not require the method of joints. The three equilibrium equations suffice.

There are 3 unknowns: horizontal reaction at A, vertical reaction at A, vertical reaction at B.

There are 3 equations: $$\sum F_x = 0$$, $$\sum F_y = 0$$, $$\sum M = 0$$

One solution approach is to take moment about A to solve for vertical reaction at B. Then sum vertical forces to solve for vertical reaction at A. With no horizontal external load, the horizontal reaction at A must be zero.

## Internal Forces

Should you desire to solve for the internal forces, note that the method of joints is intended to be used for structures that can be represented as pin-connected trusses (i.e. we can assume the joints do not support moment and members carry only axial load). Since you indicate that Joints A and D are fixed, you'll need to consider whether method of joints is appropriate.

Further, the work you have shown here does not really resemble the method of joints. In general, for this method, draw the free body diagram of each joint, showing any external loads and then the unknown axial forces of the members, which will be along the axis of the member. Use $$\sum F_x = 0$$ and $$\sum F_y = 0$$ to solve for the member axial forces. Proceed along the truss.

Method of joints is primarily a hand calculation method. If you're trying to develop a general-purpose solution algorithm for structures that include both trusses and frames, method of joints isn't what you're looking for - perhaps look into the direct stiffness method.

A note on assessing truss determinacy:

Let b = number of members and r = number of support reactions. Let j = number of joints

$$Unknowns = b + r = 8$$

In two dimensions, we can write two equilibrium equations per joint.

$$Equations = 2j = 8$$

$$Unknowns = Equations \therefore$$ truss is determinate

Thus, were this a pin connected truss, the method of joints could be used to solve for the member axial forces.

Image i have uploaded has the answer there is no need to have free body daigram of every link it simply solved in the image uploaded.

• It is always much better to use mathjax equations rather than images like these. – Teo Protoulis Jul 12 '20 at 11:37