method of joints truss
A truss is a frame of straight bars pinned together at both ends. Think of a bridge or a roof built from triangles: the bars meet at pins rather than being welded rigid, and loads sit only at the joints, so each bar is a two-force member. It can only pull or push along its own length, and that one fact makes a truss easy. The method of joints uses it directly. Cut the truss free at the pins and look at the joints one at a time. At one joint every bar force and every external load meets at a single point, so the joint is simply a particle in equilibrium, and the equations you already know are ΣFx=0, ΣFy=0. Solve one joint and carry those bar forces to the next joint, walking across the whole structure. Start by seeing why each bar takes force only along its own line.
The whole method rests on one property of a bar. Pick one bar: a truss bar is pinned at both ends and carries no load in between, so exactly two forces act, one at each pin. For those two forces to balance, that is for ΣF=0 and ΣM=0 on the bar, they must be equal in size, opposite in direction, and lie on the line that joins the two pins. So the only thing a truss bar can do is push or pull along its own axis, and it cannot take a sideways force or a bending moment the way a beam can. This is why a truss is so efficient and so easy to solve: each bar collapses to a single unknown, the axial force inside it.
The axial force in a bar comes in two kinds. Toggle the two: when the end forces pull the bar outward and the bar stretches, that is tension and it is taken as positive, and when they push inward and the bar is squeezed, that is compression and it is taken as negative. By Newton's 3rd law the bar pushes or pulls back on the joint with an equal and opposite force, so the direction flips at the joint: a tension member pulls the joint toward itself, and a compression member pushes the joint away. The trick when you draw a joint is to assume every unknown bar is in tension and to draw the arrow pointing away from the joint, along the bar. If the computed value comes out negative, that bar is really in compression, so you let the sign decide and you do not have to guess the direction.
Now lift out one joint. Pick one and cut the bars that meet there: each cut bar leaves its axial force behind, and any external load or support reaction acts there too. All of these forces pass through the joint and meet at one point, so there is no moment equation to write, and the joint is simply a particle, which leaves only the two equations a particle in equilibrium follows, ΣFx=0 and ΣFy=0. So you can solve a joint that has at most two unknown bar forces. Look at the top joint C: the load pushes down and the two diagonal bars push up and outward to hold it, two unknowns and two equations, so it solves. The whole truss is just this small problem repeated at every joint.
Put it together on a simple triangular truss. A pin on the left, a roller on the right, and a load hanging at the top joint, and dragging the load shows how the bar forces respond. Look at the supports first: by symmetry each carries half the load, so each vertical reaction is P2. Move to the top joint, where only two bars are unknown, and the two diagonals have to push up to carry the load, so both are in compression, each P(2 sinθ). Drop to a support joint and the bottom bar has to take the horizontal push of the diagonal, so that bar is pulled tight in tension, P(2 tanθ). The pattern shows: the diagonals are squeezed and the bottom chord stretches, and dragging the load larger makes every force grow in exact proportion.
One last trick saves real work. Some bars carry no force at all, and toggling the two cases shows the rule. First, at a joint where only two bars meet and there is no load, if the two bars are not in a straight line both must be 0, because nothing is there to cancel two different directions. Second, at a joint where two bars lie in a straight line and a third comes in at an angle, if there is no load the lone angled bar is 0 and the two collinear bars pass the force straight through. Finding these zero-force members first takes bars out of the puzzle before you even start the arithmetic. That completes the method of joints: find the reactions, start at a joint with two unknowns, write ΣFx=0 and ΣFy=0, carry each result forward, and cut the work with symmetry and zero-force members. From here the same free-body thinking extends to frames and machines as a whole.