truss calculations (in static)
In our most recent unit we have learned how to calculate a truss in static. Essentially this has no meaning because a real truss/bridge would have its own weight, gravity and several other forces working against it, none of which exist in "static". However this is a necessary educational base. When you calculate a truss, there are 3 main objectives. First you have to solve for the moments, then the reaction forces and finally the force being applied to each member.
Before anyone gets too excited about solving for the forces, we have to be sure that the truss "statically determinate", this simply means that we need to figure out if it is actually possible to solve the truss and all of its variables. To do this we use the equation: 2*J=M+R, in which J equals the number of joints, or places where the members attach to one another. M is the number of members (the different lines and vectors that make up a truss). Finally, R equals the number a reaction forces determined by the joints between the truss and whatever it is attached to. A pin joint keeps the truss from moving in two directions (up/down and side to side) and is therefor is worth two reaction forces, whereas a roller joint only holds the truss in one direction so you just add in one for that. If you plug in all the variables and both sides equal each other then you can continue on solving it, if they don't then there are three things you can do: change the type of joints (pin/roller) around until they add up right, add in a member here or subtract one there. If, however, you are feeling particularly motivated you can go and play some call of duty.
After determining that the truss is, indeed, solvable we must solve for the moments, if your truss is missing one, if not then you may skip this step. To do this select one point on the bridge and use that as a measuring point. Use the equation Em=0= (put forces here). The idea behind this is that E (the sum) of m (the moments (torque)) = 0, and since all of your moments combined equal Em, which equals zero, so you solve that all your moments plus the one that your solving for (usually a pin/roller joint) equal zero. now you simply multiply all of the forces by their perpendicular distance from the measuring point, add them up and subtract that from zero. Now you have isolated your unknown and its distance and can easily solve that with division.
Now that we know our moments we can solve for the reaction forces of the truss. To do this we use two equations: Erfy=0 and Erfx=0. Erfy=0 states that the sum of all of the reaction forces in the "Y" direction equal zero, Erfx says the same but in the "X" direction. We solve these much the same as moment forces. However we don't factor in distances and we only focus in one direction at a time. Now simply input all of the given forces being applied to joints and add them up, keeping in mind that any downward force is factored as a negative number. Once you have that total and your unknown on one side with zero on the other, subtract your total and your new number equals the unknown, as the equation will plainly state. Do the same for the other direction, X or Y depending on what you started with.
After solving for reaction forces we only have the members left. From here on it's pretty much easy sailing in my opinion. Before starting you need to know that you will have to sketch a Free Body Diagram (FBD) of each joint in order to solve them. Now we start. First, create a FBD of the joint that you have the most known forces on (at least two in an ideal situation). To solve we will use two equations: Efy=0 (sum of forces in the "Y" direction equal zero) and Efx=0 (same in the "X" direction). Simply plug in your known force(s) and your unknown. This is solved in the same method as the others. If you come across an angled vector you must solve for it in both "X" and "Y" directions using cosine/sine/tangent. Continue doing this for each member at each joint. Remember that all forces must balance out to zero in every single equation, because it is in static, and static things are equal.
Before anyone gets too excited about solving for the forces, we have to be sure that the truss "statically determinate", this simply means that we need to figure out if it is actually possible to solve the truss and all of its variables. To do this we use the equation: 2*J=M+R, in which J equals the number of joints, or places where the members attach to one another. M is the number of members (the different lines and vectors that make up a truss). Finally, R equals the number a reaction forces determined by the joints between the truss and whatever it is attached to. A pin joint keeps the truss from moving in two directions (up/down and side to side) and is therefor is worth two reaction forces, whereas a roller joint only holds the truss in one direction so you just add in one for that. If you plug in all the variables and both sides equal each other then you can continue on solving it, if they don't then there are three things you can do: change the type of joints (pin/roller) around until they add up right, add in a member here or subtract one there. If, however, you are feeling particularly motivated you can go and play some call of duty.
After determining that the truss is, indeed, solvable we must solve for the moments, if your truss is missing one, if not then you may skip this step. To do this select one point on the bridge and use that as a measuring point. Use the equation Em=0= (put forces here). The idea behind this is that E (the sum) of m (the moments (torque)) = 0, and since all of your moments combined equal Em, which equals zero, so you solve that all your moments plus the one that your solving for (usually a pin/roller joint) equal zero. now you simply multiply all of the forces by their perpendicular distance from the measuring point, add them up and subtract that from zero. Now you have isolated your unknown and its distance and can easily solve that with division.
Now that we know our moments we can solve for the reaction forces of the truss. To do this we use two equations: Erfy=0 and Erfx=0. Erfy=0 states that the sum of all of the reaction forces in the "Y" direction equal zero, Erfx says the same but in the "X" direction. We solve these much the same as moment forces. However we don't factor in distances and we only focus in one direction at a time. Now simply input all of the given forces being applied to joints and add them up, keeping in mind that any downward force is factored as a negative number. Once you have that total and your unknown on one side with zero on the other, subtract your total and your new number equals the unknown, as the equation will plainly state. Do the same for the other direction, X or Y depending on what you started with.
After solving for reaction forces we only have the members left. From here on it's pretty much easy sailing in my opinion. Before starting you need to know that you will have to sketch a Free Body Diagram (FBD) of each joint in order to solve them. Now we start. First, create a FBD of the joint that you have the most known forces on (at least two in an ideal situation). To solve we will use two equations: Efy=0 (sum of forces in the "Y" direction equal zero) and Efx=0 (same in the "X" direction). Simply plug in your known force(s) and your unknown. This is solved in the same method as the others. If you come across an angled vector you must solve for it in both "X" and "Y" directions using cosine/sine/tangent. Continue doing this for each member at each joint. Remember that all forces must balance out to zero in every single equation, because it is in static, and static things are equal.