Forces in Frames
Equilibrium of Frames
A frame is a collection of structure elements, usually 1-D elements, that are connected at points (nodes). Frames are all around us in structures. If you view the structure of an airplane without its outer skin, you are looking at a frame. We may describe frames as skeletal structures that are formed of beams, bars, and/or shafts connected at points which we may call nodes. We may analyze frames to determine the internal forces at the nodes and use those to further analyze the stresses in the elements of the frame.
Note that in the above two examples, we were able to obtain a number of equations of equilibrium at least equal to the number of unknown support reactions. Consider the following figures. The following figures display different frame configurations with their corresponding free body diagrams.
Statically determinate frame
In the frame above, we were able to write down four independent equations for the equilibrium (the fifth was dependent). Thus, the system of equations was solvable for the four unknown values of the reaction forces.
Now, consider the frame below. We still have four reaction forces in this problem, however, we can only write down three independent equations of equilibrium. When we removed the pinned connection between elements AC and CB, we reduced the number of equations that we were able to obtain. Thus, the system became unsolvable. This specific system of equations will result in an infinite number of solutions.
Statically indeterminate frame
Similar to the previous case, the figure below presents a case where the number of unknown reactions became five, while, the number of equations remained at three. This case will again result in a number of variables more than the number of equations with an infinite number of possible solutions.
Statically indeterminate frame
The cases above are what we call statically indeterminate structures. This type of problems can not be solved using the equations of equilibrium only, rather, we will need to use some of the knowledge we get from the study of elastic bodies (see following chapters).
In the figure below, we have the opposite case of statically indeterminate problems. In this case, the number of equations is more than the number of unknowns because of the existence of the hinge at point C. This case will result what we may call a mechanism. As you may readily see in the sketch, when the force is applied to this frame, it will move. Moving frames need the equations of dynamic equilibrium where accelerations and velocities are involved. Mechanisms are beyond the scope of this part and will be covered in the parts involving dynamics.
Internal Forces in Frames
As for the structures discussed in section 2.8, the frames have internal forces and moments that will keep them under equilibrium. Since the frames, usually, are composed of several elements, we get to find the internal forces in the connection nodes first, then it becomes a straight-forward problem to get the internal distributions. To evaluate the internal forces at the connection nodes, we draw the free-body diagrams for each of the elements and apply the equilibrium law.