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page about trusses explains that in a structure made entirely of long
narrow struts and ties, pinned at the joints, all the forces are along
the members, making calculation very simple. Unfortunately, this
is not the whole story. In removing the complexity of the
apparently simple box-girder or beam, we have merely moved our problems
elsewhere, to the joints.
The picture below represents a part of a truss with a simple pinned joint.
If we look at the forces in a single tension member, or tie, of such a structure, we might imagine something like the picture below. Please note that this diagram is not an exact representation of the distribution of the forces: it only gives a general impression. They would be concentrated near the hole in reality.
The neatly parallel forces in the body of the tie give way to something more complicated at the joint, and much more difficult to calculate. Worse still, we have to ask what makes the forces curve like this. Yes, they follow the material, and yes, they are influenced by the presence of the pin. But how? Forces do not go around corners unprovoked. The forces in an arch or a suspension bridge cable are made to curve by the pull of gravity and by the forces in the attachments.
Where are the corresponding forces here? If we imagine the presence of the pin, as in the next picture, we can see that it pushes on the tie, which is therefore in compression, as is the pin. But we already know that the tie is in tension, so how can it be in compression as well? This can happen because the tension and the compression are at right angles. Note that the lines of tension are not really spaced uniformly: the stress is concentrated near to the hole, as the energy of the material is minimised in that condition.
How do we know that the compression and the tension are at right angles? Suppose that they are not at right angles. We can decompose one of them in into two components, one parallel to the other forces, and one at right angles. The two parallel parts will tend to cancel, leaving a remnant at right angles to the remaining component. From the diagram we see that the pin touches the tie only in a small area. The pin experiences other stresses besides compression. Look at the next picture.
At each joint, a pin passes through many bars. Between any two bars, the pin experiences shear, which it must withstand. If you visit the Tour Eiffel, you can see displays of old rivets which have been badly deformed by shear. That being so, how were the members re-aligned in order to pass new, straight rivets through the holes?
Notice the expanded ends of the bars, providing an adequate cross-section to control the stresses. Note also the absence of concave sharp corners, to reduce the probability of starting cracks.
From this simple example we see that the design and construction of joints and other attachments is a crucial factor in structures. And if joints need to move, as in mechanisms and animals, things can become very difficult. Think of the number of people who experience problems with joints, tendons and ligaments. These problems can occur in sports people when they are relatively young, because of the extreme stresses that they incur. Animals probably avoid such over-stressing because their lives depend on their continued ability to catch prey, to escape from predators, and to find, and be accepted by, a mate. In many cases they can heal minor damage.
The case of a rope around a pulley is an extreme case of what we saw earlier. Can you work out what type of stresses would be found in each part of the system?
In some circumstances, the tie can be a cable, and the forces follow a narrow path.
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