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In my lecture material on different types of structural supports, there is the above picture of a hinge. It is explained that a hinge is a type of support that exerts no moment on the rod, but prevents the end of the rod from moving. On the right, the supporting force is shown not to be necessarily parallel to the rod.

However, often in solving problems we assume that the force in this type of rod is parallel to the rod itself, such as here:

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Why is it assumed that the force exerted by the hinge against the rod is parallel to the rod, if the resultant force in a hinge is not always parallel to the rod it supports?

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If the rod is pinned at both ends, the pin joints can transmit forces, but not bending moments.

If you consider the equilibrium of the rod by taking moments about one end, the line of action of the force at the other end must be along the rod, otherwise there would be a non-zero moment acting on the rod and it would rotate.

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You've made a small misunderstanding.

The difference between pinned and fixed supports is that pinned supports do not resist rotation, and therefore do not apply concentrated bending moments on the beams. Meanwhile, fixed supports resist rotations by applying said concentrated bending moments.

Your observation is correct, though: the forces applied on the beam might not be parallel to the beam, which implies in a bending moment developing along the beam. However, the fact remains: the pinned support is only applying forces. What happens to the beam is another matter entirely.

On the other hand, fixed supports apply not only forces but also concentrated bending moments.

The consequence of this is that the beam's bending moment at the support due to the support's reactions is zero in pinned supports (since the lever arm between the force and the studied point is obviously zero) and (usually) non-zero in fixed supports (since it's equal to the support's concentrated bending moment reaction).

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