what does it mean to say for example an arc is a stistically indeterminate structure?

Answer 1

Nice question. It helps to first view this question in terms of truss structures where we only have simple members connected via joints. A statically determinate truss is in which every member can be found to be in state of either pure tension or pure compression, and that at each joint those member forces balance out, so that the joints (and therefore the members too) do not move or tend to move. For example, a simple triangle is a statically determinate structure. However, if you have a square with two diagonal members not connected to each other, then you have a case where the forces upon the member and at the joints do not have an unique solution, so that it becomes somewhat unpredictable what the practical forces would be in such a structure. This is known as a statically indeterminate structure. Now, interestingly enough, it's quite possible to design a truss in the shape of an arch that is statically determinate, that is, a "kosher" structural arch, but it's not true that anything that looks like an structural arch is statically determinate. For solid arches, such as stone arches as used in Roman structures or medieval cathedrals, we have to get into what's known as FEA, or Finite Element Analysis, which makes an approximation of a statically determinate structure that matches the size and shape of the structure, in which then the forces can be calculated. But real arches with inherent elasticity (yes, even stone can be elastic) are frequently statically indeterminate, so more complicated analysis is required to predict the behavior of such arches.

If you're mathematically minded, then the analogy would be that if you have n unknowns and n linear equations relating them, then an unique solution is possible (except in certain cases of degeneracy). However, if you have n unknowns, and p linear equations, and p < n, then an infinte range of solutions would be possible, thus making it an "indeterminate problem".

Answer 2

There is a set of 3 equations called the "equations of static equilibrium". It's a fancy name, but the idea is simple. In order for an object to be in static equilibrium (not moving, basically):

1. The sum of all horizontal forces will be zero.
2. The sum of all vertical forces will be zero.
3. The sum of all moments (rotational forces) will be zero.

These don't mean there can't be any force on it, just that they have to add up to zero.

Now, think of a bridge. Let's say it's resting just on its ends (so it has two supports), and we want to know how much weight is on each of those supports. Those weights are "unknowns", so we have to calculate them, and we use the equations of static equilibrium to do it.

In this case, there are no horizontal forces, so equation 1 doesn't give us any information, so we use #2 and 3. We have two equations and two unknowns. The rules of algebra tell us that we can solve a system of 2 equations and 2 unknowns, so we can find the weight on each support. That's statically determinate, that is, we can find all the unknowns with the equations of static equilibrium. Note that the physical properties of the bridge (like its stiffness) don't matter in this case.

Now put another support in the middle of the bridge, so it has 2 spans. Now there are 3 supports, and 3 unknowns. Since you have more unknowns than equations, you can't find all the unknowns (the force at each support) with the equations of static equilibrium. That's statically indeterminate (and if you can't find the forces at the supports, you can't find the forces on the bridge members themselves - you have to know those first). There are other ways to find all 3 unknowns, and they depend on the physical properties (mainly the stiffness) of the structure.

You can have more than 3 unknowns, too.

Clear as mud? ;-)