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 HW Helper P: 3,224 1.2. A FORMAL OUTLINE (I). Under the assumption that the idea of the force method was successfully illustrated in Section 1.1., we shall now try to generalize and formalize that idea. It first has to be mentioned that the statical constraints that are to be removed from the system do not have to be external constraints, such as supports - one can add internal hinges or roller connections to the system too, which also minimizes the degree of statical indeterminacy. This is shown in Figure 4. However, one has to add a pair of genenralized forces (forces or moments) which are to prevent a relative generalized displacement (translation or rotation) at that point. The pair of generalized forces consists of two forces with an equal magnitude and direction, but of opposite orientation. Figure 4. As mentioned before, for a system with a degree of statical indeterminacy which equals n, one has to remove (and/or add) a total number of n supports (and/or internal hinges). Of course, the resulting primary system must not be a mechanism, i.e. it has to have zero degrees of freedom. We can accomplish this in a lot of ways - it will be seen later that there are smarter ways to do that, since we shall need to sketch different internal force diagrams, which can become fairly complicated, depending on the supports we choose to remove (or the hinges we choose to add). After creating the primary system, we place the set of generalized forces $\left\{X_{1}, \cdots, X_{n}\right\}$ on the system in such a manner that each force acts at the point and in the direction of the removed support. In the example in Section 1.1., the generalized force was the reaction $R_{B}$, which was placed at the point of the removed support. (Note that the generalized forces may represent pairs of forces too, as shown in Figure 4.) After doing so, it is intuitively clear that one has to solve a system of n equations, each of which expresses a displacement condition at one of the points where the supports were removed (or where internal hinges were added - we shall, for practical reasons, from now on refer to this procedure as support removal only). These equations are often referred to as compatibility equations, since they guarantee that the primary system with the generalized forces acting on it must have the same displacements as the original system. We shall now analyse the structure of the i-th equation, which has the form: $$\sum_{k=1}^n X_{k} \delta_{ik} + \delta_{i0} = 0 \ \ \text{(1),}$$ where $X_{k}$ represents the force acting at point k, $\delta_{ik}$ represents the generalized displacement at the point i, caused by the force k, and $\delta_{i0}$ represents the generalized displacement at point i, caused by the original loads applied to the system (not the added generalized forces). In words, the total displacement at every point i has two contributions: one is from the generalized forces $\left\{X_{1}, \cdots, X_{n}\right\}$, and the other one comes from the loads applied to the original system. This total displacement must vanish at every point, which is expressed by (1). (It is important to understand that every generalized force has a contribution to the displacement at point i, i.e. at every point of the system.) We shall now present some theorems and principles which justify and explain the formulation of the n equations of compatibility, each of which has the form (1). Perhaps the most important theorem is Castigliano's (second) theorem, which states that the partial derivative of the potential energy of deformation of a linearly elastic body with respect to the i-th force acting on the body equals the displacement at the point and direction of the force, i.e. $$\delta_{i} = \frac{\partial U}{\partial F_{i}} \ \ \text{(2).}$$ The proof of this theorem shall not be presented here. We shall only mention that it relies on Betti's theorem (which can be found here), which is derived under the assumption that the principle of superposition holds. Without going more in-depth, one can intuitively assume that expression (2) will be useful in the force method, since the most important issue is to find displacements at certain points, while the generalized forces acting at those points remain the unknowns - which is why the method is called the force method. Indeed, if we set $F_{i} = X_{i}$, it can easily be seen that, for a system with a degree of statical determinacy which equals n, we shall have (in consistence with the idea of the force method laid out at the beginning of this section) n equations of the form $$\delta_{i} = 0, i = 1, \cdots, n \ \ \text{(3),}$$ which, after applying Castigliano's theorem, become: $$\frac{\partial U}{\partial X_{i}}=0, i = 1, \cdots, n \ \ \text{(3').}$$ The solution of the system is the ordered n-touple $(X_{1}, \cdots, X_{n})$, which represents the reaction forces in all the removed supports. The primary system is now in the same state of equilibrium as the original system - the equations of equilibrium can easily be solved to retrieve the values of all the other reactive forces (and internal forces, automatically), and it can easily be seen that the displacement of the system equals the displacement of the original system at every point, which almost directly follows from condition (3). The only thing left to do now is to show how the deformational potential energy of the system, referred to as U, is constructed, and how the system of equations (3') is equivalent to the system (1).