Force method – algorithm

The method of calculating statically indeterminate systems will start with the method of forces.
The force method boils down to the solution of statically determinate systems. We transform a statically indeterminable system by subtracting an excessive number of constraints, and then we introduce the same number of unknown forces instead of rejected bonds. This system is called the basic system, it is a simple way to solve frame, truss or arched systems.

In this guide I will present an algorithm for solving statically indeterminate systems with the force method, while describing each point.

  1. At the beginning, it is necessary to determine the degree of static indeterminacy of the system.
    The degree of Static indeterminate of the system (DSI) – this is the number of excessive constraints (external and internal) that should be rejected so that the system becomes statically determinable.
    DSI can be calculated from the formula.

DSI = w-3n (we must also take into account internal ties in closed chambers), where:

W – number of ties

N – number of discs

Examples of statically indeterminate schemes.

  1. Knowing the number of bonds that should be rejected, we can start building the basic layout.
    The basic system is a statically determinate system that must also meet three conditions:
  • geometric identity (conformity of dimensions)
  • kinematic identity (compatibility of displacements – canonical equations)
  • static identity (compliance of loads)

While building the basic layout, you can reject any external and internal ties, but you must ensure geometric constancy. You can build basic systems that are independently working structures (systems). An example of the basic layout.

In the place of the bonds taken, the unknown forces of Xi appear.

  1. State P – state of real loads.
    At this stage, you need the ability to draw graphs of internal forces.
    The basic system is loaded with the whole of real loads and we make graphs M_{p},\: T_{p} \: and \: N_{p}.
  1. States X_{i} – the state of loads with unit forces.
    Here, too, we need to be able to draw graphs of internal forces well.
    We reject the whole of the external load and load the basic system with unit forces (not to be confused with virtual ones) in the place and direction of the rejected bonds and draw up M_{i},\: T_{i}\: and\: N_{i} graphs.
  1. Arrangement of a system of canonical equations of the force method and calculation of elements occurring in it.
    The force method is based on the assumption that both statically indeterminate and statically determinate systems work exactly the same, although they differ in the number of notes. In particular, all displacements in both systems are the same.
    The canonical equations of force methods are equal to the displacements in places of rejected constraints, because in reality there are ties there, so these displacements are equal to 0. In these equations there are unknown Xi forces, which of these equations we calculate. For example, the system of canonical equations of the system in which two bonds were rejected will look as follows.

\left\{\begin{matrix} &0=\delta _{1P}+\delta _{11}*X_{1}+\delta _{12}*X_{2} \\ &0=\delta _{2P}+\delta _{21}*X_{1}+\delta _{22}*X_{2} \end{matrix}\right.

If DSI were 1, so if we had to reject only one bond. There is no need to arrange a system of equations. In that case, we will have only one unknown. It looks like this.

\left\{\begin{matrix} 0=\delta _{1P}+\delta _{11}*X_{1} \end{matrix}\right.

  1. The solution of the system of canonical equations of the force method.
    Calculation of unknown displacements Xi.
  1. Drawing graphs of internal forces in the real system.
    To draw graphs of internal forces of the real system, we use the superposition principle. Below is an example of the formula, for the scheme with DSI = 2.

M_{P}^{n}=M_{P}^{0}+M_{1}^{0}*X_{1}+M_{2}^{0}*X_{2}

  1. Checking the correctness of calculations.
  • Static check – cutting node and analysis of its static balance.
  • Kinematic check – calculation of displacement, and predetermined value, generally zero displacement on one of the supports.
  • Computer check.

We have reached the end of the force method algorithm.
Finally, I will discuss everything again.

  1. At the beginning, we calculate DSI – the degree of statically indubitability.
  2. We build a basic system (statically determinate) by removing an excessive number of ties.
  3. We draw graphs of internal forces from the states P, \: X_{1},\: X_{2}, \: X_{i}.
  4. We arrange a system of canonical equations. We calculate the individual deltas that appear in the system of canonical equations. We calculate them by multiplying graphs of internal forces, e.g.

    \delta_{1p} – multiplication of status graphs X1 and Xp
    \delta_{11} – multiplication of status graphs X1 and X1
    \delta_{12} = \delta_{21} – multiplication of status graphs X1 and X2 etc…

  5. The solution of the system of equations so that we know the unknown displacements, i.e. X_{1},\: X_{2},\: X_{3},etc …
  6.  Drawing graphs of internal forces in the real system.
  7. Checking the correctness of calculations.

That’s enough for the theory, I invite you to the next guide to become familiar with the example of the task of the force method.