Force measurement method

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The force method (PER) is a common method of calculation of the structural analysis for the calculation of statically indeterminate systems. In practice, it is being replaced by bar statics programs , which usually use displacement methods. The force measurement method is taught because of its clarity at HTLs (high school) as well as in engineering studies.

An (externally) statically indeterminate system has at least one support reaction ( support forces and internal forces ) more than equilibrium conditions (sum of vertical forces, sum of horizontal forces, sum of moments around any pole ), i.e. H. the degree of freedom of the system is less than or equal to -1. Therefore the equilibrium conditions are not sufficient to calculate the support reactions. The additional conditions are derived from the deformations.

The force measurement method was expanded in the 19th century, especially by Heinrich Müller-Breslau in Berlin, for frame structures. Christian Otto Mohr in Dresden, on the other hand, favored the path size method (deformation method), which led to a scientific dispute. Both procedures are mathematically equivalent (dual to each other) as Georg Prange proved in 1916. At that time, the force measurement method was predominantly used, later with the development of the finite element method in structural engineering and the use of computers, this changed in favor of the displacement measurement method.

method

A pocket calculator and writing materials are sufficient to determine a solution or to check the plausibility of existing solutions , while “manual calculations” are often sufficient for manageable systems.

  1. The structure is converted into a statically determined main system by removing ties . The inner ties are reduced by inserting imaginary joints (often moment joints, but also shear and normal force joints), the outer ones by releasing one or more support connections.
  2. It is important to ensure that no kinematic (displaceable) system is created, because it cannot be used for calculations (the system of equations to be created in the next step becomes unsolvable due to the determinant ≠ 0), nor is it useful to build. The counting formula cannot be relied on here, since kinematic systems can very well fulfill this formula to zero. The immovability must either be based on the recognition of the basic structure (beam on two supports, if necessary with a cantilever; towing beam; three- hinged frame ) and / or construction principles (1st and 2nd law of formation for trusses), or via a contradiction in the construction of the displacement figure ( pole plan ) can be derived.
  3. It is also advantageous if the main system generated is as similar as possible to the initial system in terms of its load-bearing behavior (e.g. do not turn a continuous beam with five fields into a beam on two supports by removing the four inner bearings, but rather by inserting joints over the inner bearings a beam on two supports with four towers attached).
  4. By removing ties, compatibility is violated; This means that deformations occur in the main system that are impossible in the original system. In order to resolve this contradiction, a force variable is introduced for each loosened bond . This has to return the deformation calculated on the main system so that the compatibility is restored.
    In principle, a deformation condition is formulated for every surplus bond in order to obtain the binding force from it, i.e. H. A linear system of equations is set up with the force variables as unknowns.
  5. When the linear system of equations has been solved and the force values ​​have been found, the entire structure can be calculated using the equilibrium conditions.

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