Introduction to the displacement method

The displacement method is more complex than the force method, so before presenting the algorithm, I will describe it with a small introduction that will contain very important information.
The calculation is very similar, as in the force method, but the sense of physical quantities in the equations is different. It can be said that the displacement method is the reverse of the force method. To better understand the method of displacement, it is best to compare it with the force method.

Force method Displacement method
Unknowns Supernatural forces (nodes) Displacements of nodes
Canonical equations Displacements at the place of rejected ties (supports) Reactions at the place of added ties
It determines the number of unknowns Degree of static indeterminable (DSI) – this is the number of constraints that it has

should be rejected.

The degree of kinematic indeterminable (DKI) – this is the number of bonds that

should be entered

 

Below are general principles and rules that prevail in the displacement method and the basic concepts.

  1. The influence of normal forces is neglected in case of deformations and displacements of bars.
  2. The movement of the node consists of: V – vertical movement, U – horizontal movement and \varphi – rotation. The bar motion, describes the angle \psi, depending on V and U.
  3. Displacements V, U and \varphi are small in relation to the dimensions of the structure, this assumption makes it possible to omit changes in the location of nodes at height.

Degree of kinematic indeterminable (DKI) – this is the number of independent rotations of rigid nodes connecting rods and independent angles of rotation of the bars themselves needed to unambiguously determine the deformation of this system. It is defined by a pattern.

DKI = \Sigma \varphi + \Sigma \psi, where:

\varphi – number of rigid nodes in the structure,
\psi – number of independent rotations of angles ψ, calculated as the difference:
\Sigma \psi = 3n - w (the number of degrees of freedom of the system), it is converted into a kinematic chain, at each rigid node we put the articulation.

Construction of the basic layout of the displacement method.

The real system is kinematically indeterminate, so it has DKI > 0. The basic layout of the displacement method consists in adding to the real system as many additional nudes (supports) to obtain a kinematically determinate system (all displacements are known), so: DKI = 0.

After proper preparation of the basic system, all system rods should have a rigid-rigid or rigid-articulated bar diagram. For such bars, solutions for different load cases should be available, and transformation patterns are used for this. These patterns are compiled in many textbooks, but the two basic ones will be presented in this guide.

Equations of the kinematic chain.

These are equations that allow to calculate the value of the rotation angle of bars \varphi in the case when the value of one of these angles is known.

The most commonly used transformation patterns.

  1. Rigid-rigid rod.

M_{ij}=\frac{2EJ}{L_{ij}}+(2\varphi _{i}+\varphi _{j}-3\psi _{ij})

M_{ji}=\frac{2EJ}{L_{ij}}+(2\varphi _{j}+\varphi _{i}-3\psi _{ij})

2. Rigid-articulated rod.

M_{ij}=\frac{3EJ}{L_{ij}}+(\varphi _{i}-\psi _{ij})

I will end the introduction to the displacement method. In order to continue learning, I invite you to the guide named displacement method – method und example.