Mechanical Engineering

Statically Determinate System and Statically Indeterminate System

When there are more unknown forces than the number of equations available on a rigid body at equilibrium, the rigid body is called as a statically indeterminate system. For 2D rigid body equilibrium, there are always only 3 equations available to calculate for unknown forces. The 3 equations are sum of forces in the x-direction and y-direction, as well as the sum of moment equation. For example, if there are 4 unknown forces on a 2D rigid body at static equilibrium, the rigid body is a statically indeterminate system because there are only 3 equations available for force analysis. Unlike statically determinate system where force analysis can be completed just by taking the moment and force summations along x and y, solving for unknown forces in statically indeterminate system require stress-strain analysis in the element of forces. Element of forces on a rigid body is typically a metal that pulls on the beam like a rope.

Statically determinate system is a system where there the number of unknown forces are equal or lower than the number of equations available to calculate the unknown forces

As a first step of the force calculations on a statically indeterminate system, we must analyze the deformation of the entire system. Then the deformation analysis is used to derive the expression for the relationship between the change in length of the elements present in the system. Once the expression is derived, we can figure out the expression for the strain of the element of forces because the change in their length is the product of strain and initial length. Strain can be induced by both the forces and temperature changes. Once the relationship expression is expanded and simplified after the strain analysis, then the rest is just calculating the unknown forces by plugging in the known variables. There are also questions on statically indeterminate systems that requires calculation of change in length of a certain part of the system due to the applied force(s) or calculation of stress in the element of force(s).

When there are no unknown moments involved in the system, then one can solve for the forces directly with the deformation analysis When there are unknown moments involved, then Macaulay’s step function or Superposition method is required to solve for the unknown forces and/or moment(s) and it involves some type of deformation analysis as well.