Stiffness Matrix in Finite Elements Theory

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Stiffness Matrices are a fundamental part of Finite Element Analysis. These matrices always define inherent properties of the system being studied in aerospace engineering applications.

In many complex engineering problems there are some basic unknowns. If these unknowns were found, the entire model structural behavior can be predicted. These Basic unknowns and/or the Field variables which are encountered in the complex engineering problems are displacements.

In a discrete continuum, these unknowns are infinite. But the finite element procedure reduces such unknowns to a finite number by separating the solution region into small components called elements and by expressing the unknown variables in terms of approximating functions (Either Interpolating functions/Shape functions) within each element. Approximating functions are defined in terms of field variables of Noted points called nodes or nodal points in the equation.

After selecting elements and the nodal unknowns, the very next step in finite element analysis is to assemble element properties for each individual element in the model. For example, In mechanics of solids , we have to find the force-displacement i.e. stiffness characteristics of each individual element.

Mathematical Formulation of Stiffness Matrix

 Mathematically this relationship is of the form, to assemble element equation to form global system

[K]{U} = {F}

Where,

[K] = Element Stiffness or Property Matrix

{U} = Nodal Displacement Vector of the element

{F} = Nodal Force Vector

  • Once the element stiffness matrices is calculated then it is assembled into a global stiffness matrix for the entire model.
  • The nodal forces are related to the nodal displacement vectors by the element stiffness matrix.

How Element Stiffness Matrix Assembled to Global Stiffness Matrix

If the element stiffness matrices are assembled into a global stiffness matrix. The loads are also assembled to a global load vector form. This results in the following matrix equation for the overall structure.

 

stiffness to global

 

An important step in the displacement method is the formulation of the element stiffness matrix, in which each element in a finite element method is represented by an element stiffness matrix [K]e.

The principal characteristics of a FEM are encompass in the element stiffness matrix. In a structural finite element, stiffness matrix contains the model geometric and material behavior information that indicates the resistance of the element to undergo deformation when subjected to loading. This type of deformation may include axial, bending, shear, and torsional effects.

In a FE used in nonstructural analyses , similar as fluid-flow and heat exchange, the word stiffness matrix is also used in finite element theory, therefore the matrix shows the resistance of the element to change when subjected to external influences or applied loads.

Element properties are used to assemble global properties/structure properties to get system equations is, sum of element stiffness matrix and nodal displacement vector, which is equal to nodal force vector. Then the boundary conditions are imposed. The solution of these simultaneous equations give the nodal unknowns.

After the boundary conditions are imposed. The solution of these simultaneous equations give the nodal unknowns. By using these nodal values extra calculations are made to get the required values e.g. stresses, strains, moments, etc., in aerospace engineering complex problems.

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Hello, I am an aircraft structural analyst with industrial experience and a master degree on aerospace structures. Currently working for an aerospace company as a stress
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