Introduction to the Stiffness (Displacement) Method Definition of the Stiffness Matrix

UNIDAD 2 Introduction to the Stiffness (Displacement) Method Definition of the Stiffness Matrix

stiffness matrix is a matrix such that

Dr. Gerardo Pérez

Análisis por Elemento Finito: Julio-Diciembre 2015

Derivation of the Stiffness Matrix for a Spring Element

Linear spring element with positive nodal displacement and force conventions

Dr. Gerardo Pérez


General steps to derive the stiffness matrix for the spring element

Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


local stiffness matrix for the element

Dr. Gerardo Pérez


This step applies for structures composed of more than one element such that

Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Boundary Conditions

The structural system is unstable.

Structure will be free to move as a rigid body and not resist any applied loads.

Boundary conditions

Homogeneous boundary conditions

Nonhomogeneous boundary conditions

Dr. Gerardo Pérez


homogeneous boundary conditions

Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


nonhomogeneous boundary conditions

Dr. Gerardo Pérez


Dr. Gerardo Pérez


Some properties of the stiffness matrix

1. [K] is symmetric, as is each of the element stiffness matrices.

2. K is singular, and thus no inverse exists until sufficient boundary conditions are imposed to remove the singularity and prevent rigid body motion.

3. The main diagonal terms of [K] are always positive.

Dr. Gerardo Pérez


Example 1

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Example 1 Determine the nodal displacements, the forces in each element, and the reactions.

Example 2

For the spring assemblage shown in Figure, obtain (a) the global stiffness matrix, (b) the displacements of nodes 2–4, (c) the global nodal forces, and (d) the local element forces. Node 1 is fixed while node 5 is given a fixed, known displacement . The spring constants are all equal to k = 200 kN/m.

Dr. Gerardo Pérez


Example 3 Determine the nodal displacements, the forces in each element, and the reactions.

Dr. Gerardo Pérez


Potential Energy Approach to Derive Spring Element Equations

•  minimum potential energy

•  more general

•  involves nodal and element equilibrium equations along with the stress/strain law for the element

•  involves nodal and element equilibrium equations along with the stress/strain law for the element

•  principle of virtual work

elastic materials

other classes of problems

field problems

Dr. Gerardo Pérez


The total potential energy in the finite element formulation

The concepts of potential energy and of a stationary value of a function

Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


For the linear-elastic spring subjected to a force of 1000 lb shown in Figure, evaluate the potential energy for various displacement values and show that the minimum potential energy also corresponds to the equilibrium position of the spring.

Dr. Gerardo Pérez


Dr. Gerardo Pérez


Dr. Gerardo Pérez


Introduction to the Stiffness (Displacement) Method Definition of the Stiffness Matrix

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