Introduction to the Stiffness (Displacement) Method Definition of the Stiffness Matrix
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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
Análisis por Elemento Finito: Julio-Diciembre 2015
General steps to derive the stiffness matrix for the spring element
Dr. Gerardo Pérez
Análisis por Elemento Finito: Julio-Diciembre 2015
local stiffness matrix for the element
Dr. Gerardo Pérez
Análisis por Elemento Finito: Julio-Diciembre 2015
This step applies for structures composed of more than one element such that
Dr. Gerardo Pérez
Análisis por Elemento Finito: Julio-Diciembre 2015
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
Análisis por Elemento Finito: Julio-Diciembre 2015
nonhomogeneous boundary conditions
Dr. Gerardo Pérez
Análisis por Elemento Finito: Julio-Diciembre 2015
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
Análisis por Elemento Finito: Julio-Diciembre 2015
Example 1
Dr. Gerardo Pérez
Análisis por Elemento Finito: Julio-Diciembre 2015
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
Análisis por Elemento Finito: Julio-Diciembre 2015
Example 3 Determine the nodal displacements, the forces in each element, and the reactions.
Dr. Gerardo Pérez
Análisis por Elemento Finito: Julio-Diciembre 2015
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
Análisis por Elemento Finito: Julio-Diciembre 2015
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
Análisis por Elemento Finito: Julio-Diciembre 2015
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
Análisis por Elemento Finito: Julio-Diciembre 2015
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