# Chapter1 - Finite Element Method in Structural Mechanics...

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Finite Element Method in Structural Mechanics Lecture notes for ENGR 653 by Dr. K. H. Ha Professor of Engineering Concordia University ©Copyright by K. H. Ha, 1994

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Chapter 1 1 Chapter 1 Fundamental Concepts 1. System Stiffness Equation At the system level: R N × 1 = K N × N r N × 1 + R o N × 1 (1.1) where R = system nodal force vector, K = system stiffness matrix, r = system nodal displacement vector, R o = equivalent nodal force vector, N = number of system’s degrees of freedom (sDOF). The primary unknown r may be solved from (1) after imposing the boundary conditions. Questions : 1. How is Eq.1.1 useful? Ans : With K , R and R o known, we use Eq.1.1 to compute the nodal displacements r and then the elements' stresses. 2. What is r ? Ans : It is the set of N nodal displacements capable of describing any arbitrary deformed configuration of the system. When a particular set of r is found for given sets of R and R o , the solution is nearly complete. 3. How is the system stiffness matrix K established? Ans: It is assembled from the elements' stiffness matrices k . The derivation of the element stiffness matrix k is a major objective of finite element theory. 3. Where does R come from? Ans: R is the vector of external forces applied directly at the nodal points. 4. Where does R o come from? Ans: R o is the vector of nodal forces that is equivalent to externally applied non-nodal influences such as elements' thermal expansion or forces applied on the elements. 5. How is 'equivalent' defined? Ans : 'Equivalent' is defined in terms of energy. When the system is subject to an arbitrary deformation mode defined by the nodal displacements δ r , the work done by (- R o ) is equal to the work done by the non-nodal forces. This definition allows R o to be found. In finite element terminology, R o is called the consistent load vector . 6. What is meant by "imposing the boundary conditions"?
Chapter 1 2 Ans : Since r and R may be prescribed at certain (boundary) nodal points, these prescribed values are the boundary conditions that must be imposed on the solution of Eq.1.1. Note that whenever r i is prescribed, R i is unknown, and whenever R j is prescribed, r j is unknown. 7. Why does Eq.1.1 represent the set of nodal equilibrium equations? Ans : The left-hand-side is the set of externally applied nodal forces. The right-hand-side is the set of elements' internal nodal forces which are caused by the nodal displacements r and the non-nodal effects. The balance between internal forces and external forces at the nodes assures nodal equilibrium. 8. I am still lost! The above is a very brief review of the subject of matrix analysis of structures. Ignore it for the moment, proceed to Section 2, study Example 1.1, and then read it again. 2. Principle of Virtual Work Rewrite Eq.1.1 as: ( R R o ) = Kr (2.1) The left-hand-side may be viewed as the combined external nodal forces, and the right-hand- side as the internal nodal forces caused by r alone.

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Chapter1 - Finite Element Method in Structural Mechanics...

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