Finite Element Method An Introduction (One Dimensional Problems) by Tarun Kant tkant@civil.iitb.ac.in www.civil.iitb.ac.in/~tkant Department of Civil Engineering Indian Institute of Technology Bombay Powai, Mumbai 400076
Elastic Spring 1
Elastic Spring One Spring: 2 degrees of freedom (dofs) We wish to establish a relationship between nodal forces and nodal displacements as: 2
Elastic Spring 3
Elastic Spring 4
Linear response superpose two independent solutions In matrix form, Nodal force vector; Nodal displacement vector 5
Two Springs 6
Two Springs 7
Two Springs 8
Two Springs 9
Two Springs 10
Some Properties of K 1. Sum of elements in any column is 0 equilibrium 2. K is symmetric 3. singular : no BC s 4. All terms on main diagonal positive. If this were not so, a +ive nodal force P i could produce a corresponding ive u i. 5. If proper node numbering is done, K is banded. 11
An Alternative Procedure 12
where, 1, 2, 3 :Global and Local node numbers 1, 2 :Element numbers Our aim is to compute a 3x3 K matrix of the two spring assemblage 13
Continuity (Compatibility) conditions Equilibrium of nodal forces 14
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Direct Stiffness Method -Assembly 18
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Axial Rod 20
Axial Rod Where k = stiffness of spring Direct Method Simple discrete elements 21
Variational Method -Energy Method 22
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Strains 25
Constitutive Relation Minimum Potential Energy The Total Potential Energy pi is given by: where, U is strain energy stored in the body during deformation and W is the work done by the external loads We have established In the above, we have discretized displacement and strain expressions 26
Variational Statement 27
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Uniform Load 29
Axial Rod To determine Governing Equation, 1. Equilibrium equation: 2. Strain-displacement relation: 3. Constitutive relation: Using the above relations, we have 30
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Weighted Residual Method We need to minimize the weighted residue in order to develop our appropriate method 32
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Thus, we obtain the discrete governing equation. 35
Transformation Local Coordinate p 1 A, E, L p 2 u 1 u 2 x AE AE p1 L L u1 p2 AE AE u2 L L 36
Transformation (contd.) In many instances it is convenient to introduce both local and global coordinates. The local coordinates are always chosen to represent an individual element. Global coordinates on other hand, are chosen for the entire system. Usually, applied loads, boundary conditions, etc. are described in global coordinates 37
Stiffness Equation in Local Coordinates in Expanded Form Y y q 2, v 2 Q 2, V 2 p 2 u 2 x q 1, v 1 2 P 2, U 2 P 1, U 1 θ x p 1 1 u1 Q 1, V 1 We have introduced q 1 and q 2 and v 1 and v 2 q 1 and q 2 do not exist since a truss element can not withstand a force normal to its axis. 38
Stiffness Equation in Local Coordinates in Expanded Form (Contd.) Y y q 2, v 2 Q 2, V 2 p 2 u 2 x q 1, v 1 2 P 2, U 2 P 1, U 1 θ x p 1 1 u1 Q 1, V 1 p1 1 0 1 0 u1 q 1 0 0 0 0 AE v1 p 2 L 1 0 1 0u 2 q 0 0 0 0 v 2 2 39
Stiffness Equation in Local Coordinates in Expanded Form (Contd.) In a compact form the matrix equation in a local coordinates can be expressed as p ku in which, p u p q p q 1 1 2 2 u v u v 1 1 2 2 t t 40
Transformation of Coordinates Transformation is required when local coordinates for description of elements change from element to element, e.g., 5 4 10 2 3 8 4 9 1 6 11 2 7 1 3 6 5 6 Nodes 11 Elements 41
Transformation of Coordinates Y (Contd.) y q 2, v 2 Q 2, V 2 p 2 u 2 x q 1, v 1 2 P 2, U 2 P 1, U 1 θ x p 1 1 u1 Q 1, V 1 At Node -1 p P cos Q 1 1 1 1 1 1 sin q -P sin Q cos 42 At Node - 2 p P cos Q 2 2 2 2 2 2 sin q -P sin Q cos
Transformation of Coordinates (Contd.) Let sin s and cos c, then combining the above equations, we can write, p1 c s 0 0 P1 q 1 s c 0 0 Q1 p 2 0 0 c s P 2 q 0 0 s c Q 2 2 in compact form : p T. P 43
Transformation of Coordinates (Contd.) in which, p P p q p q 1 1 2 2 P Q P Q 1 1 2 2 and T is called transformation matrix, which transforms the global nodal forces, P into local nodal forces, p. Since, displacements are also vectors like forces, a similiar transformation rule exists for them too, i. e., u= TU in which, u U u v u v 1 1 2 2 U V U V 1 1 2 2 t t t t 44
Transformation of Coordinates (Contd.) We can also write P = T -1 p Since T is an orthogonal matrix, T = T -1 t 45
Stiffness Equation in Global Coordinates P = T = = t p t T k u t T k T U = K U in which, t K T kt AE cs s cs s 2 2 L c cs c cs 2 2 cs s cs s 2 2 c cs c cs 2 2 46
Stiffness Equation in Global Coordinates (Contd.) 2 2 c cs c cs 2 2 AE cs s cs s K 2 2 L c cs c cs 2 2 cs s cs s This is stiffness matrix of the truss element as shown in the figure with reference to the global X, Y coordinates. The above formulation for K first appeared in : Turner MJ, Clough RW, Martin HC and Topp LJ (1956), Stiffness and deflection analysis of complex structures, J. Aeronautical Sciences, 23(9), 805-824. 47
Stiffness Equation in Global Coordinates (Contd.) A derivation of is given in, K without involving transformation matrix, T Martin HC (1958), Truss analysis by stiffness consideration, Trans. ASCE,123, 1182-1194. End 48