2 DEFINITION OF SPRING ELEMENT kWhat is the relation between forces acting on spring ends and displacements of these ends? Or, using FEA terminology, what is the relation between nodal displacements and forces acting on these nodes?To formulate this relation we must first assume displacement pattern in-between nodes. Let’s assume linear displacement:We assume linear displacement along the length of element. This is our displacement interpolation function.0 ≤ x ≤ LThe displacement function has as many coefficients as the element has degrees of freedom, in this case 2
3 DEFINITION OF SPRING ELEMENT We now want to express displacement u along the element as a function of nodal displacementsThis is the relation between nodal displacements and displacements along the length of elements (i.e. in-between nodes)
4 DEFINITION OF SPRING ELEMENT The same can be written in matrix form:whereare called shape functions
5 DEFINITION OF SPRING ELEMENT kLet’s say that element experiences force TThis force causes nodal displacementsandRelation between force and nodal displacementsForce acting on node 1Force acting on node 2This matrix relates nodal displacements of spring element to forces acting on nodes of that element.This matrix is called element stiffness matrix
6 DEFINITION OF SPRING ELEMENT Element stiffness matrix describes relation between nodal loads and nodal displacement of an element.This is matrix is:SquareSymmetricSingular (determinant equals zero)Positive Diagonal Terms
7 F2 F2 ONE SPRING THIS IS CALLED A BAR ELEMENT 1 .2.F2Find displacements of point 2; find reactions in point 1 when force F2 is acting on node 2This problem lends itself well toward discretization with one spring element:1 .2.F ReactionF2F2
8 . 3 F3 k1 k2 TWO SPRINGS IN SERIES 1 .. 32 .F3Find displacements of point 3; find reactions in points 1 and 2This problem lends itself well toward discretization with two spring elements:Solution stepsFormulate stiffness matrix for individual elementsExpand individual stiffness matrices so that they are associated with all degrees of freedom in the systemAssemble global stiffness matrixSolve for displacements and reaction forces
9 TWO SPRINGS IN SERIESRelation between nodal displacements and forces in element 1Relation between nodal displacements and forces in element 2or
10 TWO SPRINGS IN SERIESRelation between nodal displacements and forces in element 1int he expanded formexpandedRelation between nodal displacements and forces in element 2 in the expanded formexpanded
11 TWO SPRINGS IN SERIESExpanded matrices of elements 1 and 2 can now be added to form the global stiffness matrixRelation between nodal displacements and nodal forces has been formed:Global stiffness matrixVector of nodal displacementsVector of nodal loadsThese matrixes been expanded before global stiffness matrix could be assembled.The process in which individual matrices are expanded and then added to form global stiffness matrix is called Direct Stiffness Method also known as the displacement method or matrix stiffness method
12 Direct Stiffness Method TWO SPRINGS IN SERIESThe process in which individual matrices are expanded and then added to form global stiffness matrix is calledDirect Stiffness Methodalso known as the displacement method or matrix stiffness method
13 TWO SPRINGS IN SERIES In our case node 1 and node 2 are fixed: This is A set of three equations with three unknowns: d3, F1, F2
14 THREE SPRINGS IN SERIES k1k2k31 ..34 .2 .F1F2Find displacements of points 3 and 4; find reactions in points 1 and 2.This problem lends itself well toward discretization with three spring elements.Solution stepsFormulate stiffness matrix for individual elementsExpand individual stiffness matrices so that they are associated with all degrees of freedom in the systemAssemble global stiffness matrixSolve for displacements and reaction forces
15 THREE SPRINGS IN SERIES Note that:Therefore these matrixes must be expanded before global stiffness matrix can be assembled.
16 THREE SPRINGS IN SERIES Stiffness matrix of element 1 before expansionk1k2k31 ..34 .2 .And after expansionF1F2
17 THREE SPRINGS IN SERIES Stiffness matrix of element 2 before expansionk1k2k31 ..34 .2 .And after expansionF1F2
18 THREE SPRINGS IN SERIES Stiffness matrix of element 3 before expansionk1k2k31 ..34 .2 .F1F2And after expansion
19 THREE SPRINGS IN SERIES Stiffness matrices can now be added to assemble global stiffness matrixTherefore, the relation between all nodal displacements and all nodal forces is:Vector of nodal loadsVector of nodal displacementsGlobal stiffness matrix
20 THREE SPRINGS IN SERIES Known nodal displacements (displacements boundary conditions):d1=0 d2=0Known nodal loads (force boundary conditions):f3 = - F1 f4 = F2This is a set of four linear algebraic equations with fours unknowns:d3, d4 nodal displacementsf1, f3 reaction forces
21 THREE SPRINGS IN SERIES/PARALLEL Before expansion:Element 1Element 2Element 3
22 THREE SPRINGS IN SERIES/PARALLEL And after expansion
23 STIFFNESS MATRIX CHARACTERISTICS SymmetricThis means that kij = kji. This is always the case when the displacements are directly proportional to the applied loads.SquareThe number of rows are equal to the number of columns in the matrix.SingularThe element stiffness matrix is singular (the determinate of the matrix is equal to zero) since no constraints (prescribed displacements and/or rotation) have been applied.Positive Diagonal TermsAll the terms in the main diagonal (upper left to lower right) must be positive. If kii is negative then the force and it's corresponding displacement would be oppositely directed, which is physically unreasonable. If kii = 0, then the displacement would produce no reaction force resisting it, which would imply that the structure is unstable.
24 WHY IS STIFFNESS MATRIX SYMMETRIC? BETTI’S THEOREM* (RECIPROCITY THEOREM)Betti’s theorem, discovered by Enrico Betti in 1872 states that for all linear elastic structures subject to two sets of forces Pi and Qi, the work done by the set P though the displacement produced by set Q is equal to the work done be the set Q through displacements produced by set P.*A theorem is a statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms
25 WHY IS STIFFNESS MATRIX SYMMETRIC? Betti’s theorem exampleConsider a beam on which two points 1 and 2 have been defined. First we apply force P at point 1 and measure the vertical displacement of point 2. Then we remove force P and apply force Q at point 2.Betti’s reciprocity theorem states that:PQ12ΔP1ΔQ1
26 Number of unknowns equals the number of equations WHY IS STIFFNESS MATRIX SQUARE?Number of unknowns equals the number of equations
27 WHY IS STIFFNESS MATRIX SINGULAR? Displacement boundary conditions have not yet been defined
28 WHY ARE ALL TERMS ON THE MAIN DIAGONAL MUST BE POSITIVE? Negative terms would indicate negative stiffness.Zero terms would indicate no stiffness in given direction.
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