Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method
Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix method for computing the member forces and displacements in structures DSM implementation is the base of most commercial and open-source finite element software Based on the displacement method (classical hand method for structural analysis) Formulated in the 1950s by Turner at Boeing and started a revolution in structural engineering
Institute of Structural Engineering Page 3 Goals of this Chapter DSM formulation DSM software workflow for linear static analysis (1 st order) 2 nd order linear static analysis linear stability analysis
Institute of Structural Engineering Page 4 Chapter 2a The Direct Stiffness Method: Linear Static Analysis (1 st Order)
Institute of Structural Engineering Page 5 Computational Structural Analysis Y X Physical problem Continuous mathematical model strong form Discrete computational model weak form Modelling is the most important step in the process of a structural analysis!
Institute of Structural Engineering Page 6 System Identification (Modelling) Y 3 4 6 4 1 3 5 2 1 2 X 5 6 Global coordinate system Nodes Elements Boundary conditions Loads Node numbers Element numbers and orientation
Institute of Structural Engineering Page 7 Deformations System Deformations Nodal Displacements System identification nodes, elements, loads and supports deformed shape (deformational, nodal) degrees of freedom = dofs
Institute of Structural Engineering Page 8 Degrees of Freedom Truss Structure Frame Structure u i u i u i = ( u dx, u dy ) dof per node u i = ( u dx, u dy, u rz ) 7 * 2 = 14 dof dof of structure 8 * 3 = 24 dof
Institute of Structural Engineering Page 9 Elements: Truss 1 dof per node u x DX P 1 P 2 N X/Y = local coordinate system u x = displacement in direction of local axis X P 1 2 1 P 2 1 1 2 2 1 2 DX = displacement of truss end compatibility const. equation equilibrum 2 1 p = k u p : (element) nodal forces k : (element) stiffness matrix u : (element) displacement vector
Institute of Structural Engineering Page 10 Elements: Beam 3 dof per node u x DX u y DY u y RZ u x = displacement in direction of local axis X u y = displacement in direction of local axis Y k u
Institute of Structural Engineering Page 11 Elements: Global Orientation local global = cos sin 0 0 0 0 sin cos 0 0 0 0 0 0 1 0 0 0 0 0 0 cos sin 0 0 0 0 sin cos 0 0 0 0 0 0 1-1 = T u glob = u = k glob = k = T u loc k loc
Institute of Structural Engineering Page 12 Beam Stiffness Matrix FX S = FY S = MZ S = FX S = FY S = MZ E = UX S UY S UZ S UX E UY E UZ E k 11 k 12 k 13 p p k 22 k 23 symm. is ie k 33 k 14 k 15 k 16 k 24 k 25 k 26 k 34 k 35 k 36 k 44 k 45 k 46 k 55 k 56 k 66 k iss k ise uis k k u ies iee p = k u ie E UX E =1 S FY S e.g. k 24 = reaction in global direction Y at start node S due to a Element stiffness matrix in global orientation unit displacement in global direction X at end node E
Institute of Structural Engineering Page 13 Nodal Equilibrum 3 6 f 4 r4: Vector of all forces acting at node 4 4 5 2 r4 = - k 6ES u 3 + contribution of element 6 due to start node displacement u 3 - k 6EE u 4 + contribution of element 6 due to end node displacement u 4 - k 5EE u 4 + contribution of element 5 due to start node displacement u 4 - k 5ES u 2 + contribution of element 5 due to start node displacement u 2 external load f 4 Equilibrum at node 4: r 4 = - k 5SE u 2 -k 6ES u 3 - k 5EE u 4 - k 6EE u 4 + f 4 = 0
Institute of Structural Engineering Page 14 Global System of Equations r 1 = - u 1 k 1EE + k 3SS + k 4SS u 2 u 3 u 4 k 3SE k 4SE + f 1 = 0 3 6 4 4 5 1 3 2 r 2 = - k 3ES k 2EE + k 3EE + k 5SS k 5SE + f 2 = 0 1 2 r 3 = - k 4ES k 4EE + k 6SS k 6SE + f 3 = 0 r 4 = - k 5ES k 6ES k 5EE + k 6EE + f 4 = 0 - K U + F = 0 F = K U
Institute of Structural Engineering Page 15 Global System of Equations F = global load vector = Assembly of all fe K = global stiffness matrix = Assembly of all ke U = global displacement vector = unknown F = K U = equilibrium at every node of the structure
Institute of Structural Engineering Page 16 Solving the Equation System What are the nodal displacements (= U ) for a given structure (= stiffness matrix K ) due to a given load (= load vector F )? K U = F left multiply K -1 K U = K -1 F K -1 U = K -1 F Inversion possible only if K is non-singular (i.e. the structure is sufficiently supported = stable)
Institute of Structural Engineering Page 17 Beam Element Results 1. Element nodal displacements Disassemble u from resulting global displacements U 2. Element end forces Calculate element end forces = p = k u 3. Element stress and strain along axis Calculate moment/shear from end forces (equilibrium equation) Calculate curvature/axial strain from moments/axial force 4. Element deformations along axis Calculate displacements from strain (direct integration)
Institute of Structural Engineering Page 18 1. Adjust global load vector Lateral Load 2. Adjust element stresses f = local load vector => add to global load vector F e.g. bending moment M: M due to u M due to f M diagram
Institute of Structural Engineering Page 19 Linear Static Analysis (1 st order) Workflow of computer program 1. System identification: Elements, nodes, support and loads 2. Build element stiffness matrices and load vectors 3. Assemble global stiffness matrix and load vector 4. Solve global system of equations (=> displacements) 5. Calculate element results Exact solution for displacements and stresses