Stiffness Methods For Systematic Analysis Of Structures

Stiffness Methods for Systematic Analysis of Structures(Ref: Chapters 14, 15, 16)The Stiffness method provides a very systematic way of analyzing determinate and indeterminatestructures.RecallForce (Flexibility) Method Convert the indeterminate structure to adeterminate one by removing some unknownforces / support reactions and replacing themwith (assumed) known / unit forces. Using superposition, calculate the force thatwould be required to achieve compatibilitywith the original structure. Unknowns to be solved for are usuallyredundant forces Coefficients of the unknowns in equations tobe solved are "flexibility" coefficients.Displacement (Stiffness) Method Express local (member) force-displacementrelationships in terms of unknown memberdisplacements. Additional steps are necessary to determinedisplacements and internal forces Can be programmed into a computer, buthuman input is required to select primarystructure and redundant forces. Directly gives desired displacements andinternal member forcesExample:Overall idea: Express FM in terms of displacements of I and J Assemble ALL members and enforceEQUILIBRIUM to find displacements.StiffnessMethod Page 1 Using equilibrium of assembled members,find unknown displacements. Unknowns are usually displacements Coefficients of the unknowns are "Stiffness"coefficients. Easy to program in a computer

Member and Node Connectivity:Degrees of Freedom ( Kinematic Indeterminacy)StiffnessMethod Page 2

Global and Local (member) co-ordinate axesIn order to relate: Global displacements with Local (member) deformations, and Local member forces back to Global force equilibrium,we need to be able to transform between these 2 co-ordinate axes freely:Transformation of Vectors (Displacements or Forces) between Global and Local coordinatesStiffnessMethod Page 3

Local (Member) Force-Displacement RelationshipsThese LOCAL (member) force-displacement relationships can be easily established for ALLthe members in the truss, simply by using given material and geometric properties of thedifferent members.StiffnessMethod Page 4

ASSEMBLY of LOCAL force-displacement relationships for GLOBAL EquilibriumThe member forces that were expressed in the LOCAL coordinate system,cannot be directly added to one another to obtain GLOBAL equilibrium of the structure.They must be TRANSFORMED from LOCAL to GLOBAL and then added together toobtain the global equilibrium equations for the structure which will allow us to solve for theunknown displacements.StiffnessMethod Page 5

ASSEMBLY of LOCAL force-displacement relationships for GLOBAL EquilibriumNow ALL the member force-displacement relationships can beASSEMBLED (Added) together to get Global equilibrium:Note that "q" are forces on members, so to get forces on nodes we must take "-q".Each one of the 10 equations above must sum to ZERO for global equilibrium.StiffnessMethod Page 6

Solution of unknown displacements at "free dofs" and reactions at "specified dofs"Rearranging:StiffnessMethod Page 7

MATLAB Code for 2D Truss Analysis using the Stiffness MethodStiffnessMethod Page 8Input File

MATLAB Code for 2D Truss Analysis using the Stiffness Method (Continued)Calculation of Local and Global Element Stiffness MatricesStiffnessMethod Page 9

ExampleSupport at node 1 settles down by 25mm.Determine the force in member 2.AE 8x106 NScreen clipping taken: 4/9/2014 9:37 AMScreen clipping taken: 4/9/2014 9:37 AMScreen clipping taken: 4/9/2014 9:37 AMKglobal Kglobal Solution:Displacements:Reactions:Displacement of member 2Force in Member 2StiffnessMethod Page 10

Inclined Support ConditionsSometimes, the support conditions are not oriented along global x-y axis.In these cases, one must transform specific components of the global equilibrium equations tomatch the orientation of the inclined supports so that the boundary conditions can be enforcedcorrectly.4mExample3mDegrees of freedom 3 and 4 need to be rotated to 3'' and 4''StiffnessMethod Page 11

ExampleFind displacements and reactions.Assume EA 1KGSolution:KG 'StiffnessMethod Page 12

Effect of Temperature Changes and Fabrication ErrorsChanges in lengths of truss members due to temperature or fabrication errorscan also be accommodated in the analysis by applying equivalent nodal forcesthat would result from these changes.If a member has change in length L (either due to fabrication error or due totemperature L α T L) then the equivalent nodal forces that will need to beapplied to the truss will be:StiffnessMethod Page 13

ExampleMember 2 is too short by 0.01 m.Determine the force in member 2.AE 8x106 NKglobal SolutionForce in member 2:StiffnessMethod Page 14

Space (3D) Truss AnalysisFor space (3D) trusses, all the same concepts of 2D truss analysis still hold.The main differences are: 3 dofs per node Transformation matrix becomes 3x3Coordinate TransformationStiffnessMethod Page 15

ExampleStiffnessMethod Page 16

Stiffness method for BeamsThe overall methodology of the stiffness methods is still thesame for problems involving beams:1. Define the geometry of the problem in terms of nodes and elements2. Set up the degrees of freedom: transverse displacements and rotations at nodes3. Define the loading and boundary conditions as externally applied forces and moments, anddegrees of freedom that are fixed / specified.4. Set up element force-displacement relations qM KM . dM(local and global coordinate systems are the same)5. Assemble forces and moments from all elements in terms of unknown globaldisplacements and rotationsSolve by partitioning the free and specified degrees of freedom as usual.Nodes Elements and Degrees of FreedomStiffnessMethod Page 17

Element force-displacement relationshipStiffnessMethod Page 18

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Sample MATLAB codeStiffnessMethod Page 22

PlottingElement CalculationsStiffnessMethod Page 23

Assembly and Global solutionKA KB Assembly of global stiffness matrix:KG Solution:StiffnessMethod Page 24Load:

ExampleSupport B settles by 1.5 in.Find the reactions and draw the Shear Force andBending Moment Diagrams of the beam.E 29000 ksi ; I 750 in4K1 K2 K3 Assembled Kglobal Solution:StiffnessMethod Page 25

Distributed Loads along the length of the elementBeams with distributed loads along the lengthcan be solved by the stiffness method usingfixed-end moments as follows:ExampleDetermine reactions.E 29000 ksi; I 510in4K2 K3 Global system to solve:StiffnessMethod Page 26

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Stiffness Method for Frame StructuresFor frame problems (with possibly inclined beam elements),the stiffness method can be used to solve the problem bytransforming element stiffness matricesfrom the LOCAL to GLOBAL coordinates.Note that in addition to the usual bending terms, we will alsohave to account for axial effects. These axial effects can beaccounted for by simply treating the beam element as a trusselement in the axial direction.StiffnessMethod Page 28

Transformation from Local to Global coordinatesEach node has 3 degrees of freedom:ButThus transformation rules derived earlier for trussmembers between (X, Y) and (X', Y') still hold: QrotTNote:Transformation matrix T definedabove is the same as QrotT definedin the provided MATLAB code.QrotConverting Local co-ordinates to Global:(Qrot)(Qrot)(Qrot)(Qrot)StiffnessMethod Page 29

Element Stiffness Matrix in GLOBAL coordinates:Substituting the transformation relations (l) and (2) intoLOCAL force (moment) - displacement (rotation) relationships (L):TmatrixT (in MATLAB code)(QrotT)Tmatrix (in MATLAB code)(Qrot)(QrotT)Thus, similar to trusses:Example:StiffnessMethod Page 30(Qrot)

Frame 2D MATLAB Code:StiffnessMethod Page 31

PlottingStiffnessMethod Page 32

Frame Element CodeScreen clipping taken: 4/30/2014 8:40 AMStiffnessMethod Page 33

Example:Element Stiffness matrices:(local coordinates)StiffnessMethod Page 34(Global Co-ordinates)

Global structural stiffness matrix (15 15) :Solution:StiffnessMethod Page 35

Stiffness Method for Frame Structures For frame problems (with possibly inclined beam elements), the stiffness method can be used to solve the problem by transforming element stiffness matrices from the LOCAL to GLOBAL coordinates. Note that in addition to the usual bendin