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Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16) The Stiffness method provides a very systematic way of analyzing determinate and indeterminate structures. Displacement (Stiffness) Method Express local (member) force-displacement relationships in terms of unknown member displacements.
Using equilibrium of assembled members, find unknown displacements.
- Unknowns are usually displacements Coefficients of the unknowns are "Stiffness" coefficients.
Recall Convert the indeterminate structure to a determinate one by removing some unknown forces / support reactions and replacing them with (assumed) known / unit forces.
Using superposition, calculate the force that would be required to achieve compatibility with the original structure.
Unknowns to be solved for are usually redundant forces
Coefficients of the unknowns in equations to be solved are "flexibility" coefficients.
Force (Flexibility) Method Directly gives desired displacements and internal member forces
- Easy to program in a computer Additional steps are necessary to determine displacements and internal forces
Can be programmed into a computer, but human input is required to select primary structure and redundant forces.
Example:
- Express FM in terms of displacements of I and J Assemble ALL members and enforce EQUILIBRIUM to find displacements.
Overall idea:
Degrees of Freedom ( Kinematic Indeterminacy) Member and Node Connectivity:
Local (Member) Force-Displacement Relationships These LOCAL (member) force-displacement relationships can be easily established for ALL the members in the truss, simply by using given material and geometric properties of the different members.
ASSEMBLY of LOCAL force-displacement relationships for GLOBAL Equilibrium The 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 to obtain the global equilibrium equations for the structure which will allow us to solve for the unknown displacements.
Solution of unknown displacements at "free dofs" and reactions at "specified dofs" Rearranging:
MATLAB Code for 2D Truss Analysis using the Stiffness Method Input File
Support at node 1 settles down by 25mm. Determine the force in member 2. AE = 8x10^6 N Example Screen clipping taken: 4/9/2014 9:37 AM Screen clipping taken: 4/9/2014 9:37 AM Screen clipping taken: 4/9/2014 9:37 AM Kglobal = Solution: Displacements: Reactions: Kglobal = Force in Member 2 Displacement of member 2
Inclined Support Conditions Sometimes, the support conditions are not oriented along global x-y axis. In these cases, one must transform specific components of the global equilibrium equations to match the orientation of the inclined supports so that the boundary conditions can be enforced correctly. Example 3m 4m Degrees of freedom 3 and 4 need to be rotated to 3'' and 4''
Effect of Temperature Changes and Fabrication Errors Changes in lengths of truss members due to temperature or fabrication errors can also be accommodated in the analysis by applying equivalent nodal forces that would result from these changes. If a member has change in length ∆L (either due to fabrication error or due to temperature ∆L= α ∆T L) then the equivalent nodal forces that will need to be applied to the truss will be:
Member 2 is too short by 0.01 m. Determine the force in member 2. AE = 8x10^6 N Example Solution Kglobal = Force in member 2:
Example
Stiffness method for Beams The overall methodology of the stiffness methods is still the same for problems involving beams:
- Define the geometry of the problem in terms of nodes and elements
- Set up the degrees of freedom: transverse displacements and rotations at nodes Define the loading and boundary conditions as externally applied forces and moments, and degrees of freedom that are fixed / specified.
- Set up element force-displacement relations q M = K M. d M (local and global coordinate systems are the same) Assemble forces and moments from all elements in terms of unknown global displacements and rotations
Solve by partitioning the free and specified degrees of freedom as usual. Nodes Elements and Degrees of Freedom
