A Recap of Stiffness by Definition and the Direct Stiffness Method

1. A Recap of Stiffness by Definition and the Direct Stiffness Method

2. Farther Down the Yellow Brick Road.. Virtual Work � works best for Determinate Structures Force Method � also known as the Method of Consistent Displacements Slope Deflection � uses compatibility equations How does a computer program like SSTAN or LARSA calculate deflections and stresses in a structural system? Numerical efficiency demands subroutines that are capable of looking at any structure. Today, I will introduce the Direct Stiffness Method for Truss Systems. Next week, when Dr Gurley returns, he will delve into the beam method. The beam method is very similar to today�s lecture because it only adds rotations at the joints to the mix.Virtual Work � works best for Determinate Structures Force Method � also known as the Method of Consistent Displacements Slope Deflection � uses compatibility equations How does a computer program like SSTAN or LARSA calculate deflections and stresses in a structural system? Numerical efficiency demands subroutines that are capable of looking at any structure. Today, I will introduce the Direct Stiffness Method for Truss Systems. Next week, when Dr Gurley returns, he will delve into the beam method. The beam method is very similar to today�s lecture because it only adds rotations at the joints to the mix.

3. Our Emphasis This Week: Trusses..

4. Stiffness Kij = the amount of force required at i to cause a unit displacement at j, with displacements at all other DOF = zero A function of: System geometry Material properties (E, I) Boundary conditions (Pinned, Roller or Free for a truss) NOT a function of external loads

5. From Strength of Materials..

6. Go to the Board..

7. From Stiffness by Definition We can create a stiffness matrix that accounts for the material and geometric properties of the structure A square, symmetric matrix Kij = Kji Diagonal terms always positive The stiffness matrix is independent of the loads acting on the structure. Many loading cases can be tested without recalculating the stiffness matrix Stiffness by Definition only uses a small part of the information available to tackle the problem Before the break, Dr Gurley introduced a new method to you called Stiffness by Definition. Unlike classical methods, a large component of the overall calculation could be created and reused. Once, you know the geometry, boundary conditions and material properties, you can construct a stiffness matrix that is independent of any loading. Remember virtual work and the force method? The loading was entirely germane to the calculation. Not here. Stiffness by Definition opened up new doors to efficient calculation. But, it only uses a portion of the information available to solve the problem. We have to rely upon other methods to find reactions and shear & moment diagrams.Before the break, Dr Gurley introduced a new method to you called Stiffness by Definition. Unlike classical methods, a large component of the overall calculation could be created and reused. Once, you know the geometry, boundary conditions and material properties, you can construct a stiffness matrix that is independent of any loading. Remember virtual work and the force method? The loading was entirely germane to the calculation. Not here. Stiffness by Definition opened up new doors to efficient calculation. But, it only uses a portion of the information available to solve the problem. We have to rely upon other methods to find reactions and shear & moment diagrams.

8. Stiffness by Definition Only Considers.. To reiterate the point: Stiffness by Definition only solves the unknown displacements. The rest is up to you. Keep in mind, for every DOF, we get one equation to work with. Wouldn�t it be nice if we had more equations that exposed the movement or lack thereof of any node with the loads acting on the structure and the reactions. What if we had a method that could tell us everything.. We do.. To reiterate the point: Stiffness by Definition only solves the unknown displacements. The rest is up to you. Keep in mind, for every DOF, we get one equation to work with. Wouldn�t it be nice if we had more equations that exposed the movement or lack thereof of any node with the loads acting on the structure and the reactions. What if we had a method that could tell us everything.. We do..

9. A Better Method: Direct Stiffness In Stiffness by Definition, we examine the Degrees of Freedom to build a square matrix with a size equal to the unfrozen DOFs. Every node in a truss can potentially travel in the x or y direction. By including all of the nodes, our stiffness matrix swells into a square matrix with a size equal to twice the amount of nodes on the truss system. Remember: For a known displacement, we will always have a reaction. If something is holding the DOF in place, then it will equal zero.In Stiffness by Definition, we examine the Degrees of Freedom to build a square matrix with a size equal to the unfrozen DOFs. Every node in a truss can potentially travel in the x or y direction. By including all of the nodes, our stiffness matrix swells into a square matrix with a size equal to twice the amount of nodes on the truss system. Remember: For a known displacement, we will always have a reaction. If something is holding the DOF in place, then it will equal zero.

10. A Simple Comparison In Stiffness by Definition, you build the stiffness matrix for the two released DOFs and calculate the loading scenario acting on those nodes. In Direct Stiffness, you build a stiffness matrix and loading matrix for the entire structure�all nodes�and calculate the unknown displacements. Using those displacements, you find the reactions.In Stiffness by Definition, you build the stiffness matrix for the two released DOFs and calculate the loading scenario acting on those nodes. In Direct Stiffness, you build a stiffness matrix and loading matrix for the entire structure�all nodes�and calculate the unknown displacements. Using those displacements, you find the reactions.

11. Node Naming Convention In Stiffness by Definition, you build the stiffness matrix for the two released DOFs and calculate the loading scenario acting on those nodes. In Direct Stiffness, you build a stiffness matrix and loading matrix for the entire structure�all nodes�and calculate the unknown displacements. Using those displacements, you find the reactions.In Stiffness by Definition, you build the stiffness matrix for the two released DOFs and calculate the loading scenario acting on those nodes. In Direct Stiffness, you build a stiffness matrix and loading matrix for the entire structure�all nodes�and calculate the unknown displacements. Using those displacements, you find the reactions.

12. Stiffness by Definition vs Direct Stiffness

13. The Fundamental Procedure

14. To continue.. You need your Direct Stiffness � Truss Application Handout to follow the remaining lecture. If you forgot it, look on your neighbor�s, please I have your new homework (if you don�t have it already) Go to http://www.ce.ufl.edu/~kgurl for the handout

15. Overview Our goal at present is to simplify the structure. We do this by subdividing the system into a set of finite elements. For the truss, the finite element under consideration will be a single element with endpoints or nodes.Our goal at present is to simplify the structure. We do this by subdividing the system into a set of finite elements. For the truss, the finite element under consideration will be a single element with endpoints or nodes.

16. Element Stiffness Matrix in Local Coordinates Think of the element as two nodes connected by a spring at rest. If you move node 1 towards node 2, the spring will enter compression. If you move node 1 away from node 2, the spring will enter tension. Before the structure is loaded, the element is neither in compression nor tension. After the structure is loaded, the nodes will move. The amount of stress in the member is directly proportional to the difference in the new element length minus the original member length. With trusses, this is all that we need to consider for the scope of the course. How do we measure the stiffness of the member in question? We use a matrix to describe a pair of equations that hold the relationship between the internal force, the stiffness and the displacement. Explain how to sum forces�.Think of the element as two nodes connected by a spring at rest. If you move node 1 towards node 2, the spring will enter compression. If you move node 1 away from node 2, the spring will enter tension. Before the structure is loaded, the element is neither in compression nor tension. After the structure is loaded, the nodes will move. The amount of stress in the member is directly proportional to the difference in the new element length minus the original member length. With trusses, this is all that we need to consider for the scope of the course. How do we measure the stiffness of the member in question? We use a matrix to describe a pair of equations that hold the relationship between the internal force, the stiffness and the displacement. Explain how to sum forces�.

17. Element Stiffness Matrix in Local Coordinates Think of the element as two nodes connected by a spring at rest. If you move node 1 towards node 2, the spring will enter compression. If you move node 1 away from node 2, the spring will enter tension. Before the structure is loaded, the element is neither in compression nor tension. After the structure is loaded, the nodes will move. The amount of stress in the member is directly proportional to the difference in the new element length minus the original member length. With trusses, this is all that we need to consider for the scope of the course. How do we measure the stiffness of the member in question? We use a matrix to describe a pair of equations that hold the relationship between the internal force, the stiffness and the displacement. Explain how to sum forces�.Think of the element as two nodes connected by a spring at rest. If you move node 1 towards node 2, the spring will enter compression. If you move node 1 away from node 2, the spring will enter tension. Before the structure is loaded, the element is neither in compression nor tension. After the structure is loaded, the nodes will move. The amount of stress in the member is directly proportional to the difference in the new element length minus the original member length. With trusses, this is all that we need to consider for the scope of the course. How do we measure the stiffness of the member in question? We use a matrix to describe a pair of equations that hold the relationship between the internal force, the stiffness and the displacement. Explain how to sum forces�.

18. Element Stiffness Matrix in Local Coordinates Cont.. Kij = is the Force of Moment at DOF i required to hold a unit deflection at node j while all other DOFs are fixed. Kij = is the Force of Moment at DOF i required to hold a unit deflection at node j while all other DOFs are fixed.

19. Displacement Transformation Matrix Structures are composed of many members in many orientations We must move the stiffness matrix from a local to a global coordinate system

20. How do we do that? Meaning if I give you a point (x,y) in Coordinate System Z, how do I find the coordinates (x�,y�) in Coordinate System Z�

21. To change the coordinates of a truss.. Each node has one displacement in the local system concurrent to the element (v1 and v2) In the global system, every node has two displacements in the x and y direction

22. Displacement Transformation Matrix Cont.. The relationship between v and r is the vector sum: v1 = r1*cos Qx + r2*cos QY v2 = r3*cos Qx + r4*cos QY

23. Displacement Transformation Matrix Cont..

24. Force Transformation Matrix

25. Element Stiffness Matrix in Global Coordinates Let�s put it all together.. We know that the Internal force = stiffness * local displacement (S = k * v) Units: Force = (Force/Length) * Length local disp = transform matrix * global disp (v = a * r) Substitute local displacement Internal force = stiffness * transform matrix * global disp (S = k * a * r) Premultiply by the transpose of �a� aT * S= aT * k * a * r and substitute R = aT * S to get R = aT * k * a * r

26. Element Stiffness Matrix in Global Coordinates Cont.. is an important relationship between the loading, stiffness and displacements of the structure in terms of the global system

27. Element Stiffness Matrix in Global Coordinates Cont.. Let�s expand all of terms to get a Ke that we can use.

28. Element Stiffness Matrix in Global Coordinates Cont..

29. Assembly of the Global Stiffness Matrix (KG)

30. Assembly of the Global Stiffness Matrix (KG)

31. The Entire Local Stiffness Matrix in Global Terms

32. Assembly of the Global Stiffness Matrix (KG)

33. Solution Procedure

34. Solution Procedure cont..

35. Solution Procedure cont..

36. Solution Procedure cont..

37. Internal Member Force Recovery

38. Internal Member Force Recovery Cont..

39. Conclusion We solved Element Stiffnesses Unknown Displacements Reactions Internal Forces I will cover another example in the laboratory

40. Matrices.. Matrices allow the engineer to consolidate large quantities of information into a collection. Matrices are simple. Matrices are linear. You won�t see x2 or ln(x). Look to the slide.. EXPLAIN..Matrices allow the engineer to consolidate large quantities of information into a collection. Matrices are simple. Matrices are linear. You won�t see x2 or ln(x). Look to the slide.. EXPLAIN..

56. Stiffness by Definition vs Direct Stiffness