Part 3 Trusses （构架）

1. Part 3Trusses（构架） Objective: to know the basic concepts in finite element formulation of trusses and ANSYS program • Definition of a truss • Finite element formulation • Space trusses • ANSYS program • Examples using ANSYS • Verification of results

2. 3.1 Definition of a truss • What is a truss? An engineering structure consisting of straight members connected at their ends by means of bolts, rivets, pins, or welding. Plane truss: members lie in a single plane, and as well as forces. Weights of members are negligible compared to those of the applied loads. If weights are considered, half weight of each member is applied to the connecting joints.

3. Examples of statically determinate and statically indeterminate problem

4. 10-bar planar truss structure 15-bar planar truss structure

5. 25-bar spatial truss structure

6. 2.2 Finite element formulation（公式化） • A single member when it is subjected to force F, how about the deflection? • The average stresses in any two-force member are given by Combining above equations A two-force member subjected to a force F

7. • A small balcony truss with five nodes and six elements is as follows. • From this truss, consider isolating a member with an arbitrary orientation. • Two frames of reference are required to describe truss problems: A global coordinate system and a local frame of reference • A fixed global coordinate system, XY(1) to represent the location of each joint (node), and to keep track of the orientation of each member (element), using angle such as .

8. • To apply the constraints and the applied loads in terms of their respective global components • To represent solution: the displacement of each joint in global directions • The relationship between the local (element) descriptions and the global descriptions

9. • The relationship between global and local displacements In matrix form: {U}=[T]{u} where {U} and {u} represent the displacements of nodes i and j with respect to the global XY and the local xy frame of references, respectively. [T] is the transformation matrix.

10. • In a similar way, the local and global forces may be related according to the equations • In matrix form: {F}=[T]{f} where the matrix elements are components of forces acting at nodes i and j with respect to global coordinates and local components of the forces at nodes i and j

11. • For the problem, the displacements and the forces in the local y-direction are zero. • The forces and displacements act only in the local x-direction. • The relationship between the local internal forces and displacements through the stiffness matrix So, {f}=[K]{u}

12. After substituting for {f} and {u} in terms of {F} and {U} is the inverse of the transformation matrix [T] So, The relationship between the applied forces, the element stiffness matrix and the global deflection of the nodes of an arbitrary element

13. The stiffness matrix • Assembling or connecting • Elemental stiffness matrices • Applying boundary conditions and loads • Solving fro displacements • Obtaining other information, such as normal stresses

14. Example 3.1 • Determine the deflection of each joint under the loading

15. • Preprocessing phase 1. Discretize the problem into nodes and elements 2. Assume a solution that approximates the behavior of an element

16. 3. Develop equations for elements Figure 3.7

30. To obtain

33. 3.3 Space truss • A three-dimensional truss is often called a space truss

34. • Global displacement of an element • Orientation of a member where

35. Solution process the same as the 2D truss

36. 3.4 ANSYS program • Entering ANSYS – ANSYS Product Launcher • Unix platform Dialog box

37. • Working directory • Jobname: be used as the prefix • Go to Run button • Program Organisation • Two level: Begin level (gateway) and Processor level The organisation of ANSYS

38. • Three distinct steps: • Processing: Using the PREP7 processor, you provide data such as the geometry, materials, and element type to the program • Solution: Using the Solution processor, you can define the type of analysis, set boundary conditions, apply loads, and initiate finite element solutions • Postprocessing: Using POST1 (for static or steady-statuc problem) or POST26 (for transient problems), you review the results of your analysis through graphical displays and tabular listings.

39. • The Graphical User Interface (GUI) – to communicate with ANSYS • Layout of the GUI: six main regions, or windows as shown in figure

40. • Utility Menu: Contains utility functions that are available throughout the ANSYS session, such as file controls, selecting, and graphics controls. Exit • Main Menu: Contains the primary ANSYS functions – left mouse used • Toolbar: Conduct commands and functions • Input window: Allow you to type in commands directly All previously type-in commands also appear in this window for easy reference and access • Graphics window: A window shows drawn graphics • Output window: Receives text output from the program

41. 3.5 Example using ANSYS • ANSYS offers two types of elements for the analysis of trusses: LINK1 and LINK8 1) LINK1 a 2D spar – two nodes and two degrees of freedom (UX, UY) for plane truss problems 2) LINK8 a 3D soar – 3 degrees of freedom (UX, UY, UZ) at each node • To determine the deflection of each joint under the loading shown in the figure

42. 3.6 Verification of Results • There are various ways to verify the findings 1. Check the reaction forces To obtain the reaction forces and the external forces to check for statics equilibrium 2. The sum of the forces at each node should be zero 3. Pass an arbitrary section through the truss

43. Summary Should know • the underlying assumptions in truss analysis • the significance of using global and local coordinate system in describing problem, a transformation matrix • the difference between the elemental stiffness matrix and global stiffness matrix, and to know how to assemble elemental matrices to obtain a truss’s global stiffness matrix • how to apply the BC and loads to a global matrix to obtain the nodal displacement solution • how to obtain the internal forces and stresses in each member • how to use the ANSYS software: pre, sol and post • how to verify the results of the truss analysis

44. Tutorial Problems Chapter 3:3, 5, 6, 8

45. Part 4Axial Members, Beams and Frames Objective: to know the analysis of members under axial loading, beams and frames • Members under axial loading • Beams • Finite element formulation of beams • Finite element formulation of frames • 3D beam element • An example using ANSYS • Verification of results

46. FEA of Beam to Column Bolted Connections Extended end plate connection Connection loading

47. Elevation on the Testing Frame

48. Mesh FEA supports and loading

49. von Mises stress contours E2

50. FEA deformed mesh model E3 Test E3 just prior to bolt failure