1. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche University of Illinois-Chicago Chapter 8 TRUSSES-A Finite Element Approach

2. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.1 Introduction to Truss Analysis 8.1 Introduction to truss analysis Trusses are used in many engineering applications including bridges, buildings, and towers and support structures. In a truss, we are faced with a structure where the displacements, translations, or compressions of any truss member vary linearly with the applied forces. That is, any increment in displacement is proportional to the force causing it to deform. All deformations are assumed small, so that the resulting displacements do not significantly affect the geometry of the structure and hence do not alter the forces in the members. In this case Hooke’s Law is preserved and the theory of elasticity is used to search for solutions of the truss. Most often, the truss design requires that its member be tested for tension, compression, stress, and strain relations. The applied loads are then tested against the possible yield stress to determine their evaluated limits and the overall stability of the truss.

3. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.1 Introduction to Truss Analysis Figure 8.1 A planar truss subject to vertical loads (P 1, P 2 P 3 represent arbitrary) loads

4. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.2 Finite Element Formulation Trusses are typical structures in which the finite-element method can be best illustrated. We know that FEM relies on (a) discretizing the finite element of the system, (b) developing the mathematical relationships between the forces and displacements, stresses and strains, etc., for a given element, and (c) formulating the general problem through an assembly procedure of all the elements to solve the given problem

5. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.2 Finite Element Formulation or Figure 8.2 A typical truss element F1F1 F2F2 Consider an element of an arbitrary truss, as shown in Figure 8.1. It is subjected to either tension or compression, as is the case for all the truss elements. Let us label the element’s ends 1 and 2, and, consequently, call the corresponding forces F 1 and F 2 (Fig 8.2). (8.1) (8.2) (8.3)

6. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 and where The nodal force vector for the element The nodal displacement vector The element stiffness matrix 8.2 Finite Element Formulation or Relative displacement can be written as From Equation (8.3) (8.4) (8.5) (8.6) (8.7) (8.8)

7. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 we define the element force components as follows: Figure 8.3 A truss element making an angle  with the x-axis 8.2 Finite Element Formulation Let us consider an orientation of a truss element, as shown in Figure 8.3. (8.9) (8.10)

8. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.2 Finite Element Formulation Given that the relative displacement u is along the unit vector ν, then where and then by substitution of Equations (8.13) and (8.12) into Equation (8.11) and making use of Equations (8.3), (8.9), and (8.10), we obtain an expression for each nodal force in terms of the local displacement, the orientation  and the element stiffness K. (8.11) (8.13) (8.14) (8.15) (8.16) (8.17) (8.12)

9. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.2 Finite Element Formulation Writing Equations (8.14) through (8.17) in matrix form yields where s and c are abbreviations for sin  and cos , respectively, and k is the stiffness constant. We can write Equation (8.18) in more compact form as where and (8.18) (8.21) (8.19)

10. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 For  =0, The local stiffness matrix is simply 8.2 Finite Element Formulation Which checks with Equation (8.8). Note how the zero rows and columns are simply used to expand the local stiffness matrix given by Equation (8.8) to account for the zero forces and displacements along the y-axis. (8.22)

11. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.3 Properties of the Local Stiffness Matrix where we can see that the partitioned matrices A, B, C, and D are such that First, we observe that the local stiffness matrix is symmetric and that its coefficients are functions of cos  and sin . In addition, let the local stiffness be partitioned as follows: It is evident from the partition and this relationship stated above that we can deduce the following criteria to build the local stiffness: we only need to know sub matrix 1, and then sub matrix B is obtained by pre multiplying A by -A; C and D are then obtained from A and B, respectively. (8.23) (8.24)

12. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.4 Global Stiffness Matrix 8.4 GLOBAL STIFFNESS MATRIX The method that is illustrated in what follows to obtain the global stiffness matrix is one that Huston and Passerelo have developed. It shows how the building of the global stiffness matrix can be done by a simple strategy in which connectivity tables are used to identify the truss elements and their joints. The method is as follows. Step 1. Consider an arbitrary truss, as shown in Figure 8.4. First, label the truss elements and joints in an arbitrary fashion, as shown in Figure 8.4. There are five joints (1, 2, …, 5) and seven elements ([1], [2],…, [7]). Step 2. We proceed to develop three tables that basically store geometrical information about the truss. Table 8.1 has the truss-joint/matrix-column matching, where pairs starting from 1 develop the column numbers, 2.

13. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 Table 8.2 identifies the connection joints to all the elements of the truss. 8.4 Global Stiffness Matrix

14. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 TABLE 8.1 TRUSS –JOINT/MATRIX COLUMN MATRIX JointColumnNumbers 112 234 356 478 5910 TABLE 8.2 ELEMENTS Vs JOINT NUMBERS Truss element e-------------------------- N 1 ------------------------ N 2 112 213 334 414 524 645 725 8.4 Global Stiffness Matrix

15. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 TABLE 8.3 TRUSS ELEMENTS Vs k ij e i,j from k ij e 1 2Truss 3 Elements 4 E 5 6 7 1 1151 3 7 3 22262484 33577799 44688810 8.4 Global Stiffness Matrix

16. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 Where And 8.4 Global Stiffness Matrix (8.25) (8.26) (8.27) Global forces and displacements:

17. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.4 Global Stiffness Matrix where is the local element stiffness matrix given by (8.23) For  =90 (8.28) (8.29) Thus, the [K] in Equation (8.27) is the assembled global stiffness matrix obtained from the assembly of individual element stiffness matrices, that is,

18. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 For element 5 For element 2For element 1 8.4 Global Stiffness Matrix (8.33) The global stiffness is found to be :

19. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.4 Global Stiffness Matrix

20. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.5 Solution of the Truss Problem 8.5 SOLUTION OF THE TRUSS PROBLEM,,,. Thus, (8.35) (8.36) For the truss shown in Figure (8.4) we have: Roller joint at 2 in y-dir Force P in the x-dir at joint 5

21. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 (8.38) The displacement boundary conditions are: 8.5 Solution of the Truss Problem

22. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 (8.39) Substituting [F] and [U] from Equations (8.36) and (8.38) into Equation (8.25), we obtain the general equations governing the truss force/displacement equilibrium conditions. 8.5 Solution of the Truss Problem

23. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 (8.41)(8.40) Note how in Equation (8.39) the unknowns are in the global force array as well as in the joint displacement vector. A typical strategy to solve such a problem in which the unknowns are on both sides of the equation is to solve for the U’s first by partitioning the matrices such that the force vector is completely in terms of the known forces. Eliminating the reaction forces does the partitioning. The resulting equation is : 8.5 Solution of the Truss Problem

24. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 The solutions are found to be : Once the equations are solved for the displacements, reactions R 1x, R 1y, and R 2y can be evaluated by Pre-multiplying the corresponding terms of [K] and [U] in Equation (8.39). (8.42) 8.5 Solution of the Truss Problem

25. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 Figure 8.5 A simple free-body diagram of 2D truss subject to loading The answers obtained from the FEM analysis as described before can be checked by simply taking the free-body diagram for the truss as shown in Figure 8.5. 8.5 Solution of the Truss Problem

26. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 We get Writing the equilibrium equations (8.43) (8.44) Which checks with the FEM solution. 8.5 Solution of the Truss Problem

27. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces 8.6 EVALUATION OF THE LOCAL FORCES The internal forces are those that are either compressing or causing the truss elements in tension. To find the components of the forces acting at each end, we use the previously computed global displacements and local stiffness matrix. From Equation (8.21), we have (8.45)

28. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces For joint 2:For joint 1: For joint 3: Let the local displacement be written as u e ij, where e is the element number of the truss, i denotes either end 1 or end 2 of the element, and j assigns the direction x or y to the end displacements. Assuming that all elements of the truss undergo the same displacements at each joint, we then write the following : (8.48) (8.46)(8.47)

29. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces For joint 4: For joint 5 The nodal displacements for any particular element can be found from the relationships between the global displacements and the local displacement by using Equation (8.46) to (8.50). Subsequent substitution of these values for a particular element in Equation (8.45) and multiplying by the corresponding stiffness matrix terms yield the nodal forces. The signs of these forces indicate whether the member is in tension or compression. (8.49) (8.50)

30. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 Example 8.1 Analysis of a Three-Element Truss Use the finite-element method to solve for the truss in the following figure a) Find the global stiffness matrix. b) Solve for the reaction forces. c) Solve for the member forces and determine whether a truss element is in tension or compression. 8.6 Evaluation of the Local Forces The first step in our analysis is to label the truss for the joint numbers and link numbers as shown in the Figure (8.6). The second step is to compute the local stiffness matrices for each member using Equation (8.21). Solution :

31. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 Figure 8.6 A three-element truss subject to loading 8.6 Evaluation of the Local Forces F 1y F 2y P = 100 lb F 1x

32. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces The element stiffness matrices are :

33. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces Where A 1,A 2,A 3 and L 1,L 2,L 3 are the areas of cross section and lengths of the members of the truss, respectively, and E is the young’s modulus of elasticity. And L 1 = 1, such that Now let us construct Tables 8.4 to 8.6 (see earlier Tables 8.1 to 8.3), which help in arriving at the global stiffness matrix K.

34. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 Table 8.4 Joint NumberColumn Number 112 234 356 Table 8.5 ElementsJoint Number 113 212 323 Table 8.6 K ij 123 1113 2224 3535 4646 8.6 Evaluation of the Local Forces

35. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces Using the local stiffness matrices and transferring the entries with the help of Table 8.6 we can arrive at the global stiffness matrix: Zeros in the force vector indicate that the forces in the x and y directions at joints 2 and 3 are zeroes because of the roller and free joint, respectively.

36. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces Applying the displacement boundary conditions, U 1x =0, U 1y =0, U 2y =0, and eliminating the corresponding rows and columns, we get Solving for the unknowns, we obtain

37. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces The reaction forces can be computed using Equation (8.43) as

38. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces These results can be verified using the free-body diagram of the truss. Therefore, members 1 and 3 are in tension, whereas member 2 is in compression.

39. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces The member forces are obtained from the local element force-displacement relationship: The global and local displacement at each joints are related as follows :

40. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.6 Evaluation of the Local Forces Using the local stiffness already computed and given by Equation (8.51) to (8.53), we obtain the element: The forces acting on element 1 clearly show that it is in tension as predicted. Similarly, we can obtain the magnitude of the forces and directions for elements 2 and 3.

41. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.7 Stress Analysis In the analysis of truss the main objective is to decide whether the truss elements are designed to sustain the load they support. For that we need to evaluate the stress or average stress in each element. The latter is an indication as to whether the tension or compression can be sustained. Let the element stress be given by Now we can compute the stress for each element of the truss in previous example 8.4. For each element we can write (8.51) (8.52)

42. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 And the results are 8.7 Stress Analysis

43. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.8 Force, and Displacement Incidence Matrices 8.8 Force and Displacement Incidence Matrices The global forces and local forces as well their corresponding global and local displacements can be shown to have special relationships that can be found by means of incidence matrices. These matrices are derived by examining the global and local displacement relation at the nodes and joints of the truss. Similarly the global or local displacement relation for joint 2 and 3 can be expressed as

44. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.8 Force and Displacement Incidence Matrices Writing the above relations in a matrix form yields Where

45. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 denotes the incidence matrix for element (e). Element 3 Element 2 Element 1 We can develop a relationship between the global forces and the local forces by noting that 8.8 Force and Displacement Incidence Matrices (8.53) (8.54)

46. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.8 Force and Displacement Incidence Matrices And from the global Force/Displacement relation We rewrite [F] as function of the local displacement by substituting equation (8.54) into the above equation The above global force represents the contribution of element e to the global force vector [F]. Hence the total force vector is obtained by summing the contribution of all the elements such that From the local force/displacement we rewrite the global force equation as (8.55) (8.56) (8.57) (8.58) (8.59)

47. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 where [k] -1 =[k] T. Let us define 8.8 Force and Displacement Incidence Matrices Substituting the local displacements as function of the incidence matrix and local forces we obtain Then we write the relationship between the local forces and global forces as For each element we can derive the corresponding local forces from the existing information on [F] used in the global formulation of the FE problems. This is done quite interactively if large codes are needed. (8.60) (8.61) (8.62) (8.63)

48. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.9 3-D Analysis of trusses The analysis of 3D truss is similar to the 2D case except the element stiffness must be developed for an arbitrary element in space. Consider such an element as shown in Figure 8.7 z y x    i Figure 8.7: The angle formed by a member with the x,y,z-axis. j

49. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 8.9 3-D Analysis of trusses The direction cosine with respect to each axis are given by where L is the length of the member and is given by (8.64) (8.65) (8.66) (8.67)

50. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 8 For a local element the force/displacement relation is given by 8.9 3-D Analysis of trusses Note how the nodes are represented by the subscripts 1 & 2 and the superscript denotes the element number. (8.68) (8.69) where