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Chapter four Trusses

Chapter four Trusses. 4-1 INTRODUCTION A two dimensional truss A truss consists only of two-force members. in a truss it is required that all loads are applied only at the joints. 4-2 PLANE TRUSSES. Local and global coordinate systems:

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Chapter four Trusses

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  1. Chapter fourTrusses Introduction to Finite Elements in Engineering

  2. 4-1 INTRODUCTION A two dimensional truss A truss consists only of two-force members. in a truss it is required that all loads are applied only at the joints. Introduction to Finite Elements in Engineering

  3. 4-2 PLANE TRUSSES Local and global coordinate systems: a)a Local coordinate system b)a global coordinate system Introduction to Finite Elements in Engineering

  4. The Relationship between q’ and q is developed as follows: Transformation matrix: Formulas for calculating l and m: Introduction to Finite Elements in Engineering

  5. Element Stiffness Matrix: a)In local coordinate system: The truss element is a one-dimensional element when viewed in the local coordinate system. b)In global coordinate system: consider the strain energy in local coordinates: Introduction to Finite Elements in Engineering

  6. The strain energy in global coordinates: The element stiffness matrix in global coordinates: the element stiffness matrices are assembled to obtain the structural stiffness matrix. Introduction to Finite Elements in Engineering

  7. Stress calculations: Introduction to Finite Elements in Engineering

  8. Example 4.1 (a)Determine The element stiffness matrix for each element. (b) Assemble the structural stiffness matrix for the entire truss. (c)Using the elimination approach , solve for the nodal displacement. (d)Recover the stresses in each element. (e)Calculate the reaction forces. Introduction to Finite Elements in Engineering

  9. Element stiffness matrices for element 1: Introduction to Finite Elements in Engineering

  10. Element stiffness matrices for element 2,3 and 4: Introduction to Finite Elements in Engineering

  11. b)Assembling the structural stiffness matrix K: For example: Introduction to Finite Elements in Engineering

  12. (c)Using elimination approach: Boundary conditions are: The nodal displacement vector can therefore be written as: Introduction to Finite Elements in Engineering

  13. (d)Stress calculations: The stress in members 1 and 2 is given by: Introduction to Finite Elements in Engineering

  14. (e)Support reactions: We need to determine the reaction forces along dof’s 1,2,4,7 and 8,whitch correspond to fixed supports. R=KQ-F Introduction to Finite Elements in Engineering

  15. Temperature effects: since a truss element is simply a one-dimensional element when viewed in the local coordinate system , the element Temperature load in the local coordinate system is given by: Where the initial strain associated with a temperature change is given by: since the potential energy associated with Temperature load vector is the same in magnitude whether measured in the local or global coordinate systems , we have: Introduction to Finite Elements in Engineering

  16. Example 4.2 Determine the nodal displacement and element stresses as a result of temperature increase in bars 2 and 3. Introduction to Finite Elements in Engineering

  17. Solution: The stiffness matrix for the truss has already been developed in example 4.1 , but the temperature load vector needs to be assembled. Introduction to Finite Elements in Engineering

  18. The element stresses can now be obtained from eq. 4.24. for example: Introduction to Finite Elements in Engineering

  19. Three-dimensional trusses: Local and global coordinate systems Introduction to Finite Elements in Engineering

  20. Transformation matrix: Element stiffness matrix: Formulas for calculating l , m and n: Introduction to Finite Elements in Engineering

  21. Temperature effects: Introduction to Finite Elements in Engineering

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