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Structure Analysis I Chapter 9

Structure Analysis I Chapter 9. Deflection Energy Method. Energy Method. External Work. When a force F undergoes a displacement dx in the same direction as the force, the work done is If the total displacement is x the work become. The force applied gradually. Energy Method.

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Structure Analysis I Chapter 9

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  1. Structure Analysis IChapter 9

  2. Deflection Energy Method Dr. Mohammed Arafa

  3. Energy Method External Work When a force F undergoes a displacement dxin the same direction as the force, the work done is If the total displacement is x the work become The force applied gradually

  4. Energy Method External Work If P is already applied to the bar and that another force F` is now applied The work done by P when the bar undergoes the further deflection ` is then

  5. The work of a moment is defined by the product of the magnitude of the moment M and the angle then If the total angle of rotation is the work become: The moment applied gradually Dr. Mohammed Arafa

  6. Energy Method Strain Energy – Axial Force N = internal normal force in a truss member caused by the real load L = length of member A = cross-sectional area of a member E = modulus of elasticity of a member

  7. Energy Method Strain Energy – Bending

  8. Principle of Virtual Work Work of External Loads Work of Internal Loads Virtual Load Real displacement

  9. Method of Virtual Work: TrussesExternal Loading 1 = external virtual unit load acting on the truss joint in the stated direction of n = internal virtual normal force in a truss member caused by the external virtual unit load  = external joint displacement caused by the real load on the truss N = internal normal force in a truss member caused by the real load L = length of member A = cross-sectional area of a member E = modulus of elasticity of a member

  10. Method of Virtual Work: Trusses Temperature • 1 = external virtual unit load acting on the truss joint in the stated direction of • n = internal virtual normal force in a truss member caused by the external virtual unit load •  = external joint displacement caused by the temperature change. •  = coefficient of thermal expansion of member • T = change in temperature of member • L = length of member

  11. Method of Virtual Work: Trusses Fabrication Errors and Camber 1 = external virtual unit load acting on the truss joint in the stated direction of n = internal virtual normal force in a truss member caused by the external virtual unit load  = external joint displacement caused by fabrication errors  L = difference in length of the member from its intended size as caused by a fabrication error. Dr. Mohammed Arafa

  12. Example 1 The cross sectional area of each member of the truss show, is A = 400mm2 & E = 200GPa. a) Determine the vertical displacement of joint C if a 4-kN force is applied to the truss at C

  13. Solution A virtual force of 1 kN is applied at C in the vertical direction

  14. Dr. Mohammed Arafa

  15. Example 2 Text book Example 8-14 Determine vertical displacement at C A = 0.5 in2 E = 29 (10)3 ksi

  16. Example 3

  17. Method of Virtual Work: Beam 1 = external virtual unit load acting on the beam in the stated direction of Δ m = internal virtual moment in a beam caused by the external virtual unit load Δ= external joint displacement caused by the real load on the truss M = internal moment in a beam caused by the real load L = length of beam I = moment of inertia of cross-sectional E = modulus of elasticity of the beam

  18. Method of Virtual Work: Beam Similarly the rotation angle at any point on the beam can be determine, a unit couple moment is applied at the point and the corresponding internal moment have to be determine

  19. Example 4 Determine the displacement at point B of a steel beam E = 200 Gpa , I = 500(106) mm4 Dr. Mohammed Arafa

  20. Solution

  21. Another Solution Real Load Virtual Load

  22. Example 5

  23. Example 6 Determine the Slope  at point B of a steel beam E = 200 Gpa , I = 60(106) mm4

  24. Solution Virtual Load

  25. Real Load

  26. Another Solution Real Load Virtual Load

  27. Example 7

  28. Example 8

  29. Example 9

  30. Example 4 Determine both the horizontal deflection at A

  31. Solution Real Load Virtual Load

  32. Dr. Mohammed Arafa

  33. Virtual Strain Energy Caused by: Axial Load n = internal virtual axial load caused by the external virtual unit load N = internal normal force in the member caused by the real load L = length of member A = cross-sectional area of the member E = modulus of elasticity of the material

  34. Virtual Strain Energy Caused by: Shear

  35. Virtual Strain Energy Caused by: Shear = internal virtual shear in the member caused by the external virtual unit load V = internal shear in the member caused by the real load G= shear modulus of elasticity for the material A = cross-sectional area of the member K = form factor for the cross-sectional area K=1.2 for rectangular cross sections K=10/9 for circular cross sections K=1.0 for wide-flange and I-beams where A is the area of the web.

  36. Virtual Strain Energy Caused by: Torsion

  37. Virtual Strain Energy Caused by: Torsion t= internal virtual torque caused by the external virtual unit load T= internal shear in the member caused by the real load G= shear modulus of elasticity for the material J = polar moment of inertia of the cross-sectional L = member length Dr. Mohammed Arafa

  38. Virtual Strain Energy Caused by: Temperature

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