External Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is

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1 Structure Analysis I Chapter 9

2 Deflection Energy Method

3 External Work Energy Method When a force F undergoes a displacement dx in the same direction i as the force, the work done is du e = F dx If the total displacement is x the work become U e x = F dx 0 1 U e = 2 P The force applied gradually

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

5 The work of a moment is defined by the product of the magnitude of the moment M and the angle dθ then du e = M dθ If the total angle of rotation is θ the work become: θ U e 0 = M dθ U e 2 1 = Mθ The moment applied gradually

6 Energy Method Strain Energy Axial Force σ = Eε σ = ε = N A L = NL AE P = N U = 1 P 2 2 N L U i = 2AE 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 dθ = U = 1 2 M dx EI Mθ du U i i = L = 0 2 M dx 2EI 2 M dx 2EI

8 Principle of Virtual Work P = Work of Etrnl External Loads u δ Work of Internal Loads Virtual Load 1. = u. dl Real displacement

9 Method of Virtual Work: Trusses External Loading 1. = u. dl 1. = n. NL AE 1 = external virtual unit load acting on the truss joint in the stated t 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. = n α T L 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. α = 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. = n L 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.

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 1. = n. NL AE A virtual force of 1 kn is applied at C in the vertical direction

14 Member n (KN) N (KN) L (m) nnl AB AC CB Sum nnl AE AE ( kn 1. = = = 6 2 ( m ) 6 ( kn / m = m = mm ) )

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

16

17

18 Example 3

19

20 Method of Virtual Work: Beam = 1. u. dl M d θ = dx EI L M 1.. = m EI dx 0 1 = t l i t l it l d ti th b i th t t d di ti f 1 = external virtual unit load acting on the beam in the stated direction of m = internal virtual moment in a beam caused by the external virtual unit load = 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

21

22 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 m θ L θ 1( KN. m). θ = 0 m M EI dx

23 Example 4 Determine the displacement at point B of a steel beam E = 200 Gpa, I = 500(10 6 ) mm 4

24 Solution x dx x dx x x dx M m L 6 6 ) 6 ( ) 1 ( = = = = m EI EI EI dx EI m 0 15 ) 15(10 ) 15( m EI 0.15 ) )(10 500(10 ) 200(10 ) ( ) ( = = =

25 Another Solution Real Load Virtual Load B B = = 200( m ) 500(10 6 )(10 12 ) 7.5

26 Example 5

27

28 Example 6 Determine the Slope θ and displacement at point B of a steel p p p beam E = 200 Gpa, I = 60(10 6 ) mm 4

29 Solution Virtual Load

30 Real Load Real Load 3 3 ) 3 ( 1) ( ) 3 ( (0) = = + = = L x x dx dx x dx x dx M m θ d ) 3(5 ) 3( = = + = = EI EI EI EI dx EI m θ θ θ rad ) 60(10 ) 200(10 2 ) ( ) ( 6 6 = = θ B

31 Another Solution Real Load Virtual Load θ = B = 1 = EI EI 112 = (10 9 ) rad

32 Example 7

33

34

35

36 Example 8

37

38

39 Example 9

40

41 Example 9b Determine the horizontal deflection at A

42 Solution Real Load Virtual Load

43 A = = EI EI EI 1250 = m = (10 )

44 Virtual Strain Energy Caused by: Axial Load U = U n nnl AE 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

45 Virtual Strain Energy dy = γ dx γ = τ / G dy = τ / G dx ( ) Caused by: Shear τ = K ( V / A) V dy = K dx GA du = νdy = νk V / A dx S ( ) L νvν V US = K dx GA 0

46 Virtual Strain Energy Caused by: Shear L νv U S = K dx GA 0 ν= 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.

47 Virtual Strain Energy γ = cd θ / dx γ = τ / G Tc τ = J τ Caused by: Torsion γ τ T d θ = dx = dx = dx c Gc GJ tt du t = tdθ = dx GJ U t = ttl GJ

48 Virtual Strain Energy Caused by: Torsion U t = ttl GJ 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

49 Virtual Strain Energy Caused by: Temperature d θ = U temp α T m c L 0 dx mα α T = c m dx

50 Virtual Strain Energy Caused by: Temperature U temp L = 0 m α T c m dx m= internal virtual moment in the beam caused by the external virtual unit load or unit moment α = coefficient of thermal expansion Tm = change in temperature between the mean temperature and the temperature at the top or the bottom of the beam c = mid-depth of the beam

51 Example 10

52

53

54

55 Example 11

56

2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)?

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