CE 160 Virtual Work Beam Example. + - θ C

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Nelson Collins
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1 E 6 Virtual Work eam Example. Vukazich all 26 2 k 3 k δ E θ + θ - D E 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft The beam shown is pin supported at point ; roller supported at points, D, and (note that roller supports resist movement both up and down); and has internal hinges at points and E. Neglecting the weight of the beam, for the point loads applied at points and as shown find using the method of virtual work:. The vertical deflection of the hinge at point E; 2. The slope of the deformed shape just to the left of the hinge at ( - ); 3. The slope of the deformed shape just to the right of the hinge at ( + ); 4. The vertical deflection of the hinge at point E due if the pin at settles.3 in, the roller at settles.8 in, and the roller at D settles.4 in. alculate the deflection due to support settlement only (i.e. do not consider the applied loads). Note that this is the same beam and loading that we have considered for example problems earlier in the semester. or the beam: EI = 2,, k-in 2, EI DE =,, k-in 2, and EI E = 5,, k-in 2. E 6 Virtual Work eam Example Problem

2 Real Problem is the same for questions, 2, and 3. Vukazich all 26 We found the Moment Diagram for the hinged beam earlier in the semester. 2 k 3 k D E 4.9 k 8.82 k 4.32 k 5.4 k 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft 28.8 k-ft k-ft -36 k-ft D E. Virtual System to measure δ E E D 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft E 6 Virtual Work eam Example Problem 2

3 Vukazich all 26..D of Virtual System and the virtual shear and moment diagrams (you should be able to verify these are the correct diagrams). D E ft 5 ft 5 ft 2 ft 5 ft 2 ft V Q -.8 D E 2. ft -2. ft D E E 6 Virtual Work eam Example Problem 3

4 Vukazich all 26 Use Table 4 in text to evaluate the virtual work product integrals that come from the principle of virtual work: δ = EI M M dx Integrate over segment using Table 4 in the text: 2. ft 2 ft 5 ft k-ft rom Table 4: for c a: a c 3 6ad M M L a = c = 2 ft d = 5 ft L = 27 ft M = 2 ft M 3 = k-ft a c 3 6ad M M L = = = k-ft 3 = -,973,49.2 k-in 3 E 6 Virtual Work eam Example Problem 4

5 Integrate over segment DE using Table 4 in the text: Vukazich all ft 5 ft 2 ft D E 28.8 k-ft rom Table 4: for c a: a c 3 6ad M M L a = c = 5 ft d = 2 ft L = 27 ft M = -2 ft M 3 = 28.8 k-ft a c 3 6ad M M L = = = -3.4 k-ft 3 = -5,374,77.2 k-in 3 Divide by EI δ =,973,49.2 k-in3 5,374,77.2 k-in3 2,, k-in 2 +,, k-in 2 δ E =.5487 in in =. 86 in Result is negative, so deflection is opposite the direction of the unit load upward E 6 Virtual Work eam Example Problem 5

6 2. Virtual Problem to find the slope to the left of the hinge at point Vukazich all 26 Virtual System to measure θ - E D 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft..d of Virtual System and the virtual shear and moment diagrams (you should be able to verify these are the correct diagrams) D E.8333/ft.8333/ft 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft.8333/ft V Q D E. D E E 6 Virtual Work eam Example Problem 6

7 Vukazich all 26 Use Table 4 in text to evaluate the virtual work product integrals that come from the principle of virtual work: θ = M EI M dx Integrate over segment using Table 4 in the text:. 2 ft rom Table 4: k-ft 3 M M L M = M 3 = k-ft L = 2 ft 3 M M = = k-ft 2 = -33,868.8 k-in 2 E 6 Virtual Work eam Example Problem 7

8 Integrate over segment using Table 4 in the text: Vukazich all ft rom Table 4: k-ft 2 M M L M = M 3 = k-ft L = 5 ft 2 M M = = -44. k-ft 2 = -63,54. k-in 2 Divide by EI θ = 33,868.8 k in 63,54. k in + 2,, k in 2,, k in θ = =. 487 radians Result is negative, so rotation is in the opposite direction of the unit moment clockwise E 6 Virtual Work eam Example Problem 8

9 3. Virtual Problem to find the slope to the right of the hinge at point Vukazich all 26 Virtual System to measure θ + E D 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft..d of Virtual System and the virtual shear and moment diagrams (you should be able to verify these are the correct diagrams) D E.8333/ft.5/ft.6667/ft 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft V Q.6667/ft /ft D E D E E 6 Virtual Work eam Example Problem 9

10 Vukazich all 26 Use Table 4 in text to evaluate the virtual work product integrals that come from the principle of virtual work: θ = M EI M dx Integrate over segment using Table 4 in the text: -. 2 ft 5 ft k-ft rom Table 4: for c a: a c 3 6ad M M L a = c = 2 ft d = 5 ft L = 27 ft M = - M 3 = k-ft a c 3 6ad M M L = = = k-ft 2 = 76,24.8 k-in 2 E 6 Virtual Work eam Example Problem

11 Integrate over segment D using Table 4 in the text: Vukazich all ft D 28.8 k-ft rom Table 4: 6 M M L M = - M 3 = 28.8 k-ft L = 5 ft 6 M M = = -72. k-ft 2 = -,368. k-in 2 Divide by EI θ = 76,24.8 k in,368. k in + 2,, k in,, k in θ = =. 277 radians Result is positive, so rotation is in the same direction of the unit moment counterclockwise E 6 Virtual Work eam Example Problem

12 4. Real Problem is the given support settlements Vukazich all 26 Pin support at settles.3 in downward, Roller support at settles.8 in downward, Roller support at D settles.4 in downward. Virtual System to measure δ E E D 2 ft 5 ft 5 ft 2 ft 5 ft 2 ft..d of Virtual System and the virtual support reactions (you should be able to verify these are the correct support reactions) D E ft 5 ft 5 ft 2 ft 5 ft 2 ft or this problem there is no internal virtual work due to bending deformation since we are not considering the applied loads: δ + R δ = δ.3 in in.8.4 in = Note that the external virtual work for the pin support settlement at point and the roller support at D are negative due to the real support settlement and the virtual vertical reaction being in opposite directions. δ.3 in +.44 in.72 in = δ E =. 42 in (negative, so deflection is in the opposite direction of virtual load - up) E 6 Virtual Work eam Example Problem 2

13 Vukazich all 26 Product Integral Table from Textbook E 6 Virtual Work eam Example Problem 3

Deflections. Question: What are Structural Deflections?

Deflections. Question: What are Structural Deflections? Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the