Internal Internal Forces Forces

Transcription

1 Internal Forces ENGR 221 March 19, 2003

2 Lecture Goals Internal Force in Structures Shear Forces Bending Moment Shear and Bending moment Diagrams

3 Internal Forces and Bending The bending moment, M. Moment

4 Internal Forces and Bending The shear force, V. Moment

5 Example Internal Forces in a Determine the internal forces in member ACF at point J and in member BCD at point K. Frame Problem

6 Example Internal Forces in a Frame Problem F x Determine the forces at = 0 = R Ex Ex R = 0 N F = 0 = 2400 N + R + R M y Ey F E R + R = Ey F F 2400 N ( ) ( )( ) = 0 = R 4.8 m 2400 N 3.6 m R = 1800 N F R = Ey 600 N R Ex R Ey R F

7 Example Internal Forces in a Look at the member BCD Frame Problem F = 0 = 2400 N + F + F M y By Cy B F + F = By Cy Cy 2400 N ( ) ( )( ) = 0 = R 2.4 m 2400 N 3.6 m R = Cy By 3600 N R = 1200 N

8 Example Internal Forces in a Look at the member ACF Frame Problem F y = 0 = 3600 N N + F = 1800 N Ay F Ay

9 Example Internal Forces in a Look at the member ABE Frame Problem Fy = 0 = 1800 N N N Fy = 0N

10 Example Internal Forces in a Take a look at section ACF at point J 5.4 m θ = tan = m Frame Problem 1 o Which section would you like to have to compute the moments? θ

11 Example Internal Forces in a Frame Problem Take a look at section ACF at point J F M = M 1800 N 1.2 m = 0 J M = 1800 N 1.2 m J J F = 2160 N-m ( ) ( ) = F ( o ) J 1800 N cos x = N ( o ) F y = VJ N sin V = N J

12 Example Internal Forces in a Frame Problem Determine the internal forces in ember BCD at point K. K M = M N 1.5 m = 0 K K M = 1200 N 1.5 m F K K = 1800 N-m F = 0 = x = 0N y K F k ( ) F = V 1200 N V = 1200 N ( )

13 Beams Definition A beam is defined as a structural member designed primarily to support forces acting perpendicular to the axis of the member. The principal i difference between beams and the axially loaded bars and torsionally loaded shafts is in the direction of the applied load.

14 Beams Supports A beam have a variety of supports. - roller ( 1-DOF) - pinned ( 2-DOF) - fixed ( 3-DOF)

15 Beams Loadings A beam have a variety of loads. - point loads - distributed loads - applied moments

16 Beams Types A beam can be classified as statically determinate beam, which means that t it can be solved using equilibrium equations, or it is...

17 Beams Types A beam can be classified as statically indeterminate beam, which h can not be solved with equilibrium equations. It requires a compatibility condition.

18 Beams Types A combination beam can be either statically determinate or indeterminate. i t These two beams are statically determinate, because the hinge provides another location, where the moment is equal to zero.

19 Shear and Bending moment Diagrams In order to generate a shear and bending moment diagram one needs to Draw the free-body diagram Solve for reactions Solve for the internal forces (shear, V, and bending moment, M)

20 Beam Sections A beam with a simple load in the center of the beam. Draw the free-body diagram. F = 0 y M = 0 z

21 Beam Sections Starting from the left side a take a series of section of the beam a compute the shear and bending moment of the beam.

22 Beam Sections The plot of the resulting series of shear and bending moment are the shear and bending moment diagrams. The technique is a cutting method.

23 Example Shear and Bending Moment Diagram Obtain the shear and bending moment diagram for the beam.

24 Example Shear and Bending Moment Diagram Draw the free body diagram and solve for equilibrium. i F = 0 = 20 kn + R 40 kn + R y By D RBy + RD = 60 kn M B ( ) ( ) R ( ) = 0 = 20 kn 2.5 m 40 kn 3.0 m m R = 14 kn D R = 46 kn By D

25 Example Shear and Bending Moment Diagram Look at the sections 1-1. F = 1-1 V = kn M = M = ( ) 20 kn x

26 Example Shear and Bending Moment Diagram Look at the section 2-2. F = V 2 2 = 20 kn M = 22 = M kn ( 2.5m ) M = 50 kn-m 2-2

27 Example Shear and Bending Moment Diagram Look at the section at 3-3. F = V = 20 kn + 46 kn y 3 3 V = kn MB = 0= M kn ( 25m 2.5 ) M = 50 kn-m 3 3

28 Example Shear and Bending Moment Diagram Look at the section 4-4. y Fy = V4 4= 20 kn + 46 kn V4 4= 26 kn M ( ) ( ) == M + 20 kn 5.5 m 46 kn 3.0 m M 4-4 = 28 kn-m

29 Example Shear and Bending Moment Diagram Look at section 5-5. F = V = 20 kn + 46 kn 40 kn y 5 5 V 5 5= 14 kn M5-5 = 0 = M kn ( 5.55 m ) 46 kn ( 3.0 m ) M = kn-m

30 Example Shear and Bending Moment Diagram Look at section 6-6 F = V = 20 kn + 46 kn 40 kn y 6 6 V = 14 kn 6 6 M ( ) ( ) ( ) = 0 = M + 20 kn 7.5 m 46 kn 5.0 m + 40 kn 2.0 m M 6-6 = 0 kn-m

31 Example Shear and Bending Moment Diagram Look at section end Fy = Vend = 20 kn + 46 kn 40 kn + 14 kn V = end 0kN Mend = 0= Mend + 20 kn ( 7.5 m ) 46 kn ( 5.0 m ) + 40 kn ( 2.0 m ) M = end 0 kn-m

32 Example Shear and Bending Moment Diagram Draw the shear and bending moment diagrams. Location (m) Shear (kn) Moment (kn-m)

33 Class Shear and Bending Moment Diagram Draw the shear and bending moment diagram.

34 Relations between Load, Shear, and Bending Moment Look at a section F = = V w x V + V y 0 V = w x V d V = w { = w x dx x 0 d ( )

35 Relations between Load, Shear, and Bending Moment Multiply by dx d V = w dx Integrate over V c to V d V d x d V c d V = w dx Vd Vc = w dx x c x d x c

36 Relations between Load, Shear, and Bending Moment Integrate over V c to V d V d V c x d V = w dx d xd V d V = c w d x x The difference of the shear is the area under the load curve between c and d. x c c

37 Relations between Load, Shear, and Bending Moment Look at a section x x M = 0 = M + w x C 2 V x + M + M ( x ) 2 ( ) M = V x w 2 M x d M = V w { = V x 2 0 dx x

38 Relations between Load, Shear, and Bending Moment Multiply by dx d M = V dx Integrate over M c to M d M d M c x d M = V dx d x M M V dx d c x d = c x c

39 Relations between Load, Shear, and Bending Moment Integrate over M c to M d M M d c x d M = V dx d x xd M = d M c V d x x The difference of the moment is the area under the load curve between c and d. c c

40 Example - Shear and Bending Moment Diagram Draw the shear and bending moment diagram.

41 Example - Shear and Bending Draw free-body diagram and use equilibrium equations. F = 0 = R wl+ R y A B R + R = wl A Moment Diagram B L M A = 0 = wl + RBL 2 wl wl RB = & RA = 2 2

42 Example - Shear and Bending Shear diagram. Moment Diagram wl F ( ) y = V x = wx 2 Note that the area under the load diagram.

43 Example - Shear and Bending Remember that dm M V dx = Where will the maximum moment occur? Moment Diagram

44 Example - Shear and Bending The maximum will occur where dm M dx = Moment Diagram The maximum moment is the positive area under the curve 0 M 1 wl L wl = =

45 Example - Shear and Bending The moment equation M x = ( ) ( ) Moment Diagram V x dx 2 x x w wl = ( ) 2 2 Note that the slope of the moment diagram is equal to the shear.

46 Example - Shear and Bending Moment Diagram Draw the shear and bending moment diagram

47 Example - Shear and Bending Moment Diagram Free-body diagram R Ax F x = 0 = R Ax ( ) R Ay F = 0= R 20 kn/m 6 m + R y Ay C R + R = 120 kn Ay C R C M A = 0 = 20 kn/m ( 6 m )( 3 m ) + R C ( 9 m ) R = 40 kn & R = 80 kn C Ay

48 Example - Shear and Bending Look at the shear V V A B = 80 kn Moment Diagram ( ) = 80 kn 20 kn/m 6 m = 40 kn ( ) Vx = 80 kn 20 kn/m x = 0 kn x = 4 m V = 40 kn - C + V C = 40 kn + 40 kn = 0 kn

49 Example - Shear and Bending Look at the shear diagram V V A B = 80 kn Moment Diagram ( ) = 80 kn 20 kn/m 6 m = 40 kn ( ) Vx = 80 kn 20 kn/m x = 0 kn x = 4 m V = 40 kn - C + V C = 40 kn + 40 kn = 0 kn

50 Example - Shear and Bending Find the moments M 0 m = 0 kn-m M M M 4 m 6 m 9m 1 = 80 kn 4 m 2 = 160 kn-m Moment Diagram ( )( ) 1 = 160 kn-m kn 2 m = 120 kn-m ( )( ) ( )( ) = 120 kn-m + 40 kn 3 m = 0 kn-m

51 Example - Shear and Bending Draw the moment diagram M 0 m = 0 kn-m M M M 4 m 6 m 9m 1 = 80 kn 4 m 2 = 160 kn-m Moment Diagram ( )( ) 1 = 160 kn-m kn 2 m = 120 kn-m ( )( ) ( )( ) = 120 kn-m + 40 kn 3 m = 0 kn-m

52 Class Shear and Bending Moment Diagram Draw the shear and bending moment diagram for the know reactions

53 Class Shear and Bending Moment Diagram Draw the shear and bending moment diagram

54 Class Shear and Bending Moment Diagram Draw the shear and bending moment diagram

55 Homework (Due 3/26/03) Problems: 8-15, 8-20, 8-37, 8-41, 8-51

56 Bonus Problem Shear and Bending Moment Diagram Draw the shear and bending moment diagram

57 Bonus Problem Shear and Bending Moment Diagram Draw the shear and bending moment diagram