1. 2008 Architecture LJM1 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

2. 2008 Architecture LJM2 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration methof of determining the beam deflections 7.4 Superposition methof of determining the beam deflections 7.5 Statically inderminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

3. 2008 Architecture LJM3 Highlights ： Feflecion calculation of beams and plane frames Objectives ：① Stiffness check ； ② Solution of Statically Inderminate Problems 7.1 Introduction

4. 2008 Architecture LJM4 ⑴ Deflection, w ： vertical displacement of centriod of the cross-section. ⑵ Rotational angle, , ： Angle rotated about neutral axis 2. Basic amounts measuring the beam deflection 3. Relation between w and  ： 1. Deflection curve ： axis of the beam after deformation, smooth ever-curve F x  C w C1C1 w w =w (x) 7.1 Introduction

5. 2008 Architecture LJM5 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

6. 2008 Architecture LJM6 1. Differential equation of deflection curve 挠曲线近似微分方程。 小变形 w x M>0 w x M<0 7.2 Differential equation of deflection curve

7. 2008 Architecture LJM7 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

8. 2008 Architecture LJM8 1. Integrate the equation: 2.Boundary conditions F AB C F D For uniform straight beams: Equation of rotational angle Deflection equation 7.3 Integration methof of determining the beam deflections

9. 2008 Architecture LJM9 Discussions ： ① Suitable to small deformation members, linear-elasticity materials. ② Advantage: Find deflection and rotational angle of arbitrary sections; Weakness: Troublesome Continuity condition ： Smooth condition ： P AB C wcwc 7.3 Integration methof of determining the beam deflections

10. 2008 Architecture LJM10 F D a a a B A C M Fa/2 x Ex.7.1 Draw the deflection curve of the cantilever beam ， EI. Solution ： 1. Basic foundation Determine curve ， Obey B.C and C.C 2. Draw sketch of the deflection curve D C G A w 7.3 Integration method of determining the beam deflections

11. 2008 Architecture LJM11 Ex.7.2 Determine the deflection curve |w| max |θ| max of the cantilever beam.  Bending equation  Differential equation and integration Solution: F l x w x B A 7.3 Integration methof of determining the beam deflections

12. 2008 Architecture LJM12  Find constant from B.C. X = 0: θ A =0, C=0 w A =0, D=0 Substitution B.C. into the above equations, gets: ④ Maximum deflection and rotational angle at B. （ ） F l w wBwB x B A x 7.3 Integration methof of determining the beam deflections

13. 2008 Architecture LJM13 F B C Ex 7.3 Discuss the deformation of the simple-supported beam shown. x1x1 l w A Solution: F A =Fb/l, F B =Fa/l (2) List bending-moment equations Portion AC : x2x2 Protion BC : (1) Find constraint reactions a b x 7.3 Integration methof of determining the beam deflections FAFA FBFB

14. 2008 Architecture LJM14 (3) List differential equations and then integrate For portion CB: For portion AC : 7.3 Integration methof of determining the beam deflections

15. 2008 Architecture LJM15 (4) Determine integration constants by B.C and C.C Continuity conditions: (a) (b) (c) (d) Boundary conditions: 7.3 Integration methof of determining the beam deflections

16. 2008 Architecture LJM16 Upon solving the four equations simultaneously, find AC: CB: F B C A 7.3 Integration methof of determining the beam deflections

17. 2008 Architecture LJM17 (5) Discussion When a>b, θ B > θ A Maximum deflection: When a>b, θ A 0. Thus, the point of θ 1 (w 1 ' )=0 occurs in the longer segment of the beam. w max Maximum rotational angle: x0x0 7.3 Integration methof of determining the beam deflections F B C A

18. 2008 Architecture LJM18 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

19. 2008 Architecture LJM19 1. Superposition by loads ： 2. Superposiyion using analysis of portion-by-portion Where F I is generalized force, including force and couple 7.4 Superposition method of determining the beam deflections

20. 2008 Architecture LJM20 Ex.7.4 Find deflection at C and rotational angle at A by superposition. Solution ：  Exerting load alone  Deformation caused by one load q F AB C aa  Superposition F = A B q + AB 7.4 Superposition method of finding deflections

21. 2008 Architecture LJM21 F l B A q MeMe B A MeMe q B A F B A + + w = w Me +w q +w F Ex.7.4 Find deflection and rotational angle at B by superposition. ( ) 7.4 Superposition method of determining the beam deflections Solution ：

22. 2008 Architecture LJM22 l B A Principle of portion-by-portion analysis = + Fl a A B C C B Fa w2w2 F M=Fa w1w1 Basic consideration ： Deflection equalization Basic theory ： Force transition Basic results ： Application directly 7.4 Superposition method of finding deflections

23. 2008 Architecture LJM23 A C q B F a a/2 A q F Ay F By (a) B F a/2 (b) A C B wBwB Ex.7.5 Combined beam AC 。 EI ， F=qa 。 Find w B and θ B. 解 7.4 Superposition method of finding deflections

24. 2008 Architecture LJM24 Ex.7.6 Determine w C.  For infinitesimal portion dx  From Table of deflection of beams.  Superposition q0q0 0.5L x dxdx b x f C 7.4 Superposition method of finding deflections Solution ：

25. 2008 Architecture LJM25 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

26. 2008 Architecture LJM26 1. Take equalization system 2. List compatibility equation 3. Introduce physical law to get supplementary 4. Solve the equation to get redundant reaction A F B A F B q q Redundant reactions( 多余支反力 ) F By Redundant restraints( 多于约束 ) 7.5 Statically indeterminate beams Solution ：

27. 2008 Architecture LJM27 A F l/2 A F By F B MAMA A F Solution ： 相当系统 E.E ① Take equalization equation ② Compability condition ③ Fond redundant reaction Change S.ID.P to S.D.P ！ Ex.7.7 Find reactions of the beam ④ Find other reactions A F By F MAMA F Ay ( ) 7.5 Statically indeterminate beams

28. 2008 Architecture LJM28 B C A F D E Example7.8 Two cantilever beams of AD and BE are joined by a steel rod CD. Determine the deflection of the cantilever beam AD, at D due to a force F applied at E. EI EA l l l l 7.5 Statically indeterminate beams

29. 2008 Architecture LJM29 Solution: (1) Set up a equivalent system (2) Compatible condition w D =w C - △ rod (a) w C = (w C ) p - (w C ) FD Where wcwc △ rod B C F E Fig.b FDFD A D FDFD wDwD Fig.a 7.5 Statically indeterminate beams

30. 2008 Architecture LJM30 B C A F D E FDFD wcwc △ rod wDwD Substitute w D, w C and △ rod into Eq.(a): (3) Find the deflection of D, w D 7.5 Statically indeterminate beams

31. 2008 Architecture LJM31 = q0q0 l A B l MAMA B A Ex.7.9 ： Shown is the beam AB of length l, EI, subjected to uniform load q. Draw M- diagram. M A, F B -- 多余约束反力。 q0q0 EI q0q0 L FBFB A B 7.5 Statically indeterminate beams Solution ： (1) Set up a equivalent system Different Equivalent systems!

32. 2008 Architecture LJM32  Compatible equation + q0q0 L FBFB A B = FBFB A B q0q0 A B  physical relations  Supplementary equation  M-diagram 7.5 Statically indeterminate beams M

33. 2008 Architecture LJM33 Solution ：  Set up E.S MAMA Ex.7.9 ： Plot Bending-moment diagram of simple-supported beam AB shown. B A  Geometry equation  Reactions ：  Bending-moment Diagram M ㈠ ㈩ l A B EI q0q0 FBFB FAFA MAMA B A 7.5 Statically indeterminate beams Have ：

34. 2008 Architecture LJM34 Solution ： 相当 系统 E.S. ① Equivalent system ② Compatible equation Ex.7.10 Determine maximum deflection of the cantilever beam AB shown. Suppose that EI for the two beams are equal. ④ Maximum deflection A FRFR C F B FRFR AC C B A l/2 F 7.5 Statically indeterminate beams

35. 2008 Architecture LJM35 = Ex.7.11 Determine axial load in rod BC for the structure shown. q0q0 FBFB FBFB A + q0q0 A L BC q0q0 L A B C EI 7.5 Statically inderminate beams ① Equivalent system ② Compatible equation Solution ： E.S.

36. 2008 Architecture LJM36 = L BC x f q0q0 L FBFB A B C FBFB A B + q0q0 A B  Physical relations  Supplementary  Others （ Reactions, stresses, deflections, and so on. ） 7.5 Statically inderminate beams

37. 2008 Architecture LJM37 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

38. 2008 Architecture LJM38 [w]: Allowable deflection ， [  ]: Allowable rotational angle  Check the stiffness;  、 Determine allowable loads. 1. Stiffness conditions of beams Three types of stiffness calculation  、 Design sections; 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness

39. 2008 Architecture LJM39 Strength ： Stiffness ： Methods ： → 3S are related to internal forces and properties of the cross-section Reducing Bending moment M Enhancing Inertia moment I or Modulus of section W Select materials rationally Stability ： 2. Methods of enhancing deformations of beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness

40. 2008 Architecture LJM40 Ex.7.12 The hollow circular rod AC of in-diameter d=40mm and out-diameter D = 80mm. E = 210GPa ， [δ]= 10 -5 m at C. [  ] = 0.001. at C, F 1 = 1kN, F 2 = 2kN. Check the stiffness of the overhanging beam. F2F2 B l=400 P2P2 A C a=100 200 D F1F1 B + F2F2 M = F1F1 + F2F2 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Analysis of deformations

41. 2008 Architecture LJM41  From Table of deformations F2F2 B C + + = ( ) C F1F1 A BD The overhang beam C F2F2 B D A M 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Solution ：

42. 2008 Architecture LJM42  Deformations by superposition  Check the stiffness ( ) 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness