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Ch.5 INTERNAL FORCES IN BENDING; PROPERTIES OF MATIRIALS

Mechanacs of Materials. Ch.5 INTERNAL FORCES IN BENDING; PROPERTIES OF MATIRIALS. Chapter 5 Internal forces in bending; Properties of matirials. 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments

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Ch.5 INTERNAL FORCES IN BENDING; PROPERTIES OF MATIRIALS

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  1. Mechanacs of Materials Ch.5 INTERNAL FORCES IN BENDING; PROPERTIES OF MATIRIALS LIU Jiemin @SJZU 2008

  2. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  3. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  4. 5.1 Introduction 5.1.1 Concept of bending 1. Bending: Members subjected to external forces and moments perpendicular to the axle,and the axle become to curve. Bending deformation 2. Beam：Member subjected mainly to bending deformation LIU Jiemin @SJZU 2008

  5. 5.1 Introduction 3. Example LIU Jiemin @SJZU 2008

  6. F1 q F2 Axle M 5.1 Introduction Symmetric bending---special case of plane bending Cross-section Longitudinal symmetric plane LIU Jiemin @SJZU 2008

  7. 5.1 Introduction Non-symmetric bending— No longitudinal symmetric surface or external forces not in longitudinal symmetric plane. For us, Discuss mainly symmetric bending including stresses, deformations, strength, and stiffness LIU Jiemin @SJZU 2008

  8. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  9. 5.2 Types of beams and reactions MA FAx FAy 5.2.1 Supporters and reactions ①Hinged supporter Two reactions，one horizontal freedom. ②roll-supporter One reaction，two freedoms. ③Fixed end Three reactiona, zero freedom. LIU Jiemin @SJZU 2008

  10. 5.2 Types of beams and reactions q F — — Distributed force Concentrated force M — Couple q(x) 5.3.2 Three types of statically determined beams ①Simplly supported beam ②Cantilever bean ③Overhanging beam LIU Jiemin @SJZU 2008

  11. 5.2 Types of beams and reactions q Ex.5.1 Shown is the circular cylinder tank ofL=5m，inside diameter D=1m，thicknesst =10mm，density of steel ρs= 7.8g/cm³，density of liquid ρl= 1g/cm³; height of the liquid surface h=0.8m，extensional length a=1m，Plot mechanical sketch of the tank. Solution： LIU Jiemin @SJZU 2008

  12. 5.2 Types of beams and reactions q — 均布力 LIU Jiemin @SJZU 2008

  13. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  14. F F a m m FAx FAy FB x 5.3 Shearing force and bending moment 5.3.1 Internal forcs in bending Ex.5.1 Knowing：F，a，l.Find：internal forces on the section x from A. A B l Solution：①Find reactions A B LIU Jiemin @SJZU 2008

  15. F F A B m FAx FAy m FAy FB FB x 5.3 Shearing force and bending moment ②Find internal forces Shearing force Fs A Internal forces M Bending moment C Fs 1. Bending moment：M Moment acting on the cross-section and the moment vector perpendicular to the axis. M C LIU Jiemin @SJZU 2008

  16. 5.3 Shearing force and bending moment 2. Shearing force Fs: Internal force acting on the cross-section. 3. Convention of sense of internal forces Fs and M: ①Fs: Fs is positive when moment of Fs to the arbitrary internal point is clock-wised. Fs(+) Fs(-) Fs(+) Fs(-) ②M: Positive when M makes the infinitesimal portion to be down-convex. M(+) M(–) M(+) M(–) LIU Jiemin @SJZU 2008

  17. FBy F FAy ME Me MA+ l l E E MA+ A FsA+ FAy FAy FsA+ 5.3 Shearing force and bending moment Ex.5.2 Find：internal forces on the section A+, E and D- Solution:1. Reactions D E B A FAy=2F, MA=Fl l/2 2.Fs and M on the A+ FsA+=-FAy=-2F MA+=Me-FAy×∆=Fl 3. Fs and M on the E FsE=-FAy=-2F ME=Me-FAy=0 LIU Jiemin @SJZU 2008

  18. FBy F FAy F l l D FsD- 5.3 Shearing force and bending moment 4. Fs and M on the D D E B A l/2 FsD-=F MD-=0 MD- LIU Jiemin @SJZU 2008

  19. FBy F FAy 5.3 Shearing force and bending moment Method 2: Simple cross-section method D E B Fs,A+=-2F MA+=Fl A l/2 Fs,E=-2F ME+=Me-FAy×.5l=0 Fs,D=-2F+FBy=F MA+=Me-FAy ×2l+FBy ×l=0 A+ E D- LIU Jiemin @SJZU 2008

  20. q x1 x2 ql a b 5.3 Shearing force and bending moment Ex.2：Find Fs and M on sections 1-1 and 2-2(l=a+b) 1-1 2-2 Solution：1. Reactions 2. Fsand M LIU Jiemin @SJZU 2008

  21. 5.3 Shearing force and bending moment 弯曲内力的简便求法 （1）剪力：任意截面上的剪力都等于该截面一侧所有外力 在剪力方向上投影的代数和。截面左侧向上的外力在该截面 上引起正剪力，向下的外力引起负剪力；而截面右侧向下的 外力在该截面上引起正剪力，向上的外力引起负剪力。 （2）弯矩：任意截面上的弯矩都等于该截面一侧所有外力对该截面形心力矩的代数和。截面左侧顺时针力偶在该截面上引起正弯矩，反时针力偶在该截面上引起负弯矩；所有向上外力都引起正弯矩，向下外力都引起负弯矩，不管它们是位于左侧还是右侧。 LIU Jiemin @SJZU 2008

  22. 2 2 ql 1 l l 1 5.3 Shearing force and bending moment 二、例题 例1：求图示梁1--1、2--2截面处的内力。 q 解：由直接法求内力。 LIU Jiemin @SJZU 2008

  23. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  24. 5.4 Equations of Fs and M ° Diagrams of Fs and M 1. Equations of internal forces： 剪力方程 弯矩方程 2. Diagrams of Fs and M： Curves expressed by equations of shearing forces Fs Diagram Curves expressed by equations of bending mpment M Diagram LIU Jiemin @SJZU 2008

  25. MA Fs(x) FAy M(x) + - 5.4 Equations of Fs and M°Diagrams of Fs and M A B Ex.5.1 The cantilever. Find Equs. of Fs and M; Plot Fs and M diagrams. F l Solution：① Reactions: x ② Equations of Fs and M Fs F x FL ③ Plot Fs, M diagrams x M LIU Jiemin @SJZU 2008

  26. F L MA Fs(x) F FAy x + M(x) x - FL x - M(x) FL 5.4 Equations of Fs and M°Diagrams of Fs and M 讨论：①剪力常量水平线 弯矩线性斜直线， 集中力处有突变， 集中力偶处也然。 ②土建部门，弯矩图的M轴向下,其余领域，+M向上 LIU Jiemin @SJZU 2008

  27. F b a FAy B A Fs l A M FB FAy FAy A Fs F M x2 5.4 Equations of Fs and M°Diagrams of Fs and M Ex.5.2 Knowing：F，a，b，l. Set up Fs, M Equs. Plot Fs, M diagrams. Solution：①Reactions C ② Internal force Equs. AC： CB： A LIU Jiemin @SJZU 2008

  28. a FAy Fs(x) x FB Fa/l - + x Fab/l M(x) 5.4 Equations of Fs and M°Diagrams of Fs and M F b ③Plot internal forces B A C Fb/l + LIU Jiemin @SJZU 2008

  29. FAy M Fs FBy FAy FAy 5.4 Equations of Fs and M°Diagrams of Fs and M Ex.5.2 Knowing：q，l. Set up Fs, M Equs.. Plot Fs, M diagrams. q B A Solution：①Reactions l ② Internal force Equs. Simple cross-section Method A LIU Jiemin @SJZU 2008

  30. q B A FAy l Fs(x) x + FBy - + x ql2/8 M(x) 5.4 Equations of Fs and M°Diagrams of Fs and M ③Plot Fs and M diagrams. ql/2 ④Discussion: 内力图与荷载的关系 均布载荷作用段，剪力图为斜直线。弯矩图为抛物线，两端内力是关键。载荷如何关凸凹，试请诸位自判断。 ql/2 + LIU Jiemin @SJZU 2008

  31. 5.4 Equations of Fs and M°Diagrams of Fs and M 均布载荷与剪力什么关系？ 剪力和弯矩什么关系？ 分布载荷、剪力和弯矩什么关系？ LIU Jiemin @SJZU 2008

  32. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  33. q M+dM Fs+dFs dx dx x 5.5 Relations among Fs, M and q 5.5.1 Relations among Fs, M and q q B A Fs M LIU Jiemin @SJZU 2008

  34. 5.5 Relations among Fs, M and q 5.5.2 Plot Fs, M diagrams using relations among Fs, M and q 图的形状： 1。梁段上无载荷， Fs图水平，M图斜直线 2。梁段上载荷均布， Fs图斜直线，M图抛物线 作图步骤： 1。求支反力 2。求控制截面（集中力，集中力偶，分布　　 　载荷始末，剪力为零处）的内力。 3。利用Fs、M与q间的微分关系作图 LIU Jiemin @SJZU 2008

  35. FAy Fs(x) x + FBy - x M(x) 5.5 Relations among Fs, M and q q Ex.5.5.1 Plot Fs, M diagrams. Ｃ B A Solution：①Reactions ② FＳand M on controlling sections section Fs M D + A LIU Jiemin @SJZU 2008

  36. FBy FAy a/2 5.5 Relations among Fs, M and q Ex.5.5.2 Plot Fs, M diagrams. qa2 qa qa q B Solution：①Reactions A D C a a a ②FＳand M on controlling sections qa qa/2 section Fs M qa/2 3qa/2 5qa/2 qa2 - qa2 qa2 9qa2/8 LIU Jiemin @SJZU 2008

  37. Chapter 5 Internal forces in bending; Properties of matirials 5.1 Introduction 5.2 Types of beams and reactions 5.3 Shearing forces and bending moments 5.4 Shear-Force and Bending-Moment diagrams 5.5 Relations among shear force, bending moment and loads 5.6 Internal forces in frames and curve beams LIU Jiemin @SJZU 2008

  38. q F 5.6 Internal forces in frames and curve beams 1. Frames, Plane frames, Internal forces 2. Curved beams，Plane curved beams, Internal forces LIU Jiemin @SJZU 2008

  39. Pa Pa x1 x2 0.5Pa 5.6 Internal forces in frames and curve beams Exa.6 Drawing M-diagram of the frame ABC shown. Solution: (1) Find reactions (2) List bending equations Portion BC,M(x2)= -a+Px2=P(x2-a) Portion AB, M(x1)= -Px1 (3) Draw moment diagram of the frame a C A P P 1.5a B LIU Jiemin @SJZU 2008

  40. F1 F2 F1a F1 F1a – + + 5.6 Internal forces in frames and curve beams Ex. 7 Plot FN, Fs,, M diagrams of the frame. F1 a F2 C B FN Fig. l A FSFig. M Fig. F1a+ F2 l LIU Jiemin @SJZU 2008

  41. F 轴线 5.6 Internal forces in frames and curve beams Ex.8 Plot Fs and M diagrams of the curved beam. A F B R O Solution：Internal forces in arbitrary section: 内力图画法：弯矩图画在受拉侧，轴力图和剪力图标正负。 FR LIU Jiemin @SJZU 2008

  42. MA FAy Fs(qa) x a 1 1/2 1 3/2 2 x M(qa2) 5.6 Internal forces in frames and curve beams Ex.5 Plot Fs and M diagrams. qa2 qa Solution：①Reactions q D A B C a 2a a ②Fs and M on controlling sections Sec. Fs M 2 1 ③ Fs,Ｍ diagrams LIU Jiemin @SJZU 2008

  43. Q(x) x L M(x) 5.6 Internal forces in frames and curve beams [例5.7] 图示梁的内力方程并画出内力图。 q0 解：①求支反力 RA RB ②内力方程 ③根据方程画内力图 x LIU Jiemin @SJZU 2008

  44. x M(x) 5.6 Internal forces in frames and curve beams [例4] 图示梁的内力方程并画出内力图。 L q M(x) 解：①写出内力方程 x Q(x) Q(x) x – qL ②根据方程画内力图 LIU Jiemin @SJZU 2008

  45. L L L L L L P 5.6 Internal forces in frames and curve beams [例9] 用叠加法画图示简支梁的弯矩图。 P PL M PL + 0 P x M1 0.5PL + x 0.5P 0.5PL 0.5P M2 PL + x - 0.5PL 0.5P 0.5P LIU Jiemin @SJZU 2008

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