☻ 1.0 Axial Forces 2.0 Bending of Beams M M Now we consider the elastic deformation of beams (bars) under bending loads. M M www.engineering.auckland.ac.nz/mechanical/MechEng242

Application to a Bar Fn Mt Ft Mn Normal Force: Bending Moment: S.B. Shear Force: K.J. Ft Torque or Twisting Moment: Mn

Examples of Devices under Bending Loading: Atrium Structure Excavator Yacht Car Chassis

2.0 Bending of Beams sx 2.1 Revision – Bending Moments (Refer: B,C & A – Sec’s 6.1,6.2) 2.2 Stresses in Beams (Refer: B,C & A –Sec’s 6.3-6.6) x sx Mxz P 2.3 Combined Bending and Axial Loading (Refer: B,C & A –Sec’s 6.11, 6.12) P1 P2 2.4 Deflections in Beams (Refer: B,C & A –Sec’s 7.1-7.4) 2.5 Buckling (Refer: B,C & A –Sec’s 10.1, 10.2)

RECALL… 2.1 Revision – Bending Moments Q (SFD) (BMD) M (Refer: B, C & A – Chapter 6) Last year Jason Ingham introduced Shear Force and Bending Moment Diagrams. 3m 12 kN Q (SFD) M (BMD)

P Mxz RAy RBy A B Mxz Consider the simply supported beam below: (Refer: B, C&A – Sections 1.14, 1.15, 1.16, 6.1) x y Radius of Curvature, R P Deflected Shape A B RBy RAy Mxz Mxz What stresses are generated within, due to bending?

Recall: Axial Deformation W P A B RBy Mxz RAy Bending Recall: Axial Deformation W u Load (W) Extension (u) Bending Moment (Mxz) Curvature (1/R) Axial Stiffness Flexural Stiffness

sx is NOT UNIFORM through the section depth Axial Stress Due to Bending: Mxz=Bending Moment y sx (Tension) sx (Compression) Mxz x sx=0 Beam Unlike stress generated by axial loads, due to bending: sx is NOT UNIFORM through the section depth sx DEPENDS ON: (i) Bending Moment, Mxz (ii) Geometry of Cross-section

Mxz Qxy Qxy=Shear Force Sign Conventions: y Mxz=Bending Moment x -ve sx Mxz Qxy +ve sx +VE (POSITIVE) “Happy” Beam is +VE “Sad” Beam is -VE

P Mxz=P.L RAy=P P.L Mxz P Qxy P Mxz Mxz Qxy Qxy Example 1: Bending Moment Diagrams P y Mxz=P.L A B x RAy=P L P.L Qxy Mxz x P P Mxz Qxy Mxz Qxy Q & M are POSITIVE

Shear Force Diagram (SFD) Bending Moment Diagram (BMD) P y L P.L x Mxz Qxy A B x P Qxy +ve P Shear Force Diagram (SFD) Mxz Bending Moment Diagram (BMD) -ve -P.L To find sx and deflections, need to know Mxz.

P P A A C B Qxy Mxz a b y Example 2: Macaulay’s Notation a x x Where can only be +VE or ZERO.

P A C B Mxz BMD: y a b x x (i) When 1 (ii) When 2 Eq. 1 Eq. 2 +ve A B 1 (ii) When 2 Mxz BMD: Eq. 1 Eq. 2 +ve A B C

wL2 Mxz= 2 RAy=wL wL2 Mxz wx 2 Qxy wL Distributed Load w per unit length Example 3: Distributed Load y x Mxz= wL2 2 A B x RAy=wL L wx wL2 2 Qxy Mxz wL

Mxz BMD: L x -ve -wL2 2

Summary – Is anything Necessary for Revision Generating Bending Moment Diagrams is a key skill you must revise. From these we will determine: Stress Distributions within beams, and the resulting Deflections Apart from the revision problems on Sheet 4, you might try these sources: B, C & A Worked Examples, pg 126-132 Problems, 6.1 to 6.8, pg 173 Jason Ingham’s problem sheets: www.engineering.auckland.ac.nz/mechanical/EngGen121