1 0 Axial Forces 2 0 Bending of

☻ 1. 0 Axial Forces 2. 0 Bending of Beams Now we consider the elastic deformation of beams (bars) under bending loads. M M www. engineering. auckland. ac. nz/mechanical/Mech. Eng 242 MECHENG 242 Mechanics of Materials Bending of Beams

Application to a Bar Normal Force: Fn Fn Bending Moment: S. B. Mt Mt Shear Force: Ft Ft Torque or Twisting Moment: K. J. Mn Mn MECHENG 242 Mechanics of Materials Bending of Beams

Examples of Devices under Bending Loading: Atrium Structure Excavator Yacht Car Chassis MECHENG 242 Mechanics of Materials Bending of Beams

2. 0 Bending of Beams 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) sx sx P x Mxz 2. 3 Combined Bending and Axial Loading P 1 Mxz (Refer: B, C & A – Sec’s 6. 11, 6. 12) P 2 2. 4 Deflections in Beams 2. 5 Buckling (Refer: B, C & A –Sec’s 7. 1 -7. 4) (Refer: B, C & A –Sec’s 10. 1, 10. 2) MECHENG 242 Mechanics of Materials Bending of Beams

2. 1 Revision – Bending Moments RECALL… (Refer: B, C & A – Chapter 6) Last year Jason Ingham introduced Shear Force and Bending Moment Diagrams. 12 k. N 3 m 3 m Q (SFD) 0 (BMD) M 0 MECHENG 242 Mechanics of Materials Bending of Beams

Consider the simply supported beam below: (Refer: B, C&A – Sections 1. 14, 1. 15, 1. 16, 6. 1) y Radius of Curvature, R P x B A Mxz RAy Mxz Deflected Shape Mxz MECHENG 242 Mechanics of Materials Mxz RBy What stresses are generated within, due to bending? Bending of Beams

P A W u RAy Recall: Axial Deformation Load (W) B Mxz RBy Bending Moment (Mxz) Axial Stiffness Extension (u) MECHENG 242 Mechanics of Materials Flexural Stiffness Curvature (1/R) Bending of Beams

Axial Stress Due to Bending: y x Mxz=Bending Moment Mxz sx (Compression) Mxz sx=0 Beam sx (Tension) 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 MECHENG 242 Mechanics of Materials Bending of Beams

Sign Conventions: Mxz Qxy=Shear Force y Qxy Mxz=Bending Moment x -ve sx +VE (POSITIVE) “Happy” Beam is +VE MECHENG 242 Mechanics of Materials “Sad” Beam is -VE Bending of Beams

Example 1: Bending Moment Diagrams Mxz=P. L P A y x B L RAy=P P. L Mxz P Mxz Qxy x P Mxz Qxy Q & M are POSITIVE MECHENG 242 Mechanics of Materials Bending of Beams

P L P. L Mxz B A P P Mxz x Qxy y +ve Shear Force Diagram 0 (SFD) 0 Bending Moment Diagram -P. L -ve (BMD) To find sx and deflections, need to know Mxz. MECHENG 242 Mechanics of Materials Bending of Beams

P Example 2: Macaulay’s Notation a A y b C B x x a A P Mxz Qxy Where MECHENG 242 Mechanics of Materials can only be +VE or ZERO. Bending of Beams

P a A C y b x B x 1 (i) When 0 (ii) When BMD: 2 Mxz Eq. 1 Eq. 2 +ve 0 A MECHENG 242 Mechanics of Materials C B Bending of Beams

Distributed Load w per unit length Example 3: Distributed Load y x w. L 2 Mxz= 2 A B L x RAy=w. L 2 2 Mxz wx Qxy w. L Mxz Qxy MECHENG 242 Mechanics of Materials Bending of Beams

Mxz BMD: L 0 -ve x -w. L 2 2 MECHENG 242 Mechanics of Materials Bending of Beams

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/Eng. Gen 121 MECHENG 242 Mechanics of Materials Bending of Beams