2 0 Bending of Beams 2 1 2

2. 0 Bending of Beams ☻ 2. 1 ☻ 2. 2 Revision – Bending Moments Stresses in Beams sx sx P x ☻ 2. 3 Mxz Combined Bending and Axial Loading P 1 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. 4 Beam Deflection (Refer: B, C & A–Sec 7. 1, 7. 2, 7. 3, 7. 4) Recall: THE ENGINEERING BEAM THEORY 2. 4. 1 Moment-Curvature Equation v (Deflection) y x A B NA NA A’ B’ If deformation is small (i. e. slope is “flat”): MECHENG 242 Mechanics of Materials Bending of Beams

R and (slope is “flat”) B’ A’ Alternatively: from Newton’s Curvature Equation v R if x MECHENG 242 Mechanics of Materials Bending of Beams

From the Engineering Beam Theory: Flexural Stiffness Mxz Curvature Bending Moment Recall, for Bars under axial loading: Flexural Stiffness MECHENG 242 Mechanics of Materials Axial Stiffness Extension Bending of Beams

Curvature Since, Slope Deflection Where C 1 and C 2 are found using the boundary conditions. Curvatur e Slope Deflectio n R MECHENG 242 Mechanics of Materials Bending of Beams

v = Deflection Example: P. L y P L A B v x v. Max P Deflected Shape x P. L Mxz P MECHENG 242 Mechanics of Materials Qxy Bending of Beams

P To find C 1 and C 2: Boundary conditions: (i) @ x=0 (ii) @ x=0 Equation of the deflected shape is: v. Max occurs at x=L MECHENG 242 Mechanics of Materials Bending of Beams

2. 4. 2 Macaulay’s Notation P a b Example: y L x x P Mxz Qxy MECHENG 242 Mechanics of Materials Bending of Beams

Boundary conditions: (i) @ x=0 (ii) @ x=L From (i): From (ii): Since (L-a)=b Equation of the deflected shape is: MECHENG 242 Mechanics of Materials Bending of Beams

To find v. Max: v. Max occurs where (i. e. slope=0) Assuming v. Max will be at x<a, when This value of x is then substituted into the above equation of the deflected shape in order to obtain v. Max. Note: if P v. Max MECHENG 242 Mechanics of Materials Bending of Beams

2. 4. 3 Summary After considering stress caused by bending, we have now looked at the deflections generated. Keep in mind the relationships between Curvature, Slope, and Deflection, and understand what they are: • Curvature • Slope • Deflection Apart from my examples and problems: • B, C & A Worked Examples, pg 185 -201 Problems, 7. 1 to 7. 15, pg 207 MECHENG 242 Mechanics of Materials Bending of Beams