1. Bending of Beams MECHENG242 Mechanics of Materials 2.3 Combined Bending and Axial Loading 2.0 Bending of Beams 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) P1P1 P2P2 ☻ 2.2 Stresses in Beams 2.1 Revision – Bending Moments x xx M xz xx P ☻ ☻

2. Bending of Beams MECHENG242 Mechanics of Materials 2.4 Beam Deflection (Refer: B, C & A–Sec 7.1, 7.2, 7.3, 7.4) 2.4.1 Moment-Curvature Equation Recall: THE ENGINEERING BEAM THEORY y x NA AB A’B’ If deformation is small (i.e. slope is “flat”): v (Deflection)

3. Bending of Beams MECHENG242 Mechanics of Materials A’ B’ Alternatively: from Newton’s Curvature Equation and (slope is “flat”) v x R if R

4. Bending of Beams MECHENG242 Mechanics of Materials From the Engineering Beam Theory: Flexural Stiffness Bending Moment Curvature M xz Flexural Stiffness Recall, for Bars under axial loading: Axial Stiffness Extension

5. Bending of Beams MECHENG242 Mechanics of Materials CurvatureSlopeDeflection Since, Curvature Slope Deflection Where C 1 and C 2 are found using the boundary conditions. R

6. Bending of Beams MECHENG242 Mechanics of Materials x y P B L A P M xz Q xy Example: x P P.L v = Deflection v v Max Deflected Shape

7. Bending of Beams MECHENG242 Mechanics of Materials 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

8. Bending of Beams MECHENG242 Mechanics of Materials ab L 2.4.2 Macaulay’s Notation y x Example: Q xy M xz P x P

9. Bending of Beams MECHENG242 Mechanics of Materials Boundary conditions: (i) @ x=0 (ii) @ x=L From (i): From (ii): Since (L-a)=b Equation of the deflected shape is:

10. Bending of Beams MECHENG242 Mechanics of Materials This value of x is then substituted into the above equation of the deflected shape in order to obtain v Max. To find v Max : v Max occurs where (i.e. slope=0) Assuming v Max will be at x<a, when P v Max Note: if

11. Bending of Beams MECHENG242 Mechanics of Materials 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