1. Lecture 5-6 Beam Mechanics of Materials Laboratory Sec. 3-4 Jiangyu Li University of Washington Mechanics of Materials Lab

2. Inclined Load Notice the sign convention: positive Mz compress upper part, negative stress; positive My extend front part, positive stress!

3. Inclined Load Stress Neutral axis

4. Asymmetrical Beam The origin of y and z axes must be placed at centroid C; orientation is arbitrary.

5. Sign Convention for Curvature Similar equation apply to Bending toward z axis Note difference with sign convention in bending moment

6. Asymmetric Beam If z is a principal axis, M y =0, bending in x-y plane, analogous to a symmetric beam When z axis is the neutral axis;

7. Asymmetric Beam If y is a principal axis, M z =0, bending in x-z plane, analogous to a symmetric beam When y axis is the neutral axis;

8. Asymmetric Beam When an asymmetric beam is in a pure bending, the plane in which the bending moments acts is perpendicular to the neutral surface only if the y and z axes are principle centroidal axes and the bending moment acts in one of the two principle plane. In such case, the principle plane in which bending moment acts becomes the plane of bending and the usual bending theory is valid

9. Analysis of Asymmetric Beam Locating the centroid, and constructing a set of principal axes Resolving bending moment into M y and M z Superposition

10. Principle Axes

11. Analysis of Asymmetric Beam A channel section C 10x15.3 c=0.634 I y =2.28 in 4, I z =67.4 in 4 y A =5.00 in, z A =-2.6+0.634=-1.966 in Calculating bending stress Locating neutral axis

12. Analysis of Asymmetric Beam

13. Normal Stress in Beam

14. Curved Beams Neutral axis is no longer the centroidal axis Positive M

15. Curved Beam

16. Curved Beams Curvature is large, e is small, r n is cloase to r c Recover to straight beam

17. Curved Beam Pay attention to the sign of s

18. Curved Beam Pay attention to the sign of s

19. Read Mechanics of Materials Lab Sec. 4 4.26(e), 4.72 posted online Assignment