ME 202 STRENGTH OF MATERIALS SPRING 2014 HOMEWORK 4 SOLUTIONS

Transcription

1 ÇANKAYA UNIVERSITY MECHANICAL ENGINEERING DEPARTMENT ME 202 STRENGTH OF MATERIALS SPRING 2014 Due Date: 1 ST Lecture Hour of Week 12 (02 May 2014) Quiz Date: 3 rd Lecture Hour of Week 12 (08 May 2014) HOMEWORK 4 SOLUTIONS NOTE: In the first 7 questions, Graphical solution (Mohr s Circle) will not be used, but the second 7 question will be through Mohr s Circle, and you are free in the the last The state of stress at a point in a member is shown on the element. Determine the stress components acting on the inclined plane AB. 1/23

2 2. Determine the equivalent state of stress on an element at the same point oriented 30 counterclockwise with respect to the element shown. Sketch the results on the element. 2/23

3 3. The state of stress at a point is shown on the element. Determine (a) the principal stress and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case. 3/23

4 4. A point on a thin plate is subjected to the two successive states of stress shown. Determine the resultant state of stress represented on the element oriented as shown on the right. 4/23

5 5. The stress acting on two planes at a point is indicated. Determine the shear stress on plane a a and the principal stresses at the point. 5/23

6 6. The wood beam is subjected to a load of 12 kn. If a grain of wood in the beam at point A makes an angle of 25 with the horizontal as shown, determine the normal and shear stress that act perpendicular and parallel to the grain due to the loading. 6/23

7 7. The 3-in. diameter shaft is supported by a smooth A thrust bearing at A and a smooth journal bearing at B. Determine the principal stresses and maximum in-plane shear stress at a point on the outer surface of the shaft at section a-a. 7/23

8 8. The state of stress at a point in a member is shown A on the element. Determine the stress components acting on the plane AB. 8/23

9 9. Determine the equivalent state of stress if an element is oriented 45 clockwise from the element shown. 9/23

10 10. Determine the equivalent state of stress which represents (a) the principal stress, and (b) the maximum in-plane shear stress and the associated average normal stress. For each case, determine the corresponding orientation of the element with respect to the element shown. 10/23

11 11. Determine the principal stress, the maximum in-plane shear stress, and average normal stress. Specify the orientation of the element in each case. 11/23

12 12. Determine the equivalent state of stress if an element is oriented 25 counterclockwise from the element shown. 12/23

13 13. Draw Mohr s circle that describes each of the following states of stress. 13/23

14 14. The post has a square cross-sectional area. If it is fixed supported at its base and a horizontal force is applied at its end as shown, determine (a) the maximum in-plane shear stress developed at A and (b) the principal stresses at A. 14/23

15 15. Prove that the sum of the normal strains in perpendicular directions is constant. 15/23

16 16. The state of strain at the point has components of ϵx = 180(10-6 ), ϵy = - 120(10-6 ), and ᵧxy = - 100(10-6 ). Use the strain-transformation equations to determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case specify the orientation of the element and show how the strains deform the element within the x y plane. 16/23

17 17. The state of strain on an element has components ϵx = -400(10-6 ), ϵy = 0, ᵧxy = 150(10-6 ). Determine the equivalent state of strain on an element at the same point oriented 30 clockwise with respect to the original element. Sketch the results on this element. 17/23

18 18. The strain at point A on the bracket has components ϵx = 300(10-6 ), ϵy = 550(10-6 ) ᵧxy = -650(10-6 ), ϵz = 0. Determine (a) the principal strains at A in the x y plane, (b) the maximum shear strain in the x y plane, and (c) the absolute maximum shear strain. 18/23

19 19. The 60 strain rosette is attached to point A on the surface of the support. Due to the loading the strain gauges give a reading of ϵ a = 300(10-6 ),ϵ b = (10-6 ), andϵ c = (10-6 ). Use Mohr s circle and determine (a) the inplane principal strains and (b) the maximum inplane shear strain and the associated average normal strain. Specify the orientation of each element that has these states of strain with respect to the x axis. 19/23

20 20. The 60 strain rosette is mounted on a beam. The following readings are obtained from each gauge: ϵ a = 250(10-6 ), ϵ b = -400(10-6 ), ϵ c = 280(10-6 ). Determine (a) the in-plane principal strains and their orientation, and (b) the maximum in-plane shear strain and average normal strain. In each case show the deformed element due to these strains. 20/23

21 21. If the 2-in.-diameter shaft is made from brittle material having an ultimate strength of σ ult = 50 ksi, for both tension and compression, determine if the shaft fails according to the maximum-normal-stress theory. Use a factor of safety of 1.5 against rupture. 21/23

22 22. The state of stress acting at a critical point on the seat frame of an automobile during a crash is shown in the figure. Determine the smallest yield stress for a steel that can be selected for the member, based on the maximum- shear-stress theory. 22/23

23 23. A bar with a circular cross-sectional area is made of SAE 1045 carbon steel having a yield stress of σ Y = 150 ksi. If the bar is subjected to a torque of 30 kip. in. and a bending moment of 56 kip.in., determine the required diameter of the bar according to the maximum-distortion-energy theory. Use a factor of safety of 2 with respect to yielding. 23/23