ÇANKAYA UNIVERSITY – MECHANICAL ENGINEERING DEPARTMENT

ME 202 – STRENGTH OF MATERIALS – SPRING 2014

HOMEWORK 4 SOLUTIONS Due Date: 1ST Lecture Hour of Week 12 (02 May 2014) Quiz Date: 3rd Lecture Hour of Week 12 (08 May 2014) !

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 9.

1. 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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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

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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.

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

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

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

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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.

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

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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.

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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.

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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.

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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 ϵa =

300(10 - 6),ϵb = - 150 (10-6), andϵc = - 450 (10 -

6). Use Mohr’s circle and determine (a) the in-plane principal strains and (b) the maximum in-plane 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.

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20. The 60° strain rosette is mounted on a beam. The following readings are obtained from each gauge:

ϵa= 250(10-6), ϵb= -400(10-6), ϵc = 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.

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21. If the 2-in.-diameter shaft is made from brittle

material having an ultimate strength of 𝜎ult = 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.

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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.

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23. A bar with a circular cross-sectional area is made of SAE 1045 carbon steel having a yield stress of 𝜎Y =

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.

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