from barber chapter 2; questions: 2.3, 2.7, 2.9, 2.17 from boresi chapter 1; questions: 1.24, 1.28, 1.29, 1.33 The questions are from the pdf's that I have attached and there's one more pdf file attt

2.2. The stress at a point is de ned by the components xx=7 ksi, yy=10 ksi, xy= yx=−20 ksi. Sketch the corresponding Mohr’s circle and hence nd the maximum in-plane shear stress and the orientation of the planes on which it acts. Illustrate your answer with a sketch of an appropriately rotated rectangular element, taking care to indicate the correct direction for the shear stress on these planes.

2.3. The stress at a point is de ned by the components xx=8 ksi, yy=−10 ksi, xy= yx=−4 ksi. Find the principal stresses 1, 2and the inclination of the plane on which the maximum principal stress acts to the x-plane. 2.4. The stress at a point is de ned by the components xx=0 MPa, yy=100 MPa, xy= yx=−40 MPa. Find the principal stresses 1, 2and the inclination of the plane on which the maximum principal stress acts to the x-plane. 2.5. The principal stresses at a given point are 1=−50 MPa, 2=−50 MPa. Sketch Mohr’s circle for this state of stress and hence determine the maximum in-plane shear stress. Comment on your results.

2.6. The principal stresses at a given point are 1=10 ksi, 2=5 ksi. Sketch Mohr’s circle for this state of stress and hence determine the maximum in-plane shear stress.

2.7. The principal stresses at a given point are 1=10 MPa, 2=−100 MPa. Sketch Mohr’s circle for this state of stress and determine the normal stress on a plane in- clined at an angle to the principal plane 1. Hence nd the range of values of for which the normal stress is tensile.

2.8. The stress at a point is de ned by the components xx=220 MPa, yy=220 MPa, zz=0 MPa, xy=−80 MPa, yz=40 MPa, zx=0 MPa. Find the three principal stresses and the direction cosines of the plane on which the maximum tensile stress acts.

2.9. The stress at a point is de ned by the components xx=120 MPa, yy=−20 MPa, zz=20 MPa, xy=60 MPa, yz=0 MPa, zx=0 MPa. Find the three principal stresses and the direction cosines of the plane on which the maximum tensile stress acts.

2.10. The stress at a point is de ned by the components xx=0 ksi, yy=0 ksi, zz=0 ksi, xy=−3ksi, yz=−3 ksi, zx=−3 ksi. Find the three principal stresses and the direction cosines of the plane on which the maximum tensile stress acts.

2.11. The stress at a point is de ned by the components xx=20 ksi, yy=8 ksi, zz=−15 ksi, xy=0 ksi, yz=4 ksi, zx=16 ksi. Find the three principal stresses and the direction cosines of the plane on which the maximum tensile stress acts.

2.12. The strain energy stored per unit volume in a body subjected to a uniform state of stress de ned by the principal stresses 1, 2, 3is 90 2 Material Behaviour and Failure U0= 1 2( 1e1+ 2e2+ 3e3), where e1,e2,e3are the corresponding principal strains. Use the stress-strain relations (1.7–1.9) to obtain an expression for U0in terms of the stress invariants and the material properties. Hence show that if I1=0, U0=− I2 2G . 2.13*. Show that I2< 1 3I21 for all states of stress.

2.14*. Establish a condition that must be satis ed by the six stress components xx, yy, zz, xy, yz, zx if one of the principal stresses is to be zero and the other two negative — i.e. 1=0, 2<0, 3<0. Hint: Solve the problem rst in terms of the stress invariants using equations (2.25–2.27) and then substitute for the invariants in the nal conditions using (2.17– 2.19).

2.15. If the radii of the three Mohr’s circles in Figure 2.5 (a) are denoted by 1, 2, 3 respectively, show that 21+ 22+ 23= 1 2 I21−3I2 . 2.16. The stress at a point is de ned by the components xx=120 MPa, yy=300 MPa, zz=70 MPa, xy=−80 MPa, yz= zx=0 MPa, so that zz is a princi- pal stress. Find the other two principal stresses and sketch the three Mohr’s circles.

Hence determine the magnitude of the maximum shear stress.

2.17. The stress at a point is de ned by the components xx=−20 ksi, yy=5 ksi, zz=2 ksi, xy=5 ksi, yz= zx=0 ksi, so that zzis a principal stress. Find the other two principal stresses, sketch the three Mohr’s circles and hence determine the magnitude of the maximum shear stress.

2.18. The stress at a point is de ned by the components xx=85 MPa, yy=320 MPa, zz=100 MPa, xy=−40 MPa, yz=50 MPa, zx=−20 MPa. Find the three principal stresses, sketch the three Mohr’s circles and hence determine the magnitude of the maximum shear stress.

2.19. The stress at a point is de ned by the components xx=5 ksi, yy=0 ksi, zz=−8 ksi, xy=−4 ksi, yz=2 ksi, zx=−2 ksi. Find the three principal stresses, sketch the three Mohr’s circles and hence determine the magnitude of the maximum shear stress.