Find the axial stress (Subscript[\[Sigma], z]) and axial strain (Subscript[\[Epsilon], z]) for a thick-walled circular pipe subjected to an internal pressure Subscript[p, i] for the following three cases: a) the pipe is long and open (plane strain) b) the pipe is short and open (plane stress) c) the pipe is is closed Express the above solutions in terms of the pressure Subscript[p, i], the the pipe inner and outer radii and the material constants.

Repeat the preceding problem using sx= 5.5 MPa. ??y= 4 MPa. and txy= 3.2 MPa.

Plastic bar of diameter d = 32 mm is compressed in a testing device by a Force P = 190 N that is applied as shown in the figure. (a) Determine the normal and shear stresses acting: on all faces of stress elements oriented at (1 ) an angle 8 = 00, (2) an angle ?? = 22.5s, and (3) an angle ?? = 45°. In each case, show the stresses on a sketch of a properly oriented element. What are smaxtmax (b) Find smax and tmax in the plastic bar if a re-cantering spring of stiffness k is inserted into the testing device, as shown in the figure. The spring stillness is 1/6 of the axial stiffness of the plastic bar.

A brass plate with a modulus of elastici ty E = 16 X 106 psi and Poisson’s ratio a = 0.34 is loaded in biaxial stress by normal stresses r and cry. (sec figure). A strain gage is bonded to the plate at an angle 4 = 350 If the stress o is 10,700 psi and the strain means used by the gageise = 390 X 106.whatisthcmax- imum in-plane shear stress (Tr, ),. and shear strain (>‘ ).? What is the maximum shear strain (y ) in the x-: plane? What is the maximum shear strain (y ).. in the y-r plane?

The strains for an element of material in plane strain (see figure) are as follows: x = 480 ×10-6. y = 140 × l0-6, and xy = —350 x 10”. Determine the principals strains and maximum shear strains, and show these strains on sketches of properly oriented elements.

The Poisson ratio formula for an auxetic structure given in the figure is given as follows:H = 8 cm L = 4 cm α = 50 ° t = 5 mmThe strain value of the structure in the same direction as a result of the pull applied in the X axis from the bottom was measured as 0.26 mm / mm. Under the same loading condition:a) Calculate the strain on the Z axis of the structure.b) In this case, what is the strain type of the structure in the Z axis?

1.4-7 The data shown in the table below were obtained from a tensile test of high-strength steel. The test specimen had a diameter of 13 mm and a gage length of 50 mm (see figure for Prob. 1.4-3). At fracture, the elongation between the gage marks was 3.0 mm and the minimum diameter was 10.7 mm. Plot the conventional stress-strain curve for the steefor the steel and determine the proportional limit, modulus of elastics of elastic- ity (i.e., the slope of the initial part of the stress-strain,tress-strain curve), yield stress at 0.1% offset, ultimate stress, percent, elongation in 50 mm, and percent reduction in area. 'ess, percent area. TENSILE-TEST DATA FOR PROB. 1.4-7 Elongation (mm) 0.005 0.015 0.048 Load (kN) 5 10 30 50 0.084 60 0.099 64.5 0.109 67.0 0.119 68.0 0.137 69.0 0.160 70.0 0.229 72.0 0.259 76.0 0.330 84.0 0.584 92.0 0.853 100.0 1.288 112.0 2.814 113.0 Fracture

3.3 A steel wire of length L = 20 cm is subjected to nonuniform heating such that the exten- sional strain is £x=k(x/L), where k= 2 x 102 and x is the distance of a point from one end of the wire. Find the length of the deformed wire. Ans: 20.1 cm.

The stress- strain curve for four materials are shown in figure below. Which material (A/B/C/D) has 6 SS D 30 7- Highest UTS 8- Lowe UTS 34 Stress O Strain

1.6-7 A wire of length L = 2.5 m and diameter d = 1.6 mm is stretched by tensile forces P = 600 N. The wire is made of a copper alloy having a stress- strain relationship that may be described mathemat- ically by 124,020ɛ 0 s8s 0.03 (o = MPa) 1+ 300ɛ in which e is nondimensional and o has units of MPa. (a) Construct a stress-strain diagram for the material. (b) Determine the elongation of the wire due to the forces P. (c) If the forces are removed, what is the permanent set of the bar? (d) If the forces are applied again, what is the proportional limit?