3. (20 Points) Determine the stress state and the corresponding von Mises cquivalent stress of the point A subjected to the plane-strain condition, knowing that the following strains have been obtained by the use of the rosette shown: ε₁ = -50 x. Assume that Young's modulus and Poisson's ratio of the material are 200 GPa and 0.3, respectively.

1. (10 Points) A 6-m long by 50-mm diameter rod of aluminum alloy (E = 70 GPa, a = 22.5 x /℃, and v = 0.346) is attached at the ends to supports that yield to permit a change of 1.0 mm in the rod when stressed. When the temperature is 35℃, there is no stress in the rod. After the temperature of the rod drops to - 20 ℃, determine (a) the maximum normal stress in the rod and (b) the change in diameter of the rod.

2. (10 Points) The brass pipe AD is fitted with a jacket used to apply a hydrostatic pressure of 3.5 MPa to portion BC of the pipe. Knowing that the pressure inside the pipe is 0.7 MPa, determine the maximumn normal stress in the pipe.

4. (20 Points) The structural tube shown has a uniform wall thickness of 8 mm. Knowing that the 60-kN load is applied 4 mm above the base of the tube, determine the stress state at point a.

5. (20 Points) Knowing that the beam AD is made of a solid steel bar. Determnine (a) the slope at point B, (b) the deflection at point A. Use Young's modulus as 200 GPa.

6. (20 Points) A concrete beam is reinforced by three stee! rods placed as shown. The modulus of elasticity is 20 GPa for the concrete and 200 GPa for the steel. Using an allowable stress of 9 MPa for the concrete and 140 MPa for the steel, determine the largest allowable positive bending moment in the beam.

(b) The solid shaft is subjected to a torque, bending moment, and shear force as shown in Figure 4. Determine the principal stresses acting at points A and B. (16%)

(a) Please draw graphs to show the variation of stress Mohr's circle for a steel bar under axial constant amplitude fatigue loading. The ratio of minimum stress σmin to maximum stress σmax is -0.5. (9%)

3. Determine the deflection of end C of the 150-mm-diameter solid circular shaft (Figure 3). The shaft is made of steel having a modulus elasticity of E = 200 GPa. (25%)

2. As shown in Figure 2, the allowable working stresses for the manufactured wood beam are 10 MPa in bending and 1 MPa in shear. Determine the maximum intensity w0 of the distributed loading that the beam can safely support. (25%)

(b) 下圖1-b所示的組件由橫截面積為400 mm²的鋁管AB組成。一直徑為10 mm 的鋼桿連接到剛性軸環上,並穿過管子。如果對鋼桿施加了80 kN 的拉 伸載荷,試決定鋼桿端 C 的位移。已知鋼桿和鋁管的楊氏係數分別為 Est = 200 GPa, Eal = 70 GPa。(15%)

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