Tutorial 3: Stress Analysis PDF

Tutorial 3: Stress analysis 1. An aircraft tow bar is positioned by means of a single hydraulic cylinder connected by a 25-mm-diameter steel rod to two identical arm-and-wheel units DEF, see Fig. 1(a). The mass of the entire tow bar is 200 kg, and its center of gravity is located at G. For th...

Tutorial 3: Stress analysis 1. An aircraft tow bar is positioned by means of a single hydraulic cylinder connected by a 25-mm-diameter steel rod to two identical arm-and-wheel units DEF, see Fig. 1(a). The mass of the entire tow bar is 200 kg, and its center of gravity is located at G. For the position shown, determine the normal stress in the rod. Ans: -4.97 MPa. (a) (b) Figure 1: See questions 1 and 2. 2. In many situations it is known that the normal stress in a given direction is zero. For ex- ample, σz = 0 in the case of the thin plate shown in Fig. 1(b). For this case, which is known as plane stress, show that if the strains ε x and ε y have been determined experimentally, we can express σx , σy , and ε z as follows: ε x + νε y   σx = E (1a) 1 − ν2 ε y + νε x   σy = E (1b) 1 − ν2   ν εz = − (ε x + ε y ) (1c) 1−ν 3. Members AB and CD are 30-mm-diameter steel rods, and members BC and AD are 22- mm-diameter steel rods, see Fig. 2(a). When the turnbuckle is tightened, the diagonal member AC is put in tension. Knowing that a factor of safety with respect to buckling of 2.75 is required, determine the largest allowable tension in AC. Use E = 200 GPa and consider only buckling in the plane of the structure. Ans: FAC 147.3 kN. 4. (a) An axial force of 60 kN is applied to the assembly shown in Fig. 2(b) by means of rigid end plates. Determine (a) the normal stress in the brass shell, (b) the corresponding deformation of the assembly. (b) The length of the assembly decreases by 0.15 mm when an axial force is applied by means of rigid end plates, see Fig. 2(b). Determine (a) the magnitude of the applied force, and (b) the corresponding stress in the steel core. Ans: P=101.6 kN and σS = 100 MPa 5. The brass shell (αb = 20.9×10−6 /◦ C) is fully bonded to the steel core (αs = 11.7×10−6 /◦ C), see Fig. 2(c). Determine the largest allowable increase in temperature if the stress in the steel core is not to exceed 55 MPa. 1  (a) (b) (c) Figure 2: See questions 3, 4 and 5. 6. Each of the four vertical links connecting the two rigid horizontal members in Fig. 3(a) is made of aluminum (E = 70 GPa) and has a uniform rectangular cross section of 10 mm×40 mm. For the loading shown, determine the deflection of points (a) E, (b) F, and (c) G. Ans: δE = 80.4 µm, δF = 209 µm, δG = 390 µm (b) (c) (d) (a) Figure 3: See questions 6, 7, 8 and 9. 7. A Bronze bar is fastened between a Steel bar and an Aluminum bar as shown in Fig. 3(b). Axial loads are applied at the positions indicated. Find the largest value of P that will not exceed an overall deformation of 3.0 mm, or the following stresses: 140 MPa in the Steel, 120 MPa in the Bronze, and 80 MPa in the Aluminium. Assume that the assembly is suitably braced to prevent buckling. Use the Young’s modulus of steel, Aluminum and Brass as 200 GPa, 70 GPa, and 83 GPa, respectively. 8. A steel bar of length 2.5 m with a square cross section 100 mm on each side is subjected to an axial tensile force of 1300 kN, see Fig. 3(c). Assume that E = 200 GPa and ν = 0.3. 2  Determine the increase in volume of the bar. 9. Determine the dilatation e and the change in volume of the 200-mm length of the rod shown in Fig. 3(d) if (a) the rod is made of steel with E = 200 GPa and ν =0.30, (b) the rod is made of Aluminium with E = 70 GPa and ν =0.35. 10. At room temperature (20 ◦ C) a 0.5-mm gap exists between the ends of the rods shown in Fig. 4(a). At a later time when the temperature has reached 140 ◦ C, determine (a) the normal stress in the aluminum rod, (b) the change in length of the aluminum rod. (a) (b) (c) Figure 4: See questions 10 and 11. 11. A 60-mm cube is made from layers of graphite epoxy with fibers aligned in the x direction. The cube is subjected to a compressive load of 140 kN in the x direction. The properties of the composite material are Ex = 155.0 GPa, Ey = 12.10 GPa, Ez = 12.10 GPa, νxy = 0.248, νxz = 0.248, and νyz = 0.458. Determine the changes in the cube dimensions, knowing that (a) the cube is free to expand in the y and z directions, refer to Fig. 3(b); and (b) the cube is free to expand in the z direction, but is restrained from expanding in the y direction by two fixed friction less plates, see Fig. 4(c). 3