A member of a 3D truss is located between the origin of rectangular coordinates and point P (1,1,0.5) m, as shown in Figure P3.1. If the member remains straight and end P experiences small displacements, u=3 mm, v = 2 mm, and w=-1.5 mm, find the average extensional strain of the member.
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- A wine of length L = 4 ft and diameter d = 0.125 in. is stretched by tensile forces P = 600 lb. The wire is made of a copper alloy having a stress-strain relationship that may be described mathematically by =18,0001+30000.03(=ksi) in which is nondimensional and has units of kips per square inch (ksi). (a) Construct a stress-strain diagram for the material. (bj 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?A member of a 3D truss is located between the origin of rectangular coordinates and point P (1,1,0.5) m, as shown in Figure P3.1. If the member remains straight and end P experiences small displacements, u=3 mm, v= 2 mm, and w=-1.5 mm, find the average extensional strain of the member. FIGURE P3.1 0.5 0. (u, v, w) (3,2,-1.5) mm P₁ P(1.1, 0.5)H.W.10 Rigid bar ABCD is supported by two bars as shown in Figure P2.7. There is no strain in the vertical bars before load P is applied. After load P is applied, the normal strain in bar (2) is mea- sured as -3,300 µm/m. Use the dimensions L1 = 1,600 mm, L2 = 1,200 mm, a = 240 mm, b = 420 mm, and c = 180 mm. Determine (a) the normal strain in bar (1). (b) the normal strain in bar (1) if there is a 1 mm gap in the connection at pin C before the load is applied. (c) the normal strain in bar (1) if there is a 1 mm gap in the connection at pin B before the load is applied. (1) a Rigid bar B L (2)
- 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?Q4 Figure Q4(a) shows the stress-strain curves of material A and material B. A circular rod of material A (diameter 3 cm, length 10 cm) is bonded to a circular rod of material B (diameter 4 cm, length 20 cm) as shown in Figure Q4(b). a) Based on the stress-strain curves shown determine the yield stress (a,) and the elastic modulus (E) of material and material B. b) Assuming that the bond is rigid, determine the force P required to stretch the composite rod by 0.3 mm? c) Assuming a Poisson's ratio value of 0.33, determine the change in diameter of the rod made from material B for the same P as determined in part (b). 300 Material A 200 Material B 100 0.2 0.4 0.6 0.8 1.0 Strain x10 FIGURE Q4(a) B P 20 cm 10 cm FIGURE Q4(b) Stress / MPaA rectangular block is 7 cm long in the x-direction, 5 cm in the y-direction, and 9 cm long in the z-direction. The block is subjected to a triaxial loading of three uniformly distributed forces as follows: 15 kN in the x- direction, 21 kN in the y-direction, and 18 kN in the z-direction. Calculate the strain in the direction y if Poisson's ratio v = 0.30 and having a modulus of elasticity of E = 150 GPa. y 7 cm 9 cm 21 kN 15 kN 5 cm 18 kN
- A bar of titanium alloy of length 120mm and square cross – section, 7.5 mm x 7.5 mm, is pulled axially by a force of 15 kN. Find the percentage decrease in thickness if E = 106 GN/m2 and v = 0.33 m/s. V= velocity A bar of aluminum alloy of rectangular section75 mm x 20 mm and 800 mm long is stretched by an axial force of 150 kN. Find the volumetric strain, the actual change in volume and the percentage reduction in cross – sectional area of the bar. Take E = 70 GN/m2 and v = 0.34 m/s.V=velocity4 A rectangular steel beam is held between two rigid, unyielding walls 2.25 m apart. The modulus of elastic- ity of the steel is 210 GPa, the coefficient of thermal expansion is 11.7 × 10-6 1/°C, and the cross-sectional area of the beam is 6.5 cm². If the beam temperature is increased by 30°C, what is the change in stress in the beam?4. The section of peripheral nerve in neck is under tension of force F=0.18N due to sudden movement that causes large deformation of the nerve. As shown in the below figure, the nerve consists of two structures: fascicles and epineurium with modulus of elasticity of Efascicles=Ef2=MPa and Eepineurium=Ee 0.5MPA. Note: you can model all three fascicles as one part with the cross section of Afascicles=A=3((200×10“)²)= 12 n x10*. Use compatibility condition (fascicles and epineurium deform together) b) Determine the deformation of the peripheral nerve (L=50mm). Note: make sure to include the entire solution including the internal forces in fascicles and/or epineurium. d) If the nerve can tolerate strain up to 0.2, determine if the nerve got injured or not due to this sudden movement Peripheral nerve Under tension AF 50mm Fascicles F 400μm Epineurium 400um 400µm Cross section of peripheral nerve 1000um (Answer: 8 9.4mm )
- 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 mm The 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?a) The total strain energy for a bar of length L. is given by the equation 2EA How would the equation change if the bar is tapered as shown in Figure Q4 (a)? Cakulation is not required. Figure Q4 (a) b) The frame ABCD with uniform diameter 40 mm and length L- 500 mm shown in Figure Q4 (b) is in the horizontal (xy) plane. One end of the frame (4) is fixed to a rigid wall and the other end (D) is subjected to a vertical force F-2 kN. Given that the frame is made of a material with E-80 GPa and G-30 GPa. Determine: i) The strain energy stored in the frame member CB (ignore energy due to shear). i) The vertical deflection at point D using Castigliano's second theorem (consider bending effecet only). Figure Q4 (b)A large thin rectangular plate is subjected to certain traction boundary conditions on its thin edges. The following stress function is proposed for this problem: 2 p(x,y) = A x¹ + B x²y² + Cy² (i) use the biharmonic equation to find restrictions, if any, on values of A, B, and C (ii) calculate all stress components (iii) calculate all strain components in terms of A, B, and C as well as the Young modulus and Poisson's ratio E and y, respectively. (iv) check that the equilibrium equations are satisfied (v) determine the traction boundary conditions at x=+a and y=+b