Elasticidade-Basica-Exercicios-Resolvidos-e-Problemas

42 Basic elasticity Note that P could have been found directly in this particular case from the axial strain Thus from the first of Eqs 152 σx Eεa 70 000 1000 106 70 Nmm2 as before References 1 Timoshenko S and Goodier J N Theory of Elasticity 2nd edition McGrawHill Book Company New York 1951 2 Wang C T Applied Elasticity McGrawHill Book Company New York 1953 3 Megson T H G Structural and Stress Analysis 2nd edition Elsevier 2005 Problems P11 A structural member supports loads which produce at a particular point a direct tensile stress of 80 Nmm2 and a shear stress of 45 Nmm2 on the same plane Calculate the values and directions of the principal stresses at the point and also the maximum shear stress stating on which planes this will act Ans σI 1002 Nmm2 θ 2411 σII 202 Nmm2 θ 11411 τmax 602 Nmm2 at 45 to principal planes P12 At a point in an elastic material there are two mutually perpendicular planes one of which carries a direct tensile stress at 50 Nmm2 and a shear stress of 40 Nmm2 while the other plane is subjected to a direct compressive stress of 35 Nmm2 and a complementary shear stress of 40 Nmm2 Determine the principal stresses at the point the position of the planes on which they act and the position of the planes on which there is no normal stress Ans σI 659 Nmm2 θ 2138 σII 509 Nmm2 θ 11138 No normal stress on planes at 7021 and 275 to vertical P13 Listed below are varying combinations of stresses acting at a point and referred to axes x and y in an elastic material Using Mohrs circle of stress determine the principal stresses at the point and their directions for each combination σx Nmm2 σy Nmm2 τxy Nmm2 i 54 30 5 ii 30 54 5 iii 60 36 5 iv 30 50 30 Ans i σI 55 Nmm² σII 29 Nmm² σI at 115 to x axis ii σI 55 Nmm² σII 29 Nmm² σII at 115 to x axis iii σI 345 Nmm² σII 61 Nmm² σI at 795 to x axis iv σI 40 Nmm² σII 60 Nmm² σI at 185 to x axis P14 The state of stress at a point is caused by three separate actions each of which produces a pure unidirectional tension of 10 Nmm² individually but in three different directions as shown in Fig P14 By transforming the individual stresses to a common set of axes x y determine the principal stresses at the point and their directions Ans σI σII 15 Nmm² All directions are principal directions P15 A shear stress τxy acts in a twodimensional field in which the maximum allowable shear stress is denoted by τmax and the major principal stress by σI Derive using the geometry of Mohrs circle of stress expressions for the maximum values of direct stress which may be applied to the x and y planes in terms of the three parameters given above Ans σx σI τmax τmax² τxy² σy σI τmax τmax² τxy² P16 A solid shaft of circular crosssection supports a torque of 50 kNm and a bending moment of 25 kNm If the diameter of the shaft is 150 mm calculate the values of the principal stresses and their directions at a point on the surface of the shaft Ans σI 1214 Nmm² θ 3143 σII 464 Nmm² θ 12143 P17 An element of an elastic body is subjected to a threedimensional stress system σx σy and σz Show that if the direct strains in the directions x y and z are εx εy and εz then σx λe 2Gεx σy λe 2Gεy σz λe 2Gεz where λ νE1 ν1 2ν and e εx εy εz the volumetric strain P18 Show that the compatibility equation for the case of plane strain viz ²γxyxy ²εyx² ²εxy² may be expressed in terms of direct stresses σx and σy in the form ²x² ²y²σx σy 0 P19 A bar of mild steel has a diameter of 75 mm and is placed inside a hollow aluminium cylinder of internal diameter 75 mm and external diameter 100 mm both bar and cylinder are the same length The resulting composite bar is subjected to an axial compressive load of 1000 kN If the bar and cylinder contract by the same amount calculate the stress in each The temperature of the compressed composite bar is then reduced by 150C but no change in length is permitted Calculate the final stress in the bar and in the cylinder if E steel 200000 Nmm² E aluminium 80000 Nmm² α steel 0000012C and α aluminium 0000005C Ans Due to load σ steel 1726 Nmm² compression σ aluminium 691 Nmm² compression Final stress σ steel 1874 Nmm² tension σ aluminium 91 Nmm² compression P110 In Fig P110 the direct strains in the directions a b c are 0002 0002 and 0002 respectively If I and II denote principal directions find εI εII and θ Ans εI 000283 εII 000283 θ 225 or 675 23 A displacement field in a body is given by u c2x y² v cx² 3y² where c 10⁴ Subsequent to the loading determine a the length of the sides AC and AD b the change in the angle between sides AB and AD and c the coordinates of point C 24 Determine the state of strain on an element positioned at 0 2 1 where ε 10⁴ The displacement field and strain distribution in a member have the form u cx y z² v 2cyz w 2cy 25 What relationship connecting the constants as bs and cs make the foregoing expressions for displacement and strain distribution in a member have the form εx a₀ a₁xy a₂y² εy b₀xy b₁y² b₂xy² γxy c₀xy c₁xy² c₂y² Redo Prob 24 for the following system of strains εx a₀ a₁y a₂y² εy b₀ b₁y b₂y² γxy c₀ c₁y c₂y² 26 A 100by 150mm rectangular plate QABC is deformed into the shape shown by the dashed lines in Fig P26 All dimensions shown in the figure are in millimeters Determine a the strain components εxx εyy and γxy b the principal strains and the direction of the principal axes 27 Calculate the principal strains and their orientations at point A of the deformed rectangular plate by using Fig P27 28 As a result of loading the rectangle shown in Fig P28 deforms into a parallelogram in which sides OA and BC shorten 0003 mm and rotate 500µ 228 For a given steel E 200 GPa and ν 03 determine at a point within this material is given by 400 0 0 300 400 0 200 300 400 229 For a material with G 80 GPa and E 200 GPa determine the strain tensor for a state of stress given by 0 100 200 μ 300 400 400 5 15 20 4 10 5 MPa 230 The distribution of stress in an aluminum machine component is given in megapascals by σx y 2z2 τxy 3x2 τzy x z τxz z3 y Calculate the state of strain of a point positioned at 1 2 4 Use E 70 GPa and ν 03 231 The distribution of stress in a structural member is given in megapascals by Eq d Example 124 Chater 1 Calculate the strains at the specified point Qx12 for E 200 GPa and ν 025 232 A railroad rail of type E 10 mm is subjected to b0x a 300 mm b 400 mm and thickness t 13 mm is subjected to maximum shear stresses as shown in Fig P232 Calculate the change in a the length AB b the volume of the plate Figure P232 Problems Chapter 2 Strain and StressStrain Relations 221 A tensile test is performed on a 12 mmdiameter aluminum alloy specimen γ 035 using a 50 mmgage length When an axial tensile load reaches a value of 16 kN the gage length has increased by 010 mm and the elastic modulus of elasticity E is obtained within this load range to be calculated on the bar 222 A long testing specimen is subjected to tensile loading The increase in length resulting from a load of 9 kN is 0025 mm for an original length l0 of 75 mm What are the τx and τz strains and stresses Cal Secs 27 through 10 223 A 50mmsquare plate is subjected to stresses shown in Fig P223 What deformation are BD Express the solution in terms of E ν 03 using two approaches a Determine the components of the transformation of strain b determine the stresses on planes perpendicular to the radial lines using the generalized Hookes law 224 A uniform pressure P acts over the entire straight edge of a large plate Fig P224 Determine the distance to a nail that is marked by the surface of P according to the loading in terms of Poissons ratio v and P as required to measure that strain at a critical point on the surface of a loaded beam The readings σx 100 Gpa εx 50 μe principal 100 θ 45 and angles σy 120 GPa and ν 03 225 The following are Fig P224 θ 15 with forces and their directions Use E 200 GPa and ν 03 226 A 30 hook μ 250 με and εz 200 με for θs 75 β 30 and θ 5 Determine strain components sεx and εz 227 The strains measured at a point on the surface of a machine element are εx 400 μ εy 300 μ and ε 50 μ for θ 30 θ 30 and Figure P224 Figure P223 x 50 mm y 56 mm 5 MPa 10 MPa 20 MPa 0 5 10 15 0 10 15 20 4 5 30 89