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Unformatted text preview: Torsion of a Circular Cylinder Shailendra P. Joshi University of Houston 1 Torsion of a circular cylinder Let small rotation angle at Assume no body forces act. Fully solve this problem in 3D 2 Torsion of a circular cylinder Let small rotation angle at P What are the displacements at a given point P(x1, x2, x3) due to a rotation? 3 Torsion of a circular cylinder Let small rotation angle at P What are the displacements at a given point P(x1, x2, x3) due to a rotation? 3 Torsion of a circular cylinder Let small rotation angle at P What are the displacements at a given point P(x1, x2, x3) due to a rotation? 3 Torsion of a circular cylinder Let small rotation angle at P What are the displacements at a given point P(x1, x2, x3) due to a rotation? 3 Strains Knowing axial strains shear strains 4 Strains Knowing axial strains shear strains 4 Strains Knowing axial strains shear strains 4 Strains Knowing axial strains shear strains 4 Strains Knowing axial strains shear strains 4 Strains Knowing axial strains shear strains Similarly, 4 Strains Knowing axial strains shear strains Similarly, 4 Strains Knowing axial strains shear strains Similarly, We note that 4 Non-zero strain components Non-zero stress components Is this a possible state of stress? 5 Non-zero strain components Non-zero stress components Is this a possible state of stress? 5 Non-zero strain components Non-zero stress components Is this a possible state of stress? 5 Check for equilibrium i=1 à automatically satisfied i=2 i=3 6 Check for equilibrium i=1 à automatically satisfied i=2 i=3 6 Check for equilibrium i=1 à automatically satisfied i=2 i=3 6 Check for equilibrium i=1 à automatically satisfied i=2 i=3 6 Check for equilibrium i=1 à automatically satisfied i=2 i=3 6 Check for equilibrium i=1 à automatically satisfied Twist per unit length i=2 i=3 6 Check for equilibrium i=1 à automatically satisfied Twist per unit length i=2 i=3 For equilibrium, the increment in angular rotation should be a constant 6 Traction boundary conditions 7 Traction boundary conditions 1. Lateral surface 7 Traction boundary conditions 1. Lateral surface 7 Traction boundary conditions 1. Lateral surface 7 Traction boundary conditions 1. Lateral surface 7 Traction boundary conditions 1. Lateral surface 7 Traction boundary conditions 1. Lateral surface 7 Traction boundary conditions 1. Lateral surface 0 7 Traction boundary conditions 1. Lateral surface 0 7 Boundary conditions continued… 2. End faces (consider the right face) 8 Boundary conditions continued… 2. End faces (consider the right face) 8 Boundary conditions continued… 2. End faces (consider the right face) 8 Boundary conditions continued… 2. End faces (consider the right face) 8 Boundary conditions continued… 2. End faces (consider the right face) 8 Boundary conditions continued… 2. End faces (consider the right face) 8 These surface tractions will result in forces/ moments 1. Resultant forces 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 0 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 0 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 0 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 0 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 0 2. Resultant moments 9 These surface tractions will result in forces/ moments 1. Resultant forces 0 2. Resultant moments 9 Twisting moment Twist per unit length Polar second moment of area At 10 Twisting moment Twist per unit length Polar second moment of area At Thus, the stress state is 10 Twisting moment Twist per unit length Polar second moment of area 11 Twisting moment Twist per unit length Polar second moment of area Just as we wrote the flexure formula, we write the torsion formulas - 11 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 Equivalent representation of the torsion formulas • Consider a point P(x2, x3) • At P, both and are present, in general • We may represent these two shear stresses by an equivalent shear stress , in terms of the radial distance (r) from the center of the circular crosssection. • is the resultant of and Important characteristics of this representation • varies linearly with r, because and vary linearly with x2 and x3 • at r = a (radius of the bar), • Along the x2 axis, • Along the x3 axis Thus, the two torsion formulas can be combined into a single expression, in terms of and r 12 ...
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