Chapter2, Thick Walled Cylinder

 

CHAPTER TWO THICK-W THICK-WALLED ALLED CYLINDERS CYLI NDERS and SPINNING DISKS *

Important concepts and equations in MECH 202

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Calculation of mechanical and thermal stresses in thick-walled cylinders and disks

1.1 The Theory of Elasticity (MECH 202) Method – Equilibrium Equations and boundary Conditions and Strain-Displacement Relations 1.2 Equilibrium Equations, Stress, Strain and Displacement formulas for thick- walled walled cylinders cylinders and spinning disks disks 1.3 Thick-walled cylinder under Pressure, Compound Cylinders 1.4 Stresses in spinning disks 1.5 Thermal stress in cylinders and disks

 

1.1 THE THEORY THEORY OF ELASTICITY (MECH 202) METHOD •

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The deformation mode does not have to be described in order to solve a problem The solution satisfies (1) condition of equilibrium at every point (2) continuity of displacement field (3) loading and support conditions (boundary condition)

 

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Equilibrium equations and boundary conditions

(1) In general stresses are functions of the coordinates

 

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Equilibrium equations and boundary conditions

(2) Consider the equilibrium of a differential element

 

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Equilibrium equations and boundary conditions

(3) The equilibrium of forces in X and Y directions requires (t – thickness of the element)

(4) Differential equations of equilibrium

 

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Equilibrium equations and boundary conditions

(5) Boundary conditions

 

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Strain-displacement relations

FIGURE 2.3.1. Lines OA and OB are drawn on a body in its strain-free state. As a result of  loading, configuration OAB is converted to O’A’B’. Equations 2.3.3 are derived from this sketch.

 

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Strain-displacement relations

(2) Normal strain

(3) Shear strain

 

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Equations in polar p olar coordinates

 

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Saint-V Sain t-Vena enant' nt'ss Princi Principle ple

If a system of forces acting on a small region of an elastic solid is replaced by another force system, acting within the same region and having the same resultant force and moment as the first system, then stresses change appreciably only in the neighborhood of the loaded region.

 

Illustration ation of Saint-Venant's Saint-Venant's principle. When distributed distributed pressure FIGURE 2.2.3. Illustr loads (a) Are replaced by staticall statically y equivalent concentrated loads (b) Stresses Stresses change considerably considerably near theloads (shaded regions), but change little elsewhere.

 

1.2 EQUILIBRIUM EQUATIONS, STRESS, STRAIN AND DISPLACEMENT FORMULAS FOR THICK-WALLED THICK-WALLED CYLINDERS AND SPINNING DISKS

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ThinThi n- and thic thick-w k-walle alled d cylinders

 

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Equilibrium equations for for cylinder and disk

FIGURE 3.2.1. (a) Cross section of a thick-walled cylinder, under internal pressure P i and external pressure Po , or plan view of a disk spinning with constant angular velocity ω. (b) Force that act on a differential element of dimensions r dθ by dr by h, wher where e h is th the e (constant) thickness of the disk or a typical length along the cylinder. The mass density is ρ.

 

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Strain, displacement and elastic ela stic stress formulas

 

C1, C2 are constants of integration, integration, they are determined by boundary bou ndary conditions.

 

1.3 THICK-WALLED CYLINDER UNDER PRESSURE, COMPOUND CYLINDER THICK-WALLED CYLINDER UNDER PRESSURE, COMPOUND CYLINDER •

Thick-walled cylinder under pressure (w = 0)

(b) stress formulas By using above boundary conditions the constants C1 and C2 are found to be

(a) Internal pressure

a = 3b

(b) External pressure

a = 3b

 

Finally,, we have stress Finally stress formulas (Lamé solution)

FIGURE 3.4.2. Stress σθ at r= b due to internal pressure only. The upper curve is the Lamé solution. The lower curve is the thinwalled thinw alled equation, equation, σθ = Pib/t. The middle curve uses the mean radius R = (a + b)/2 in place of b in the thin-walled equation.

FIGURE 3.4.3. Pressurized holes in a flat body of

arbitrary contour. contour. If holes are widely separated and not close to an edge, stresses at the holes due to pressure P are much like stresses in a

very thick pressurized cylinder.  

(1) (1)

If t/b t/b < 0.1 0.1 (t (thin hin-w -wall all), ), the the str stres esss solu solutio tion n σθ for thin-wall is almost the same as thick-wall cylinder cy linder..

(3)

In a cylind cylinder er w with ith closed closed ends ends,, axia axiall for force ce is provi provided ded b by y pre pressu ssure re aga agains instt the en end d caps, the axial stress due to internal and external pressure load is

(4) The largest shear stress is on the inside surface, at r = b,

 

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Compound cylinder (1) More efficient use of material

(2) Shrink-fitting The contact pressure Pc due to shrink-fitting depends on the radius mismatch Δ, i.e., Ur(outer-cylinder) (Pc , …) Ur(inner-cylinder) (Pc , …) = Δ •

FIGURE 3.6.1. (a) Stresses produced by shrink-fit contact pressure Pi at thin interface of a compound cylinder cylinder.. (b) Stresses produced by the combination of shrink-fit and internal pressure. Dashed lines represent stresses due to Pi alone. alo ne.

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Once Pc is known, the residual residual stresspressure in th the e 2 cylinder cylinders s can be calculate calculated by Lamé solution Superposing the stresses of internal Pi → more efficient use ofd material

 

1.4 STRESSES IN SPINNING DISKS OF CONSTANT THICKNESS

* A small central hol hole e doubles the stress over

FIGURE 3.7.1. Stresses σ and σ in elastic spinning disks of

the case of no hole!

FIGURE 3.7.1. Stresses σr and σθ in elastic spinning disks of constant thickness. thickness. (a) Solid disk. (b) Disk with a central hole.

 

· Example

FIGURE 3.7.2. Two days days to connect a disk and a shaft. The second way, with a solid disk and a discontinuous shaft, s haft, results in lower stresses. Larger flanges (dashed lines) allow bol bolts ts to be placed pla ced farther from the axis, where disk stresses are lower lower..

A steel disk and a solid shaft are connected by shrink-fitting. To determine: (1) What are the stresses at standstill; (2) At what speed the shrink-fit will loosen; and what are the stresses at this speed; (3) What are the largest σ θ and contact pressure at half this speed.

 

1.5 THERMAL STRESS IN CYLINDERS AND DISKS