LAME’S EQUATIONS THICK CYLINDER CLASS NOTES FOR MECHANICAL ENGINEERING

LAME’S EQUATIONS

 THICK CYLINDER CLASS NOTES

FOR MECHANICAL ENGINEERING

These equations deal with the hoop and

radial stresses variation in a thick cylinder.

Various cases considered are thick cylinder

subjected to fluid pressure only

internal/external or internal and external.

Lame’s equation determine the maximum

stresses and their location in the thick

cylinder.

 

Lame’s Equations are

σ_h = β/r² + α

σ_r = β/r² – α

Where α and β are Lame’s constants

There is a thick vessel subjected to internal fluid pressure p_i at the inner radius r_i. It is under external fluid pressure p_o at the outer radius r_o.

We want to find the equations for σ_h and σ_r at any radius. It at any radius between r_i and r_o.

Fig. Thin Cylindrical element  Fig. Longitudinal Failure

Consider a long open ended thick walled cylinder  in Fig (a) and Fig.(b) subjected inside pressure p_i and external pressure p_o. Inside pressure is high.

Considering the failure of the cylinder lengthwise.

DERIVATION LAME’S EQUATIONS

Fig. a  Consider two concentric sections at radius r and r + dr of length ‘l’ .

Cut this thin ring into two halves.

Fig. b

Consider its equilibrium under the forces acting due to σ_h and σ_r on the element

Applying the equation of equilibrium to the half ring element,

Sum of downward forces= Sum of upward forces

σ_h dr l+ σ_h dr l + (σ_r+ dσ_r) 2 (r +dr) l= σ_r 2rl

On simplification, we get

2 σ_h dr = –2dr σ_r –2rdσ_r—2dr dσ_r

Divide by 2dr and neglect dr dσ_r

We get

σ_h  = — σ_r  –r (dσ_r/dr)                                 (1)

ASSUMPTIONS IN LAME’S EQUATIONS

 That a transverse plane remains a transverse plane before and after the fluid pressures are applied.

As per this assumption, longitudinal strain has to be constant.

ϵ_long = σ_l/E — μσ_h/E +μ σ_r/E = constant

σ_l  is constant and small. σ_l/E becomes extremely small,

therefore can be neglected.

Therefore — μσ_h/E +μ σ_r/E = constant

Taking  out –μ/E, we get

σ_h — σ_r = constant= 2α                   (2)

(2α has been assumed to be constant because of convenience)

Substitute from eq (1) in eq (2), we get

σ_r + 2α = — σ_r  –r (dσ_r/dr)

2 σ_r+ r (dσ_r/dr) =– 2α

Multiply each term by r

2r σ_r+ r²(dσ_r/dr) =– 2α r

Write it in the below form

d(r² σ_r)/dr =– 2α r

Integrating, we get

r² σ_r = —  α r² + β

σ_r =  β/r²  — α                                  (3)

Put this value in eq(1), we get

σ_h =  β/r²  + α                               (4)

Eqs.(3) and (4) are Lames equations.

Lame’s Constants in Lame’s equations

The values of the Lame’s constants α and β can be found from the physical boundary conditions. These physical boundary conditions are applied to eq (3 ), we get

At  r=r_i,  σ_r = p_i+              p_i = β/r_i²  — α

At  r=r_o, σ_r =p_o                        p_o= β/r_o²  — α

α = (p_ir_i² –p_or_o²) /(r_o² –r_i²)

β = (r₁² r₂²[p_i—p_o]) /(r_o² –r_i²)

Substitute the values of α and β in lame’s equations (3) and (4),

we get

σ_h =  (r₁² r₂²[p_i—p_o])/[r²(r_o² –r_i²)]+ (p_ir_i² –p_or_o²) /(r_o² –r_i²)

On simplification, we get

σ_h =  [(p_ir_i² –p_or_o²) + r₁² r₂²/(r_o² –r_i²)r²]/ (r_o² –r_i²)      (5)

σ_r =  [(p_or_o² –p_ir_i²) + r₁² r₂²/(r_o² –r_i²)r²]/ (r_o² –r_i²)      (6)

From equations (5) and(6)

it is evident that the stresses are inversely proportional to the square of the radius. These stresses vary parabolically.

Maximum Stresses

From eq (5) it is evident that the maximum value of σ_h will occur at the innermost radius. It is given by

σ_hmax = [p_i(r_i²+r_o²)—2p_or_o²)]/ (r_o² –r_i²)

From eq (6)

It is evident that the maximum value of σ_r is larger of p_i or p_o. In our analysis it has been assumed that p_i > p_o. Therefore σ_r will also be maximum at the innermost radius.  Its value is equal to p_i.

σ_{r max} = p_i   with p_i >  p_o   

Longitudinal Stress

Longitudinal stress, σ_L = [p_ir_i²—p_or_o²)]/ (r_o² –r_i²).

This contain no variable. Hence it is constant. It is small as compared to the maximum values of σ_hmax and σ_{r max} . Hence σ_L is neglected.

Variations of hoop and radial stresses in a thick shell when plotted.

Fig. Variation of hoop and radial stresses in a thick under internal & external pressures

Hoop stress is tensile. It is maximum at the inner radius. Radial stress is compressive. It is also maximum at the inner radius with p_i >p_o. Maximum hoop stress is much greater than maximum radial stress. Hence it is designed on the basis of hoop stress.

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