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§Chapter 6

Nonlinearity Important features of shell behavior require the inclusion of nonlinear terms in the equations. In general, this introduces considerable complexity into the analysis. However, a relatively simple approximation can be used, referred to as the "prestress effect", which yields quantitative information about the stability limit of a shell with compressive stress and the stiffening effect of tensile stress. An understanding of some large displacement effects in a shell can also be obtained with the simple approximation of an inverted "dimple". §6.1 Prestress Effect The moderate rotation theory is derived from the variational formulation in section 2.12.2. This produces two additional nonlinear terms in the coefficient matrix. The first (2.12.19) gives a product of the meridional stress resultant Ns and the rotation, which is the "prestress" term. A second term (2.12.20) gives the square of the rotation, which can be of importance but which will not be considered in this section. One advantage in placing the general shell equations in matrix form (2.11.22), is that nonlinear effects become easier to handle. The relationship between radial and axial components, and normal and tangential components has often been used. However, the angle between the normal component ζ and the z axis is only ϕ before deformation. To take into account this deformation, one should use ϕ +χ . As a consequence, by using (2.4.1) and (2.4.2) the tangential resultant is: Nϕ = H cos(ϕ + χ ) + V sin(ϕ + χ ) (6.1.1) For small deformations, χ is small in comparison to ϕ . Thus,

Q = H sin(ϕ +χ ) – V cos(ϕ +χ )= H ( sinϕ cosχ + cosϕ sinχ ) + V (cosϕ cosχ – sinϕ sinχ )= (H sinϕ –V cosϕ ) + (H cosϕ +V sinϕ )χ + O (χ 2) (6.1.2)

This equation has the same interaction between the tangential load and rotation, as in the classical beam theory that Euler beam buckling is based on. From one perspective, this is a nonlinearity. In beam theory, on the other hand, the rotation has little effect on the axial load. The resulting equation is thus linear and provides, in a simple manner, the classical stability limit. Because of this term, shells behave in a similar manner. One can also consider the compatibility conditions (2.7.3)

121

d hd s

= εs cos ϕ +χ – χ sin ϕ +χ = εs cosϕ – χ sinϕ – ε s sinϕ + χ cosϕ χ + O χ 2 (6.1.3) The second term is this equation is a real nonlinearity and causes many problems in the analysis of shells. In contrast to beams or plates, the thin shell collapses before the classical stability load is reached. We have little time .... In the moment equilibrium equation (2.4.12) there is an additional nonlinear term

d ds

(r Ms ) = –Mθ cosϕ +r Q = – Mθ cosϕ + r H sinϕ –V cosϕ +r H cosϕ –V sinϕ χ (6.1.4)

In dimensionless form

1λ

d dχ

M s( E t c )e

= . . . + 2ρ E t cr2

* χ

λ

(6.1.5) where the term ρ is defined as

ρ = H cosϕ +V sinϕ r2

2 E t c (6.1.6)

The matrix equation (4.1.4):

1λ

ddx

y= a0 + 1λ

a1 + 1λ 2

a2 • y – b (6.1.7)

is modified only in that the new factor ρ appears in the matrix a0

a 0 =

0 sinϕ * 2ρ Etcr 2

* 0

0 0 0 E t r 2

*

1 E t 3 *

0 0 0

0 0 – sinϕ * 0

(6.1.8) This factor does not affect the first two terms of the asymptotic series (γ0,γ1) in the particular solution that provides the radial load H and displacement h. Consequently equation (4.2.13) is still valid. Applying this to (6.1.6) results in

122

ρ = V r

2 E t c sin2ϕ (6.1.9)

The solution of the homogeneous equation will be influenced by this factor. This complementary solution (4.3.1) results in the recursion relations (4.3.3-5):

a0 – ξ ' I • α0 = 0a0 – ξ ' I • α1 = – a1•α0 + α' 0

. . . . (6.1.10) Thus ρ has an influence on the eigenvalue ξ '. The condition that the determinant is equal to zero yields the polynomial for the eigenfunction:

ξ ' c r2 *4 – 2ρ ξ ' c r2 *

2 + 1 = 0 (6.1.11) The solution is

ξ ' c r2 *2 = ρ ± ρ 2 – 1 (6.1.12)

For values of ρ between -1 and 1, one uses the following relationship for γ ρ = cosγ (6.1.13) The solution can be expressed as the following

ξ ' c r2 * 2 = cos γ ± i 1 – cos2γ

= e ± i γ (6.1.14) which gives the eigenvalues: ξ ' c r2 * = ± e ±i φ /2 (6.1.15)

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ξ '

–ξ '

–ξ '

ξ '

γ

ρ < – 1 ρ = 0

ρ = 1

ρ > 1

Figure 6.1 - Locus of the eigenvalues in the complex plane. The arrows show the direction of the change in the eigenvalues as ρ increases. For negative values ρ < – 1, all roots are on the imaginary axis. For ρ > 1, all roots are real.

The behavior of the roots in the complex domain is shown in Fig. 6.1. The previous linear results correspond to the value of ρ = 0. The angle γ is π /2 and the roots lie on the line where the angle γ /2 = 45 degrees. The absolute value of ξ ' c r 2 * is always 1, for values of ρ between -1 and 1, but the angle changes. As the root ξ ' moves down

towards the real axis, its complex conjugate ξ ' moves up likewise. This applies to – ξ ' and – ξ ' as well. While ρ increases to the value of 1, γ has a limiting value of 0 ; ξ '

and ξ ' coalesce on the real axis, along with – ξ ' and – ξ ' . All roots are pure real numbers for large values of ρ . When ρ is equal to – 1, then ξ ' and ξ ' coalesce on the imaginary axis. All roots are pure imaginary numbers when ρ is less than – 1. Thus, with enough large tangential stress in the particular solution, such as a balloon, the solution of the homogeneous equation is purely exponential. With enough large compression from the particular solution, the solution of the homogeneous equation will be oscillating. §6.2 Stability With an oscillating solution, one can satisfy the homogeneous boundary conditions and obtain a solution of arbitrary magnitude, which indicates an instability in the system. Without further discussion, the value of ρ = – 1 is the classical stability limit of a spherical and cylindrical shell with external pressure.

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A shell with constant external pressure load has a membrane or particular solution of:

V = p r

2 (6.2.1) where ρ is found from (6.1.6):

ρ = p r 2

2

4 E t c (6.2.2)

The critical pressure can be found by solving for pressure, giving

pcr =

4 ρ E t c r 2

2 (6.2.3)

Shell

Beam

PlatePressure

Displacement

pcr

Figure 6.2 - Stability characteristics of beams, plates and shells. The beam and plate still maintain a load-carrying capability after the critical load has been reached. This is not the case for thin shells, under many types of loading, as indicated by the sketch in Fig. 6.2. In general, a large, nonlinear deformation or a "snap-through" may occur when the load is substantially less than the classical critical value. The exact value depends on the state of imperfection in the shell, and has been the subject of considerable research. §6.3 Edge Zone Stiffness In addition to stability with the negative values of ρ , there is also interest in the behavior of shells when ρ is positive. The eigenvector is as before

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α0 = α0(x )

– ξ ' 2 E t3/sinϕ *

1ξ '

E t / r 2 *

– ξ ' 1sin ϕ *

1

(6.3.1) The parameter ρ does not explicitly appear because it only influences one element in the matrix a0 and the eigenvector can be determined by the remaining part. Of course, ρ has a influence on ξ '. As before, one can calculate the influence coefficients. At the point where s = se, the solution is

Y =

y1y2y3y4

= Re C

– ξ ' 2

1 / ξ '

–ξ '

1

1 + O λ -1

(6.3.2) where

ξ ' = ei γ / 2 ρ = cosγ

cos γ /2 = 1 + cos γ2 = 1 + ρ

2 (6.3.3) This results in y4 = Re C (6.3.4) y3 = –ReC eiγ / 2 = –Re C cosγ /2 + Im C sinγ /2 (6.3.5) y2 = ReC e–iγ / 2 = Re C cosγ /2 + Im C sinγ /2

= y4 cosγ /2 + y3 + y4cosγ /2

sinγ /2 sinγ /2

= 2y4 cosγ /2 +y3 = 2 y4 1+ρ +y3 (6.3.6)

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y1 = – ReC ei γ = – Re C cosγ + Im C sinγ

= –y4 cosγ +

y3+y4cosγ /2sin γ/2 sinγ

= –y4

sin γ/2 sinγ cosγ2 – cosγ sinγ2 + 2y3cosγ /2 (6.3.7)

y1 = y4 + 2y3 1+ρ (6.3.8) The influence of the parameter ρ is now clearly shown

y1y2

= 2 1+ρ 11 2 1+ρ • y3

y4 (6.3.9) With positive values of ρ , the shell is stiffer against movement. With negative values, the shell is more flexible. Especially when ρ reaches the value of – 1, the classical stability limit, the two stiffness terms ( k11 and k22) are equal to zero. One can prove that this result is also applicable when ρ > 1. The parameter ρ has no effect on the k21 element. In the problem of a pressure vessel with built-in walls, one gets y3 = 0, y4 c = y4 p y1 c = y4 c = – y4 p The nonlinear parameter ρ , has little effect on this problem. This is in agreement with the results in Fig. 6.3. Only with large positive values of ρ is the solution of the linearized equations not valid. Practically speaking, however, a shell must be very thin (i.e. a balloon) before ρ becomes larger than 1 without the appearance of nonlinearities in the material behavior.

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2 4 6 80-1

Linear

Nonlinear

30°

ρ

StressR/t = 100

Figure 6.3 - Stress as a function of ρ for the fixed edge shell and for the pressurized shell with a discontinuous slope. (From P. Van Dyke, AIAA J. Nov. 1966, p.2045)

Now consider the problem of slope discontinuity. In this case the radial force of the particular solution is discontinuous.

y2 c = – 12 Δy2 p

y3 c = 0

y4 c = 1

k22 y2 c

y1 c = k12

k22 y2 c

y1 c = 1

1+ρ y1 c ρ =0

The discontinuity effect is reduced with a large internal pressure. When an external pressure is applied and ρ reaches the value of – 1, the stress will increase without bound. The edge boundary bending stress due to a discontinuity in the curvature of the meridian has a similar dependence on ρ . In this case the maximum bending stress is not at the point of discontinuity, but rather in the boundary layer. §6.4 Shear Deformation

128

The analysis of transverse shear deformation in shells is easily formulated using the previously established equations in matrix form. Shear deformation by itself is not a nonlinear effect. Here, however, we consider the interaction with the pressurization effect. From (2.8.20) and (2.5.3) one obtains the following expression for the unit transverse shear force:

Q = E t

µ χ – –1

sin ϕ dhds

+ εs cot ϕ (6.4.1)

where E/µ is the effective transverse shear modulus. This equation is the difference between the rotation of the normal χ and the rotation of the reference surface of the shell β. Ιn homogeneous elastic materials, µ is equal to about 3. This additional term only shows up in the a2 matrix and thus has minor significance. With a composite shell with a relatively soft matrix, µ is large and may be of magnitude of λ 2. Hence the matrix a2 gets the element

a24 0 = µ /λ 2

sin2ϕE t

*

(6.4.2) and

b0= V cosϕ

E t e 1000

+ µ

λ 2sinϕE t

*0001 (6.4.3)

The equation of the eigenvalues is

ξ ' r 2t *

4– 2ρ + µ

λ 2 t

r 2 * ξ ' r 2t * 2+ 1 + 2ρ

µ

λ 2 t

r 2 * = 0

(6.4.4) The solution is

ξ ' =

1+ 2ρ µ

λ 2 cr 2

* 4

r 2c * ei γ /2

(6.4.5) where

129

cosγ = ρ + µ

2λ 2 c / r 2 *

1 + 2ρ µ

λ 2c

r 2

*

(6.4.6) The eigenvector is

α0 = α0 x

– ξ ' 2 E t c 2sinϕ

*1 –

µ

λ 2 ξ ' 2 r 2* 2

1ξ '

E tr 2

*

– ξ ' 2 1sinϕ *

1 – µ

λ 2 ξ ' 2 r 2* 2

1

(6.4.7) The resulting edge stiffness coefficients at the point se are

y 1y 2

= 1f + µ /λ 2

2f f +ρ + µ / 2λ 2 1

1 2 f +ρ + µ / 2λ 2 • y3

y4

(6.4.8) where

f = 1 + 2ρ µ /λ 2 The stability limit is cosγ = – 1, consequently the critical value of the prestress parameter is:

ρ = – 1 –

µ

2λ 2c

r 2*

(6.4.9) The smaller the transverse shear stiffness, the larger will be the value of µ and the weaker the shell shall be. The effect of the transverse shear on discontinuity stress can be seen most easily without the prestress term. Setting this equal to zero: ρ = 0

130

yields the value of the dimensionless stiffness coefficients:

k12 = 11 + µ /λ 2

k12k22

= 1

1 + µ

2λ 2 which are exactly the values of these coefficients with the inclusion of transverse shear deformation divided by their values when transverse shear deformation is ignored. The first of these gives the stress at the edge of a clamped pressure vessel, while the second gives the edge bending stress at a slope discontinuity. The curves are shown in Fig. 6.4.

1 2 3

1

µ

λ 2

σs B σs B µ = 3

Figure 6.4 - Effect of transverse shear deformation on edge stress. Shown is the ratio of the maximum bending stress with and without the transverse shear deformation included, for the case of a slope discontinuity in a pressurized spherical cap and for the case of clamped edge. The transverse shear flexibility actually decreases the discontinuity stress.

131