The Nonlinear Instability Modes of Dished Shallow Shells under Circular Line Loads

This paper investigated the nonlinear stability problem of dished shallow shells under circular line loads. We derived the dimensionless governing differential equations of dished shallow shell under circular line loads according to the nonlinear theory of plates and shells and solved the governing differential equations by combing the free-parameter perturbation method FPPM with spline function method SFM to analyze the nonlinear instability modes of dished shallow shell under circular line loads. By analyzing the nonlinear instability modes and combining with concrete computational examples, we obtained the variation rules of the maximum deflection area of initial instability with different geometric parameters and loading action positions and discussed the relationship between the initial instability area and the maximum deflection area of initial instability. The results obtained from this paper provide some theoretical basis for engineering design and instability prediction and control of shallow-shell structures.


Introduction
The dished shallow shell is a thin shallow shell that is composed by circular plate and shallow conical shell.The instability phenomenon of dished shallow shell is usually regarded as a control signal in automatic control systems of instruments.So, the theoretical analysis of nonlinear instability characters of dished shallow shell is necessary for the engineering design and application.The nonlinear stability problem of shallow shell has been focused and studied by many scholars, and some research results have been achieved.Liu and Chen 1, 2 applied the modified iteration method to study the nonlinear stability problem of dished shallow shells.Chakrabarti et al. 3 investigated the nonlinear stability of a shallow unsymmetrical heated orthotropic sandwich shell of double curvature with orthotropic core.Tani 4 studied the large deflection instability problem of truncated conical shells under compound loads by using finite difference method.Xu et al. 5 studied the nonlinear stability problem of truncated conical shell with variable thickness under uniformly distributed loads by applying modified iteration method.Ramsey 6 and H. Wang and J.-K.Wang 7 applied perturbation method to investigate the plastic instability problem of conical shells under axial pressure and the nonlinear instability problem of thin shallow conical shell under uniformly distributed loads.The existing research mainly focused on the characteristic equation that was constructed by the external load and the center deflection.The reason for the fact that they choose the center deflection to construct the characteristic equation is that they believe the instability characteristics firstly appear at the center point of the shallow shell, and they can discuss the overall stability according to this characteristic equation.Very few people discussed the nonlinear local stability problem of shallow shell, which is where does the shallow shell under external loads start to lose stability?i.e., where is the initial instability area? .Only Zheng and Chen 8, 9 have studied the nonlinear local stability of dished shallow shell.However, so far, researchers have not presented studies of nonlinear instability modes of dished shallow shell or investigated the internal relationship between the initial instability area and the instability modes.
In this paper, the free-parameter perturbation method 10 and spline function method 11 are combined to analyze the nonlinear instability modes of dished shallow shell under circular line loads.The relationships between the maximum deflection area of initial instability and the geometrical parameter and loading position are studied when the simply supported dished shallow shell start to lose stability, and the internal relationships between the initial instability area and the initial instability modes are investigated.The research results obtain some valuable and significant conclusions for engineering application and theoretical research and provide some basis for engineering design and instability prediction and control.

Dimensionless Basic Equations
The dished shallow shell is shown in Figure 1.The radius of the bottom circular plate plane is a, the radius of the upper circular plate is b 2 , the thickness of the dished shallow shell is h, and the radius of axisymmetric circular line load is b 1 .
Choose the dimensionless quantities as follows: where w denotes deflection, E denotes Young's modulus, N r denotes radial membrane force, r denotes radial coordinate value, v denotes Poisson's ratio, P is load parameter, and f denotes the vector height of dished shallow shell: f a • tgα.Introducing Heaviside step function:

2.2
The dimensionless governing differential equations of the dished shallow shells under circular line loads 1 are

The Free-Parameter Perturbation Expansion of Dimensionless Equation
Expand the dimensionless load P , angle θ, and radial membrane force S as the following forms with respect to the perturbation parameter ε: when the coefficients of ε 1 are equal, when the coefficients of ε 2 are equal,

3.5
when the coefficients of ε 3 are equal, According to the above derivation method, the four and more than four order power stepwise approximate equations can be obtained, but they will not be discussed here.
The corresponding boundary conditions for 3.4 -3.6 can be expressed as follows: where i 1, 2, 3.
Assume that ϕ i ρ and ψ i ρ are the functions that satisfy the following equations: 3.9 3.10 ϕ i ρ and ψ i ρ satisfy the following boundary conditions: where i 1, 2, 3. We can prove the following formulas are correct according to 3.8 -3.11 :

3.12
Substituting θ i and S i i 1, 2, 3 into 3.2 and 3.3 yields

3.14
In 3.13 and 3.14 , the constant coefficients P i i 1, 2, 3 and perturbation parameter ε are unknown quantities.P i i 1, 2, 3 can be determined by giving ε a specific definition according to the traditional perturbation method, but we will not give ε a specific definition in this paper.

Spline Function Solution to Functions ϕ i ρ and ψ i ρ
Cubic multiple nodes spline function is applied to solve 3.8 -3.10 .Cubic multiple nodes spline function was widely applied to solve nonlinear equations 11 .

Mathematical Problems in Engineering
Transforming 3.8 into integral forms yields Substituting 4.2 into 4.1 yields the following nonlinear integral equations: where Because F ρ and K ρ, ξ are continuous on interval 0, 1 and square interval 0 ≤ ρ, ξ ≤ 1, respectively, the consistent approximation of F ρ and K ρ, ξ can be obtained by using polynomials on the two intervals.
We use K ρ, ξ and F ρ to replace F ρ and K ρ, ξ approximately, then the consistent approximation of F ρ and K ρ, ξ are as follows: and F i denote the function values of corresponding nodes.φ i ρ and φ j ξ are cubic multiple nodes spline functions.
Assume that the solution of ϕ 1 ρ is 4.9 Substituting 4.8 -4.9 into 4.3 and making coefficients of φ i ρ i 0, 1, . . ., N are equal yields the following linear equations: where ϕ 1 ρ i can be obtained by solving linear equations 4.10 .Substituting ϕ 1 ρ i into 4.9 can obtain an approximate solution of ϕ 1 ρ .We can obtain the approximate solution of 4.2 by using the same method

4.12
Adopting the same method and the data that have been figured out, we also can obtain solutions of 3.9 and 3.10 .Therefore, all the approximate solutions of ϕ i ρ and ψ i ρ are as follows: 13

The Determination of Dimensionless Critical Load
Assume that the dimensionless critical load P and deflection W satisfy the following equations: Mathematical Problems in Engineering W ρ can be obtained according to 3.13 where Substituting 5.2 into 4.1 , and omitting high-order minuteness that more than four orders, and comparing with 3.1 yield expressions of α i ρ :

5.4
ϕ i ρ i 1, 2, 3 can be figured out according to 4.13 and values of α i ρ i 1, 2, 3 can be obtained by substituting concrete values of ρ into 5.3 and 5.4 .Then, we can obtain the characteristic equation 5.1 that is determined by deflections of different points on shell surface.Therefore, we obtained the solution of 3.13 and 3.14 such as 5.1 while did not determine the perturbation parameter ε.Now, we can calculate the critical geometric parameter k cr and critical load P cr according to the following steps.
Firstly, substituting concrete values of k into extremum condition dP/dW 0 yields The corresponding formula of critical force is For the dished shallow shell whose truncated conical ratio γ and geometric condition k are determined values, each value of ρ have a corresponding group of concrete α i ρ i 1, 2, 3 .Then we can find values of k that satisfy the following condition according to trial method.
Here, k, namely 5.6 , is significant.That is, only k ≥ k cr with 5.6 is significant, and the instability phenomenon is existent, namely, the jumping phenomenon of dished shallow shell happen.
Comparing all corresponding critical loads of values of ρ, the minimum critical load is the critical load for initial instability.

The Determination of Deflection of Each Point under Dimensionless Loads
For the dished shallow shell whose truncated conical ratio γ and geometric condition k are determined values, the elastic characteristic equation of dimensionless load and deflection 5.1 is permanently significant when P is equal or less than the initial instability critical load.
For the dished shallow shell whose geometric condition is k, each value of ρ has a corresponding group of concrete α i ρ i 1, 2, 3 .Then, we can construct a concrete function P with respect to W ρ .
From the analysis of 5.1 , we know that for concrete value of ρ, α i ρ i 1, 2, 3 is a determined value.Here, if P is a determined value, 5.1 is a standard simple cubic equation.Therefore, we can obtain the deflection value W ρ according to the solving method of simple cubic equations supplied in paper 11 .That is, we can get the deflection of each point when the external load is determined.
If we figure out the deflection of each point when the geometric condition is k and the external load is initial instability critical load, we can determine the corresponding initial instability mode of the dished shallow shell under determined geometric condition and initial instability critical load.
If we figure out the deflection of each point under corresponding external load when the geometric condition is k and the external load equal or less than the initial instability critical load, we can obtain the deflection curve of the dished shallow shell under specific geometric condition and external load.

Computational Examples and Analysis of Numerical Results
In the following computational example, the number of fitting point is M N 100 and the Poisson's ratio is v 0.3.
For the dished shallow shell under circular line load p whose boundary condition is simply supported, we take γ 0.3 and γ 0.2, 0.3, 0.5, then v 1 0.3, v 2 0.3.We can figure out λ 1 0.538461, λ 2 1.857143 according to 4.7 .The characteristic equation 5.1 constructed by deflection of arbitrary point is The corresponding P cr − ρ curves of different values of k are shown in Figures 2 a , 3 a , and 4 a while truncated conical ratio γ 0.3 and γ 0.2, 0.3, 0.5, respectively.The abscissa value denotes the radius value of the pint, where the perturbation parameter is selected and the ordinate value denotes the corresponding value of critical load in Figures 2 a , In order to explain Figures 2 a and 2 b , we listed the initial instability and maximum deflection position of the dished shallow shell under different geometric parameters in Table 1.
We can obtain the following conclusions from Figures 2 a and 2 b and Table 1.
1 When the circular line load acted on the circular plate section, the initial instability area of dished shallow shell moved from the center of circular plate to the edge of dished shallow shell with the increase of k while 3 ≤ k ≤ 7. The initial instability area of dished shallow shell moved from the edge of dished shallow shell to the edge of circular plate with the increase of k while k > 7.
2 With the stepwise increase of external load, the increase amplitude of deflection of each point enlarged, but the area of maximum deflection almost did not move, and the deflection where the circular line load acted on did not fluctuate markedly.The maximum deflection area of dished shallow shell under external load appeared at the center of circular plate ρ 0 while k ≥ 3.
3 When the external load got close to the initial instability critical load, the increase amplitude of deflection of each point near the initial instability area was significant.
The maximum deflection area of initial instability appeared at the center of circular plate ρ 0 while k ≥ 3.
As the supplementary specification data for Figures 3 a and 3 b , the initial instability and maximum deflection position of the dished shallow shell under different geometric parameters are listed in Table 2.
We can obtain the following conclusions from Figures 3 a and 3 b and Table 2.
1 When the circular line load acted on the edge of circular plate, the initial instability area of dished shallow shell moved from the loading action position to the edge of dished shallow shell with the increase of k while 4 ≤ k ≤ 7, and the initial instability area of dished shallow shell moved from the edge of dished shallow shell to the center of circular plate with the increase of k while k > 7.
2 With the stepwise increase of external load, the increase amplitude of deflection of each point was enlarged, but the maximum deflection area almost did not move, and the deflection where the circular line load acted on did not fluctuate markedly.
The maximum deflection area of dished shallow shell under external load appeared at the center of circular plate ρ 0 while k ≥ 4.
3 When the external load got close to the initial instability critical load, the increase amplitude of deflection of each point near the initial instability area was significant.
The maximum deflection area of initial instability appeared at the center of circular plate ρ 0 while k ≥ 4.       We can obtain the following conclusions from Figures 4 a and 4 b and Table 3.
1 When the circular line load acted on the conical shell section, the initial instability area moved from the loading action position to the center of circular plate with the increase of k while 4 ≤ k ≤ 7; the initial instability area appeared at the center of circular plate while 7 ≤ k ≤ 9; the initial instability area moved from the center to the edge of circular plate with the increase of k while 9 < k ≤ 12; the initial instability area moved from the edge to the center of circular plate with the increase of k while k > 12.  1 When the circular line load act on the circular plate, the edge of circular plate, and the conical shell, respectively, and the geometric parameter k is relatively small, the initial instability area of dished shallow shell appears at the loading action position, but the maximum deflection area of initial instability appear at the center of circular plate ρ 0 .
2 The initial instability area of dished shallow shell under circular line load does not always appear at the loading action position.The initial instability area of dished shallow with different γ and γ presents different rules with the variation of k when k is relatively large.The maximum deflection area of initial instability appears at the center of circular plate ρ 0 when the circular line load act on the circular plate, the edge of circular plate, and the conical shell, respectively.The maximum deflection area of initial instability appears at the loading action position when the circular line load act on the conical shell and k is relatively large.

Figure 1 :
Figure 1: Geometrical size and load of the dished shallow shell.
3 a , and 4 a .The abscissa value of the lowest point of P cr -ρ curve is the radius value of initial instability point of dished shallow shell.The ordinate value of the lowest point of P cr -ρ curve is the initial instability critical load of dished shallow shell.The corresponding W ρ -ρ curves of different values of k are shown in Figures 2 b , 3 b , and 4 b , while P increase progressively and truncated conical ratio γ 0.3 and Mathematical Problems in Engineering γ 0.2, 0.3, 0.5, respectively.The abscissa value denotes the radius value of the pint, where the perturbation parameter is selected and the ordinate value denotes the dimensionless deflection value in Figures 2 b , 3 b , and 4 b .The abscissa value of the highest point of W ρ − ρ curve is the radius value of the largest deflection point of dished shallow shell.The corresponding W ρ −ρ curve is the initial instability mode when P is initial instability critical load.

Table 1 :
The initial instability and maximum deflection position under different geometric parameters γ 0.3, γ 0.2 .

Table 3 :
The initial instability and maximum deflection position under different geometric parameters γ 0.3, γ 0.5 .