Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 793798, 21 pages
http://dx.doi.org/10.1155/2011/793798
Research Article

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

1College of Civil Engineering, Chongqing University, Chongqing 400045, China
2Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China
3Chongqing Vocational College of Architectral Engineering, Chongqing 400039, China
4Internal Trade Engineering and Research Institute, Beijing 100069, China

Received 17 August 2010; Revised 14 February 2011; Accepted 15 February 2011

Academic Editor: E. E. N. Macau

Copyright © 2011 Liu Chang-Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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.

2. 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 𝑏2, the thickness of the dished shallow shell is , and the radius of axisymmetric circular line load is 𝑏1.

793798.fig.001
Figure 1: Geometrical size and load of the dished shallow shell.

Choose the dimensionless quantities as follows:𝑤𝑊=121𝜈2,𝑘=121𝜈2𝑓,𝛾=𝑏1𝑎𝑏,𝛾=2𝑎,𝜃=𝑑𝑊,𝑎𝑑𝜌𝑆=2𝜌𝑁𝑟𝐸3121𝜈2𝑟,𝜌=𝑎𝑎,𝑃=2𝑏2𝑃𝐸4121𝜈231𝜈2,(2.1) where 𝑤 denotes deflection, 𝐸 denotes Young’s modulus, 𝑁𝑟 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=𝑎tg𝛼.

Introducing Heaviside step function:𝑢(𝜌𝛾)=1(𝜌𝛾),0(𝜌<𝛾).(2.2) The dimensionless governing differential equations of the dished shallow shells under circular line loads [1] are𝐿(𝜌𝜃)=𝑃𝜌2[],1𝑆𝑘𝑢(𝜌𝛾)+𝜃𝐿(𝜌𝑆)=𝑘𝜃𝑢(𝜌𝛾)+2𝜃2,(2.3) where 𝐿()=𝜌(𝑑/𝑑𝜌)(1/𝜌)(𝑑/𝑑𝜌)().

The corresponding boundary conditions are𝜌=0𝜃=0,𝑆=0,𝜌=1𝑑𝜃𝑑𝜌+𝜈1𝜃=0,𝑑𝑆𝑑𝜌𝜈2𝑆=0,(2.4) where the values of 𝑣1 and 𝑣2 are related to the concrete boundary conditions, and for the general boundary conditions, their values are as follows [10]:

rigidly clamped edge: 𝜈1=,𝜈2=𝜈,

clamped but free slip edge: 𝜈1=,𝜈2=,

simply supported but free slip edge: 𝜈1=𝜈,𝜈2=,

simply supported: 𝜈1=𝜈,𝜈2=𝜈.

3. 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:𝑃=𝑃1𝜀+𝑃2𝜀2+𝑃3𝜀3++𝑃𝑛𝜀𝑛𝜀+𝑜𝑛+1,(3.1)𝜃=𝜃1𝜀+𝜃2𝜀2+𝜃3𝜀3++𝜃𝑛𝜀𝑛𝜀+𝑜𝑛+1,(3.2)𝑆=𝑆1𝜀+𝑆2𝜀2+𝑆3𝜀3++𝑆𝑛𝜀𝑛𝜀+𝑜𝑛+1,(3.3) where 𝑃𝑖(𝑖=1,2,,𝑛) are undetermined constant coefficients and 𝜃𝑖 and 𝑆𝑖(𝑖=1,2,,𝑛) are undetermined coefficients with respect to 𝜌.

Substituting (3.1)–(3.3) into (2.3) and (2.4) and comparing the coefficient of the same order power of ε, we can obtain the following stepwise approximate equations:

when the coefficients of 𝜀1 are equal,𝐿𝜌𝜃1=𝑃1𝐹(𝜌)𝑆1𝐿𝑘𝑢(𝜌𝛾),𝜌𝑆1=𝜃1𝑘𝑢(𝜌𝛾),(3.4)

when the coefficients of 𝜀2 are equal,𝐿𝜌𝜃2=𝑃2𝐹(𝜌)𝑆2𝑘𝑢(𝜌𝛾)𝑆1𝜃1,𝐿𝜌𝑆2=𝜃21𝑘𝑢(𝜌𝛾)+2𝜃21,(3.5)

when the coefficients of 𝜀3  are equal,𝐿𝜌𝜃3=𝑃3𝐹(𝜌)𝑆3𝑆𝑘𝑢(𝜌𝛾)1𝜃2+𝑆2𝜃1,𝐿𝜌𝑆3=𝜃3𝑘𝑢(𝜌𝛾)+𝜃1𝜃2.(3.6)

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:𝜌=0𝜃𝑖=0,𝑆𝑖=0,𝜌=1𝑑𝜃𝑖𝑑𝜌+𝜈1𝜃𝑖=0,𝑑𝑆𝑖𝑑𝜌𝜈2𝑆𝑖=0,(3.7) where 𝑖=1,2,3.

Assume that 𝜑𝑖(𝜌) and 𝜓𝑖(𝜌) are the functions that satisfy the following equations:𝐿𝜌𝜑1=𝜌2𝑘𝑢(𝜌𝛾)𝜓1,𝐿𝜌𝜓1=𝑘𝑢(𝜌𝛾)𝜑1,𝐿(3.8)𝜌𝜑2=𝑘𝑢(𝜌𝛾)𝜓2𝜓1𝜑1,𝐿𝜌𝜓2=𝑘𝑢(𝜌𝛾)𝜑2+𝜑212,𝐿(3.9)𝜌𝜑3=𝑘𝑢(𝜌𝛾)𝜓3𝜓1𝜑2+𝜓2𝜑1,𝐿𝜌𝜓1=𝑘𝑢(𝜌𝛾)𝜑3+𝜑1𝜑2,(3.10)𝜑𝑖(𝜌) and 𝜓𝑖(𝜌) satisfy the following boundary conditions:𝜌=0𝜑𝑖=0,𝜓𝑖=0,𝜌=1𝑑𝜑𝑖𝑑𝜌+𝜈1𝜑𝑖=0,𝑑𝜓𝑖𝑑𝜌𝜈2𝜓𝑖=0,(3.11) where 𝑖=1,2,3.

We can prove the following formulas are correct according to (3.8)–(3.11):𝜃1=𝑃1𝜑1,𝜃2=𝑃2𝜑1+𝑃21𝜑2,𝜃3=𝑃3𝜑1+2𝑃1𝑃2𝜑2+𝑃31𝜑3,𝑆1=𝑃1𝜓1,𝑆2=𝑃2𝜓1+𝑃21𝜓2,𝑆3=𝑃3𝜓1+2𝑃1𝑃2𝜓2+𝑃31𝜓3.(3.12)

Substituting 𝜃𝑖 and 𝑆𝑖(𝑖=1,2,3) into (3.2) and (3.3) yields𝜃=𝑃1𝜑1𝑃𝜀+2𝜑1+𝑃21𝜑2𝜀2+𝑃3𝜑1+2𝑃1𝑃2𝜑2+𝑃31𝜑3𝜀3,(3.13)𝑆=𝑃1𝜓1𝑃𝜀+2𝜓1+𝑃21𝜓2𝜀2+𝑃3𝜓1+2𝑃1𝑃2𝜓2+𝑃31𝜓3𝜀3.(3.14) In (3.13) and (3.14), the constant coefficients 𝑃𝑖(𝑖=1,2,3) and perturbation parameter 𝜀 are unknown quantities. 𝑃𝑖(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.

4. Spline Function Solution to Functions 𝜑𝑖(𝜌) and 𝜓𝑖(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].

Transforming (3.8) into integral forms yields𝜑1𝜌(𝜌)=210𝐺1(𝜌,𝜉)𝜉𝑢𝜉𝛾1𝜌𝑑𝜉+210𝐺1(𝜌,𝜉)𝑘𝜉𝑢𝜉𝛾2𝜓1𝜓(𝜉)𝑑𝜉,(4.1)1𝜌(𝜌)=210𝐺2(𝜌,𝜉)𝑘𝜉𝑢𝜉𝛾2𝜑1(𝜉)𝑑𝜉.(4.2)

Substituting (4.2) into (4.1) yields the following nonlinear integral equations:𝜑1(𝜌)=𝐹(𝜌)+10𝐾(𝜌,𝜉)𝜑1(𝜉)𝑑𝜉,(4.3) where𝜌𝐹(𝜌)=210𝐺1(𝜌,𝜉)𝜉𝑢𝜉𝛾1𝐾1𝑑𝜉,(4.4)(𝜌,𝜉)=410𝑘2𝜌𝜉𝜂2𝑢𝜂𝛾2𝑢𝜉𝛾2𝐺1(𝜌,𝜂)𝐺2𝐺(𝜂,𝜉)𝑑𝜂,(4.5)𝑖1(𝜌,𝜉)=𝜌2+𝜆𝑖1,0<𝜉𝜌𝜉2+𝜆𝑖𝜆,𝜌<𝜉<1(𝑖=1,2),(4.6)1=1𝜈11+𝜈1,𝜆2=1+𝜈21𝜈2.(4.7)

Because 𝐹(𝜌) and 𝐾(𝜌,𝜉) are continuous on interval [0,1] and square interval 0𝜌,𝜉1, respectively, the consistent approximation of 𝐹(𝜌) and 𝐾(𝜌,𝜉) can be obtained by using polynomials on the two intervals.

We make equidistant node division on interval [0,1] and square interval 0𝜌,𝜉1, the node values are 𝜌𝑖 and (𝜌𝑖,𝜉𝑖), where 𝜌𝑖=𝑖/𝑁, 𝜉𝑗=𝑗/𝑀(𝑖=0,1,,𝑁;𝑗=0,1,,𝑀), and 𝑁 and 𝑀 are the divided node number.

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:𝐾(𝜌,𝜉)=𝑁𝑀𝑖=0𝑗=0𝐾𝑖𝑗×𝑁𝜙𝑖(𝜌)×𝑀𝜙𝑗(𝜉),𝐹(𝜌)=𝑁𝑖=0𝐹𝑖×𝑁𝜙𝑖(𝜌),(4.8) where 𝐾𝑖𝑗=𝐾(𝜌𝑖,𝜉𝑗),𝐹𝑖=𝐹(𝜌𝑖), 𝐾𝑖𝑗, and 𝐹𝑖 denote the function values of corresponding nodes. 𝜙𝑖(𝜌) and 𝜙𝑗(𝜉) are cubic multiple nodes spline functions.

Assume that the solution of 𝜑1(𝜌) is𝜑1(𝜌)=𝑁𝑖=0𝜑1𝜌𝑖𝜙𝑖(𝜌).(4.9)

Substituting (4.8)–(4.9) into (4.3) and making coefficients of 𝜙𝑖(𝜌)(𝑖=0,1,,𝑁) are equal yields the following linear equations:𝜑1𝜌𝑖𝑁𝑘=0𝜑1𝜌𝑘𝑎𝑖𝑘=𝐹𝑖,(4.10) where𝑎𝑖𝑘=𝑀𝑗=0𝐾𝑖𝑗𝑐𝑗𝑘,𝑐𝑗𝑘=10𝜙𝑗(𝜉)𝜙𝑘(𝜉)𝑑𝜉,(4.11)𝜑1(𝜌i) can be obtained by solving linear equations (4.10). Substituting 𝜑1(𝜌i) into  (4.9) can obtain an approximate solution of 𝜑1(𝜌). We can obtain the approximate solution of (4.2) by using the same method𝜓1(𝜌)=𝑁𝑗=0𝜓1𝜌j𝜙𝑗(𝜌).(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 𝜑i(𝜌) and 𝜓i(𝜌) are as follows:𝜑𝑖(𝜌)=𝑁𝑗=0𝜑𝑖𝜌𝑗𝜙𝑗𝜓(𝜌),(4.13)𝑖(𝜌)=𝑁𝑗=0𝜓𝑖𝜌𝑗𝜙𝑗(𝜌).(4.14)

5. The Determination of Dimensionless Critical Load and Deflection

5.1. The Determination of Dimensionless Critical Load

Assume that the dimensionless critical load 𝑃 and deflection 𝑊 satisfy the following equations: 𝑃=𝛼1(𝜌)𝑊(𝜌)+𝛼2(𝜌)𝑊2(𝜌)+𝛼3(𝜌)𝑊3𝑊(𝜌)+𝑜4,(𝜌)(5.1)

𝑊(𝜌) can be obtained according to (3.13)𝑊(𝜌)=𝑃1𝑌1𝑃(𝜌)𝜀+2𝑌1(𝜌)+𝑃21𝑌2(𝜌)𝜀2+𝑃3𝑌1(𝜌)+2𝑃1𝑃2𝑌2(𝜌)+𝑃31𝑌3(𝜌)𝜀3𝜀+𝑜4,(5.2) where 𝑌𝑖(𝜌)=𝜌1𝜑𝑖(𝜌)𝑑𝜌(𝑖=1,2,3).(5.3)

Substituting (5.2) into (4.1), and omitting high-order minuteness that more than four orders, and comparing with (3.1) yield expressions of 𝛼𝑖(𝜌):𝛼1(𝜌)=𝑌11𝛼(𝜌),2(𝜌)=𝑌13(𝜌)𝑌2𝛼(𝜌),3(𝜌)=𝑌14(𝜌)𝑌3(𝜌)+2𝑌15(𝜌)𝑌22(𝜌).(5.4)

𝜑𝑖(𝜌)(𝑖=1,2,3) can be figured out according to (4.13) and values of 𝛼𝑖(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 𝑘cr and critical load 𝑃cr according to the following steps.

Firstly, substituting concrete values of 𝑘 into extremum condition 𝑑𝑃/𝑑𝑊=0 yields 𝑊cr=𝛼2±𝛼223𝛼1𝛼23𝛼3.(5.5)

The corresponding formula of critical force is 𝑃cr=𝛼1𝑊cr+𝛼2𝑊2cr+𝛼3𝑊3cr.(5.6)

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 𝛼𝑖(𝜌)(𝑖=1,2,3). Then we can find values of 𝑘 that satisfy the following condition according to trial method. 𝛼223𝛼1𝛼3=0.(5.7) Here, 𝑘, namely (5.6), is significant. That is, only 𝑘𝑘crwith (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.

5.2. 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 𝛼𝑖(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, 𝛼𝑖(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.

6. 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 𝑣1=0.3, 𝑣2=0.3. We can figure out 𝜆1=0.538461, 𝜆2=1.857143 according to (4.7).

The characteristic equation (5.1) constructed by deflection of arbitrary point is 𝑃=𝛼1(𝜌)𝑊(𝜌)+𝛼2(𝜌)𝑊2(𝜌)+𝛼3(𝜌)𝑊3(𝜌).(6.1)

The corresponding 𝑃cr𝜌 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), 3(a), and 4(a). The abscissa value of the lowest point of 𝑃cr-𝜌 curve is the radius value of initial instability point of dished shallow shell. The ordinate value of the lowest point of 𝑃cr-𝜌 curve is the initial instability critical load of dished shallow shell.

fig2
Figure 2
fig3
Figure 3
fig4
Figure 4

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 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.

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.

tab1
Table 1: The initial instability and maximum deflection position under different geometric parameters (𝛾=0.3,𝛾=0.2).

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 3𝑘7. 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 𝑘3. (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 𝑘3.

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.

tab2
Table 2: The initial instability and maximum deflection position under different geometric parameters (𝛾=0.3,𝛾=0.3).

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 4𝑘7, 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 𝑘4. (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 𝑘4.

Likewise, in order to explain Figures 4(a) and 4(b), we listed the initial instability and maximum deflection position of the dished shallow shell under different geometric parameters in Table 3.

tab3
Table 3: The initial instability and maximum deflection position under different geometric parameters (𝛾=0.3,𝛾=0.5).

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 4𝑘7; the initial instability area appeared at the center of circular plate while 7𝑘9; the initial instability area moved from the center to the edge of circular plate with the increase of 𝑘 while 9<𝑘12; the initial instability area moved from the edge to the center of circular plate with the increase of 𝑘  while 𝑘>12.(2)With the stepwise increase of external load, the increase amplitude of deflection of each point was enlarged, but the maximum deflection area under each step load almost did not move. The maximum deflection area of dished shallow shell under external load appeared at the center of circular plate (𝜌=0) while 4𝑘<7, and the maximum deflection area appeared at the loading action position while 𝑘7.(3)When the external load was 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 4𝑘5 and 6<𝑘8; the maximum deflection area appeared at the loading action position and its adjacent area while 5<𝑘6 and 𝑘>8.

7. Conclusions

This paper obtained the nonlinear instability modes of dished shallow shell under circular line load by applying free-parameter perturbation method. By analyzing the nonlinear instability modes, we obtained the following conclusions.(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.(3)With the stepwise increase of external load, the increase amplitude of deflection of each point of dished shallow shell was enlarged. Under each step load, the maximum deflection area of dished shallow shell almost does not move, and the deflection where the circular line load act on does not fluctuate markedly, but the maximum deflection area of initial instability will move from its lateral side to itself when the circular line load act on the conical shell and 𝛾 is relatively large. The increase amplitude of deflection of each point near the initial instability area is significant when the external load is close to the initial instability critical load.(4)The maximum deflection area of dished shallow shell presents different rules with the variation of 𝑘,𝛾, and 𝛾. But, when the external load gets close to the initial instability critical load, the increase amplitude of deflection of each point near the initial instability area is significant, so the maximum deflection area of initial instability is not always the maximum deflection area under the previous load.(5)When the geometric parameter 𝑘 is a determined value, the critical load when the circular line load act on the conical shell is larger than the critical load when the circular line load act on the circular plate and the edge of circular plate, but the maximum deflection of initial instability when the circular line load act on the conical shell is smaller than the maximum deflection of initial instability when the circular line load act on the circular plate and the edge of circular plate. That is, the critical load increase with the increase of 𝛾, but the maximum deflection of initial instability decrease with the increase of 𝛾.

These conclusions provide some theoretical basis for engineering design and instability prediction and control of shallow-shell structures.

Acknowledgment

This work has been supported by the Science and Technology Program of Chongqing Municipal Education Commission: Free-Parameter Perturbation Method (Project no. KJ08A12).

References

  1. D. Liu and S.-L. Chen, “Snap-buckling of dished shallow shells under uniform loads,” Applied Mathematics and Mechanics, vol. 18, no. 1, pp. 29–36, 1997. View at Google Scholar
  2. D. Liu and S.-L. Chen, “Snap-buckling of dished shallow shells under line loads,” Applied Mathematics and Mechanics, vol. 19, no. 3, pp. 227–234, 1998. View at Google Scholar
  3. A. Chakrabarti, B. Mukhopadhyay, and R. K. Bera, “Nonlinear stability of a shallow unsymmetrical heated orthotropic sandwich shell of double curvature with orthotropic core,” International Journal of Solids and Structures, vol. 44, no. 16, pp. 5412–5424, 2007. View at Publisher · View at Google Scholar
  4. J. Tani, “Buckling of truncated conical shells under combined axial load, pressure, and heating,” Journal of Applied Mechanics, Transactions ASME, vol. 52, no. 2, pp. 402–408, 1985. View at Google Scholar
  5. J.-C. Xu, C. Wang, and R.-H. Liu, “Nonlinear stability of truncated shallow conical sandwich shell with variable thickness,” Applied Mathematics and Mechanics, vol. 21, no. 9, pp. 985–986, 2000. View at Google Scholar
  6. H. Ramsey, “Plastic buckling of conical shells under axial compression,” International Journal of Mechanical Sciences, vol. 19, no. 5, pp. 257–258, 1977. View at Google Scholar
  7. H. Wang and J.-K. Wang, “Snap-buckling of thin shallow conical shells,” Engineering Mechanics, vol. 7, no. 1, pp. 27–33, 1990. View at Google Scholar
  8. Z.-L. Zheng, B.-Q. Sun, B. Li, and S.-L. Chen, “Nonlinear local stability of dished shallow shells under uniformly distributed loads,” Journal of Chongqing University, vol. 30, no. 12, pp. 55–58, 2007. View at Google Scholar
  9. S.-L. Chen and Q.-Z. Li, “Free-parameter perturbation-method solutions of the nonlinear stability of shallow spherical shells,” Applied Mathematics and Mechanics, vol. 25, no. 9, pp. 881–888, 2004. View at Google Scholar
  10. S.-L. Chen, “Free-parameter perturbation method,” in A festschrift for the 90th birthday of Professor W.-Z. Chien, Z.-W. Zhou, Ed., pp. 35–43, Shanghai University Press, Shanghai, China, 2003. View at Google Scholar
  11. K.-Y. Ye and W.-P. Song, “Deformations and stability of spherical caps under centrally distributed pressures,” Applied Mathematics and Mechanics, vol. 9, no. 10, pp. 857–863, 1988. View at Publisher · View at Google Scholar