1 Introduction

When the cone shell with big radii need link cylinder with small radii, the cone-cylinder combined shell is usually adopted. Because of the bevel between the generatrix of cone shell and that of cylinder shell, there exist highly concentrated forces in the combined zone under uniform external pressure. The radial force aggravates the stress state of the combined zone whose direction is the same with the external pressure^[1], especially for the high submergence depth pressure structure. Thus, the measures should be taken to improve the distribution of stress in the combined zone. The cone-toroid-cylinder combined shell is done by imbedding a toroid between the cone and the cylinder, which can help the generatrix transit smoothly and reduce the longitudinal stress effectively. However the cone-toroid-cylinder combined shell does not work well with the circumferential stress of toroid's midsurface and it becomes the dominated stress. In order to reduce the circumferential stress of toroid, the strengthened frame was put forward^[2]. Chen^[3] testified the strengthened frame scenario by the turning model. It showed that strengthened frame induced the fluctuating of the longitudinal stress which was bad for the safety. Bai^[4] pointed out that increasing thickness of toroid was more effective than installing the frame in the midspan of toroid. Guo^[5-6]verified the conclusion by a series of turning model tests and the results corresponded with the theoretical results well. But it did not take the effect of machining like welding into consideration. Lü^[7]tested a concave-convex cone-toroid-cylinder combined shell under external pressure. Although the circumferential stress of the toroid was smaller than the permissible stress, the circumferential stress of the concave toroid is 6.1 times bigger than that of the convex toroid and it should appeal attentions, which implied that the circumferential stress of covcave toroid was much more dangerous. And the stress of frame beside concave toroid is much bigger than that of convex toroid^[8]. Especially during the process, initial imperfection was unavoidable, which may have influences on the mechanical property.But the study on that is seldom discussed.

2 Calculation Model

To consult the finite element calculation model designed in Refs.[4, 7]. The shells and frames use the shell 181 element in ANSYS.E=2.1×10⁵ MPa, μ=0.3, σ_s=785 MPa. l₁/R₁=0.07, l₂/R₁=0.14, t₁/R₁=0.02, t₂/R₁=0.01, t₃/R₁=0.013, t₄/R₁=0.008, I=9.2e7 mm⁴. Fig. 1 shows the dimensions of the model.

The pressure p is 6.61 MPa. One side of the cylinder shell is immovable, and the other side is only free in axial direction and is applied by the concentrated force F. F=p×πR₂²/n, n is the number of the nodes at one side of the cone shell.

3 Analysis of the Shape of Initial Imperfection 3.1 The Cause of Initial Imperfection

Welding and numerical control lathe is the main processing mean of concave cone-toroid-cylinder combined shell. And the process is shown as follows. Firstly the toroid of concave cone-toroid-cylinder combined shell and its extension is produced by numerical control lathe, which is of high precision^[9]. Compared with traditional methods welding, it can be considered to be with no initial imperfection. Secondly frame 8 and frame 9 are installed at sides of the toroid. At last the extension of the cylinder shell and the cone shell is connected respectively by circumferential welding. Weld lines are shown in Fig. 1. After factory research it can be known that fillet weld caused by frames' welding may lead to the deviation in radial direction and the initial deflection comes into being. Thus the circumferential welding seam can also cause the deviation of the shell's concave-convex^[10-13]. The location of the geometrical caused by welding is the same with weld line in Fig. 1. This paper mainly studies the influence of initial imperfection produced during the process of combined shell on stress.

3.2 The Shape of Frame's Initial Imperfection

Initial imperfection of the frame consists of deviation of verticality, waviness and initial deflection. The deviation of verticality and waviness is easy to be checked but it is difficult for the initial deflection^[14]. Therefore, the paper considers the influence of the frame's initial deflection on the stress of cone-toroid-cylinder combined shell.

Frame instability is very dangerous.Choose frame initial imperfection same with its instable deflection to analyze for safety.Based on Ref.[15], to use Eq.(1) describes the changes of the initial deflection along the circumferential direction.

$ w\left( \theta \right) = a\sin \left( {n\theta /2} \right) $

(1)

where θ denotes circumferential coordinate of the node, and a denotes the maximum amplitude of initial deflection; n stands for the number of circumferential half-wave of initial deflection.

Current specification^[16] indicates that 16 points should be adopt along the circumferential direction during the measures of initial deflection and obviously n is 16 or less.

3.3 The Shape of Shell's Initial Imperfection

The initial imperfection of shell is shell's concave-convex^{[12-13, 17]}.Regard the deformation of cylinder shell and cone shell instability as the concave-convex to increase design safety. According to Ref.[10], use Eq.(2) to show shell's concave-convex.

$ w = b\sin \left( {\frac{{{\rm{\pi }}\left( {x - {x_0}} \right)}}{l}} \right)\cos m\theta $

(2)

where x₀ is axial coordinate of the node at one side, and x represents axial coordinate of an arbitrary node; m is the number of circumferential half-wave of shell's concave-convex.

Current specification indicates that 16 points should be adopted along the circumferential direction during the measurement of concave-convex and obviously m is less than or equal 16.

4 Analysis of the Influence of Initial Imperfection on Stress 4.1 Stress of Perfect Concave cone-toroid-cylinder Combined Shell

Stress of perfect concave cone-toroid-cylinder combined shell in still water can be obtained in Table 1. According to Ref.[11], the allowable stress is made.

The following information can be obtained from Table 1: (1) Longitudinal stress in combined zone is far less than permissible stress because the toroid helps the generatrix transit smoothly and reduces the bad effect of extra longitudinal moment caused by bevel effectively; (2) The toroid makes the extra radial force distribute along the arc but the value does not decrease. Thus the toroid does not modify the circumferential stress of combined shell.σ₂^。 of the toroid is bigger than σ₁ and it becomes the dominated stress of concave cone-toroid-cylinder combined shell; (3) The circumferential stress of midsurface at the side of cylinder shell and cone shell close to the toroid is big, which is different from the law of circumferential stress distribution. The reason is that the mild distribution of high toroid's σ₂^。 makes the stress in zone next to toroid large; (4) Stress of frames besides the toroid is bigger than others, among which the largest value stress of frame 11 is still obviously less. The stress of frame 9 is approaching to permissible stress. Direction of Frame's stress is the same with circumferential stress of shell. In the combined zone of the frame and shell, frame's stress is equal to the membrane stress in midsurface of the shell, which leads to the large stress of frame at sides of the toroid. This is the first time for cone-toroid-cylinder combined shell's study that stress of frame besides toroid is dangerous enough to be concerned.

The stress of frame 9 and the circumferential stress in shell midsurface are high and they become the dominated stress of concave cone-toroid-cylinder combined shell. The dominated stress may exceed the permissible stress because of the initial imperfection produced during the process and assembling. So in the following, the effect of frame initial deflection and concave-convex of shell on the dominated stress will be discussed respectively. The results offer the reference for the design of concave cone-toroid-cylinder combined shell.

4.2 The Effect of Different Frame Initial Deflection on Stress

When frames 8 and 9 have initial deflection, the effect of different combination of initial deflection on stress is analyzed. The frame initial deflection is the same as Eq.(1) and the maximal initial deflection is equal to 0.002 75 R^[10] (R-frame radii)。

Tables 2 and 3 show circumferential stress of shell and stress of frame vary with frame's different initial deflection separately.

Tables 2 and 3 show that:(1) Circumferential stress of toroid midsurface rises for the existence of frames initial deflection, which raises shell imperfection. And the imperfection will increase circumferential stress and membrane stress under static pressure.When there 14 half waves in frame 8 and 2 half waves in frame 9, circumferential stress of toroidmidsurface reaches the maximum value 664.7 MPa, approaching permissible stress 0.85 σ_s; (2) the stress of frame is sensitive to its deflection which cause frame stress rising over the permissible stress, the reason for which is that frame deflection arouses extra bending stress. When frame 8 has 4 half waves and frame 9 has 16 half waves, stress of frame 9 is up to its maximum value 509.4 MPa, which is over permissible stress 0.6 σ_s.

4.3 The Effect of Shell Concave-convex on Stress

When only shell imperfection exists, analyze the effect of various shell concave-convex on dominated stress. Supposing the shape of shell concave-convex is the same with Eq.(3), and the largest amplitude of concave-convex is 0.15 t^[10](t is the thickness of shell).

From Tables 4 and 5:(1) Shell concave-convex increases shell's membrane stress and bending stress, thus the circumferential stress of toroid midsurface gets larger. When cone shell's concave-convex is 10 half waves and that of cylindrical shell is 12 half waves, circumferential stress of toroid midsurface reaches the largest value 657.4 MPa, smaller than permissible stress 0.85 σ_s; (2) Shell concave-convex causes deformation of frame, which creates extra moment increasing frame stress. When the concave-convex of cone shell has 16 half waves and that of cylindrical shell has 8 half waves, frame stress reaches a maximum value 477.1 MPa, larger than permissible stress 0.6 σ_s.

4.4 Effect of Frame and Shellximum Value 477 on Stress

The shape of frame and shell's imperfection is random. Certain shape of frame and shell's imperfection which has the worst effect ondominated stress is chosen to study the influence of its amplitude changing ondominated stress.

The amplitude of frame and shell's imperfection changes in constant gradient, i.e., frame deflection and shell concave-convex are applied Δ times of permissible imperfection W_c and f_c^[10]at the same time.

$ \left\{ \begin{array}{l} W = \Delta {W_c}\\ f = \Delta {f_c} \end{array} \right. $

(3)

where, Δ is the multiple of initial imperfection, W_c=0.002 75R, f_c=0.15t.

Table 6 shows that circumference stress in the middle surface varies with the value changing of imperfection and Table 7 shows that #9 frame's stress varies with the value changing of imperfection.

Remarks: Case one: frame 8 deflection is 14 half waves, frame 9 deflection is 2 half waves, concave-convex of cone is 10 half waves and that of cylinder is 12 half waves; Case two: frame 8 deflection is 4 half waves, frame 9 deflection is 16 half waves, concave-convex of cone is 10 half waves and that of cylinder is 12 half waves; Case three: frame 8 deflection is 14 half waves, frame 9 deflection is 2 half waves, concave-convex of cone is 16 half waves and that of cylinder is 8 half waves; Case four: frame 8 deflection is 4 half waves, frame 9 deflection is 16 half waves, concave-convex of cone is 16 half waves and that of cylinder is 8 half waves.

Fig. 2 shows that circumferential stress of toroid midsurface σ₂ and stress of frame 9 σ_f varies with initial imperfection amplitude.

As can be seen from Fig. 2, (1) dominated stress increases approximately linearly with initial imperfection amplitude, which illustrates that extra bending moment caused by initial imperfection has a linear relation with the imperfection amplitude; (2) when Δ=0.5, circumferential stress of toroid midsurface is lower than permissible stress 0.85 σ_s while stress of frame is larger than its permissible stress 0.6 σ_s, which shows that stress of frame is much more sensitive to initial imperfection than circumferential stress of toroid; (3) when Δ=1, both circumferential stress of toroid and stress of frame exceed the permissible stress, which is dangerous. For example, circumferential stress of toroid gets largest in case one and is more than 2.2% of the permissible stress; stress of frame 9 reaches the maximum in case two and exceeds 11.9% of permissible stress. The damage of frame is more dangerous than that of shell.Therefore initial imperfection Case two has the worst effect on concave cone-toroid-cylinder combined shell's stress, and should be avoided during the construction.

5 Study on Strengthening Methods for Concave Cone-toroid-cylinder Combined Shell with Initial Imperfection

Although concave cone-toroid-cylinder combined shell can sufficiently lower longitudinal constraint stress in combined area, circumferential stress of toroid midsurface and stress of frames that are on both sides of toroid are relatively high. What's worse, initial imperfection could increase circumferential stress of toroid midsurface and stress of frame to get over the permissible stress. In order to improve the safety of structure, concave cone-toroid-cylinder combined shell needs to be strengthened. Based on the calculation model mentioned in part 1, the strengthening scenarios are as follows: (1) Scenario one, frame 8 and frame 9 are strengthened, which improves frame's geometrical moment of inertia I to 1.155 6e8 mm⁴; (2) Scenario two, toroid shell gets thicker by Δt₃/ R₁=0.004. The weight increased in both scenarios is the same. The dominated stress under two strengthening scenarios are compared in Table 8, where the calculation model has initial imperfection as case two in Table 7.

As Table 8 shows:(1) stress of frame is reduced effectively to meet the requirement via strengthening frames 8 and 9(scenario one), and circumferential stress of toroid midsurface becomes lower at the same time; (2) through thickening the toroid (scenario two), the circumferential stress of toroid midsurface is reduced significantly, which is in accord with the conclusion in Refs.[4-6]. However, the scenario two cannot allow the stress of frame meet the requirement. Thus strengthening scenario one is the best choice to reduce dominated stress.

6 Conclusions

Circumferential stress of toroid midsurface and stress of frame in cone-toroid-cylinder combined shell are dominated stresses, which are approaching the current standard requirement. And the initial imperfection has large effect on them. The strengthening scenarios to improve the distribution of dominated stress are analyzed and compared. The following conclusions can be drawn.

(1) Circumferential stress of toroid midsurface and stress of frame are so much more dangerous than others that they should be paid close attention, especially the stress of frame 9 which was always ignored before.

(2) Dominated stress increases approximately linearly with initial imperfection amplitude, which illustrates that extra bending moment caused by initial imperfection has a linear relation with the imperfection amplitude.

(3) When the imperfection is the same with case one, it has the worst influence to circumferential stress of toroid; when the imperfection is the same with case two, it has the worst influence to frame stress.

(4) Stress of frame aside toroid is more sensitive to initial imperfection than circumferential stress of toroid midsurface. In order to improve the security of the cabin, it is better to strengthen frame on either hand of the toroid.