Thermal critical buckling temperature for hyperbolic doubly-curved shells structure composites

Jian-min SU， Yan-chun ZHAI， Shao-qing WANG， Qiang LI

(Facility Horticulture Laboratory of Universities in Shandong, Weifang University of Science and Technology, Weifang 262700,China)

Abstract: Taking the hyperbolic doubly-curved shells structure composite as the research object, the thermal buckling behavior of the composite laminated hyperbolic doubly-curved shells structure under the temperature field is studied by using the finite element Block-Lanczos research analysis method, and the influences of the laminated thickness, boundary conditions, fiber direction of the composite laminated hyperbolic doubly-curved shells on the critical buckling temperature are studied. The results show that the critical buckling temperature of the hyperbolic doubly-curved shells of composite laminates is closely related to the boundary conditions, ply thickness and fiber direction and has a certain regular distribution under uniform temperature field. Through the thermal buckling analysis of composite hyperbolic doubly-curved shells structure in this paper, an effective analysis and modeling method is provided for structural design and practical application of composite Materials

Key words: Hyperbolic doubly-curved shells, Thermal buckling, Boundary conditions, Thickness, Angle

1 Introduction

Compared with traditional metal materials, composite materials are materials with new properties and characteristics formed by two or more materials with different properties in a specific way on a macro scale. Composite materials have been widely used in aerospace, mechanical manufacturing, facility agriculture, automobile industry, and other fields in recent years due to their high strength, large rigidity, light weight, anti-fatigue, vibration reduction, high temperature resistance, and strong designability. Because composite laminated structures often have to bear complex thermal loads in the working process, it is very necessary to study the thermal buckling behavior of composite interlayer structural materials under different temperature conditions and structural parameters [1-5], but there are few studies on composite laminated hyperbolic doubly-curved shells structures. Hyperbolic doubly-curved shells are very common in our daily life and are a form of flat shells. The so-called flat shells refer to the ratiof/a≤1/5 between the vector heightFof the thin shell and the relatively short side A of the bottom surface covered by the thin shell. From the point of view of geometric structure, flat shell surface is actually a part of a common surface, such as cylindrical shell, spherical shell, and hyperbolic doubly-curved shells are all forms of flat shell. Hyperbolic doubly-curved shells have many advantages, such as excellent mechanical properties, beautiful structural design, excellent economic indicators, and wide structural span.

Tian et al．[6] studied the thermal buckling behavior of composite laminates under uniform temperature field and non-uniform temperature field, and discussed the influence of different boundary conditions and ply direction on the critical buckling temperature of laminates. D. Chronopoulos et al. [7] studied and analyzed the natural frequency changes of laminated structures under greenhouse conditions. J et al. [8] summarized the advantages of equivalent monolayer theory and delamination theory, and put forward the basic theory of sectional shear deformation of composite sandwich plates and laminated plates. Rui et al. [9] had carried out theoretical analysis and research on the damping characteristics of the new hyperbolic doubly-curved shells composite sandwich structure. Li et al．[10-11] studied and analyzed the influence of temperature change on the critical thermal buckling load of laminated plates with different thicknesses, the change of fiber laying direction angle and the change of boundary conditions on the critical buckling temperature of laminated composite plates. Liu et al．[12] established the buckling control equation of damp-heat effect under the macro-micro model, analyzed and studied the buckling behavior of composite laminates under humidity and temperature, and studied the influence law of different boundary conditions and different load conditions on the buckling behavior. Bin et al. [13] analyzed and studied the influence of humid and hot environment conditions on the bending performance, critical buckling temperature, and vibration characteristics of composite laminated plates based on classical laminated plate theory. Hiroyuki et al. [14], Ravlaeigh-Ritz method was used to analyze and study the buckling performance of composite laminates under humid and hot environments. In the research process, the elastic properties of the materials and the laws that the thermal and wet properties changed with the changes in temperature and humidity were also considered. Samuel et al. [15] studied and analyzed the thermal buckling characteristics and damping characteristics of reinforced sandwich composite plates. Literature used classical von Karman plate theory and Hamiltonian principle to derive the structural kinematics equation and the critical thermal buckling temperature formula of composite laminated plates. Through solving, the distribution laws of natural frequency and damping ratio of laminated plates are obtained. M.Biswal et al. [16] analyzed and studied the natural frequency of hyperbolic doubly-curved shells structure and the influence of ambient temperature and humidity on the natural frequency.

At present, a large number of studies have been carried out on the thermal buckling behavior of laminated plate structures in domestic and foreign data [17-20], but few studies have been carried out on the thermal buckling of composite hyperbolic doubly-curved shells structures. Composite hyperbolic doubly-curved shells structures are widely used in engineering practice. Therefore, the critical thermal buckling temperature of composite hyperbolic doubly-curved shells structures is analyzed and calculated in this paper, and the influences of factors such as ply thickness, boundary conditions and fiber direction on the critical buckling temperature under uniform temperature field are calculated by finite element BlockLanczos method, thus obtaining some regularity studies.

Fig.1 Hyperbolic doubly-curved shells

2 Theoretical analysis

2.1 Formula deduction

The lengths of the composite laminated hyperbolic doubly-curved shells structure are a, b and h. The strain-displacement relationship of any point (x,y,z) in the hyperbolic doubly-curved shells can be deduced by Von Karman's classical theory:

ε=εm+εθ+zκ

(1)

Where,εm、εθandκrespectively represent variable vectors and curvature vectors. The vibration displacement formula can be expressed as:

(2)

Where,μ,νandωare respectively component displacements in the bid system and polynomials satisfying the motion conditions. By integrating the stress at any point on the hyperbolic doubly-curved shells and the force along the thickness direction, the resultant force vector and resultant force moment vector at any point in the hyperbolic doubly-curved shells can be expressed as:

(3)

A、BandDare tensile stiffness matrix,tnsile-bending coupling stiffness matrix and bending stiffness matrix. Can be expressed as:

(4)

R(k)、Q(k)Can be expressed as:

(5)

Temperature functionfΔtis:

She now came to a space of marshy82 ground in the wood, where large, fat water-snakes were rolling in the mire, and showing their ugly, drab-colored bodies

fΔt=Δt·f(x,y)

Using Block-Lanczos finite element analysis method, the vibration characteristics of laminated hyperbolic doubly-curved shells structures are analyzed. For discrete analysis of spatial regions, according to the displacement difference formula of node elements:

(6)

Expressed as a matrix as follows：

a=Nqe{ε}=[B]qe

(7)

(8)

[B] is the strain matrix.

2.2 Verification of results

The structural parameters of Hyperbolic doubly-curved shells are as follows:E1=125 GPa，E2=5 GPa;G12=G13=2.5 GPa,G23=1 GPa,ν12=0.25,ρ=1 600 kg/m3,a/b=1,a/h=100,Rx=Ry=R

The natural frequencies of free vibration under the condition of simply supported edges are shown in Table 1.

Table 1 Natural frequencies of literature solutions and solutions

Table 1 shows that the natural frequency of the composite hyperbolic doubly-curved shells structure obtained by the calculation model in this paper decreases with the increase ofRy/b, and the natural frequency of the hyperbolic doubly-curved shells structure calculated in this paper is in agreement with the calculation results in reference and the error is within 3%. Therefore, the correctness of this model is verified.

3 Influence of ply thickness on critical buckling temperature of laminated hyperbolic doubly-curved shells structures

The thickness of each layer of the laminated hyperbolic doubly-curved shells structure is 1/1/1 mm, the chord length is 0.4 m, the radius is 0.5 m, the central angle is π/6, the four sides are simply supported, and the parameters of the anisotropic laminated composite hyperbolic doubly-curved shells structure are as follows:E1=132 GPa,E2=10.3 GPa,G12=G13=65 MPa，G23=3.91 GPa,ν12=0.25,ρ=1 570 kg/m3,α1=1.2×10-6/℃，α2=2.4×10-5/℃.

Fig.2 Model of hyperbolic doubly-curved shells

Table 2 and Fig.3 show that the thickness of the laminated hyperbolic doubly-curved shells structure has a great influence on the buckling temperature. when the chord length, radius and wrap angle are fixed, the critical buckling temperatures of mode(1,1) and mode(2,2) show an upward curve change under the condition of simple support on four sides, wherein the upward trend of mode(2,2) is more obvious than that of mode(1,1), which indicates that the thickness of the laminated hyperbolic doubly-curved shells structure under the condition of simple support on four sides has an influence on the critical buckling temperature of mode(2,2) than that of mode(1,1) The critical buckling temperatures of mode(1,1) and mode(2,2) change linearly under the condition of four-sided solid support, and the buckling temperatures of mode(1,1) and mode(2,2) change with thickness is basically the same, which indicates that the thickness of the underlying composite hyperbolic doubly-curved shells structure under the boundary condition of four-sided solid support has basically the same influence on the critical buckling temperatures of mode(1,1) and mode(2,2). From the above, it can be seen that the critical buckling temperatures of mode(1,1) and mode(2,2) of laminated hyperbolic doubly-curved shells structure increase with the increase of thickness, and the changes are very obvious, so the thickness can be increased to increase the critical buckling temperature in the structural design of composite hyperbolic doubly-curved shells.

Table 2 Influence of thickness on critical buckling temperature under different boundary conditions in hyperbolic spherical shell structure

Fig.3 Effect of thickness on critical buckling temperature

4 Influence of different boundary conditions on critical buckling temperature of laminated hyperbolic doubly-curved shells

Table 3 shows that the critical buckling temperature of laminated hyperbolic doubly-curved shells structures varies obviously with the boundary conditions. When the boundary condition is CCCC (C stands for fixed support), the critical buckling temperatures of laminated hyperbolic doubly-curved shells structures mode(1,1) and mode(2,2) are the highest, while when the boundary condition is CFFF(F stands for free edge), the critical buckling temperatures of laminated hyperbolic doubly-curved shells structures mode(1,1) and mode(2,2) are the lowest. The more fixed or simply supported edges, the higher the critical buckling temperature. When the number of fixed or simply supported edges is the same, the critical buckling temperature is higher when the fixed or simply supported edges are symmetrically constrained. Therefore, in the boundary constraint of hyperbolic doubly-curved shells structure, simply supported or fixed support constraint should be chosen as far as possible.

Table 3 Critical buckling temperature of hyperbolic spherical shell under different boundary conditions

5 Influence of laying angle on critical buckling temperature of laminated hyperbolic doubly-curved shells

5.1 Influence of laying angle on critical buckling temperature under four simply supported boundary conditions

Table 4 and Fig. 4 show that the critical buckling temperature of mode (1,1) increases with the increase of angle at 0°～45° and decreases with the increase of angle at 45°～90° when the base layer and intermediate layer of the laminated hyperbolic doubly-curved shells structure are changed under the condition of simple support. At the same time, changing the basic layer and intermediate layer is similar to changing the basic layer and intermediate layer separately. The 45 point is the highest point of the critical buckling temperature of mode (1,1).

Fig.4 Laying angle-critical buckling temperature curve under simply supported boundary conditions(mode(1,1))

Table 4 and Fig.5 show that the critical buckling temperature of mode (2,2) increases with the increase of angle-ply at 0°～45° and decreases with the increase of angle-ply at 45°～90° when the interlayer of laminated hyperbolic doubly-curved shells structure is changed under the condition of simple support. When the base layer is changed, the critical buckling temperature of its mode(2,2) decreases with the increase of angle-ply in the range of 0°～45°, and increases with the increase of angle-ply in the range of 45°～90°. At the same time, the change rule when changing the base layer and the middle layer is similar to that when changing the base layer, but the change is more obvious than that of the base layer. The 45° is the turning point of the critical buckling temperature change of mode (2,2).

Table 4 Critical buckling temperature of different laying angles under four simply supported boundary conditions

Fig.5 Laying angle-critical buckling temperature curve under simply supported boundary conditions(mode(2,2))

From this, it can be seen that the critical buckling temperatures of the mode (1,1) and (2,2) are inconsistent when the ply angles of the laminated hyperbolic doubly-curved shells structure are changed under the condition of simple support, which provides a certain reference for the structural design and practical application of the composite hyperbolic doubly-curved shells.

5.2 Influence of laying angle on critical buckling temperature under boundary condition of four-side fixed support

From Table 5 and Figs. 6 and 7, it is observed that the critical buckling temperatures of mode (1,1) and mode(2,2) increase with the increase of angle and decrease with the increase of angle at 45°～90° when the interlayer of the laminated hyperbolic doubly-curved shells structure is changed under the fixed support condition. When the base layer is changed, the critical buckling temperatures of mode(1,1) and mode(2,2) decrease with the increase of angle-ply in the range of 0°～45°, and increase with the increase of angle-ply in the range of 45°～90°. At the same time, changing the basic layer and the intermediate layer is similar to changing the basic layer.The 45° is the turning point of the critical buckling temperature changes of mode (1,1) and mode(2,2). From this, it can be seen that the critical buckling temperatures of mode (1,1) and (2,2) are basically the same and clo ser to each other when the ply angles of the laminated hyperbolic doubly-curved shells structure are changed under the condition of fixed support, and the change rule of the critical buckling temperature under the condition of fixed support is clearer than that under the condition of simple support.

Table 5 Critical buckling temperature of different laying angles under boundary conditions of four-side fixed supports

Fig.6 Laying angle-critical buckling temperature curve under Fixed supported boundary conditions(mode(1,1))

Fig.7 Laying angle-critical buckling temperature curve under Fixed supported boundary conditions(mode(2,2))

6 Conclusion

Under the boundary conditions of simple support on four sides or fixed support on four sides, the critical buckling temperature of the composite hyperbolic doubly-curved shells structure increases with the increase of the thickness, and the thickness change of the hyperbolic doubly-curved shells structure has a very obvious influence on the critical buckling temperature.

The critical buckling temperature of hyperbolic doubly-curved shells structures is obviously different with different boundary conditions. The more fixed or simply supported edges, the higher the critical buckling temperature. When the number of fixed or simply supported edges is the same, the critical buckling temperature is higher when the fixed or simply supported edges are symmetrically constrained.

No matter the ply angle of the hyperbolic doubly-curved shells structure is changed under the boundary conditions of simple support or fixed support, the variation rule of the pre-critical buckling temperature is consistent angle-ply in the range of 0°～90° (the variation curve is an upward convex curve with high middle and low sides), and the highest point of the first two-order critical buckling temperature is around angle-ply 45°.

There is an optimal ply angle to obtain a maximum critical buckling temperature for hyperbolic structures, so the optimal angle value should be selected in structural design.