Optimal Design of a Composite Cylindrical Grid Shell Loaded with External Pressure
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mal Design of a Composite Cylindrical Grid Shell Loaded with External Pressure V. V. Vasiliev Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia e-mail: [email protected] Received December 16, 2019; revised February 15, 2020; accepted February 22, 2020
Abstract—Cylindrical grid shells made of modern composite materials by automated continuous winding that have a high degree of weight perfection and are widely used in aerospace engineering are considered. The problem of using such structures as deep-submergence vehicle bodies operating not under external pressure is discussed. The problem of optimal design of cylindrical grid shells based on the criterion of minimum mass under constraints imposed on strength and stability is considered. By minimizing the safety factors for possible forms of destruction, an analytical solution that determines the optimal design parameters of the composite grid shell is obtained. Keywords: Composite grid structures, optimal design, deep-submergence vehicles DOI: 10.3103/S0025654420030152
1. INTRODUCTION. COMPOSITE GRID STRUCTURES The variant of the composite grid construction considered in this article has the form of a three-layer cylindrical shell consisting of the outer and inner composite layers with circumferential reinforcement and the middle layer formed by a system of spiral ribs made by winding a unidirectional composite material based on carbon fibers. The process of rib winding is shown in Fig. 1. The space between the ribs is filled with foam or polymer containing glass microspheres. A skin is made by ring winding. The cross-section of the wall by the plane orthogonal to the axes of the ribs is shown in Fig. 2 and the grid structure is presented in Fig. 3. A review of studies on composite grid structures used in aerospace engineering is given in [1]. In the present article, we consider the possibility of using such structures as deep-submergence vehicle bodies [2, 3]. 2. STATEMENT OF THE PROBLEM ON OPTIMAL DESIGN We consider a cylindrical shell with a radius R and length l that is loaded with uniform external pressure q. The axial and ring deformations of the momentless shell are related with pressure as follows:
B11ex + B12ey = − 1 qR, B12ex + B22ey = −qR. 2 The stiffness coefficients of the wall have the form (indices “r” and “s” refer to ribs and skin)
B11 = 2E r hbr c 4,
B12 = 2E r hbr c 2 s 2,
B22 = h(2E r br s 4 + E s hs ),
(2.1)
(2.2)
where Е are the elastic moduli and (Figs. 2 and 3)
h br , hs = s , a h We find the deformations from Eqs. (2.1) br =
ex = −qR
s = sin φ,
E s h − 2E r br (2c 2 − s 2 )s 2 , 4E r E s hbr hc 4 303
c = cos φ.
(2.3)
ey = −qR 2c − s 2 2E s hhs c 2
2
304
VASILIEV
Fig. 1.
hs/2
h
hs/2 a
br Fig. 2.
y x
M
AA a b
M l
h
A
A
br br
Fig. 3.
and obtain the stresses in the ribs and skin
qR qR(2c 2 − s 2 ) (2.4) = = − , σ E e . s s y 4hbr c 2 2hhs c 2 In a circumferentially reinforced skin, only ring stresses are applied. For the angle of the ribs that is determined by the equa
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