Shallow Prismatic Shell in a Nonuniform Temperature Field
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SHALLOW PRISMATIC SHELL IN A NONUNIFORM TEMPERATURE FIELD B. S. Khapko
UDC 559.377
We study the thermal stressed state of a shallow prismatic shell composed of two flat elements. The analytic solutions of the problems of heat conduction and thermoelasticity are obtained by the method of finite integral transformations in the form of double series. The distributions of forces and moments along the coordinate lines are analyzed for different angles of inflection of the shell.
Numerous surfaces of thin-walled structural elements applied in the building industry have kinks. Thus, the structures in the form of shallow shells composed of flat elements may have kinks of the middle surface in a certain direction along the lines of conjugation [1, 2]. Shallow shells with piecewise-smooth surfaces used in the building industry were theoretically studied under the action of forces in [3 – 5] and thermal loads in [6 – 9]. The elastic platelike systems with discontinuities of toughness, curvature, and loads were investigated in [10]. In what follows, we perform the numerical analysis of a shallow prismatic shell composed of two flat elements and subjected to thermal loading. Consider a shallow prismatic shell with sides l and b rectangular in plan (Fig. 1). The shell is composed of two rectangular platelike elements. In a Cartesian coordinate system ( x, y, z ), its middle surface is described by the equation z ( x, y ) = tan α1 y −
1 ( tan α1 − tan α 2 ) ( y − b1 + y − b1 ) , 2
(1)
where tan α1 and tan α2 are the angular coefficients of the directrices of the first and second elements (1 and 2 in Fig. 1, respectively), y = b1 is the line of conjugation of the elements (the line of kink), b =
b1 ( tan α 2 − tan α1 ) , tan α 2
and θ = π – α 2 + α 1 is the angle of inflection. This angle is assumed to be small. By using the theory of generalized functions, for the differential characteristics of the middle surface of a shallow shell with faces of the same size, we can write [11] k1 =
∂2 z = 0, ∂x 2
k12 =
∂2 z = 0, ∂x ∂y
k2 =
∂2 z = – 2 tan α1 δ ( y – b1 ). ∂y 2
The shell exchanges heat with an ambient medium with temperatures tc+ ( x, y) and tc− = 0 on the surfaces z = ± h, respectively. The heat-transfer coefficients on the surfaces of the shell are identical ( μ+ = μ – ). The shell is free of the action of forces, its edges are freely supported by vertical diaphragms, and its ends are kept at a temperature equal to zero. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 2, pp. 33–38, March–April, 2005. Original article submitted January 19, 2005. 170
1068–820X/05/4102–0170
© 2005
Springer Science+Business Media, Inc.
S HALLOW P RISMATIC SHELL IN
A
NONUNIFORM TEMPERATURE F IELD
171
Fig. 1. Schematic diagram of the shell. In dimensionless coordinates x b
η =
and
ξ =
y , b
the stationary temperature field of this shell is described by the system of equations [9] 2 k0 ΔT1 − μ1 T
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