Инженерная теория сопротивления неоднородных стержней из композиционных материалов

Инженерная теория сопротивления неоднородных стержней из композиционных материалов

ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2016. № 6

Просьба ссылаться на эту статью следующим образом:

Горбачев В.И. Инженерная теория сопротивления неоднородных стержней из компо-

зиционных материалов // Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки.

2016. № 6. C. 56–72. DOI: 10.18698/1812-3368-2016-6-56-72

ENGINEERING RESISTANCE THEORY OF HETEROGENEOUS

RODS MADE OF COMPOSITE MATERIALS

V.I. Gorbachev

vigorby@mail.ru

Lomonosov Moscow State University, Moscow, Russian Federation

Abstract

Keywords

To construct the engineering resistance theory of heteroge-

neous rods, we used an integral formula which presents the

displacement of the body points in the initial problem of the

heterogeneous body elasticity theory by means of the points

displacement in a similar problem, but for a homogeneous

elastic body (an accompanying task). The integral formula

implies an equivalent notion of displacements series in a

heterogeneous rod. The displacements are compared to the

derivatives in the accompanying homogeneous rod. We

approximately defined the points displacement of the ac-

companying rod by classical strength of materials methods

through the three components of the points displacement

vector relative to its axis. As a result, we presented the dis-

placement vector components of any point of the heteroge-

neous rod in the form of series of derivatives displacement

of the longitudinal axis of a homogeneous rod. According to

the displacement, we found the series for stresses in the

heterogeneous rod. Furthermore, by longitudinal stress we

determined the internal force factors in the heterogeneous

rod cross section — longitudinal force and two bending

moments, presented in series of derivatives of the three

components of the rod axis displacement vector. Then,

from Zhuravsky equations we derived a system of three

ordinary differential equations of infinite order with respect

to the three components of the longitudinal axis displace-

ment vector. This paper studies the so-called theory of zero-

order approximation, which takes into account only the rod

axis longitudinal deformation and curvature (kinematic

factors) to express internal force factors. The coefficients

within the kinematic factors are the effective rigidity of the

rod — longitudinal rigidity, four bending rigidities and four

rigidities of mutual influence, which are calculated after

solving the supporting planar and antiplanar problems in

cross-section of the heterogeneous rod

Heterogeneous rod, problem of the

heterogeneous body elasticity theo-

ry, theory of zero-order approxi-

mation, effective rigidity