CLC number: O343.2

On-line Access: 2024-08-27

Received: 2023-10-17

Revision Accepted: 2024-05-08

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JIANG Ai-min, DING Hao-jiang. The analytical solutions for orthotropic cantilever beams (I): Subjected to surface forces[J]. Journal of Zhejiang University Science A, 2005, 6(2): 126-131.

@article{title="The analytical solutions for orthotropic cantilever beams (I): Subjected to surface forces",

author="JIANG Ai-min, DING Hao-jiang",

journal="Journal of Zhejiang University Science A",

volume="6",

number="2",

pages="126-131",

year="2005",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.2005.A0126"

}

%0 Journal Article

%T The analytical solutions for orthotropic cantilever beams (I): Subjected to surface forces

%A JIANG Ai-min

%A DING Hao-jiang

%J Journal of Zhejiang University SCIENCE A

%V 6

%N 2

%P 126-131

%@ 1673-565X

%D 2005

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.2005.A0126

TY - JOUR

T1 - The analytical solutions for orthotropic cantilever beams (I): Subjected to surface forces

A1 - JIANG Ai-min

A1 - DING Hao-jiang

J0 - Journal of Zhejiang University Science A

VL - 6

IS - 2

SP - 126

EP - 131

%@ 1673-565X

Y1 - 2005

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.2005.A0126

**Abstract: **This paper first gives the general solution of two-dimensional orthotropic media expressed with two harmonic displacement functions by using the governing equations. Then, based on the general solution in the case of distinct eigenvalues, a series of beam problems, including the problem of cantilever beam under uniform loads, cantilever beam with axial load and bending moment at the free end, cantilever beam under the first, second, third and fourth power of

**
**

. INTRODUCTION

In this paper, we will consider the orthotropic plane problems. The general solution of two-dimensional orthotropic media expressed with two harmonic displacement functions is given at first by use of the governing equations. Then, based on the general solution in the case of distinct eigenvalues, a series of beam problems, including cantilever beam under uniform loads, cantilever beam with axial load and bending moment at the free end, cantilever beam under the first, second, third and fourth power of

Analytical solutions for various problems are obtained by the superposition principle.

. GENERAL SOLUTION FOR THE PLANE PROBLEM OF ORTHOTROPIC SOLID

Governing Eq.(

Ding et al.(

The polynomials listed in Appendix A can be chosen as harmonic functions

with

. THREE SOLUTIONS FOR CANTILEVER BEAM WITHOUT BODY FORCES

Substituting Eq.(

The boundary conditions are

Substituting Eqs.(10c), (10d) and (10e) into Eqs.(11a) and (11b) , we arrive at

Then, the unknown constants

Substituting Eq.(

The boundary conditions are

Substituting Eqs.(19b) and (19c) into Eqs.(20a) and (20b), we have

Then, the constants

We introduce the displacement function as follows

Substituting Eq.(

When

When

Then, the unknown constants

When

Then,

When

Substituting Eq.(27b) into the third of Eq.(25c), we have

Then, the constants

When

From Eq.(27b) and Eq.(25c), we have

Then,

(46)−(52). To satisfy the left boundary conditions in Eq.(25c), the solution above should be superposed on the rigid body displacement solutions as follows

APPENDIX A

Harmonic polynomials for the plane problems can be written in the following form:

* Project (Nos. 10432030, 10472102) supported by the National Natural Science Foundation of China

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