Iwinski / Neal Theory of Beams

The Application of the Laplace Transformation Method to Engineering Problems
2. Auflage 2014
ISBN: 978-1-4831-8601-6
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 160 Seiten

Theory of Beams: The Application of the Laplace Transformation Method to Engineering Problems, Second Enlarged Edition emphasizes the method used than the broad coverage of all the significant cases that may be met in engineering practice. The content of this edition is mostly the topics presented in the first edition, but are roughly doubled. This edition is divided into four chapters, wherein most of the modifications made are included in the fourth chapter. The first chapter provides an introduction of the study, followed by discussions on theory of beams. Then, specific topics on the transform of the load function; beams with transverse and axial loading; beams and free beam on elastic foundations and non-homogeneous elastic foundations; and simple beam with terminal forces and couples resting on an elastic foundation are examined. This book ends with a table presenting transforms and functions. This text will be of interest to mathematicians and engineers, as well as mathematics and engineering students.

Autoren/Hrsg.

Iwinski, T.

Neal, B. G.

Inhaltsverzeichnis

1;Front Cover;1
2;Theory of Beams: The Application of the Laplace Transformation Method to Engineering Problems;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE TO THE SECOND EDITION;8
6;PREFACE TO THE FIRST EDITION;10
7;Chapter I. INTRODUCTORY INFORMATION;12
7.1;1. General Introduction;12
7.2;2. Step functions and multi-step functions;16
7.3;3. Some remarks on the Laplace transformation method;22
8;Chapter II. THEORY OF BEAMS;25
8.1;1. Assumptions. Differential equation of the elastic curve. Load function;25
8.2;2. The elastic curve of single-span beams;37
8.3;3. Determination of static quantities for a single-span beam;57
8.4;4. Beams on three supports;60
8.5;5. The elastic curve of continuous beams;63
8.6;6. Theorem of three moments;67
8.7;7. Single-span beams on elastic supports;72
8.8;8. Continuous beams on elastic supports;77
8.9;9. The theorem of five moments;78
9;Chapter III. THEORY OF BEAMS WITH VARIABLE FLEXURAL RIGIDITY;82
9.1;1. Derivation of the differential equation of the deflection curve Shape function. Examples;82
9.2;2. Deflection curve for single-span beams;89
10;Chapter IV. PROBLEMS WITH MORE COMPLEX LOADING;103
10.1;1. The transform of the load function;103
10.2;2. Beams with transverse and axial loading;106
10.3;3. Beams on elastic foundations (Winkler's type): general solution;109
10.4;4. Free beam on an elastic foundation;111
10.5;5. Simple beam with terminal forces and couples resting on an elastic foundation;113
10.6;6. Beams on non-homogeneous elastic foundations whose elasticity varies in a stepwise manner;118
10.7;7. On the non-continuous solutions in the theory of structures;132
11;Chapter V. TABLES OF TRANSFORMS;152
12;REFERENCES;158
13;INDEX;160

Leseproben

Chapter II

THEORY OF BEAMS

Publisher Summary

This chapter presents the theory of beams. It presents an assumption where a beam of length is , and one uses the right-handed system of rectangular coordinates , , with the origin at the centroid of the left end cross-section of the beam, the -axis along the axis of the beam and - and -axes taken along the principal axes of the second moment of the cross-section, the positive direction of -axis being vertically downward. The internal elastic force system set up at a cross section of the beam, which is parallel to -plane, reduces to a transverse vertical force () (shearing force) and a couple () (bending moment). If the portion of the beam to the left of the section be removed, () and would be the forces equivalent to those that the left portion exerts on the right portion through the section, so that with () and () the equilibrium of the right portion would be preserved. It can be shown that in the regions where () is positive, the beam deflects so that its concavity is upward, that is, the elastic curve in convex with respect to the axes of coordinates.

1 Assumptions. Differential equation of the elastic curve. Load function

Consider a beam of length as shown in Fig. 5. We shall use the right-handed system of rectangular coordinates with the origin at the centroid of the left end cross-section of the beam, the -axis along the axis of the beam and - and -axes taken along the principal axes of the second moment of the cross-section, the positive direction of -axis being vertically downwards.

FIG. 5

The vector quantities (forces, force moments, bending moments, shearing forces) with which we shall be dealing in the problems forming the subject of this paper, in the majority of cases, will be entirely in the direction of one of the axes of coordinates. Then we shall be able to treat them as scalars, a single component defining completely their magnitude (the other two components being zero). If this single component is positive, we shall speak of the corresponding vector (or vector quantity) as being positive, and vice versa. This, therefore, will be the meaning of a force or a force moment being positive or negative. The positive direction for measuring the angles will be in conformity with the universal usage, i.e. that which, for example, in the -plane, which, for example, requires a right-handed screw to rotate to the tip of the screw moving along (arrowed in Fig. 5).

The internal (elastic) force system set up at a cross section of the beam (parallel to -plane) reduces to a transverse vertical force (shearing force) and a couple (bending moment). Of the two possibilities we choose that one in which and are defined by the effect of the left portion of the beam (i.e. to the left of the section in question) on to the right portion. In other words, if the portion of the beam to the left of the section be removed, and would be the forces equivalent to those that the left portion exerts on the right portion through the section, so that with and the equilibrium of the right portion would be preserved.

It can be shown that in the regions where is positive, the beam deflects so that its concavity is upwards, i.e. the elastic curve in convex with respect to the axes of coordinates (since the positive branch of -axis is directed downwards). Similarly, in the regions where is negative, the elastic curve is concave.

With these assumptions the differential equation of the deflected beam as given by the theory of elasticity

Iy?=±M(x), (2.1.1)

(2.1.1)

(where is the second moment of the cross-sectional area with respect to the centroidal axis parallel to -axis, is the bending moment and the Young’s modulus of elasticity) becomes

Iy?=-M(x)· (2.1.2)

(2.1.2)

Assuming further that no concentrated load or couple is applied within the portion under consideration, then with being the shearing force and the load per unit length of the beam we have

M(x)dx=-Q(x) (2.1.3)

(2.1.3)

or

Qdx=q(x)· (2.1.4)

(2.1.4)

We shall now consider beams having a constant second moment throughout the whole length. In this case double differentiation of eq. (2.1.2) (with the above given relations taken into account) leads to the following differential equation

IV=1Bq(x), (2.1.5)

(2.1.5)

where letter ( = const.) is used to represent the flexural rigidity of the beam.

Let us now perform the Laplace transformation of the above differential equation. By eq. (1.3.7) we obtain for the left side

4Y(s)-s3y0-s2y'0-sy?0-y'?0, (2.1.6)

(2.1.6)

where zero suffix denotes the value of the respective functions at the point = + 0, and represents the transform of the elastic curve . For the right side of eq. (2.1.5) the result of the transformation depends on the type of loading acting on the beam. Let us, therefore, consider the three cases most frequent in practice.

Case 1

Uniformly distributed load ( = = const.) on a portion of the beam between = and = (Fig. 6).

FIG. 6

Transformation of the right side of eq. (2.1.5) in this case yields

B?08e-sxq(x)dx=qB?abe-sxdx=-qsB[e-sx]ab=qsBe-asqsBe-bs· (2.1.7)

(2.1.7)

If the beam carries several (say uniformly distributed loads, each of different magnitude ( and acting on a different portion of the beam from = to = (Fig. 7) then instead of a single term in eq. (2.1.7) there will be such terms, and the transformation (after some simplifications) will result in

FIG. 7

(s)=1sy0+1s2y'0+1s3y?0+1s4y'?0++1s5BSi=1mqi(e-ais-e-bis)· (2.1.8)

(2.1.8)

Performing now the inverse transformation as indicated in eqs. (1.3.5) and (1.3.6) the equation of the elastic curve for the generalized case 1 is obtained as follows:

(x)=y0+y'0x+y?0x22!+y'?0x33!++14!BSi=1mqi<(x-ai)4>ail-14!BSi=1mqi<(x-bi)4>bil· (2.1.9)

(2.1.9)

The four unknown quantities 0, '0, ?0 and '0? appearing in this equation can be evaluated, as will be shown later, from the boundary conditions depending on the nature of the supports of the beam.

It is to be noted that the first four terms on the right side of eq. (2.1.9) form a polynomial of third degree. Its existence is conditioned by the form of the left side of the differential equation (2.1.5). The remaining terms (each a multi-step function), on the other hand, clearly depend solely upon the form of the function , i.e. upon the type of loading of the beam. This distinction in the nature of the terms on the right side of eq. (2.1.9) will enable us later directly to write down relevant formulae for the deflection as well as for other characteristic data of a beam loaded in any arbitrary manner.

Denoting the above mentioned polynomial by the symbol , eq. (2.1.9) can be written

(x)=W(x)+124BSi=1mqi<(x-ai)4>ail--124BSi=1mqi<(x-bi)4>bil· (2.1.10)

(2.1.10)

Case 2

A concentrated load applied at = (Fig. 8). Instead of this concentrated load consider a uniformly distributed load over a length a between = and = + a, its magnitude being such that ·a = , i.e. being a function of a. Now letting a tend to zero while the product · a is kept constant so that ? 8, the concentrated load may be defined as follows:

FIG. 8

a?0qa=P· (2.1.11)

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