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A theoretical and practical treatise on the strength of beams and columns
A THEORETICAL AND PRACTICAL TREATISE ON THE STRENGTH OF BEAMS AND COLUMNS
In which the ultimate and the elastic limit strength of beams and columns is computed from the ultimate and elastic limit compressive and tensile strength of the material, by means of formulas deduced from the correct and new theory of the transverse strength of materials.
BY ROBERT H. COUSINS,
E. & F. N. SPON, 1889
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A theoretical and practical treatise on the strength of beams and columns
INTRODUCTION
For more than two centuries the mathematical and mechanical laws that govern the transverse strength of Beams and Columns have received the attention of the most expert mathematicians of all countries. Galileo, in 1038, formulated and published the first theory on the subject. He was followed by such philosophers as Mariotte, Leibnitz, Bernoulli!, Coulomb, and others, each amending and extending the work of his predecessor, until the year 1824, when Navier succinctly stated the theory that is recognized to be correct at the present day, and to which subsequent writers and investigators have added but little.
This theory has neither received the endorsement of the experimenters nor of some of the theoretical writers. "Excepting as exhibiting approximately the laws of the phenomena, the theory of the strength of materials has many practical defects" (Wiesbach). "It has long been known that under the existing theory of beams, which recognizes only two elements of strength namely, the resistance to direct compression and extension the strength of a bar of iron subjected to a transverse strain cannot be reconciled with the results obtained from experiments on direct tension, if the neutral axis is in the centre of the bar" (Barlow).
During the present century much time and means have been expended in attempts to solve, experimentally, the problems that have engaged the attention of the mathematicians, and as the result of their labors we find such experimenters as Hodgkinson, Fairbairn, and others, whose names are household words in the literature of the subject, adopting empirical rules for the strength of Beams and Columns rather than the rational formulas deduced by the scientists. "For no theory of the rupture of a simple beam has yet been proposed which fully satisfies the critical experimenter” (De Volson Wood).
That we should be able to deduce the strength of Beams and Columns from the known tensile and compressive strength of the material composing them, has been apparent to many writers and experimenters on the subject, but to the present time no theory has been advanced that embodies the mathematical and mechanical principles necessary to its accomplishment. The theory herein advanced and the formulas resulting there from deduce the strength of Beams and Columns from the direct crushing and tensile strength of the material composing them, without the aid of that coefficient that has no place in nature, the Modulus of Rupture. The theory and the formulas deduced there from are in strict accord with correct mechanical and mathematical principles, and the writer believes that they will fully satisfy the results obtained by the experimenter.
The great practical benefits to be derived from the correct theory of the strength of Beams and Columns will be evident, when we consider the countless tons of metal that have been made into railroad rails, rolled beams, and the other various shapes, and that the manufacturers were without knowledge of the work to be performed by the different parts of the beam or column in sustaining the load that it was intended to carry. The best that they have been able to do is to compute the strength by the aid of an empirical quantity deduced from experiments on "similar beams." The correct theory will enable them to foretell the strength of any untried shape, and the reason for the strength of those that have been long in use, which is the " true object of theory."
This theory has neither received the endorsement of the experimenters nor of some of the theoretical writers. "Excepting as exhibiting approximately the laws of the phenomena, the theory of the strength of materials has many practical defects" (Wiesbach). "It has long been known that under the existing theory of beams, which recognizes only two elements of strength namely, the resistance to direct compression and extension the strength of a bar of iron subjected to a transverse strain cannot be reconciled with the results obtained from experiments on direct tension, if the neutral axis is in the centre of the bar" (Barlow).
During the present century much time and means have been expended in attempts to solve, experimentally, the problems that have engaged the attention of the mathematicians, and as the result of their labors we find such experimenters as Hodgkinson, Fairbairn, and others, whose names are household words in the literature of the subject, adopting empirical rules for the strength of Beams and Columns rather than the rational formulas deduced by the scientists. "For no theory of the rupture of a simple beam has yet been proposed which fully satisfies the critical experimenter” (De Volson Wood).
That we should be able to deduce the strength of Beams and Columns from the known tensile and compressive strength of the material composing them, has been apparent to many writers and experimenters on the subject, but to the present time no theory has been advanced that embodies the mathematical and mechanical principles necessary to its accomplishment. The theory herein advanced and the formulas resulting there from deduce the strength of Beams and Columns from the direct crushing and tensile strength of the material composing them, without the aid of that coefficient that has no place in nature, the Modulus of Rupture. The theory and the formulas deduced there from are in strict accord with correct mechanical and mathematical principles, and the writer believes that they will fully satisfy the results obtained by the experimenter.
The great practical benefits to be derived from the correct theory of the strength of Beams and Columns will be evident, when we consider the countless tons of metal that have been made into railroad rails, rolled beams, and the other various shapes, and that the manufacturers were without knowledge of the work to be performed by the different parts of the beam or column in sustaining the load that it was intended to carry. The best that they have been able to do is to compute the strength by the aid of an empirical quantity deduced from experiments on "similar beams." The correct theory will enable them to foretell the strength of any untried shape, and the reason for the strength of those that have been long in use, which is the " true object of theory."
CONTENTS.
- FORCES DEFINED AND CLASSED.
- RESISTANCE OF CROSS SECTIONS TO RUPTURE.
- TRANSVERSE STRENGTH.
- CAST-IRON BEAMS.
- WROUGHT-IRON AND STEEL BEAMS.
- TIMBER BEAMS.
- COMBINED BEAMS AND COLUMNS.
- INCLINED BEAMS.
- TRUSSED BEAMS.
- TRANSVERSE STRENGTH
SECTION I. General Conditions.
Coefficients or Moduli of Strength are quantities expressing the intensity of the strain under which a piece of a given material gives way when strained in a given manner, such intensity being expressed in units of weight for each unit of sectional area of the material over which the strain is distributed. The unit of weight ordinarily employed in expressing the strength of materials is the number of pounds avoirdupois on the square inch.
Coefficients of Strength are of as many different kinds as there are different ways of breaking a piece of material.
Coefficients of Tensile Strength or Tenacity is the strain necessary to rupture or pull apart a prismatic bar of any given material whose section is one square inch, when acting in the direction of the length of the bar. This strain is the T of our formulas.
Coefficient of Crushing Strength or Compression is the pressure required to crush a prism of a given material whose section is one square inch, and whose length does not exceed from one to five times its diameter, in order that there may be no tendency to give way by bending sideways. This pressure is the C of our formulas.
Elasticity of Materials. It is found by experiment that if the load necessary to produce a strain and fracture of a given kind is applied in small instalments, that before the load becomes sufficiently intense to produce rupture, it will cause a change to take place in the form of the material, and if the load is removed before this intensity of the fibre strain passes certain limits, the material possesses the power of returning to its original form. This is called its elasticity.
Elastic Limits. When the material possesses the power of recovering its exact original form without "set" on the removal of a load of a given intensity, the greatest load under which it will do this is called the limit of perfect elasticity.
The limit of elasticity as ordinarily defined and used by experimenters is that point or intensity of strain where equal instalments or increments of the applied load cease to produce equal changes of form, or where the change in form increases more rapidly than the load.
The Elastic Limit of Beams may be deter- mined by applying small equal parts of the load and noting the increase in deflection after each increase of the load, allowing sufficient time for each increase of the load to produce its full effect. "When it is found that the deflections increase more rapidly than the load, its elastic limit has been reached and passed. The relation between the elastic limit load of the beam and the elastic limit of the tensile and compressive fibre strains of the material composing the beam will be shown in the sequel, or that the elastic limit load of the beam produces the elastic limit strain for the fibres.
Working Load and Factor of Safety. The greatest load that any piece of material, used in a structure, is expected to bear is called the working load.
The breaking load to be provided for in designing a piece of material to be used in a structure is made greater than the working load in a certain ratio that is determined from experience, in order to provide for unforeseen defects in the material and a possible increase in the magnitude of the expected working load.
The factor of safety is the ratio or quotient obtained by dividing the breaking load by the working load required.
General Formula. In our first chapter we deduced rules or formulas, from which can be computed the Greatest Bending Moment that a load applied to a beam in a given manner will produce without reference to the shape of its cross-section, or to the material composing the beam.
In our second chapter principles are deduced from which can be computed the Greatest Moment of Resistance cross sections of the various shapes and material will exert at the instant of rupture, without reference to the length of the beam or to the manner in which the load may be applied.
To avoid repetition, the formulas for the Moments of Resistance are deduced in this chapter. Our knowledge of the transverse strength of beams will now be complete if we compute and make the Greatest Moment of Resistance of the cross-section of the beam equal to the Greatest Bending Moment of the applied load.
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A theoretical and practical treatise on the strength of beams and columns
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