The theory of strains in girders and similar structures

The following pages have been written at various times during such brief intervals of leisure as the author could spare from his professional duties. They are for the most part the result of experience combined with theory; it is therefore hoped that they may supply the student with what has long been a want in Engineering literature, namely, a Handbook on the Theory of Strains and the Strength of Materials giving practical methods for calculating the strains which occur in girders and similar structures. The theory of transverse strain has, indeed, been incidentally treated by writers on Mechanical Philosophy; their researches, however, have been confined to strains in plain girders, or to a few brief remarks on the more elementary forms of trussing, which, without further development, are of little practical use, and but too frequently afford a pretext for the ill-concealed contempt which so-called practical men sometimes entertain for theoretic knowledge.

A thorough acquaintance with the theory of strains and the strength and other properties of materials forms the basis of all sound engineering practice, and when this is wanting, even natural constructive talent of a high order is frequently at fault, and the result is either excess and consequent waste of material, or, what is still more disastrous, weakness in parts where strength is essential. The time has gone by when practical sagacity formed the sole qualification for high engineering success. Before the improvement of the steam engine gave rise to a new profession there were indeed some memorable names on the roll of engineers, generally self-taught mechanics, whom great natural ability had raised to pre-eminence in their profession  but practice which was formerly excusable, or even worthy of the highest commendation, would, now that knowledge has increased, be properly described as culpable waste, arising either from prejudice or ignorance.

The usual resource of the merely practical man is precedent, but the true way of benefiting by the experience of others is not by blindly following their practice, but by avoiding their errors as well as extending and improving what time and experience have proved successful. If one were asked what is the difference between an engineer and a mere craftsman, he would well reply, that the one merely executes mechanically the designs of others, or copies something which has been done before without introducing any new application of scientific principles, while the other moulds matter into new forms suited for the special object to be attained; and though experience and practical knowledge are essential for this, he lets his experience be guided and aided by theoretic knowledge, so as to arrange and proportion the various parts to the exact duty they are intended to fulfil.

The well-educated engineer should combine the qualifications of the practical man and of the physicist, and the more he blends these together, making each mould and soften what the other would seem to dictate if allowed to act alone, the more will his works be successful and attain the exact object for which they are designed. The engineer should be a physicist, who, in place of confining his operations to the laboratory or the study, exerts his energies in a wider field in developing the industrial resources of nature, and compelling mere matter to become subservient to the wants and comforts and civilization of the human race.

Transverse-strain Shearing-strain. The formulae investigated in this chapter are, unless otherwise expressed, applicable to all flanged girders whose webs are formed of bracing, or if continuous, yet so thin that the transverse strength of the web as an independent rectangular girder may be neglected without sensible error. Our knowledge of the strains in this vertical web when continuous is still imperfect. Analogy indeed leads us to conclude that they follow laws similar to those which hold good in braced girders, but in the absence of experimental proof this is to a certain degree conjecture a conjecture, however, which I feel confident my readers will share after they have had the patience to read through this book.

The mode in which a load affects a girder may be thus analysed. From experience we learn that the load bends the girder downwards and develops longitudinal strains of tension and compression in the flanges. If the semi-girder, represented in Fig. 3, be supposed divided into vertical slices or transverse sections of small thickness, the weight tends to shear or separate the section on which it immediately rests from the adjoining one. The lateral connexion of the sections, however, prevents this separation, and the second section is drawn down by a vertical force equal to the weight which tends to shear it from the third section and so on. Thus, a vertical force equal to the weight is transmitted from section to section as far as the point of support. This vertical strain has been aptly named the Shearing-strain; but few writers, until the last few years, have noticed the practical results which follow from the fact that this force can be communicated from section to section only through the medium of some diagonal strain. Respecting the exact directions of the strains which this shearing force develops in a continuous web we know nothing positively; it is probable that they assume various directions crossing each other like close lattice-work, some vertical, some diagonal, perhaps some curved. However this may be, we know that certain of them must be diagonal, since the weight, which is a vertical force, produces strains in the flanges, which are longitudinal, through the medium of the web, which in fact fulfils the part of bracing in a lattice girder. The reader will perceive that we have really three sets of forces to deal with, namely, horizontal, vertical, and diagonal forces. The latter, however, may be resolved into horizontal and vertical components, and thus we have at present only horizontal and shearing forces to consider, recollecting that the shearing-strain of any transverse section of a girder means the total vertical strain transmitted through that section, including in the term shearing strain the vertical components of diagonal strains.

Horizontal strains in braced or thin continuous webs may be neglected. When the vertical web of a girder with horizontal flanges is open-work like latticing, the shearing- strain is altogether transmitted through the bracing, the flanges being capable of conveying strains in the direction of their length only; but when the web is continuous, as in a plate-girder, there can be no doubt that a certain amount of shearing-force acts upon the flanges also, so inconsiderable, however, that we may practically neglect it. If, however, one or both flanges are curved, the whole or a considerable portion of the shearing-strain is conveyed through that part of the flange which is sloped, the amount depending upon its angle of inclination. In this case the web has less duty to perform than if the flanges were horizontal, and its sectional area may therefore be reduced. It will also be observed that the diagonal strains developed by the shearing force in a continuous web have horizontal components within the web itself, and consequently, a continuous web aids the flanges to a certain extent, for those parts of the web which adjoin the flanges share the horizontal strains in the latter, and this flange action of the web is greater the thicker the web is. When, however, the web is very thin, the total amount of this flange action of the web is small compared with the strain in the flanges themselves and may therefore be neglected without introducing any serious error. In this chapter all horizontal strains in the web are neglected.

Girder of greatest strength Areas of horizontal flanges should he to each other in the inverse ratio of their ultimate unit-strains. The distribution of a given amount of material in the flanges, so as to produce the girder of greatest strength, occurs when both flanges are simultaneously on the point of rupture, for if either flange contain more material than is required to sustain its proper strain when the other gives way, it can spare some of the surplus material to strengthen the other.

19. Girder of uniform strength Economical distribution of material. A girder of uniform strength is one in which all parts, both flanges and web, are duly proportioned to the strain which they have to bear, i.e., are equally capable of sustaining the particular strain which is transmitted through them. If such a girder were perfect, there is no reason why any one part should fail before another, since the train in each part is the same sub-multiple of the ultimate or breaking-strain of that part. The girder of uniform strength is obviously the most economical also in its proportions, for no part has a wasteful excess of material; the tensile or compressive unit-strain is constant throughout the entire length of each flange respectively, and the shearing-unit-strain in each section of the web is the same as in every other section.