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THEORY OF STRUCTURES



AND



STRENGTH OF MATERIALS.



WITH



DIAGRAMS, ILLUSTRATIONS, AND EXAMPLES.



BY

HENRY T. BOVEY, M.A., D.C.L., F.R.S.C.,

PROFESSOR OF CIVIL ENGINEERING AND APPLIED MECHANICS, M*GILL UNIVERSITY, MONTREAL.
MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS; MEMBER OF THE INSTITU-
TION OF MECHANICAL ENGINEERS; LATE FELLOW OP

QUEENS' COLLEGE, CAMBRIDGE (ENG.).



FIRST THOUSAND.




NEW YORK:

JOHN WILEY & SONS,

53 EAST TENTH STREET.

1893.



COPYRIGH^, 1893,

BV
HENRY T. BOVEY.






ROBERT DRTJMMOND,



886 Pearl Street,
New York.



444 & 446 Pearl Street,
New York.



DEDICATED



WHOSE BENEFACTIONS TO M*GILL UNIVERSITY

HAVE DONE SO MUCH TO ADVANCE THE CAUSE OF

SCIENTIFIC EDUCATION.



PREFACE.



THE present work treats of that portion of Applied
Mechanics which has to do with the Design of Structures.

Free reference has been made to the works of other
authors, yet a considerable amount of new matter has been
introduced, as, for example, the Articles on " Surface Loading"
by Carus-Wilson, " The Flexure of Columns " by Findlay, and
" The Efficiency of Riveted Joints " by Nicolson ; also my
own Articles on " Maximum Shearing Forces and Bending
Moments," " The Flexure of Long Columns," " The Theorem
of Three Moments," etc.

I am much indebted to Messrs. C. F. Findlay and W. B.
Dawson for valuable information respecting the treatment of
Cantilever Bridges, Arched Ribs, and the Live Loads on Bridges.

To Messrs. J. M. Wilson, P. A. Peterson, C. Macdonald,
and others, many thanks are due for data respecting the Dead
Weights of Bridges.

I am under deep obligation to my friend Prof. Chandler,
who has kindly revised the proof-sheets, and who has made
many important suggestions.

I have endeavored so to arrange the matter that the
student may omit the advanced portions and obtain a com-
plete elementary course in natural sequence.

At the end of each chapter, a number of Examples,



VI PREFACE.

selected for the most part from my own experience, are
arranged with a view to illustrating the subject-matter an
important feature, as it is admitted that the student who care-
fully works out examples obtains a mastery of the subject
which is otherwise impossible.

The various Tables in the volume have been prepared from
the most recent and reliable results.

A few years ago I published a work on " Applied Me-
chanics," consisting mainly of a collection of notes intended
for the use of my own students. The present volume may be
considered as a second edition of that work, but the subject-
matter has been so much added to and rearranged as to make
it almost a new book. I venture to hope that this volume
may prove acceptable not only to students, but to the profes-
sion at large.

HENRY T. BOVEY.

McGiLL COLLEGE, MONTREAL,
November, 1892.



CONTENTS.



CHAPTER I.

FRAMED STRUCTURES.

PAGE

Definitions I

Frames of Two or More Members 2

Funicular Polygon 3

Polygon of Forces 4

Line of Loads 5

Mansard Roof 6

Non-closing Polygons ... 7

Funicular Curve 10

Centre of Gravity 1 1

Moment of Inertia 12

Cranes, Jib 13

" Derrick 16

" Composite 31

Shear Legs ^ \~f

Bridge Trusses 17

Roof Trusses i >

King-post Truss 21

Incomplete Frames 27

Queen-post Truss 31

Composite Frames 32

Roof-weights 37

Wind-pressure 38

Distribution of Loads 39

Examples of Roof Trusses 41

Examples of Bridge Trusses (Fink, Bollman, Howe, Bowstring, Single-
intersection, etc) 52

Method of Sections 62

Piers 65

vii



VI il CONTENTS.



Tables of Roof -weights and Wind-pressures ............................ 67

Examples ............................................................. 7

CHAPTER II.
SHEARING FORCES AND BENDING MOMENTS.

Equilibrium of Beams ................................................ 93

Shearing Force .................................. ...................... 95

Bending Moment ................... ................................ 96

Examples of Shearing Force and Bending Moment ...................... 97

Relation between Shearing Force and Bending Moment .................. 108

Effect of Live (or Rolling) Load ........................................ in

Graphical Representation of Moment of Forces with Respect to a Point.. . 116

Relation between Bending Moments and Funicular Polygon .............. 118

Maximum Shear and Maximum Bending Moment at any Point of an Arbi-

trarily Loaded Girder ............................................. 121

Hinged Girders ....................................................... 126

Examples ........... ' ................................................ 131

CHAPTER III.
GENERAL PRINCIPLES, ETC.

Definitions. . . ....................................... ................ 140

Stress, Simple ........................................................ 140

" Compound .................................................... 140

Hooke's Law ........................................................ 141

Coefficient of Elasticity ................................... ............ 141

Poisson's Ratio ........................ . .............................. 142

Effect of Temperature ................................................. 142

Specific Weight ....................................................... 143

Limit of Elasticity ................................................... 143

Breaking Stress ....... ............................................... 147

Dead and Live Loads ................................................. 143

Repeated Stress Effect ................................................ 145

Wohler's Experiments ................................................ 145

Testing of Metals .................................................... 147

Launhardt's Formula .................................................. 159

Wey ranch's Formula ............... ................................... 153

Unwin's Formula ......................................... . ......... 159

Flow of Solids ....................................................... 162

Work, Internal and External ........................................... 168

Energy, Kinetic and Potential ......................................... 167

Oblique Resistance .................................................. 169

Values of k .......................................................... 174

Momentum. Impulse .................. . ............................ 176

Angular Momentum .................................................. 177



CONTENTS. . IX

PAGE

Useful Work. Waste Work 178

Centrifugal Force 181

I m pact 1 84

Extension of a Prismatic Bar 189

Oscillatory Motion of a Weight at the End of a Vertical Elastic Rod 190

Inertia 198

Balancing 198

Curves of Piston Velocity 205

Linear Diagrams of Velocity 206

Curves of Crank-effort. . . 207

Curves of Energy 207

Fluctuation of Energy 207

Tables of Strengths, Elasticities, and Weights of Materials 210

Tables of the Breaking Weights and Coefficients of Bending Strengths of

Beams 213

Table of the Weights and Crushing Weights of Rocks, etc 214

Table of Expansions of Solids 215

Examples 216



CHAPTER IV.

STRESSES, STRAINS, EARTHWORK, AND RETAINING WALLS.

Internal Stresses 235

Simple Strain 235

Compound Strain 236

Principal Stresses 240

Curves of Maximum Shear and Normal Intensity 240

Combined Bending and Twisting Stresses 244

Combined Longitudinal and Twisting Stresses 247

Conjugate Stresses 247

Relation between Principal and Conjugate Stresses 247

Ratio of Conjugate Stresses 250

Relation between Stress and Strain 251

Rankine's Earthwork Theory 255

Pressure against a Vertical Plane 257

Earth Foundations 258

Retaining Walls 260

Retaining Walls. Conditions of Equilibrium 260

Rankine's Earthwork Theory applied to Retaining Wall? 264

Line of Rupture 265

Practical Rules respecting Retaining Walls 267

Reservoir Walls 271

General Case of Reservoir Walls 275

General Equations of Stress 276

Ellipsoid of Stress 281

Stress-strain Equations 281



X CONTENTS.

PAGE

Isotropic Bodies 283

Relation between A , A, and G 285

Traction 287

Torsion 288

Work done in the Small Strain of a Body (Clapeyron) 292

Examples 294



CHAPTER V.
FRICTION.

Friction 300

Laws of Friction 300

Inclined Plane 301

Wedge , 302

Screws . 306

Endless Screw 309

Rolling Friction 310

Journal Friction 312

Pivot 316

Cylindrical Pivot 316

Wear 318

Conical Pivot 319

Schiele's Pivot {anti-friction) 320

Belts and Ropes 321

Brakes 323

Effective Tension of a Belt 324

Effect of High Speed 325

Slip of Belts 326

Prony's Dynamometer 327

Stiffness of Belts and Ropes 327

Wheel and Axle 329

Toothed Gearing 331

Bevel- wheels 335

Efficiency of Mechanisms 335

Table of Coefficients of Journal Friction 336

Examples 337

CHAPTER VI.

t

TRANSVERSE STRENGTH OF BEAMS.

Elastic Moment 340

Moment of Resistance 340

Neutral Axis 340

Transverse Deformation 344

Coefficient of Bending Strength 344



CONTENTS. XI

PAGE

Equalization of Stress 349

Surface Loading 350

Effect of Bending Moment in a Plane which is not a Principal Plane 354

Springs 355

Beams of Uniform Strength 358

Flanged Girders 365

Classification of Flanged Girders 365

Equilibrium of Flanged Girders , 366

Moments of Inertia of I and other Sections 371

Design of a Girder of I-section 381

Deflection of Girders 384

Camber 387

Stiffness 389

Distribution of Shearing Stress 391

Beam acted upon by Forces Oblique to its Direction 396

Similar Girders 401

Allowance to be made for Weight of Beam 405

Examples 407



CHAPTER VII.
TRANSVERSE STRENGTH OF BEAMS {Continued.}

General Equations 428

Interpretation of the General Equations 432

Examples Cantilever 435

Girder upon Two Supports 439

" fixed at One End and resting upon Support at the

Other 442

" " fixed at Both Ends 445

" " upon Two Supports not in the same Horizontal Plane. 446

" Neutral Axis of Arbitrarily Loaded Girders 448

" Cantilever with Varying Section 455

" Girder Encastre at the Ends 458

Springs 456

Work done in bending a Beam 460

Transverse Vibrations of a Beam supported at Both Ends , 461

Imperfect Fixture 461

Continuous Girders 463

Theorem of Three Moments 463

Swing-bridges 470

Maximum Bending Moment at the Points of Support of a Continuous

Girder of n Spans 475

General Theorem of Three Moments 484

Comparative Merits of Continuous Girders 486

Examples 490



Xll CONTENTS.

CHAPTER VIII.
PILLARS.

PAGE

Classification of Pillars 513

Form " " 514

Failure " " 515

Uniform Stress 516

Uniformly Varying Stress 517

Hodgkinson's Formulae 520

Gordon's Formula 522

Values of the Coefficients (a and f) in Gordon's Formula 523

Graphical Representation of Strength of Pillars 524

Rankine's Modification of Gordon's Formula 526

Formula for Safe Working Stress 526

Value of " Radius of Gyration " for Different Sections 526

American Iron Columns 532

Long Thin Pillars 534

Long Columns of Uniform Section (Euler's Theory) 538

Resistance of Columns to Buckling (Weyrauch's Theory) 550

Baker's Formulae .... 549

Flexure of Columns , 554, 557

Examples 563

CHAPTER IX.

TORSION.

Definition 568

Coulomb's Laws 568

Torsional Strength of Shafting 569

St. Venant's Results 572

Torsional Rupture 572

Resilience of Shafting 574

Effect of Combined Bending and Twisting 574

Distance between Bearings for Shafting 575

Efficiency of Shafting 577

Spiral Springs 577

Figures illustrating the Distortion produced by twisting Round, Square,

and Rectangular Iron Bars 5850, 585^

Examples 580

CHAPTER X.
CYLINDRICAL AND SPHERICAL BOILERS.

Cylinders 586

Efficiency of Riveted Joints in Boilers 587



CONTENTS. X1U

PAGE

Thick Hollow Cylinder 588

Spherical Shells 591

Practical Formulae 592

Examples 594

CHAPTER XL
BRIDGES.

Classification 597

Curved and Horizontal Flanges 597

Depth of Girders (or Trusses) 597

Position of Platform 598

Comparative Advantages of Two, Three, and Four Main Girders 600

Dead Load 600

Live Load 600

Trellis (or Lattice) Girder 600

Warren Girder 603

Howe Truss 61 1

Single-intersection Truss 616

Double-intersection Truss 616

Whipple Truss 618

Linvillc Truss 618

Post Truss 618

Quadrangular Truss 618

Bowstring Truss 618

Bowstring Truss with Isosceles Bracing 624

Bowstring Suspension-bridge (Lenticular Truss) 626

Cantilever Trusses 627

Curve of Cantilever Boom 634

Deflection of Cantilevers 638

Rollers 639

Wind-pressure 651

Regulations respecting Wind-pressure 653

Lateral Bracing 654

Chords 655

Stringers 656

Maximum Allowable Working Stresses 657

Camber 659

Rivet Connections between Flange and Web 660

Eye-bars, Pins, and Rivets 661

Steel Eyebars . 665

Rivets 666

Dimensions of Rivets 667

Strength of Punched and Drilled Plates 668

Riveted Joints 668

Theory of Riveted Joints 671



XIV CONTENTS.



Covers 675

Efficiency of Riveted Joints 676

Tables of Weights of Actual Bridges 682-687

Table of Loads for Highway Bridges 688

Examples 689-702

CHAPTER XII.

SUSPENSION-BRIDGES.

Cables 703

Anchorage 704

Suspenders . . 706

Curve of Cable (catenary) 706

Link " Cable" 709

Length of Cable 712

Weight of Cable 713

Deflection due to Change of Length 714

Pressure upon Piers 718

Stiffening Truss 719

Stiffening Truss hinged at the Centre 725

Suspension-bridge Loads 730

Modifications of the Suspension-bridge proper 731

Examples 734~739

CHAPTER XIII.
ARCHED RIBS.

Definitions 740

Equilibrated Polygon (Line of Resistance) 741

Polygon of Pressures 743

Linear Arch 743

Conditions of Equilibrium 745

Joint of Rupture 747

Minimum Thickness of Abutment 749

Empirical Formulae 750

Linear Arch in Form of a Parabola 750

" " " " " Transformed Catenary 750

" " " " " Circular Arc 753

" " " " " Elliptic Arc 753

Hydrostatic Arch 757

Geostatic Arch , 759

General Arch Theory 760

Arched Ribs 762

Bending Moment and Thrust at any Point of an Arched Rib 763

Rib with Hinged Ends 764



CONTENTS. XV

PAGE

Semicircular Rib with Hinged Ends ' 765

Graphical Determination of the Thrust at any Point of an Arched Rib 767

Rib in Form of a Circular Arc 769

Rib with Fixed Ends * 771

" " " " in Form of a Circular Arc 773

" " " " " ' " Semicircle 775

" " " " " " " Parabola 775

Effect of a Change of Temperature 777, 786

Deflection of an Arched Rib 780, 802

Elementary Deformation of an Arched Rib 781

Rib of Uniform Stiffness 788

Parabolic Rib of Uniform Depth and Stiffness 789, 795, 800

Arched Rib of Uniform Stiffness with Fixed Ends 804

Stresses in Spandril Posts and Diagonals 804

Maxwell's Method of Determining the Stresses in a Framed Arch 806

Examples 809-812




THEORY OF STRUCTURES.



CHAPTER I.
FRAMES LOADED AT THE JOINTS.

I. Definitions. Frames are rigid structures composed oi
straight struts and ties, jointed together by means of bolts,
straps, mortises and tenons, etc. Struts are members in com-
pression, ties members in tension, and the term brace is applied;
to either.

The external forces upon a frame are the loads and the
reactions at the points of support, from which may be found
the resultant forces at the joints. The greatest care should be
exercised in the design of the joints. The resultant forces
should severally coincide in direction with the axes of the
members upon which they act, and should intersect the joints
in their centres of gravity. Owing to a want of homogeneity
in the material, errors of workmanship, etc., this coincidence is
not always practicable, but it should be remembered that the
smallest deviation introduces a bending action. Such an
action will also be caused by joint friction when the frame is
insufficiently braced. The points in which the lines of action
of the resultants intersect the joints are also called the centres
of resistance, and the figure formed by joining the centres of
resistance in order is usually a polygon, which is designated the
line of resistance of the frame.

The position of the centres should on no account be allowed
to vary. It is assumed, and is practically true, that the joints
of a frame are flexible, and that the frame under a given load



2 THEORY OF STRUCTURES.

i

does not sensibly change in form. Thus an individual mem.
her is merely stretched or compressed in the direction of its
length, i.e., along its line of resistance, while the frame as a
whole may be subjected to a bending action.

The term truss is often applied to a frame supporting a
weight,

2. Frame of Two Members. OA, OB are two bars
jointed at O and supported at the ends A, B. The frame in






FIG.



FIG. 2.



FIG. 3.



Fig. I consists of two ties, in Fig. 3 of two struts, and in Fig.
2 of a strut and a tie.

Let P be the resultant force at the joint, and let it act in
the direction OC. Take OC equal to P in magnitude, and
draw CD parallel to OB. OD is the stress along OA, and CD
is that along OB.

Let the angle AOB a, and the angle COD = /3.

Let S, , 5 2 be the stresses along OA, O, respectively.



S, _ OD _ sin(<x ft)
~P ~~OC~ sin a~



, S, _ CD _ sin ft
P ~ OC ~ sin a



3. Frame of Three or More Members- Let A.A^A^ . . .

be a polygonal frame jointed at A l9 A tt A 91 . . . Let P l ,
/*,, P 3 , . . . be the resultant forces at the joints A lt A 9 , A 3 ,
. . . , respectively. Let S lt S a , 5 3 , . . . be the forces along
A t A 9t A^A Z , . . . , respectively.

Consider the joint A^.

The lines of action of three forces, P lt S l , and S 9 , intersect
in this joint, and the forces, being in equilibrium, may be
represented in direction and magnitude by the sides of the



FRAMES LOADED AT THE JOINTS. 3

triangle Os,s 6 , in which v 6 is parallel to P lt Os J to S lt and Os 9
to 5..

Similarly,/^, S t , 5 2 maybe represented by the sides of
the triangle Os^., which has one side, Os iy common to the
triangle Os^^, and so on.

Thus every joint furnishes a triangle having a side common
to each of the two adjacent triangles, and all the triangles to-
gether form a closed polygon s^^. . . The sides of this
polygon represent in magnitude and direction the resultant




FIG. 4.



forces at the joints, and the radii from the pole O to the angles
jjVs .' represent in magnitude, direction and character,
the forces along the several sides of the frame A^A Z A^, . .
The polygon A^A^A 3 ... is the line of resistance of the
frame, and is called the funicular polygon of the forces P lt P 39
/*,... with respect to the pole O.

The two polygons are said to be reciprocal, and, in general,
two figures in graphical statics are said to be reciprocal when
the sides in the one figure are parallel or perpendicular to cor-
responding sides in the other.

A triangle or polygon is also said to be the reciprocal of a
point when its sides are parallel or perpendicular to correspond-
ing lines radiating from the point. Thus the triangle Os^ is



4 THEORY OF STRUCTURES.

the reciprocal of the point A lt and the polygon
is the reciprocal of the point O.

If more than two members meet at a joint, or if the joint is
subjected to more than one load, the resulting force diagram
will be a quadrilateral, pentagon, hexagon, . . . according as
the number of members is 3, 4, 5, ... or the number of loads
2* 3 4, *

In practice it is usually required to determine the stresses
in a number of members radiating from a joint in a framed
structure. If the reciprocal of the joint can be drawn, its
sides will represent in direction and magnitude the stresses in
the corresponding members.

Corollary. The converse of the preceding is evidently true.
For if a system of forces is in equilibrium, the polygon of
forces -V 2 .y 3 . . . must close, and therefore the polygon which
has its sides respectively parallel to the radii from a pole O to
the angles s 1 , s 2 , s s , . . . and which has its angles upon the
lines of action of the forces, must also close.

EXAMPLE I. Let O be a joint in a framed structure, and
let Os 1} Os^, Os s , ... be the axes of the members radiating
from it. The polygon A^A^A^ ... is the reciprocal of O, the
side A^A^ representing the stress along Os lt the side A^A 3 that
along Os z , etc.

Ex. 2. Let the resultant forces at the joints be paral-
lel. The polygon of forces becomes the straight line v 5 ,




FIG. 6.



FIG. 7.



FRAMES LOADED AT THE JOINTS. 5

which is often termed the line of loads. Thus, the forces P s ,
/>..../>, are represented by the sides s,s^ s y s 3 , . . . v*> which
are in one straight line closed by s^ t and s & s t , representing the
remaining forces P l and P t , and the triangles Os t s tt Os^ 3 , . . .
are the reciprocals of the points A t , A,. . . . Draw OH per-
pendicular to s t s t . The projection of each of the lines Os lt
Oft, Os 3 , . . . perpendicular to s^ is the same and equal to
OH, which therefore represents in magnitude and direction
the stress which is the same for each member of the frame.

Let a lf # 2 , a 3 , . . . be the inclinations of the members
A,A^ t A^AS, . . . respectively, to the line of loads. Then

OH Hs l tan a l = Hs & tan a b ;
.-. Off (cot a ; -f cot a 6 ) = ffs 1 + ffs, s.s,



and OH, in direction and magnitude, is equal to the stress
common to each member. Also, the stress in any member,
e.g., A^AS Os, Offcosec <x t .

Corollary. Let the resultant forces at the joints A lt A 6
be inclined to the common direction of the remaining forces,
and act in the directions shown by the dotted lines. Let P/,
P t ' be the magnitudes of the new forces ; draw s^' parallel to
the direction of P t ' so as to meet Os 6 in s e ' ; join s^s^. Since
there is equilibrium, s 6 's 6 must be parallel to the line Sj
of action of P t ' . Thus, s^s^ is the force polygon.

Ex. 3. The forces, or loads, />, P t , . . . P 6 are
generally vertical, while P lt P 6 are the vertical re-
actions of the two supports.

Suppose, e.g., that A t A t . . . A 6 is a rope or chain
suspended from the points A lt A 6 , in a horizontal
plane and loaded at A^A S . . . with weights P 9 , P tt . . .
The chain will hang in a form dependent upon the
magnitude of these weights. The points H and S 6
will coincide, and Off will represent the horizontal

riG. 8.

tension of the chain.

Let the polygon A^A^ . . . A t be inverted, and let the rope
be replaced by rigid" bars, A.A^ A^A Z . . . The diagram of




O THEORY OF STRUCTURES.

forces will remain the same, and the frame will be in
equilibrium under fat given loads. The equilibrium, however,
is unstable as the chain, and consequently the inverted frame
will change form if the weights vary. Braces must then be
introduced to prevent distortion.





-H



FIG. 9.



FIG. 10.



Tal^e the case of a frame DCBA . . . symmetrical with
respect to a vertical through A, and let the weights at A, B, C,
. . . be W lt W^ W z , . . . , respectively.

Drawing the stress diagram in the usual manner, OH rep-
resents the horizontal thrust of the frame.

The portions s^^, s^s 3 , ... of the line of loads give a
definite relation between the weights for which the truss will
be stable. The result may be expressed analytically, as
follows :

Let arj, or a , of,, . . . be the inclinations of AB, BC, CD, . . . ,
respectively, to the horizontal.

Let the horizontal thrust OH = H. Then



It ^ = IV,= IV 3 = . ..

cot af 1 = 3 cot or 2 = 5 cot of a = . . .

If there are two bars only, viz., AB, BC, on each side of the
vertical centre line, the frame will have a double slope, and in
this form is employed to support a Mansard roof.



FRAMES LOADED A T THE JOINTS. ?

4. Non-closing Polygons. Let a number of forces P 19
P 9 , P 3 , . . . act upon a structure, and let these forces, taken in
order, be represented in direction and magnitude by the sides
of the unclosed figure MNPQ . . . This figure is the unclosed




polygon of forces, and its closing line TM represents in direction
and magnitude the resultant of the forces P l , P 2 , P 3 , . . .

For PM is the resultant of P^ and P^ and may replace
them ; QM may replace PM and P 3 , i.e., P lt .P 9 , and />,; and



so on.



Take any point O and join OM, ON, OP, . . .

Draw a line AB parallel to OM and intersecting the line of
action of P l in any point B. Through B draw BC parallel to
ON and cutting the line of action of P^ in C. Similarly, draw
CD parallel to OP, DE to OQ, EF to OR, . . . The figure
ABCD ... is called the funicular polygon of the given forces
with respect to the pole O. The position of the pole O is arbi-



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