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WORKS BY 
 
 L. A. WATERBURY 
 
 PUBLISHED BY 
 
 JOHN WILEY & SONS 
 
 Cement Laboratory Manual. 
 
 A Manual of Instructions for 
 the Use of Students in Cement 
 Laboratory Practice. 12mo, 
 vii +122 pages, 28 figures. 
 Cloth, $1.00. 
 
 A Vest = pocket Handbook of 
 Mathematics for Engineers. 
 
 vi + 91 pages, 61 figures. Mo- 
 rocco, $1.00 net. '' 
 
A VEST-POCKET 
 
 
 
 _j 
 < 
 
 HANDBOOK 
 
 O C 
 
 <T 5 
 
 OF O 
 
 MATHEMATICS 
 
 s 
 
 FOR 5 
 
 ENGINEERS 
 
 BY 
 L. A. WATERBURY, C.E. 
 
 U 
 
 Professor of Civil Engineering, University of 
 
 5'S v 
 
 OF THE \ 
 
 UNIVERSITY ) 
 
 , or 
 
 FIKST EDITION 
 
 FIRST THOUSAND 5 - 
 
 ji 
 
 NEW YORK 
 
 JOHN WILEY AND SONS 
 
 LONDON: CHAPMAN & HALL, LIMITED 
 
 1908 
 
GENERAL 
 
 COPYRIGHT, 1908, 
 BY 
 
 L. A. WATERBURY 
 
 
 Stanhope Ipress 
 
 F. H. GILSON COMPAN1 
 BOSTON. U.S.A. 
 
PREFACE. 
 
 Tliis handbook is intended as a reference 2 oc 
 
 book, for the use of those who have studied > m 
 
 or are studying the branches of mathematics < 
 
 usually taught in engineering courses. It is < } 
 
 not intended for a text book, and does not, ^ 
 
 therefore, attempt to prove many of the _j 
 
 formulae which are given. < m 
 
 Most of the material in this book was z 3 
 
 obtained from the following sources: algebra S Q 
 
 from Hall & Knight's Algebra (Macmillan {Jl ^ 
 
 Co.) ; trigonometry from Bowser's Trig- -jj 
 onometry; analytic geometry from Candy's 
 
 Analytic Geometry ; calculus from Taylor's ; 
 
 Differential and Integral Calculus ; theoret- < ' 
 
 ical mechanics from Church's Mechanics of o D 
 
 Engineering ; and mechanics of materials H 
 
 from Merriman's Mechanics of Materials ; - o 
 
 to all of which the writer is very much in- s= 
 
 debted and from all these Authors he has ' < a 
 
 received permission to use the material. The - 
 
 reader is referred to these works for the proof ; ^_ 
 
 and explanation of the various formulae. - t; 
 
 L. A. W. _ h 2 
 TUCSON, ARIZ., March, 1908. 
 
 179775 
 
CONTENTS. , > 
 
 PAGE j- 
 
 ALGEBRA . . 1 2 
 
 H O 
 
 TRIGONOMETRY 5 
 
 Plane Triangles 8 
 
 Spherical Triangles 9 ^ 
 
 > UJ 
 
 ANALYTIC GEOMETRY 11 ^5 
 
 Transformation of Coordinates. ... 11 % \n 
 
 The Straight Line 13 ~ 
 
 The Circle 14 
 
 The Parabola 15 < 
 
 The Ellipse 15 | 
 
 The Hyperbola 16 : 5 
 
 The Cycloid 17 = j 
 
 Miscellaneous Curves 17 ~ '3 
 
 Solids 18 
 
 DIFFERENTIAL CALCULUS 20 j w 
 
 INTEGRAL CALCULUS 23 ^ 
 
 UJ Q 
 
 THEORETICAL MECHANICS 36 b J 
 
 - o 
 
 Statics: -- 
 
 Equilibrium of Forces 38 ^ tf 
 
 Center of Pressure 39 - ^ 
 
 Center of Gravity 40 : < 
 
 Moment of Inertia 42 5 { 
 
 Product of Inertia 44 I ^ 
 
 Radius of Gyration 45 
 
 Ellipsoid of Inertia 45 * 
 
 Dynamics: w ! 
 
 Velocity and Acceleration .... 46 
 
 Falling Bodies 46 : - 
 
 Impact 46 
 
 Virtual Velocities 47 5 [ 
 
 Curvilinear Motion 47 
 
 Projectiles 48 
 
 v 
 
VI CONTENTS 
 
 PAGE 
 
 Translation 49 
 
 Rotation 49 
 
 Center of Oscillation 50 
 
 Pendulum 51 
 
 Work, Energy, Power 51 
 
 Friction 51 
 
 MECHANICS OF MATERIALS 53 
 
 Direct Stress 54 
 
 Eccentric Loads 55 
 
 Equation of Neutral Axis 58 
 
 Kernel or Core-Section 58 
 
 Section Modulus Polygons 59 
 
 Diagonal Stresses 61 
 
 Pipes, Cylinders, Spheres 62 
 
 Riveted Joints 63 
 
 Beams: 
 
 Vertical Shear 64 
 
 Shearing Stresses 65 
 
 Bending Moment 65 
 
 Theorem of Three Moments .... 66 
 
 Flexural Stresses 67 
 
 Table of Beams 68 
 
 Struts and Columns: 
 
 Euler's Formula 81 
 
 Rankine's Formula '. 82 
 
 Ritter's Formula 83 
 
 The Straight Line Formula .... 83 
 
 Engineering News Formula .... 84 
 
 Eccentrically Loaded Columns ... 86 
 
 TORSION 86 
 
ALGEBRA. 
 
 EXPONENTS AND LOGARITHMS. : 
 
 If a w = b, m = loga 6 . a m . a n = a m + w , - 
 
 .'. log x . y = logo: +logy. a TO H-a n = a m ~ n <g 
 .'. log (z-^2/) = log:r-log2/. 
 
 - 09 
 
 .'. log x 2 = 2 . log x. (a wl ) n = a w n , ^ : 
 
 /. logx ft = n .logx. a=l, : i '] 
 
 ^ < 
 
 /. log(l)=0. F 
 
 The base of the common system of loga- j w 
 
 rithms is 10. $ 3 
 
 The base of the natural system of loga- 3 Q 
 
 rithms is C d 
 
 ,4-+ . . . = 2.7182818284. 
 
 The cologarithm of a number is the loga- 
 rithm of its reciprocal. Log f-J = log x. 
 
 To transform a logarithm from base e to 
 base 10, multiply by logio e. 
 
 Log 10 e = 0.43429448. 
 
 Log. 10 = 2.30258509. 
 
 1 
 
 " log 10 ' 
 1 
 
2 ALGEBRA 
 
 QUADRATIC EQUATIONS. 
 
 
 2a 
 
 PROPORTION. 
 
 If a : b : : c : d, 
 
 a c b d 
 
 7- j* or - = - 
 b a a c 
 
 ad = bc, ^=^- d , 
 o d 
 
 a b _ cd a+b _ c+d 
 b d ab cd 
 
 ARITHMETICAL PROGRESSION. 
 
 a, a+d, a +2 d, . . . 
 Last term, L = a + (n 1) d. 
 Sum of terms, 
 
 GEOMETRICAL PROGRESSION. 
 
 o, ar, ar 2 , ar 3 , . . . 
 Last term, L = ar n ~ l . 
 Geometric mean, M = ^ab. 
 
 Sum, 
 
 rL-a 
 
 1-r r-1 
 
 For an infinite geometrical series, the sum 
 to infinity is S= _ 
 

 
 ALGEBRA 
 
 HARMONIC PROGRESSION. 
 
 b, c are in harmonic progression if 
 a _ a b 
 
 . 
 
 if _, , _ are j n arithmetical progression. 
 a b c 
 
 PERMUTATIONS AND COMBINATIONS. 
 
 ab and ba are two permutations but only 
 one combination. 
 
 The number of permutations possible of 
 n things taken r at a time is 
 
 n P r = n (n-1) (n-2) . . . (n-r + 1). 
 
 "P-=ln. 
 
 ([n =1X2X3X4 . . . Xn). 
 
 BINOMIAL THEOREM. 
 
 +b) n = a n +n . a n-1 . b 
 
 
 SERIES. 
 
 1. An infinite series in which the terms 
 are alternately positive and negative is con- 
 vergent if each term is numerically less than 
 the preceding term. 
 
4 ALGEBRA 
 
 2. An infinite series in which all of the 
 terms are of the same sign is divergent if 
 each term is greater than some finite quan- 
 tity, however small. 
 
 3. An infinite series is convergent if from 
 and after some fixed term the ratio of each 
 term to the preceding term is numerically less 
 than unity. 
 
 4. An infinite series in which all the terms 
 are of the same sign is divergent if from and 
 after some fixed term the ratio of each term 
 to the preceding term is greater than unity, 
 or is equal to unity. 
 
 5. If there are two infinite series in each 
 of which all of the terms are positive, and if 
 the ratio of the corresponding terms in the 
 two series is always finite, the two series are 
 both convergent, or both divergent. 
 
 DETERMINANTS. 
 
 |at &i| =a6 _ Q b 
 
 'O2 b 2 \ 
 
 ai 61 ci 
 
 tt 2 b 2 C 2 
 
 as 63 03 
 
 ~d\ &2 C 3 + 
 2 b 3 Ci + 
 03 . 61 . C2 
 
 CL2 bi . C3 03 . &2 Ci . 
 
 then 
 
 a^x + b%y + ciz 4- dz = 0, 
 a^x + b$y + c&z + ds = 0, 
 
 3; = -y = 
 
 -1 
 
TRIGONOMETRY. 
 
 r* C 
 
 2 
 
 Radius = 1 . > 
 
 E Q F 2 PC 
 
 AB sin 0. 
 
 ;>\ 
 
 / h- r- 
 ./ > UJ 
 
 OA=cos 0. 
 
 ^ 
 
 xp <| 
 
 \ ' J 
 
 CD = tan 0. 
 EF = cotO. o 
 
 X 
 
 N 
 
 AC J 
 
 OD = sec0. F 'g- 
 OF = cosec 0. 
 
 i! 
 
 s\ 
 
 BG = covers 0=1 sin 0. 
 
 . sin 
 tan 0=- H' 
 
 sin 2 
 
 sec 2 = i 
 cosec 2 = l+cot 2 0. 
 exsec = sec 0-1. 
 
 For in radians, 
 
 05 07 
 
 02 04 06 
 
 ~ + ~ 
 
 03 2.05 1707 
 
 tan = + T +3^5 +373-75^7 
 
TRIGONOMETRY 
 
 Sli 
 
TRIGONOMETRY 7 
 
 sin (A +B) =sin A . cos B +cos A . sin B. 
 sin (AB)= sin A . cos B cos A . sin B. 
 cos (A +B)= cos A . cos B sin A . sin',B. 
 cos (A J5) =cos A . cos 5 +sin A . sin B. 
 tan A + tan J? 
 
 1 _ tanA . tana - oS 
 
 tan A - tan B r ~ 
 
 (A-B)^ 1+tanA-tanB - ; ; j 
 
 z ^ 
 
 sin 2 A =2 . sin A . cos A. < 5 
 
 cos 2 A cos 2 -4 sin 2 A 
 
 = 2 cos 2 ,4-1 <, 
 
 2 . tan A 
 
 tan 2 A - - z A 
 I tan 2 A 
 
 /A\ _ 1 cos A 
 
 sin 3 A =3 . sin A 4 . sin 3 * 
 cos 3 A = 4 cos 3 A 3 cos A . 
 3 tan A tan 3 A 
 
 tan 3 A = 
 
 1-3 tan 2 A 
 
 sin A +sin B = 2 . sin - .cos ^ 
 
 sin A sin B = 2 cos ~ .sin pr 
 
 A+B A-B 
 cos A + cos B = 2 cos s .cos jr -- 
 
 .A+B . A-B 
 cos A cos B = 2 sm ~ . sin r 
 
TRIGONOMETRY. 
 
 sin A +sin B = tan (A +B) 
 sin A sin B tan (A B) 
 
 sin A + sin B 
 cos A + cos B 
 
 = tan (A+B). 
 
 sinA+sinB 
 cos A cos B 
 
 sin A sin B _ , , . 
 cos A +cos B ~~ 
 
 sin A sin B 
 cos A cos B 
 
 = cot 
 
 cos A 4- cos B 
 cos A cos B 
 
 rv- 
 
 PLANE TRIANGLES. 
 
 sin A +sin B 
 +sin C 
 
 = 4 cos . cos 
 
 Fig. 2. 
 
 cos A +cos B +cos C 
 
 = l+4.sin| 
 
 . C 
 
 tan A +tan B +tan 
 
 tan A . tan B . tan C. 
 
 a 
 
 sin 
 
 A sin B sin C 
 
 a 2 = &2 +c 2_2 . 6 .C . COS , 
 
 = 
 a-6 tan \ (A-B)' 
 
TRIGONOMETRY 
 
 Area = i b . c . sin A 
 _ a? sin B . sin C 
 2 . sin A 
 
 = \/s(s-a) (s-6) (s-c), 
 where s = i(a+6+c). 
 
 SPHERICAL TRIANGLES. '- 
 
 > UJ 
 
 Center of sphere is at 0. ^ S 
 
 ?c 
 /o 
 
 B 
 
 Fig. 3. 
 
 Right Spherical Triangles. Let C repre- 
 sent the right angle, 
 
 cos c = cos a . cos 6. 
 sin 6 = sin B . sin c. 
 tan a cos B . tan c. 
 tan a = tan A . sin b. 
 
 tan A . tan B = 
 
 cos c 
 
 cos A =sin B . cos a. 
 
 OBLIQUE SPHERICAL TRIANGLES. 
 
 sin a sin b sin c 
 
 7- = p> = - 77 = modulus. 
 
 sin A sin B sin C 
 
 cos a = cos b . cos c +sin 6 . sin c . cos A. 
 cos A = cos B . cos C -f sin B . sin C . cos a. 
 cot a . sin 6 = cot A . sin C + cos C . cos 6. 
 Let 
 
10 
 
 TRIGONOMETRY 
 
 then 
 
 s-c) 
 
 sin 6 . sin c 
 
 /A\ _ A/sin s . sin (s a) 
 \2 ) ~~ sin 6 . sin c 
 
 tan (-\ = \/ sin (*-fr) sin (* c) . 
 \2 / sin s . sin (s a) 
 
 sin - = V - : 
 
 sin B . sin C 
 
 /cos (S-B) . cos (S - C) 
 sin B . sin C 
 
 / _ cos S. cos (S -A) 
 cos (S B). cos (S-Cy 
 
ANALYTIC GEOMETRY. 
 
 TRANSFORMATION OF COORDINATES. 
 
 To transform an equation of a curve from 
 one system of coordinates to another system, 
 substitute for each 
 
 variable its value in ^ 
 
 < 
 
 P 
 
 terms of variables 
 of the new system. 
 
 
 ^ /p f >l 
 
 - 
 
 Rectangular Sys- * 
 tern. Old Axes Par- 
 
 
 
 : 1 
 
 allel to New Axes. X 
 
 
 
 Fig. 4. 
 
 Rectangular System. Old Origin Coincident 
 with New Origin. 
 
 Fig. 5. 
 
 x f = x . cos 9+y .sind. 
 y'=y .cosQ x . sin 6. 
 x =x f . cos y f . sin 0. 
 
 y =!/ cos O+x* .sin 
 11 
 
12 ANALYTIC GEOMETRY 
 
 Rectangular System. Old Axes not Parallel 
 to New Axes. Old Origin not Coincident 
 with New Origin. 
 
 Fig. 6. 
 
 x f = (x-K) cos + (y-k) sin 0. 
 y f = (y-k') cos O-(x-h) sin 9. 
 x = x f . cos y f .sin 0+h. 
 V =y f . cos +xf . sin +k. 
 
 Polar and Rectangular Systems. 
 
 x = p . cos 0. 
 y = P . sin 0. 
 
 P = vV+2/\ 
 
 tan 9-H. 
 
 Fig. 7. 
 
 sin0 = 
 
 cos 
 
 ^x*+v*' 
 
 v5T^' 
 
 cot = - . 
 
 y 
 
 8 ec* = ^EZ 2 . 
 cosec 5 = 
 
 /x*+tf 
 
 y 
 
 
ANALYTIC GEOMETRY 13 
 
 THE STRAIGHT LINE. 
 
 Equations of Straight Line. An equation of 
 the first degree containing but two variables 
 can always be represented by a straight line. 
 
 The equation of the straight line may as- 
 sume the following forms, for the rectangular 
 system of coordinates. 
 
 Ax+By+C=0 .... (1) 
 y = mx+k ...... (2) 
 
 in which m is the value of the tangent of the 
 angle which the line makes with the X-axis, 
 and k is the intercept on the Y-axis between 
 the line and the X-axis. 
 
 y-y' = A(x-x') ... (3) 
 
 in which x' ', y' are the coordinates of a point 
 of the line, and A is a constant. 
 
 in which x r , y 1 and x" ', y" are the coordinates 
 of two points of the line. 
 The polar equation of 
 a straight line is 
 
 p . cos (0-a)=& (5) 
 
 where k is the length of 
 the normal ON. 
 
 Distance between Two Points. The distance 
 between two points, x', y' and a/', y", is 
 equal to 
 
 The distance between two points, pi, #1, 
 and 02, 02, is equal to 
 
 ^Pl 2 + P2 2 ~ 2 Pl . p 2 . COS (0i - 2 ) 
 
 . : 
 
14 
 
 ANALYTIC GEOMETRY 
 
 Angle between Two Lines. The angle be- 
 tween two lines, y = m'x+kf and y = m"x+W ', 
 is the difference between the two angles 
 whose tangents are m' and m". 
 
 Area of Triangle. The area of the triangle 
 whose vertices are (x\, 2/1), (x 2 , 2/2), and (x 3 , 2/3), 
 is equal to 
 
 i r^ 
 
 * 3 2/3 1 
 
 THE CIRCLE. 
 
 The most general equation of the circle, 
 for rectangular coordinates, is 
 
 in which a, b are the coordinates of the cen- 
 ter of the circle, and R is the radius. 
 
 The following are special equations of the 
 circle for rectangular and polar systems of 
 coordinates. 
 
 2 R .cos B. 
 
 X 
 
 Fig. 10. 
 
 2R .sin0. 
 
ANALYTIC GEOMETRY 
 
 15 
 
 THE PARABOLA. 
 
 If the Y-axis coincides with the directrix, 
 DM, then 
 
 Q 
 
 
 7 
 
 TP 
 
 D 
 
 O 
 
 \ 
 
 1 
 
 V 
 
 If the F-axis coincides with ON, passing 
 through the vertex, then 
 
 In Fig. 12, F is the focus, OF=OD=a, and 
 L L is the latus rectum = 4 a. 
 
 Eccentricity, e -p^ =1. 
 
 THE ELLIPSE. 
 
 D e 
 
 Fig. 13. 
 
 F, F are foci. 
 
 Eccentricity, e<l. 
 
 The area of the ellipse is equal to nab. 
 
16 ANALYTIC GEOMETRY 
 
 THE HYPERBOLA. 
 
 Fig. 14. 
 
 A -A = principal hyperbola. 
 B~B = conjugate hyperbola. 
 cc asymptote. 
 
 Principal hyperbola: 
 
 Asymptotes: - ^ =n 
 a 2 b 2 
 
 Conjugate hyperbola: - - = 
 a 2 b 2 
 
 When referred to the asymptotes as axes, 
 the equations become: 
 
 Principal hyperbola: xy = 
 
 Conjugate hyperbola: xy= - ( a<i+b \ 
 
 D-D is the di- 
 rectrix. 
 
 F, F are foci. 
 
 FP 
 pQ-Ol. 
 
 Fig. (5. 
 
ANALYTIC GEOMETRY 
 THE CYCLOID. 
 
 17 
 
 Fig. 16. 
 
 x = a (0 sin 0), 
 ya (1 cos 0), 
 
 x = a , vers" 1 \\ - ^2 ay-y*- 
 THE SPIRAL OF ARCHIMEDES. 
 
 THE RECIPROCAL OR HYPERBOLIC 
 SPIRAL. 
 
 I 
 
 THE PARABOLIC SPIRAL. 
 
 THE LITUUS OR TRUMPET. 
 
 THE LOGARITHMIC SPIRAL. 
 
 log p = k . 9. 
 
 If fc = l, and logarithms to the base a are 
 employed, then the equation may be written 
 
18 ANALYTIC GEOMETRY 
 
 THE CATENARY. 
 
 
 THE CUBIC PARABOLA. 
 
 THE SPHERE. 
 
 For the origin at the center, 
 
 where R is the radius. 
 
 CONES. 
 
 The equation of the cone generated by the 
 hne, z = mx+c, rotated about the Z-axis, is 
 
 OBLATE SPHEROIDS. 
 
 The equation of the oblate spheroid gen- 
 erated by the ellipse, g + g = l, rotated about 
 its minor axis, is 
 
 PROLATE SPHEROIDS. 
 
 The equation of the prolate spheroid gen- 
 erated by the ellipse, g + ~ = 1, rotated about 
 its major axis, is 
 
 x 2 y- z"i 
 
 ^_ i _ _i_ i 
 
 A2 "*" A2 "T" Z9 A 
 
ANALYTIC GEOMETRY 19 
 
 HYPERBOLOIDS. 
 
 The equation of the hyperboloid of one 
 
 x 2 z 2 
 aappe, generated by the hyperbola, --^=1, 
 
 rotated about its conjugate axis, is 
 
 2 4. _ * = i 
 a 2 a 2 6 2 
 
 The equation of the hyperboloid of two 
 
 x 2 z 2 
 nappes, generated by the hyperbola, 2 - p = 1 , 
 
 rotated about its transverse axis, is 
 a 2 ~P ~P =1 * 
 
 THE PARABOLOID 
 
 The equation of the paraboloid of revolu- 
 tion generated by the parabola, x 2 = 4a 
 rotated about its axis, is 
 
 GENERAL EQUATION OF CONIC 
 SECTION. 
 
 The general equation of any conic section, 
 for which the Y-axis coincides with the 
 directrix and the X-axis passes through the 
 foci normal to the directrix, is 
 
 where k is the distance from the directrix to 
 the focus, and e is the eccentricity. 
 
DIFFERENTIAL 
 CALCULUS. 
 
 Variables will be represented by u, v x y 
 and 0, and constants by a, 6, m, and 'n. ' 
 
 D will be used as the sign for the deriva- 
 tive, and d as the sign for the differential. 
 
 Sm i x = angle whose sine is x. 
 
 D <&>- 
 
 dx 
 
 .'. To obtain the derivative of any func- 
 tion, drop the differential of the variable 
 from the differential of the function. 
 
 d (av)=a .dv. 
 d (u +v +x) = du +dv +dx. 
 d(x .y)=y .dx+x.dy. 
 d(u.v.x.y. ..) = (v.x. 
 
 (u.x.y . . .-)dv + ( u .v. 
 
 (u . v . x . . .) dy + . . . 
 
 20 
 
DIFFERENTIAL CALCULUS 21 
 
 dx 
 dx* = y . xV~ l . dx -\-x* . Iog x . -^ 
 
 where M = logo e. 
 
 d (&")=&*. iogb.^ 
 
 dx a =a . x a ~ l . dx. 
 
 d (sin x) = cos x .dx. 
 
 d (cos x)= -sinx . dx. 
 
 d (tan x)=sec 2 x . dx. 
 
 d (cot x) = cosec 2 x .dx. 
 
 d (sec x) =sec x . tan x . dx. 
 d (cosec x) = cosec x . cot x . dx. 
 
 d (vers x) = d (1 - cos x) = +sin x . dx. 
 d (covers x)=d (l-sinx)= -cosx.dx. j 
 dtsin-x^dx/vT^ II 
 
 d(cos- 1 x)=-dx/Vl-x 2 . z < 
 
 d (tan- 1 x)=dx/(l+x 2 ). 
 
 d(cot~ 1 x)=-dx/(l+x 2 ). j eo 
 
 d(sec- 1 x) = dx/(xV / x 2 -l). ; g 
 ^ ^ 
 
 d (vers- 1 x) = dx/v / 2 x-x 2 . 2 O 
 
 d (covers - 1 x) = - dx/v^x-x 2 . _ H s 
 
 ^ "*i 
 To differentiate a junction : M 
 
 1. Find the value of the increment of the 
 function in terms of the increments of its 
 variables ; 
 
 2. Consider the increments to be infinitesi- 
 mals, and in all sums drop the infinitesimals 
 of higher order than the first, and in the 
 
22 DIFFERENTIAL CALCULUS 
 
 remaining terms substitute differentials for 
 increments. 
 
 For the maximum value of a function the 
 first derivative is zero, and the second deriv- 
 ative is negative. 
 
 For the minimum value of a function the 
 first derivative is zero, and the second deriva- 
 tive is positive. 
 
 If -~ assumes the form - then 
 
 Fx D (Fx) 
 fx ~ D (/* ' 
 
 The radius of curvature for a curve, y = fx, is 
 
 = = ^ 
 
 da. d?y dx . d?y 
 
 (dx? 
 
 where s is length of curve. 
 
INTEGRAL CALCULUS. 
 
 I dx=x+C, where C is the constant of 
 
 integration. The constant C must be added 
 to all of the following forms. 
 
 J (dx+dy+dz . . .) = 
 
 Cdx+Cdy+fdz + ... 
 
 Cx.dx=^- 
 J n+1 
 
 dx . 
 
 x or vers x. 
 
 A*. <**-*. 
 
 J log e a 
 
 \ e* . dx e*. 
 
 I a x . log e a . dx = a*. 
 
 I sin x . dx= cos 
 
 I cos x . <c = sin x or covers x. 
 
 cosec 2 x . dx = cot x. 
 
 I sec x . tan x . dx = &ec x. 
 
 23 
 
24 INTEGRAL CALCULUS 
 
 J cosec x . cot x . dx = cosec x. 
 J tan x . dx = log (sec x). 
 
 i cot a; . dx = log (sin x). 
 
 J cosec x . dx = log (tan |) 
 J"sec x . dx = \og [tan ^ + 1)] 
 
 =4- 
 
 /(fa 1 , /x-a\ 
 
 ^^ = 2^- loS C-+i)' 
 
 =log (a; 
 
 f _ rf^ 1 , /x\ 
 
 \ - , = - . sec -1 ( - ) , 
 
 J x \/ x 2_ a 2 a \a/ 
 
 -x 2 
 
 = covers - 
 
Cf 
 
 INTEGRAL CALCULUS 
 
 C, if 
 
 25 
 
 I a . dx = a I dx. 
 
 fo-c. 
 
 \ x . dy = xy I y . 
 
 dx. 
 
 i^t! ~&&+**-i . log (a+te)]. 
 
 /a . dc ' If, , , , . , a "I 
 __^_ 2 = _|_ log(a+6x) + ___J 
 
 2 1 
 
 a+bx] 
 
 C dx I 
 
 J x(a+bx) ~ ~ a ' log 
 
 a+bx 
 
 1 _! j /a+&a:\ 
 
 a(a+6x) a 2 " 3g V a; /' 
 
 _ , 
 
 + ' g 
 
 axa?' \ a? 
 
 dx 1 . / A /b\ 
 
 rr- o= F= tan" 1 (x V -V 
 a+6x 2 Vafr > a/ 
 
 when a >0 and 6 >0. 
 
26 INTEGRAL CALCULUS 
 
 dx 1 <Sa+x ^~b 
 t , ^.2 ~ / lg -/= /==- 
 
 when a>0 and 6<0. 
 
 dx 
 
 r_dx_ x _ J_ f_c 
 
 J (a +6x 2 ) 2 2 a (a +6x 2 ) 2 a J a J 
 
 2+6x2 
 
 dx 
 
 (a +6x 2 ) n+1 2na' (a +6z 2 ) n 
 , 2n-l f dx 
 
 L^l f_ 
 
 2 no J (a 
 
 J a+6x 2 = 
 
 a; a 
 6 
 
 (a+6x 2 ) n + 1 2 rib (a 
 
 ^J_ f dx 
 
 2n6 J (a+6x 2 )*' 
 
 /dx J_ / x 2 \ 
 
 x(a+6x 2 ) 2 a S Va+6x 2 / 
 
 r ^__ = _j__^r 
 
 J x 2 (a +6x 2 ) ax a J 
 
 a+6x 
 
 f_ da; = ! C dx 
 
 J x 2 (a +6x 2 ) n + 1 a J x 2 (a +6x 2 )* 
 
 _6 C dx 
 
 aj (a+6x 2 ) n + 1 ' 
 
 6 (nP+m+1) 
 
INTEGRAL CALCULUS 27 
 
 nP+m+l 
 
 o 
 nP+ 
 
 T^i (V . (a +bx n )P~i . dx, 
 +m+lJ 
 
 a (ro + 1) 
 
 x m + n . 
 
 b (nP -\- Tfi -f- n -f- 1 ) /* 
 o (w + 1) I 
 
 ^ +1 -( 
 
 on (P -HI) 
 
 nP+m+n + 1 
 an(P-fl) 
 
 -A. f- 
 
 2a J < 
 
 1562 
 
 /a+bx . dx 
 
 105 6 3 
 
28 INTEGRAL CALCULUS 
 
 * . dx 2 x n ^a+bx 
 
 n ~ l . dx 
 
 /g. rfx _ 
 Va+6x 
 
 2 na 
 (2n+l) 6 
 
 2(2a- 
 
 fx n ~ l . 
 J V+ 
 
 /a+bx 
 when a >0, 
 
 2 
 
 or = . 
 
 when a<0. 
 dx 
 
 f 
 
 / a+bx 
 
 x nV a +bx (n-l)ax n ~ 
 
 (2n-3) 6 f dx_ 
 (2n-2) a J ^n-iVa 
 
 +62; 
 
INTEGRAL CALCULUS 
 
 29 
 
 fa; 2 *S 
 
 /z 2 . dx x*/~ - * , 2 
 
 , = - ~V a 2_ a; 2 + s 
 
 Va 2 -^ 2 2 2 
 
 f__^_ x 
 
 J (a 2 -z 2 )i a 2 V a z-a; 2 ' 
 
 f(a 2 -rc 2 )5.^ = 
 
 (5 a 2 
 
 dx 1 , ( x \ 
 
 - - -- = - . log 1 - . I 
 
 x V x z +a z a \ a + V x * + a?' 
 
 S 
 21 
 
 u. 
 
 
30 INTEGRAL CALCULUS 
 
 C dx _ 
 
 2a?x 2 2 a 3 
 
 1 . 
 log 
 
 C x 
 
 dx= |(2* 2 a ; 
 a 4 
 
 L-<fl -a cos" 1 -- 
 
 ; 2 a log - 
 
 f - 
 
 : Jvs 
 
 r _j 
 
 - log (x + ^x 2 a 
 
 , 4- log (x-l-^a^ia 2 ). 
 
 ^(2a; 2 5a 2 ) V x 2 a 2 
 
INTEGRAL CALCULUS 31 
 
 /;* 
 
 , , - = vers~ 1 -- 
 
 V2ax-x 2 a 
 
 /x m dx = x m ~ 1 ^ / 2ax-x 2 
 \/2ax-x 2 
 
 (2m- 
 
 i-l)qj^-'.<fa 
 
 dx 
 
 m 
 
 "(2m 
 
 -l C 
 
 -1) a J x m-i 
 
 dx 
 
 / 
 / 
 
 / 
 /. 
 
 ax-x* . dx 
 
 , a 2 . , x a 
 + 2 Sm 
 
 a; w V2ax-rc 2 . rfa:= - ' 
 
 (2m+l) a 
 m+2 
 
 C x m-i t 
 
 = 
 
 (2m-3)ax m 
 
 (2m 
 
 -3 C^Zcuc a? 
 -3)aJ x w ~i * d:C ' 
 
 v ax 2 +6x +c 
 
 -7= log (2 ax +6 4-2 Va v / a x 2 +6x+c). 
 
 dx 
 
 /v / ^+6^+"c . dx= 2a * +b V ax *+bx+c 
 4.a 
 
 _/&2_-4_ac\ /*_ 
 V 8a / J x 
 
32 INTEGRAL CALCULUS 
 
 C dx = _JL_ gin _ 1 / 2aa;-6 \ 
 
 7. da; = 
 
 f 
 / 
 
 4 o 
 
 6 2 +4oc 
 8a 
 
 a?dz 
 
 n: fe 
 T 2^ 
 
 ^ r 
 
 3 a 
 c. da:. 
 
 sin (2 aO. 
 
 Jsin 2 a:.da;=|-|sin 
 
 /cos 2 a; . dx= + T sin (2 a;). 
 2 4 
 
 I sin 2 a; . cos 2 x . dx= -(x - sin 4 arV 
 
 C C dx 
 
 \ sec a; . esc x . dx = \ . 
 J J sin x . cos x 
 
 = log tan x. 
 
 Csec*x.csc*x.dx= C . , dx r- 
 J J sin 2 x . cos 2 x 
 
 = tan x cot x. 
 
 m+n 
 
 H I sin" 1 " 2 x cos n x . dx, 
 
 m +nj 
 
INTEGRAL CALCULUS 33 
 
 sin w+1 a;. cos n ~ 1 x 
 
 i-i r . 
 
 , I si 
 i+nj 
 
 f+tmt 
 
 m-l C 
 
 I sm m ~ 
 
 m J 
 
 H I sin w x . cos w ~ 2 x . dx. 
 
 + ' 
 
 dx = 
 
 m 
 
 cos n x . dx - 
 sin x . 
 
 c . cos n-1 x , n 1 C 
 
 H I cos l x . dx. 
 
 n n J 
 
 /sin** x , _ 
 cos n x 
 
 sin w+1 x n m 2 rsin m x. 
 
 (n l)cos n ~ 1 x n I J cos"" 2 
 
 /cos n x , 
 . m . dx = 
 sin x 
 
 -n 2 f*cos n x 
 i \ J sin m ~ 
 
 dx 
 
 (m 1) sin" 1 " 1 x m- 
 
 /dx cos x m 2 C dx 
 
 sin m x (m 1) sin" 1 " 1 x m lj sm m ~' i x ^ ^ 
 
 3 O 
 
 /dx _ sin x n 2Cdx ^~<t 
 
 cos" re (n 1) cos"" 1 x n lj cos""" 2 x' ^g 
 
 u S 
 
 /tan"- 1 ^ /" 
 tan"x . dx = r I tan"~ 2 x . dx. 
 n L J _^_ 
 
 ont"" 1 r C V> 
 
 cot" a;, da; = , - j cot n ~*x.dx. : 
 
 l ~ J EE 
 
 -^ = g& 
 
 a+o cos x ui S 
 
 v / 
 if a 2 > 
 
 2 / A /a - b x\ 
 
 tan" 1 ! V r.tan-l, 
 
 ,2_ft2 \ a +6 2/' 
 
34 INTEGRAL CALCULUS 
 
 if a 2 <& 2 . 
 
 . sin a; . 
 
 dx 
 
 /- 
 
 x m cos x +m I re 77 *" 1 cos x dx. 
 I x m . cos x . dx = 
 
 x m sin x m I re" 1 "" 1 cos re dx. 
 
 /sin re , re 3 , re 5 re 7 , 
 
 dx=X -3l + 5H-7]7 + --- 
 
 /sin re ^ _ 1 sin x 1 fcos re a 
 m ^*^ 1" m. \ ' T I m T 
 x m m Lx m m \J x m ~ 
 
 /cos re, _. x 2 x* re 8 
 
 ~x~ '~2\2 + 4[4~"Q{6 + ' 
 
 /cosx , 1 cos re 1 /*sin re dx 
 ^ dx= ^l-^-^ij-^r- 
 
 \ resin" 1 re . dx = 
 
 ^ [(2 re 2 - 1) sin" 1 re +rc Vl-a; 2 ]. 
 
 /"*>.*- 
 
 rc n + 1 sin- 1 a; 1 fx n + l dx 
 n + l n + l J Vl-x z 
 
 /n 
 re- 
 
 rc n + 1 cos" 1 re , 1 f re n + 
 n + l 
 
INTEGRAL CALCULUS 35 
 
 x n tan" 1 x . dx 
 
 n+1 n + lj 1+z 2 
 
 dx 
 
 *1 
 
 3 O 
 
 S3 
 
 O < 
 
 z cc 
 < LJ 
 
 n 
 
 2 u. 
 O 
 
 dx 2 . xi . 
 
 = sec" 1 
 /~n _ /,2 an a 
 
34 INTEGRAL CALCULUS 
 
 =. log - 
 
 if a 2 <6 2 . 
 
 ERRATA 
 
 should read 
 
 J -'** - cos .r . , 
 
INTEGRAL CALCULUS 35 
 
 I x n tan" 1 x . dx = 
 
 n+1 
 
 ! dr. 
 
 *V""dfc-2-2--2 f *-! 
 
 i/ aj 
 
 fe a * , -1 e a * , a f e a * . 
 
 dx= - - . r - -{ -- ; -.dx. 
 J x n n-l x n ~ l n-lj x n ~ ] 
 
 /ax / \ ^ ax\ a ( cos ( nx ^ +nsin(nrc)"l 
 [ a?+n* 
 
 / 
 
 dx 
 
THEORETICAL 
 MECHANICS. 
 
 NOTATION. 
 
 A =area. 
 
 a acceleration. 
 
 a n = normal acceleration. 
 
 a t tangential acceleration. 
 
 b = breadth. 
 
 C x = component of force parallel to the 
 
 X-axis. 
 C y = component of force parallel to the 
 
 C t = component of force parallel to the 
 
 Z-axis. 
 d = depth or distance. Also the sign of 
 
 the differential. 
 F= force. 
 
 F n = normal force or component of force. 
 F t = tangential force or component of 
 
 force. 
 /= coefficient of friction. Also the sign 
 
 of a function of a variable. 
 g = acceleration due to gravity = 32.2. 
 (The exact value is 32.1808- 
 0.0821 cos 2 L, where L is the 
 latitude.) 
 h distance from center of moments to 
 
 line of force. 
 7 = moment of inertia. 
 I ff = moment of inertia referred to center 
 
 of gravity. 
 
 I gx moment of inertia about an axis 
 through the center of gravity and 
 parallel to the X-axis. 
 
THEORETICAL MECHANICS 37 
 
 7 = polar moment of inertia about the 
 
 pole 0. 
 
 7a.= moment of inertia about the Jf-axis. 
 I y = moment of inertia about the Y-axis. 
 / = moment of inertia about the Z-axis. 
 J = product of inertia. (Subscripts are 
 
 similar to those for /.) 
 K=a constant. 
 L = power. 
 M = moment of a force. 
 
 W 
 
 m = mass = 
 Q 
 
 N=SL normal force or component of a 
 
 force. 
 
 P = point considered. 
 R = resultant of a system of forces. 
 r = radius of gyration. 
 s= space. 
 T= tangential force or component of a 
 
 force. 
 f = time. 
 V volume. 
 v = velocity. 
 VQ = initial velocity. 
 vt= tangential velocity. 
 tf*= velocity parallel to the X-axis. 
 v y = velocity parallel to the F-axis. 
 W = weight. 
 w = work. 
 
 , y, z = rectangular coordinates of a point. 
 p, = polar coordinates of a point. 
 
 p = distance from pole to center of 
 
 gravity, 
 a = angle. 
 $== angle of friction. 
 
38 THEORETICAL MECHANICS 
 
 STATICS. 
 Equilibrium of Forces. 
 
 Fig. 17. 
 
 For a system of concurrent forces in equi- 
 librium in one plane: 
 
 (Cz=F cos a,Cy=F sin a, where a is the 
 angle which F makes with X-X,) 
 
 For a system 
 of non-concur- 
 rent forces in 
 equilibrium in 
 one plane : 
 
 Also, i 
 
 Fig. 18. 
 
 If three forces are in equilibrium they 
 must be concurrent or parallel. 
 
THEORETICAL MECHANICS 39 
 
 If a system of non-concurrent forces in 
 space is in equilibrium, the plane systems 
 formed by projecting the given system upon 
 three coordinate planes must each be in 
 equilibrium. 
 
 A couple consists of two equal and oppo- 
 site parallel forces acting on a rigid body at 
 a fixed distance apart. 
 
 The moment of a couple is equal to the 
 product of one force by the distance between 
 the two forces. 
 
 Center of Pressure. 
 FI, F 2 , F 3 , etc., are parallel. 
 
 Fig. 19. 
 
 If F is the force exerted by a variable 
 pressure, then 
 
 xFdx 
 
 Fdx 
 
40 THEORETICAL MECHANICS 
 
 Center of Gravity. 
 For an area, 
 
 x dx dy 
 
 dxdy 
 
 Fig. 21 
 
 fx.dx 
 
THEORETICAL MECHANICS 41 
 If x 2 -xi = 1 
 
 y= 
 
 \ y O 2 -Zi) dy 
 \ (x2-xi) dy 
 
 J y fy . dy 
 
 Fig. 22. 
 
 For a homogeneous mass, 
 
 ///**** 
 
 x- * xdm -. 
 Zdm 
 
 fSf 
 
 dx dy dz 
 
 -V => 
 
 F 
 
 Fig. 23. 
 /-//*** 
 
 fff 
 fff 
 
 zdxdydz 
 
 dxdydz 
 
42 THEORETICAL MECHANICS 
 
 Rectangular Moment of Inertia. 
 For an area, 
 
 L= C fy^dxdy. 
 
 Fig. 24. 
 
 Fig. 25. 
 
 =^dA 
 
 = j 7/ 2 . (xi-x 
 
THEORETICAL MECHANICS 43 
 If Va-yi=fx, 
 
 Fig. 26. 
 
 = ( x* (ytyi) dx 
 
 = Cx 2 . f 
 
 . fx . dx. 
 
 Fig. 27. 
 
 Polar Moment of Inertia. 
 For an area, 
 
44 THEORETICAL MECHANICS 
 Since p 2 =x z +y 2 , 
 
 Fig. 28. 
 
 For a mass, 
 
 p*dxdydz 
 
 '//; 
 
 =kfff(x*+y*)dxdydz, 
 
 Fig. 29. 
 
 where k is the weight per cubic unit divided 
 by g. 
 
 Product of Inertia. 
 J = l 
 
 . y . dx . dy. 
 Ji=Jo. f .+Akh, 
 
THEORETICAL MECHANICS 45 
 
 where J\ is the value of J referred to X X 
 and Y F, Jo.%. is the value of J for axes 
 parallel to X - X and Y Y passing through 
 the center of gravity, and h, k are the co- 
 
 Fig. 30. 
 
 ordinates of the center of gravity referred to 
 X-X and Y-Y. 
 
 (See "A Complete Analysis of General 
 Flexure in a Straight Bar of Uniform Cross- 
 Section," by L. J. Johnson, Trans. Am. Soc. 
 C. E. t Vol. LVI, 1906.) 
 
 Radius of Gyration. 
 
 - -. 
 
 A m 
 
 Ellipsoid of Inertia. 
 
 The moments of inertia about all axes 
 through any given point of any rigid body 
 are inversely proportional to the squares of 
 the diameters which they intercept in an 
 imaginary ellipsoid, whose center is the 
 given point, and whose position depends 
 upon the distribution of the mass and the 
 location of the given point. This ellipsoid 
 is the ellipsoid of inertia for the body. The 
 axes which contain the principal diameters 
 of the ellipsoid are called the principal axes 
 of the body for the given point. 
 
46 THEORETICAL MECHANICS 
 
 DYNAMICS. 
 Velocity and Acceleration. 
 
 ds 
 V = Jt' 
 
 dr cPs 
 
 Uniformly Accelerated Motion. 
 If a is constant, 
 
 2 a 
 
 Falling Bodies. 
 For a body falling in a vacuum, a = g, hence 
 
 
 Force and Acceleration. 
 
 - = 
 
 ~~ ' C 
 
 Direct Central Impact. 
 For two inelastic bodies, let 
 mi = mass of first body. 
 m2 = m ass of second body. 
 Vi= original velocity of first body. 
 v 2 = original velocity of second body. 
 v = common velocity after impact. 
 
THEORETICAL MECHANICS 47 
 
 Then v = miVl + 2V2 . 
 
 mi +m,2 
 
 For two elastic bodies having velocities 
 i and k 2 after impact, 
 
 The product of mass by its velocity is 
 momentum. 
 
 The sum of the momenta before and after 
 impact is constant. 
 
 Virtual Velocities. 
 F = force. 
 
 v = direction of motion of P. 
 du = virtual velocity of force. 
 
 fa = velocity of force. 
 
 ds 
 
 -^ = velocity of P. 
 
 F . du = virtual moment of force. 
 
 The virtual moment of a force is equal to 
 the algebraic sum of the virtual moments of 
 its components. 
 
 For a system of concurrent forces in 
 equilibrium, 2 p . du = 0, 
 
 For any small displacement or motion of 
 a rigid body in equilibrium under non-con- 
 current forces in a plane, with all points of 
 the body moving parallel to this plane, 
 
 Y 
 
 Curvilinear Motion of a Point. 
 
 ds 
 
 v Vt== dt' 
 
 " / clx /rfs\ 2 
 
 ^_ *^ 
 
 ~ 1 V y.7 
 
 Y 
 
 Fig. 3!. 
 
 -(%f+m 
 
48 THEORETICAL MECHANICS 
 
 dv _ d?s 
 t dt~ dP 
 
 = a x cos a +Oy sin a. 
 n = Oy cos a a* sin a. 
 
 where r is the radius of curvature. 
 
 F=m . a, .'. 
 
 where r is the radius of curvature. 
 
 Ft = m . a x cos a +m . Oy sin a 
 
 1 V -/* 
 
 Projectiles. 
 Neglecting resistance of air, 
 
 X = VQ COS OQ . t. 
 
 y=vosm OQ. t-*%g&, 
 -X, 
 
 2 1' 2 COS 2 OQ ' 
 
 Horizontal range, 
 
 x r = sin 2 oo, 
 
 which is a maximum for a = 45. 
 The greatest height of ascent, 
 
THEORETICAL MECHANICS 49 
 
 Translation of Rigid Body. 
 dF x a x .dm. 
 R I * dm. 
 
 Fig. 33, Fig. 34. 
 
 The resultant force must act in a line 
 through the center of gravity and parallel to 
 the direction of motion. 
 
 Rotation of a Rigid Body. 
 
 Let O be the axis of rotation. 
 6 = angular space passed over by any line 
 from O. 
 
 a = angular accelera- "*x s 
 
 tion. / X V 
 
 a) = angular velocity. 
 Then 
 
 dO 
 
 = 
 dt dt* ' 
 
 Fig. 35, 
 
 For uniform acceleration, a. = k, .'. 
 
 2a 
 
 .t. 
 
50 THEORETICAL MECHANICS 
 For a point p distant from O, 
 
 p . ai. 
 p . a. 
 
 Fig. 36. 
 
 For a mass m concentrated p distant 
 from O, 
 
 Center of Percussion or Oscillation. 
 
 If an unsupported bar upon being struck 
 at a begins to rotate about 6, then a is the 
 center of percussion for 6 as a center, and b 
 is the center of instantaneous rotation. 
 
 Fh = 
 
 dF=a . p . dm. 
 
 F=a l P . dm 
 
 = a . p.m. 
 
 ^dHv Fi e- 37 - 
 
THEORETICAL MECHANICS 51 
 
 Pendulum. 
 
 * = time of oscillation 
 from one extreme posi- 
 tion to the other. 
 
 r = radius of gyration. 
 
 Then 
 
 Work, Energy, and Power. 
 
 Work is equal to the product of the force 
 by the distance through which it acts. 
 
 Power is the rate of doing work. 
 
 1 H.P. = 33,000 ft.-lb. per min. = 550 ft.-lb. 
 per sec. 
 
 Energy is the capacity or ability to do 
 work. 
 
 K.E. = Energy of a moving body. 
 
 For rotation, 
 
 Fig. 39. 
 Angle of friction, 
 
 K.E. = /.<o2. 
 
 Friction. 
 
 F= friction. 
 N = normal force. 
 /= coefficient of fric- 
 tion. 
 F=f.N. 
 
 pi 
 
 ^-tan- 1 :. 
 
52 THEORETICAL MECHANICS 
 
 Average values of / for motion are as 
 follows: 
 
 Wood on wood 25-. 50 
 
 Metal on wood 50-. 60 
 
 Leather on metal . 56 
 
 Leather on metal, lubricated ... 0.15 
 
 Metal on metal, dry 0.15-.24 
 
 Lubricated surfaces: 
 
 Ordinary 0.08 
 
 Best 0.03-0.36 
 
 For values of / for rest add 40 per cent to 
 above values. 
 
 Fig. 40. 
 dF=f.Nds 
 
 , where 0j is in radians. 
 
MECHANICS OF 
 MATERIALS. 
 
 w 
 8 
 
 I 
 
 i 
 
 iOOOC 
 
 i-KM COO 
 
 o . 
 
 d^ 
 
 03 GQ 
 
 '"A I X* 
 
 t< 10 O iO CO 00 
 
 II 
 
 H S3 
 
 IO^OI-KNT-IC* 
 
 (N CO CO 
 
 00 O O O TH rH iH u 
 
 O O 
 
 -lTfiiO 
 
 DOOO O 00 00 W OS iO W 
 
 SllOCDO iH '-' rHrH 
 
 " -3 PL, 3 "3 U 
 
 ftB^filPill 
 
 53 
 
54 MECHANICS OF MATERIALS 
 
 NOTATION. 
 
 A= area. 
 6 = breadth. 
 d= depth. 
 
 Z=modulus of elasticity. 
 e = total deformation. 
 F=force of load. 
 / = moment of inertia. 
 /o = polar moment of inertia. 
 J = product of inertia. 
 Z = length. 
 M= moment. 
 R = resultant of forces. 
 r = radius of gyration. 
 = unit stress. 
 s = section modulus. 
 V= vertical shear. 
 JF= total weight. 
 w = weight per lineal unit, 
 A = maximum deflection. 
 e=unit deformation. 
 
 Direct Stress. 
 
 I 
 
 s 
 
 Fl 
 . 
 
 eA 
 
MECHANICS OF MATERIALS 55 
 
 Fig. 43. 
 
 Consider a section a a perpendicular to 
 axis of a bar, and take axes of coordinates 
 through center of gravity. 
 
 Let x t y coordinates of any point of 
 section. 
 
 n n = neutral axis. 
 
 v = distance of any point from line through 
 center of gravity and parallel to neutral 
 axis, positive toward P. 
 
 yo = value of v for neutral axis. 
 
 F= force or resultant of forces acting at P. 
 
 N = component of F normal to section 
 considered. 
 
 o = unit stress at center of gravity. 
 
 * The method here presented is taken 
 from a paper by L. J. Johnson, M. Am. Soc. 
 C. E., "An Analysis of General Flexure in a 
 Straight Bar of Uniform Cross Section," 
 Trans., Am. Soc. C. E., volume LVI, p. 169, 
 1906. 
 
56 MECHANICS OF MATERIALS 
 
 S = S * . v 
 
 = So (y cos a x . sin a) 
 
 jy N. x p (y xtan a) 
 A- J Iy tan a 
 
 jy N .y p (y x tan a) 
 
 Ig J tan a 
 
 jY N . p p (yx . tan a) cos 
 -4 J Iy . tan a 
 
 N . p p (y x . tan a) sin 
 
 
 /-,/. tan a 
 
 AT N(y P Iy-x p J)y+N(xjJ ae -y P J)x 
 
 lysin 6 J . cos 0) y+(I x cos ./sin 
 
 *)af| 
 
 ./V 
 
 In the above equations -r- is the portion of 
 .A 
 
 S which is direct stress, and the other term 
 is the portion due to the bending moment, 
 M = N . pp. If s represent the section 
 modulus 
 
 V(/ysin0 J . cos0) y +(/ cos Q J . sin 0) x) 
 then 
 
 N . M 
 
 NOTE. The values of the section modu- 
 lus given in the handbooks are computed 
 
 from the formula s = - , which is the value of 
 
 y 
 
MECHANICS OF MATERIALS 57 
 
 s for .7 = and for P located on Y-Y. For 
 angles and Z-bars J does not equal zero. 
 In the above equations, 
 
 tan a = 
 
 /-/. tan 
 J Iy tan 
 
 I x cot 0-J 
 J cot 0-I y 
 
 I x cos 7 . sin 
 J cos lysin 
 
 For any bar having a section which is 
 symmetrical about either axis, .7=0, and 
 the values of S become 
 
 c 
 
 = 
 
 x \ 
 
 If for a symmetrical section, P is on Y Y, 
 then sin = 1 and cos = 0, or 
 
 , 
 
 o ~:~ H 
 
 . P P . y 
 
 Fig. 44. 
 
 For a rectangular section, for which N is 
 applied on Y Y and p distant from the 
 axis of the bar, the extreme fiber stresses are 
 
 = ^1 
 
58 MECHANICS OF MATERIALS 
 
 Equation of Neutral Axis. 
 
 The equation of the neutral axis for an 
 eccentric load is 
 
 &P *i*-yp.J\ iziy - J* 
 
 V ~ \Xp.J -y p . I y ) X A (xp.J-y p .Iy-) ' 
 
 Kernel or Core-Section. 
 
 The kernel of a section (sometimes called 
 the core-section) is the area within which P t 
 the point of application of the resultant of the 
 forces, must fall in order that the stress 
 shall be of the same sign throughout the 
 section. It is the area bounded by the 
 locus of the P's corresponding to a series of 
 neutral axes touching the periphery of the 
 section but never crossing the section. For 
 every side of the section there will be an 
 apex of the kernel. If x a , y a and Xb, j/j are 
 the coordinates of a and b, which are two 
 consecutive vertices of the section, then the 
 coordinates, Xab, 2/o6, of the vertex of the kernel 
 corresponding to the side, ab, of the section 
 will be 
 
 X a b 
 
 (x a -Xb) J (ya yb) J y 
 
 = -- - - 
 
 yob 
 
 A (x a yb-xby a ) 
 If ab is parallel to X X, then 
 
 J I 
 
 Xab= ~T~17' v<*=-- A 
 
 jA. |/a A. , ya 
 
 If ab is parallel to Y Y, then, 
 
 Iy J 
 
 A.X a ' V<>b A.Xa 
 
MECHANICS OF MATERIALS 59 
 
 The radii vectores of the kernel are lengths 
 which for any need only be multiplied by 
 the area of the section (A) to give the sec- 
 tion modulus 
 
 \(I y sin0-J. 
 
 but these lengths must be considered posi- 
 tive it measured on the opposite side of O 
 from P. 
 
 Section Modulus Polygons. 
 
 In the equation S -r -\ -- (see Eccentric 
 A s 
 
 Loads), s is the section modulus. The sec- 
 tion modulus polygon is the polygon the 
 lengths of whose radii vectores are the 
 graphical representations of the values of s 
 for extreme fibers for successive values of 
 from to 360 degrees. The section modu- 
 lus polygon is a figure whose sides are parallel 
 to the sides of the kernel of the given section 
 but which lie on opposite sides of the center 
 of gravity from the sides of the kernel. 
 The most general value of s is 
 
 _ __ 
 
 (lysin J cos 0)y + (I y cos Q J . sin 0) x 
 
 For any section which is symmetrical 
 about either axis, s becomes 
 
 I y sin . y +I X cos . x 
 
 For any symmetrical section for which P 
 lies on Y Y. = 90, hence 
 
60 MECHANICS OF MATERIALS 
 
 If for any symmetrical section P lies on 
 X-X,e = 0, hence 
 
 There will be one vertex of the s-polygon 
 for each side of the polygon bounding the 
 section. If x a , y a and #5, y b , are the coordi- 
 nates of a and 6, two consecutive vertices of 
 the bounding polygon of the section, then 
 the coordinates of the vertex of the s-polygon 
 corresponding to the side ab of the bounding 
 polygon will be 
 
 If ab is parallel to X-X, 
 
 J I* 
 
 X a b= , yab = ' 
 
 ya y<* 
 
 If ab is parallel to Y-Y, 
 
 Jy J 
 
 Xab=, yab= > 
 
 X a X a 
 
 For sections symmetrical about either 
 X-X, or Y-Y, J = 0, and the values of 
 
 y a 
 
 and can be found in the handbooks 
 
 z a 
 
 issued by the steel companies, under the 
 column marked "Section Modulus." The 
 vertices can then be plotted and connected 
 by straight lines to form the s-polygon. 
 From this s-polygon the values of s for any 
 value of 9 can be obtained graphically. 
 
 The most advantageous plane of loading for 
 any section will be that having the greatest 
 value of s. 
 
MECHANICS OF MATERIALS 61 
 
 DIAGONAL STRESSES 
 
 Fig. 45. 
 
 F = axial load. 
 
 A =area of section normal to axis of bar 
 n n = any diagonal section. 
 
 = angle which n n makes with axis. 
 = unit axial stress. 
 $3 = unit shear along plane normal to axis. 
 $ n =unit tension or compression normal 
 
 to section n n. 
 $n == unit shear along section n n. 
 
 For combined direct stress and vertical shear, 
 S n = ~ (1 -cos 2 0) +S. . sin 2 9. 
 
 Sn = IT sin 2 +S, . cos 2 0. 
 
 The maximum or minimum value of S n 
 
 S 
 occurs when cot 2 = , and is 
 
 max. /S= ; 
 
 The maximum value of S 8n occurs when 
 S 
 
 tan 20= 
 
 -, and is 
 
62 MECHANICS OF MATERIALS 
 
 For axial load only, >S 8 =0, hence 
 
 n = . sin 2 = 
 
 sin 2 * 
 
 The maximum value of S n occurs when 
 = 90, and is then the unit axial stress. 
 The maximum value of S en occurs when 
 
 Q W 
 
 = 45, and is - 01-75 i 
 *j Z A 
 
 THIN PIPES, CYLINDERS, AND SPHERES. 
 
 S = unit stress in metal. 
 t = thickness of metal. 
 d = diameter. 
 ;p = unit pressure of 
 
 liquid or gas. 
 = angle which the 
 direction of P 
 makes with 
 X-X. 
 
 Fig. 46. 
 
 For the transverse stress across a longi- 
 tudinal section of a pipe or cylinder, 
 
 . cos = % p. d. 
 
 P*d 
 -' 
 
 For the longitudinal stress across a trans- 
 verse section of a pipe, or for the stress in a 
 thin hollow sphere, 
 
 o 
 
 nd.t 
 
 4t 
 
 which is one-half of the unit transverse stress 
 in a pipe having the same diameter and 
 thickness. 
 
MECHANICS OF MATERIALS 63 
 RIVETED JOINTS. 
 
 Fig. 47. 
 
 a = distance center to center of two con- 
 secutive rivets in one row. 
 d= diameter of rivet or rivet hole. 
 F stress in unriveted plate in length a. 
 
 t thickness of plate. 
 S$=unit tensile stress. 
 $ c =unit compressive or bearing stress. 
 = unit shearing stress. 
 et= efficiency of joint for tension. 
 e a = efficiency of joint for compression. 
 e t = efficiency of joint for shear. 
 m = number of shearing sections of rivets 
 in distance a. (Notice that for butt 
 joints each rivet has two shearing 
 areas.) 
 
 n number of bearing areas of rivets in 
 distance a. 
 
 F = t (a d) Sf = m . xd? . St~n t d Se. 
 
 m. x. d?S g 
 
 4 . atS t 
 n . dS e 
 
64 MECHANICS OF MATERIALS 
 
 For maximum, efficiency, make e t = e t 
 for which 
 
 4 . n . S e . t 
 m.n . SB 
 
 and a ~[l +n -3-1 1 . 
 
 For single riveted lap joints the maximum 
 efficiency is approximately 55 per cent, for 
 double riveted lap joints approximately 70 
 per cent, for triple riveted lap joints approx- 
 imately 75 per cent, and for triple and 
 double riveted butt joints approximately 80 
 per cent. 
 
 BEAMS. 
 
 Vertical Shear. The vertical shear at any 
 given section of a horizontal beam is the 
 sum of the vertical components of all of the 
 stresses at that section. The vertical shear 
 is equal to the sum of all the reactions of 
 the supports upon the left of the given sec- 
 tion minus the sum of all of the vertical loads 
 on the left of the section. 
 
 For any beam the vertical shear upon the 
 right side of the left support of any span is 
 
 where 
 
 MI = the moment at the left support, 
 M2 = the moment at the right support, 
 10 = the uniform load per lineal unit, 
 jP=any concentrated load, 
 a = the distance from the left support to F, 
 Z = the length of span. 
 
MECHANICS OF MATERIALS 65 
 
 Shearing Stresses. If F = vertical shear at 
 any section, 
 
 - 
 
 where S 9 is the average unit shear. 
 
 The actual unit vertical shear at any 
 point is equal to the unit horizontal shear at 
 that point, and may be determined by the 
 following equation: 
 
 where 6 is the breadth of the section at the 
 given point, y is the distance of the point 
 considered from the neutral axis, and c is 
 the distance from the neutral axis to the 
 extreme fiber on the same side as the point 
 considered. 
 
 The maximum value of S, occurs at the 
 neutral axis, and is 
 
 V C V 
 
 max.S.= j- -^ I y. 
 * ** o 
 
 l.b 
 
 where AI is the area of the portion of the 
 section on one side of the neutral axis, and 
 yi is the distance from the neutral axis to 
 the center of gravity of the portion of the 
 section on one side of the neutral axis. 
 
 For a rectangular section, the maximum 
 unit shear is | of the mean unit shear. 
 
 For Diagonal Shear, see Diagonal Stresses, 
 page 61. 
 
 Bending Moment. The bending moment 
 at any point for any beam is 
 
 M = Mj + V l x-$wx*-'2F (x-a), 
 
 * See "Merriman's Mechanics of Materi- 
 als," page 269. 
 
66 MECHANICS OF MATERIALS 
 
 where 
 
 M = bending moment at section considered, 
 MI = bending moment at the left support, 
 FI = vertical shear upon the right side of 
 
 the left support, 
 w = uniform load including weight of 
 
 beam, per lineal unit, 
 F=any concentrated load upon the left 
 
 of the section considered, 
 x = distance from the left support to the 
 
 section considered, 
 o = distance from left support to F. 
 
 For any beam of one span Vi is equal to 
 the reaction at the left support. 
 
 The maximum values of M occur at those 
 
 sections for which 7 =0, that is, where the 
 
 dx 
 
 shear passes through zero. 
 
 The values of M for special cases are given 
 in Table of Beams, page 68. 
 
 Theorem of Three Moments. For any two 
 consecutive spans of a continuous beam, let 
 
 MI = moment at the left support, 
 M 2 = moment at the middle support, 
 M 3 = moment at the right support, 
 Zi=length of the first span, 
 h = length of the second span, 
 1 = length of span for equal spans, 
 wi = uniform load per lineal unit on first 
 
 span, 
 w 2 = uniform load per lineal unit on second 
 
 span, 
 FI = any concentrated load on the first 
 
 span, 
 F 2 = any concentrated load on the second 
 
 span, 
 
 i= distance from first support to FI, 
 d2 = distance from middle support to F%. 
 
MECHANICS OF MATERIALS 67 
 
 Then, for uniform loads only, 
 
 ih +2 M 2 (h + 1 2 ) +M 3 1 2 = -$w 1 l l 3 -iw 2 l2*. 
 
 For equal spans with equal uniform loads, 
 
 For concentrated loads only, 
 
 Mill +2M 2 (h +fe) +M 3 1 2 
 
 Flexural Stresses. The tensile and com- 
 pressive stresses in a beam, produced by 
 bending, are the same as the stresses upon a 
 section having an eccentric load, due to the 
 moment of that load. Therefore, for pure 
 flexure the tensile and compressive stresses 
 for the extreme fibers of any section can be 
 
 determined by placing =0 in the formula 
 
 for S given under Eccentric Loads, which 
 gives 
 
 where s is the section modulus, the values for 
 which are given under Section Modulus 
 Polygons. 
 
 For combined flexure and direct stress, the 
 tensile and compressive stresses are given by 
 the formulae for Eccentric Loads. 
 
 Elastic Curves. The curve \vhich is as- 
 sumed by the neutral surface of a beam 
 under load is called the elastic curve. 
 
 The radius of curvature of the elastic curve 
 is 
 
 ff/ = dP ^dtf 
 ' M dx.d*y d*y' 
 
68 MECHANICS OF MATERIALS 
 
 from which the equation of the elastic curve 
 can be obtained, for any particular case, by 
 
 placing M equal to El , and by making 
 
 two integrations to obtain an equation in 
 terms of x and y. 
 
 The deflection of a beam at any given 
 point is obtained by substituting the par- 
 ticular value of x in the equation of the 
 elastic curve and solving for y. The maxi- 
 mum deflection occurs at the section for 
 which #y 
 
 dx~ 
 
 (For particular cases, see Table of Beams.) 
 
 TABLE OF BEAMS. 
 
 NOTE . The equations for elastic curves 
 and the values of A apply only to beams of 
 uniform section. 
 
 Beams Supported at Both Ends and Uniformly 
 Loaded. 
 
 Moment 
 
 Fig. 48. 
 
MECHANICS OF MATERIALS 69 
 
 =Rix- wx* 
 
 wlx - wx* 
 
 A when x= - , or 
 
 A== oo3 "Fr ^QQA" 
 384 til o84 
 
 Beam Supported at Both Ends and Loaded 
 with a Concentrated Load at Center of Span. 
 
 f 
 
 Ri 
 
 Moment 
 
 Fig. 49. 
 
70 MECHANICS OF MATERIALS 
 
 V=R lt or V = R 2 . 
 M = Rtx, on the left of F, 
 
 = Rix-Fx-- t on the right of F. 
 
 EI l = Fx > on the Ieft of F - 
 
 48 EIy = F (4 *'-3 Pa;), on the left of F. 
 
 A= __ 
 
 ' 48 J^/ ' 
 
 (For both uniform and concentrated loads, 
 combine the results for each.) 
 
 Beam Supported at Both Ends and Loaded with 
 a Concentrated Load Distant a from the Left 
 Support. 
 
MECHANICS OF MATERIALS 71 
 
 V=Ri, on the left of F, 
 = #2, on the right of F. 
 
 M=Rix, on the left of F, 
 
 *=Rix-F (x-a), on the right of F. 
 
 EI~~ = R l x, on the left of F, 
 
 = Ri.x-F (x-a), on the right of F. 
 Ely = - Rix* + cix 4- c 2 , on the left of F, 
 
 Fax* 
 on the right of F. 
 
 The maximum deflection (A) occurs at the 
 section for which 
 
 and is 
 
 Beam Supported at Both Ends and Loaded 
 with Several Concentrated Loads. 
 
72 MECHANICS OF MATERIALS 
 
 The maximum moment (Mm) occurs at 
 the section for which R\ 2 ^=0, that is, 
 where the vertical shear is zero. 
 
 For a system of movable loads the maxi- 
 mum moment will occur under one of the 
 loads, the loads being in such a position 
 
 , F 1 ! F * ! F 3 
 
 
 
 Fig. 51. 
 
 that the center of the span is midway be- 
 tween the center of gravity of all the loads 
 and the section at which the maximum 
 moment occurs. 
 
 The maximum deflection of a beam loaded 
 with several loads is the sum of the deflec- 
 tions produced by each load at the section 
 at which the maximum deflection for the 
 entire system of loads occurs. The deflec- 
 tions produced by each load can be obtained 
 by means of the equation of the elastic curve 
 for a single load. 
 
MECHANICS OF MATERIALS 73 
 
 Cantilever Beam with Uniform Load. 
 R l = w l=W 
 #2=0. 
 
 Moment 
 
 Fig. 52. 
 
 Of if is taken from the free end, 
 
 1 
 
 ~2 WX ' 
 
 El = i wP-' wlx + i wx*. 
 
 4 - 4 
 
 I wl* _l Wl 3 
 8 El "8 El ' 
 
74 MECHANICS OF MATERIALS 
 
 Cantilever Beam with Concentrated Load at 
 
 the Free End. 
 Ri = F. 
 
 #2=0. 
 
 3 El' 
 
 Beam Fixed at Both Ends and Uniformly 
 Loaded. 
 
 M = 
 
 - wlx 2 
 
MECHANICS OF MATERIALS 75 
 
 
 
 By placing ~ when x =0 and when x*=l t 
 
 w ( - 
 
 JLE 
 
 384 ;/ 
 
 -re 4 ). 
 
 Beam Fixed at Both Ends and Loaded at 
 the Center of the Span with a Concen- 
 trated Load. 
 
 V = Ri, on the left of F, 
 =* R 2 , on the right of F. 
 
76 MECHANICS OF MATERIALS 
 
 M = - FI + FX, on the left of F, 
 
 a & 
 
 on the right of F. 
 
 ' Moment 
 
 Mo 
 
 
 Fig. 55. 
 + |F*, on the left of F. 
 
 on the right of F. 
 
 By placing - = when a: = and when x = . 
 
MECHANICS OF MATERIALS 77 
 
 48 Ely 
 
 , on the left of F. 
 
 1 FP 
 
 '' 192 El ' 
 
 Beam Fixed at Both Ends and Loaded with 
 a Concentrated Load Distant a from the 
 Left Support. 
 
 Fig. 56. 
 
 V RI, on the left of F, 
 = #2, on the right of F. 
 
 M = Mi +Rix, on the left of F, 
 
 = MX + Rix - F (x - a), on the right of F. 
 
78 MECHANICS OF MATERIALS 
 
 7 = Afi + Rix, on the left of F. 
 
 6 Ely = 3 M !X 2 +# i: r 3 , on the left of F. 
 
 The maximum deflection (A) occurs at 
 2al 
 
 the section for which x - 
 
 l+2a 
 
 El (I +2 a) 2 3 El (I +2 a) 3 
 
 Continuous Beam with Uniform Loads. 
 u?j=load per lineal unit on Zj. 
 w 2 = load per lineal unit on 1 2 , etc. 
 TFi = total load on Zi. 
 TF 2 = total load on I 2 . t etc. 
 
 R n =V n . 
 
 For a continuous beam supported at the 
 ends, 
 
 O+2M 2 (h 
 

 MECHANICS OF MATERIALS 79 
 M 2 1 2 +2 M 3 (l a 
 
 = - w 2 l 2 z j wda-, etc. 
 
 = - w n - 2 l n - 2* r w-i/n-i 2 . 
 
 From the above simultaneous equations 
 *, . . M n -\ can be determined. 
 
 1*2 t R 3 t R 3 
 
 *i zi -F ti r ^H 
 
 Moment X^X 
 
 V 
 
 Fig. 57. 
 
 For equal spans with equal uniform load 
 over the entire beam, the ends of the beam 
 resting upon supports, the moment at any 
 support is KwP or KWl, and the vertical 
 shear is Nwl or NW, where K and N have 
 the values given in the following table: 
 
80 MECHANICS OF MATERIALS 
 
 2 
 
 o o o o o 
 
 -S HS 
 
 a 
 
 <O 
 
 J4 
 
 a 
 
 
 
MECHANICS OF MATERIALS 81 
 
 For a continuous beam with fixed ends con- 
 sider an imaginary span to be added at each 
 end of the beam, with the free ends resting 
 upon supports. Then write the equation of 
 three moments for each two consecutive 
 spans, making 1 = for the first and last 
 spans, and compute the moments at the 
 supports as shown above. 
 
 Continuous Beams with Concentrated Loads. 
 
 Determine the moments at the supports in 
 a similar manner to that employed for con- 
 tinuous beams with uniform load, employing 
 the equation of three moments for concen- 
 trated loads. 
 
 STRUTS AND COLUMNS. 
 
 Euler's Formula, 
 
 Fig. 58. 
 
 .-Vf .sin-CO-cr 
 y=a . sin (*V -) . 
 
82 MECHANICS OF MATERIALS 
 
 7 J / Tf 
 
 Since y = a when x - , - v TTT must equal 
 2 2 til 
 
 = 7t z E \j\ , for round ends. 
 
 For one end round and the other end fixed, 
 
 4. 
 replace I by - 1 and TT by 2 n, which gives 
 
 o 
 
 o 
 
 For 6oiA ends /ixed, replace Z by - I and ^ 
 
 by 3 it, in the formula for round ends, which 
 gives 
 
 Rankine's Formula. (Sometimes called 
 Gordon's Formula.) 
 
 Fig. 59. 
 
 From the formula for eccentric loads for 
 a symmetrical section (page 57), the maxi- 
 mum stress will be 
 
MECHANICS OF MATERIALS 83 
 
 where y is the distance from the neutral 
 axis to the extreme fiber. 
 
 But, I^Ar 2 , M = Fa and a = K- , where 
 
 y 
 
 K is some constant depending upon charac- 
 ter and condition of the column. Hence 
 
 Cambria handbook gives the following 
 values of K for steel columns: 
 1 
 
 36,000 
 
 for both ends fixed, 
 
 24^00 f r one end fixed, 
 
 , for pin connected ends. 
 
 18,000 
 
 The above values are to be used with 
 following values of S for ultimate strength: 
 5 = 50,000 for medium steel. 
 S = 45,000 for soft steel. 
 
 Hitter's Formula. Hitter's formula is the 
 
 same as Rankine's formula except that the 
 o 
 
 expression is used for K , in which S e is 
 nE 
 
 the elastic limit of the material, and n is 
 
 g 
 
 equal to n 2 for round ends, n? for one end 
 
 round and one end fixed, and 4 rc 2 for both 
 ends fixed. 
 
 The Straight Line Formula. The straight 
 line formula is 
 
 where C is a constant depending upon the 
 character and condition of the column. 
 
84 MECHANICS OF MATERIALS 
 
 Merriman gives the value of C in the 
 above equation to be 
 
 C = Sf ( ^ J \ t 
 
 which is obtained by making the straight line 
 a tangent to the curve for Euler's formula 
 
 passing through the point S for - = 0. 
 
 The following values of S and C for allow- 
 able stresses are given in Cooper's Specifica- 
 tions for Railroad Bridges, 1906. 
 
 5 = 10,000, C = 45, 
 
 for live load on chords, 
 5 = 20,000, C = 90, 
 
 for dead load on chords, 
 5= 8,500, C = 45, 
 
 for live load on posts of through 
 
 bridges, 
 5 = 17,000, (7=90, 
 
 for dead load on posts of through 
 
 bridges, 
 5= 9,000, C7=40, 
 
 for live load on posts of deck 
 
 bridges, 
 5 = 18,000, (7=80, 
 
 for dead load on posts of deck 
 
 bridges, 
 5=13,000, (7 = 60, 
 
 for wind load on lateral struts. 
 
 Engineering News Formula. The Engin- 
 eering News, Vol. 57, No. 1, Jan. 3, 1907, 
 gives the following formula: 
 
 which is the same as Rankine's formula given 
 on page 87, allowing the eccentricity a to 
 
MECHANICS OF MATERIALS 85 
 
 remain in the formula instead of substituting 
 K-. The value of a to be used may be 
 
 considered to represent the eccentricity due 
 to imperfection in manufacture (since it is 
 impossible to obtain the ideal straight 
 column), plus the additional eccentricity 
 due to the failure to obtain an axial load. 
 The proper value of a to obtain correctly 
 proportioned columns might be determined 
 empirically by experiment, or it may be 
 determined by comparison with column 
 formulae in use which have been found to 
 give correct results. 
 
 For any formula of the Rankine type, 
 
 y 
 
 In the article above mentioned the values 
 of a for a number of formulae in use are 
 thus computed, the mean values being as 
 follows: 
 
 a = 0.000051-, for steel, 
 
 y 
 
 a = 0.000177- , for cast iron, 
 
 72 
 
 a = 0.000164- , for timber. 
 For any formula of the straight line type 
 CAlr 
 
 ~ 
 
 In the article above mentioned the values 
 of a for a number of formulae of the straight 
 
 line type have been computed, using- =0.8, 
 the mean values being as follows: 
 
 a =0.0053 I, for steel, 
 
 a = 0.0015 I, for cast iron. 
 
 a =0.0044 I, for timber. 
 
86 MECHANICS OF MATERIALS 
 
 Eccentrically Loaded Columns. To the 
 quantity a in the Engineering News formula 
 add the eccentricity of the load at the end 
 of the column, that is 
 
 '-ll 1 
 
 (e+an/l 
 
 where e = eccentricity of load at the end of 
 the column. 
 
 To determine the maximum stress of an 
 eccentrically loaded column by Rankine's 
 formula replace a in the above formula by 
 
 p 
 its equivalent K - , which gives 
 
 TORSION. 
 
 Circular Sections. 
 Twisting moment, M = Fa. 
 
 Circular Sections 
 
 Fig. 60. Fig. 61. 
 
 Resisting moment, M r I SdA, where 
 S is the shearing stress at the extreme fiber. 
 M =M r , or 
 
 M ^ 
 M ~ R ' 
 
 where 7 is the polar moment of inertia. 
 
MECHANICS OF MATERIALS 87 
 
 For a solid round shaft 7? = - nR 3 , hence 
 
 K 2, 
 
 Non-Circular Sections. (Taken from Mer- 
 riman's "Mechanics of Materials.") For 
 non-circular sections the above formulae are 
 only approximate. 
 
 For an elliptical section whose major axis 
 is m and whose minor axis is n the maximum 
 stress is 
 
 WFa 
 
 For a rectangular section whose long side 
 is m and whose short side is n, the maximum 
 stress is 
 
 , 9 Fa 
 
 Transmission of Power. The horse-power 
 which is transmitted by a shaft is 
 
 " 
 
 i 
 550X12 
 
 where a = moment arm in inches, 
 
 <>== number of revolutions per sec. 
 
 But, Fa =^, hence 
 
INDEX 
 
 PAGE 
 
 Acceleration 46 
 
 Analytic Geometry 11 
 
 Arithmetical Progression 2 
 
 Beams 64 
 
 Continuous Beams 78 
 
 Coefficients for Continuous Beams . . 80 
 
 Table of Beams 68 
 
 Theorem of Three Moments 66 
 
 Bending Moment 65 
 
 Belt, Friction of 52 
 
 Binomial Theorem 3 
 
 Calculus 20 
 
 Catenary, The 18 
 
 Center of Gravity . , 40 
 
 Center of Pressure 39 
 
 Circle, The 14 
 
 Columns 81 
 
 Combinations and Permutations .... 3 
 
 Cones, Equation of 18 
 
 Conic Sections, General Equation of . . 19 
 
 Core Sections 58 
 
 Couple, Definition of 39 
 
 Curves, Elastic , . . . 67 
 
 Cycloid, The 17 
 
 Cylinders, Stresses in 62 
 
 Deflection of a Beam 68 
 
 Determinants , . . . . 4 
 
 Differential Calculus 20 
 
 Differentiation 21 
 
 Dynamics 46 
 
 Elastic Curves 67 
 
 Ellipse, The 15 
 
 Ellipsoid of Inertia 45 
 
 Energy . , 51 
 
90 INDEX 
 
 PAGI 
 
 Equilibrium , o 
 
 Exponents ] 
 
 Falling Bodies tj 
 
 Force ^ 
 
 Friction r,] 
 
 Friction of Belt 55 
 
 Geometric Progression 5 
 
 Harmonic Progression 2 
 
 Hyperbola, The ie 
 
 Hyperboloids ig 
 
 Impact 4(3 
 
 Integral Calculus 23 
 
 Kernel of a Section 58 
 
 Logarithms 1 
 
 Materials, Strength of 53 
 
 Maximum Value of a Function 22 
 
 Mechanics, Theoretical 3i} 
 
 Minimum Value of a Function .... 22 
 
 Moment, Bending 65 
 
 Moment of Inertia - 42 
 
 Moment, Maximum for Concentrated 
 
 Loads 70 
 
 Motion of a Point, Curvilinear 47 
 
 Neutral Axis, Equation of 58 
 
 Parabola, The 15 
 
 Parabola, The Cubic '. . 18 
 
 Paraboloids 19 
 
 Pendulum 51 
 
 Permutations and Combinations .... 3 
 
 Pipes, Stresses in 62 
 
 Plane Triangles 8 
 
 Power 51 
 
 Power, Transmission of 87 
 
 Product of Inertia 44 
 
 Progression 2 
 
 Projectiles 48 
 
 Proportion 2 
 
 Quadratic Equations 2 
 
 Radius of Curvature 22 
 
 Radius of Curvature of Beams 67 
 
INDEX 91 
 
 PAGE 
 
 Radius of Gyration 45 
 
 Riveted Joints 63 
 
 Rotation 49 
 
 Section Modulus 59 
 
 Series 3 
 
 Shear, Diagonal 65 
 
 Shear, Vertical 64 
 
 Shearing Stresses 65 
 
 Sphere, The 18 
 
 Spherical Triangles 9 
 
 Spheroids 18 
 
 Spirals 17 
 
 Statics 38 
 
 Straight Line, The 13 
 
 Stresses 
 
 Combined Stresses 67 
 
 Diagonal Stresses 61 
 
 Direct Stresses 54 
 
 Eccentric Stresses 55 
 
 Flexural Stresses 67 
 
 Stresses in Pipes, Cylinders, and 
 
 Spheres 62 
 
 Struts 81 
 
 Taylor's Theorem 22 
 
 Theorem of Three Moments 66 
 
 Torsion 86 
 
 Transformation of Coordinates .... 11 
 
 Transmission of Power 87 
 
 Triangles, Solution of 8 
 
 Trigonometry 5 
 
 Velocity and Acceleration 46 
 
 Velocities, Virtual 47 
 
 Vork and Energy 51 
 
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