UC-NRLF SB Eh LIVE-LOAD STRESSES IN RAILWAY BRIDGES WITH FORMULAS AND TABLES BY GEORGE E. BEGGS, A.B., C.E. Assistant Professor of Civil Engineering in Princeton University; Associate Member of the American Society of Civil Engineers; Member of the Society for the Promotion of Engineering Education FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, INC. LONDON : CHAPMAN & HALL, LIMITED 1916 IV PREFACE The author wishes to acknowledge his indebtedness to the American Bridge Company for material assistance, and in particular to Mr. 0. E. Hovey, Assistant Chief Engineer of this company, for his encouragement and help. The author also desires to acknowledge the valuable suggestions made in the revision of the original text by Professor F. H. Constant, of the Civil Engineering Department of Princeton. To Professor William H. Burr of Columbia University, the writer is permanently indebted for the logical and thorough instruction received from him as a student. G. E. B. PRINCETON UNIVERSITY December, 1915. CONTENTS ARTICLE I. ARTICLE II. ARTICLE III. ARTICLE IV. ARTICLE V. ARTICLE VI. ARTICLE VII. ARTICLE VIII. ARTICLE IX. ARTICLE X. Influence Lines. Definition and Use PAGE 1 Sum and Rate of Variation of Ordinate-load Products between Two Diverging Lines Sum and Rate of Variation of Ordinate-load Products for any Influence Line. Position of Loading for Maximum Live-load Stress Girder Bridge without Panels. General Formulas for Reactions, Shears, and Bending Moment with its Rate of Variation General Formulas for Pier Reaction and its Rate of Variation. Illustrative Problem Girder Bridge with Panels. General Formulas for Live-load Stresses and their Rate of Variation. Illustrative Problem . . Through Pratt Truss. General Formulas for Live- load Stresses and their Rate of Variation. Illus- trative Problems Three-hinged Arch. Application of the General Method to the Calculation of Live-load Stresses 13 23 27 31 48 Equivalent Uniform Loads 54 Method of Calculating Table of Load Sums and Moment Sums for any Standard Loading. Illus- trative Example 57 ARTICLE XI. Summary of Formulas 59 Tables 1 to 21 67 LIVE-LOAD STRESSES ARTICLE I. INFLUENCE LINES. DEFINITION AND USES. INFLUENCE lines are useful in determining the position of live load on a bridge to produce maximum effect. They offer also a convenient method of deriving general alge- braic formulas for stresses and rules for maximum when the general relations between influence lines and algebraic formulas are once understood; and in the case of the more complex problems of skew bridges, arches, cantilever bridges, etc., the influence lines themselves serve as a most direct method for the determination of the maximum live-load stresses. An influence line may be defined as a line showing the variation in any function caused by a single unit load as it moves across the bridge. Vertical loads only will be considered. The function may be a reaction, bending moment, shear, stress, deflection, or any quantity what- soever at a given part of a bridge, provided that its value is a function of the position of the unit load on the bridge. Refer to Fig. la. Consider the span AB, and let Z be any function at the fixed position C on the span L. If the load unity moves across the span AB and the value of Z be calculated for each position of the unit load and its value z plotted below the corresponding position of this load as an ordinate from a horizontal base line, the locus of the plotted points will be the influence line for Z. For example, if Z be the bending moment at the fixed section C in a beam of span L, the influence line will be as shown in Fig. Ib. In plotting influence lines, ordinates repre- l 2 LP'E-LOAD STRESSES sen ting positive quantities are plotted above the base line; and negative, below. In case the influence line consists of several straight segments, it is necessary to determine the value of the ordinates only where the influence line has a change of direction; i.e., at the salient points. For example, FIG 1. the points A, C, and B are the salient points of the influence line in Fig. Ib. The value of Z caused by a single load w is equal to wz, if z is the influence ordinate below w. The value of Z caused by a series of loads w^ w Zj w 3 , etc., is . . (1) LIVE-LOAD STRESSES 3 where Zi, z 2 , z 3 , etc., are the influence ordinates below the corresponding loads. It will be convenient to speak of such a quantity as wz as an ordinate-load product. Formula (1) therefore may be expressed thus: Z = Sum of ordinate-load products. The area between the influence line and the base line is called the influence area. It may be shown that the value of Z caused by a uniform load on the bridge is pro- portional to the area A z of the influence line between the ordinates at the extremities of the uniform load. If the uniform load in Fig. la has an intensity of q per unit of length, the load in the length dx equals q dx, and the influ- ence of this elementary load on the value of Z is zq dx, where z is the influence ordinate below q dx. Summing up for the length of the uniform load, z = If a series of equal loads w is on the span, the value of Zis Z = zwz = w2z ........ (3) If a series of unequal loads, w\, w 2 , etc., is multiplied by the corresponding ordinates of an influence line or a por- tion of an influence line which has a constant ordinate z, as in Fig. Ic, the value of Z is Z = z (wi + w 2 +...)= zSw = zW (4) where W equals the sum of these loads. If a series of unequal loads is multiplied by the corre- sponding ordinates of an influence line or a portion of an influence line consisting of two diverging lines, as shown in Fig. Id, the value of Z, or the sum of the ordinate load products, and the rate at which Z varies as the loading advances, are given by the two theorems that follow. The slope of a line is defined at the beginning of Art. 2. 4 LIVE-LOAD STRESSES Theorem I. The sum of the ordinate-load products between two di- verging lines equals the difference between the slopes of the two lines multiplied by the sum of the moments of the loads about the intersection of these lines. In symbols, this is stated as Z = C a M a V ....... (5) * Theorem II. The rate at which the sum of the -ordinate-load prod- ucts between the two diverging lines increases as the load- ing moves away from the intersection of these lines equals the difference between the slopes of the two lines multiplied by the sum of the loads. In symbols, this is stated as dZ = CJU_ C M, (ga) dx dx dx The proofs of these theorems follow in the next article. -. ARTICLE II. SUM AND RATE OF VARIATION OF ORDINATE-LOAD PRODUCTS BETWEEN THE TWO DIVERGING LINES. CONSIDER the diverging lines DAB and AC in Fig. 2. Use the following notation: w = any vertical load. z = ordinate below w in the angle BAC. Z = 2w n z n = sum of ordinate-load products. Salient Point FIG. 2. M a = 2w n x n = moment sum of all loads to left of A a about A. W a = 2w n = load sum of all loads to left of A a. S R = slope of line DA = tangent of angle which DA makes with the horizontal. 6 LIVE-LOAD STRESSES S L = slope of line AC = tangent of angle which AC makes with the horizontal. C a = (S L S R ) = length of ordinate unit distance x n from A. Slopes are counted numerically positive when upward to the left. The sign of C a (called the coefficient at salient point A) is, accordingly, negative when AC diverges below DA produced to the left of A. The value of C a may be determined graphically as -- or it may be figured algebra- x n ically as (S L S R ). Proof of Theorem I, or that Z = C a M a . Consider the load w n distant x n from the salient point a. By the similar triangles AEF and AGH, Ca 1.00 Xn 9 Therefore, w n z n = C a w n x n (A) Summing up all of the ordinate-load products, Z = 2w n z n = C a 2w n x n = C a M a (5) Proof of Theorem II, or that -^ = C a W a . From equation (A) above, the increase in the ordinate- load product w n z n for an advance dx n of the load is w n dz n = C a -w n -dx n . Summing up the increases of all the ordinate-load products and noting that dx is the same for all loads, dZ = 2w n dz n = C a dx-2w n = C a -W a -dx. Dividing by dx, -r- = C a W a = | ARTICLE III. SUM AND RATE OF VARIATION OF ORDINATE-LOAD PRODUCTS FOR ANY INFLUENCE LINE. POSITION OF LOADING FOR MAXIMUM LIVE-LOAD STRESS. AN influence line of a general type is shown in Fig. 3, this one in t particular being for the member U^L* of the Salient Internals C)rflinates Slopes Coefficients Influence Line for U 3 L 4 of 189' Three Hinged Arch Moment Sums t-f T- FIG. 3. arch shown in Fig. 15. It is assumed that the ordinates at all salient points and the intervals between these points are known. Ordinates and slopes are counted positive or nega- tive as already defined. The slope of any segment of the 7 8 LIVE-LOAD STRESSES influence line equals the ordinate at the left minus the ordinate at the right end of this segment divided by the corresponding interval. The coefficient C at any salient point equals the slope of the segment at the left minus the slope of the segment at the right of this point. The sub- tractions in each case are made algebraically. It should be remembered, as has already been pointed out in Art. 2, that the value of any coefficient C may also be measured graphically from an influence line which has been drawn to scale. For example, in Fig. 3 the value of the coefficient C 2 = and C 4 = - Xi X 4 The algebraic calculation of the coefficients at all sa- lient points of the influence line in Fig. 3 is given below. If it be assumed that this influence line has been drawn to scale, the signs of the numerical values of the slopes and coefficients will be as given in the parentheses. i = -* 2 (+) Ci = - *, (-) S 2 = "~7~ * ( ) C 2 = Si S 2 ( + ) 4 = S 3 - S 4 Cs = s 4 - (+) A numerical evaluation of the slopes and coefficients for this influence line is given in Fig. 15 of Art. 8, which the reader should check in order to understand completely the method of procedure. These coefficients should also be checked by the graphical method as already explained. 2 59 For example, in Fig. 15 the value of C 2 = -~- = .0863. It will be noted in the algebraic calculation of the coeffi- cients C at all salient points that each slope enters once LIVE-LOAD STRESSES 9 as positive and once as negative. Therefore the sum of all coefficients equals zero. SC = ..... ..... (6) This formula serves as a check on the values of the coef- ficients which have been determined either by calculation or by graphical measurement. The general formulas for the sum of the ordinate-load products for any influence line (viz., with several salient points such as the one shown in Fig. 3) may be arrived at by considering the two contiguous sloping sides of the influence line meeting at each salient point as two diverg- ing lines. The entire influence line is thus made up of pairs of diverging lines (see Fig.- 3) to each pair of which for- mula (5) may be directly applied. Thus in Fig. 3, Ordinate-load products in \dfc = CiMi ( ) " [c0e = C 2 M 2 (+) " \eha = C 3 M 3 (-) " = CM, (+) (+) The signs of the CM's are + or according to the signs of the coefficients, for the M' s are always positive. Summing up the above equations and observing that the ordinate-load products cancel one another except between the influence line fghkm and its base line fom, it follows that the sum of the ordinate-load products for the influence line, or the live-load stress, is S = dM^ + C 2 M Z + . . . = ZCM. .... (7) The letter S represents in general any stress or sum of ordinate-load products for any influence line, while Z stands for the sum of ordinate-load products for any geometrical figure. The rate at which S varies as the load advances a dis- tance dx equals 10 LIVE-LOAD STRESSES dS , --- -j ----- h c c a# But by formula (5a) this becomes = dTFi + C 2 TF 2 + . . . = SCT7. ;,v.. (8) Wi, W 2 , etc., = sum of all of the loads to the left of points 1, 2, etc., respectively, whether on the span or not. Mi, Mz, etc., = moment of the same loads about points 1, 2, etc., respectively, whether on the span or not. The above formulas (6), (7), and (8) apply equally well when the loading is headed from left to right instead of from right to left, the latter being the more usual way. In applying these formulas, however, it will save confu- sion not to reverse the loading, but to turn the influence line end for end, for this operation changes neither the values nor the signs of the coefficients C. The stress S 2CM is related to its derivative - -- = ax 2CTP in the same way that any function is related to its i Q derivative. Thus, if the value of -j- passes through zero as the loading advances, the stress itself may have reached any one of four conditions; namely, 1. Numerically maximum positive value. 2. " minimum " " 3. " maximum negative " 4. minimum In practice it is desirable to find the positions of load- ing to satisfy the first and third conditions. This may be done by proceeding as directed below. It is assumed in stating the following rules that the live load is advancing from right to left. In case the live load advances from left to right, the wheel will be tried first to the left and LIVE-LOAD STRESSES 11 then to the right of a salient point. In other words, dx is always an increment in the same direction as the loading advances. Rule 1. To determine the position of loading to give a maximum positive stress, place the live load on the part of the bridge corresponding to the positive portion of the influence line. Try a wheel first immediately to the right of a salient point that has a negative coefficient and then just to the left of this point. Calculate the value of -7- = ctx I,CW for each of these successive positions of loading. If j & the sign of -7- changes from + to , a position of load- ed ing for maximum positive stress is determined. Rule 2. To determine the position of loading to give a numerically maximum negative stress, place the live load on that part of the bridge corresponding to the negative portion of the influence line. Try a wheel first immedi- ately to the right of a salient point that has a positive coef- ficient and then just to the left of this point. Calculate 7Q the value of -5 = 2CTF for each of these successive posi- tions of loading. If the sign of -r- changes from to +, a position of loading for numerically maximum negative stress is determined. It will be noted that the negative coefficients C occur at those salient points where the angles of the influence line point upward, while the positive coefficients C occur at those salient points where the angles point downward. It is unnecessary to seek a position of loading for maxi- mum positive stress by placing a wheel successively to the right and to the left of any salient point which has a posi- tive coefficient; for if ^ = 2(7TF be + when the wheel is to the right of this point, it would have a still larger + 12 LIVE-LOAD STRESSES value when the wheel is to the left of the point. A change, TO therefore, of -7- from + to would not result. Similarly, it may be shown to be unnecessary to seek a numerically maximum negative stress by trying wheels at any salient point which has a negative coefficient. Formulas (7) and (8) are the general formulas for the solution of the sum of the ordinate-load products of an influence line and the rate of change of this sum, and are applicable to any form of influence line. They give at once a definite solution of the position of a set of loads produc- ing maximum positive and negative stresses in any member of any truss or girder for which an influence line can be drawn and the values of such stresses. The method is particularly advantageous in the case of statically indeter- minate structures, such as two-hinged and no-hinged arches, swing bridges, continuous girders, etc., where general ana- lytical criteria for the positions of loads producing maxi- mum stresses cannot readily be expressed and where such maximum stresses have had to be found by assuming posi- tions of loadings and scaling the influence-line ordinates under all the loads, a laborious process and one open to much liability of mechanical inaccuracy. In applying the present method to the simple forms of girders and trusses (viz., the statically determinate struc- tures where the ordinates of the influence lines are readily expressible algebraically) it will generally be more conve- nient to transform formulas (7) and (8) in each case whereby the coefficients C may be expressed in terms of the geo- metric proportions of the truss or girder. This, in the fol- lowing articles (4 to 7 inclusive), we shall proceed to do for the case of girder bridges (with and without panels), pier reactions, and through Pratt trusses with curved or hori- zontal chords. The general method will, however, be ap- plied directly to the case of the three-hinged arch in Art. 8, which will serve as a typical example of the application of the method to any influence line. ARTICLE IV. GIRDER BRIDGE WITHOUT PANELS. IN Fig. 4 is shown a girder bridge without panels. The live load has advanced beyond the span, this being the Point (( l FlG. 4. most general case. Formulas for the end reactions and for the bending moment and shear at any section will be developed. 13 14 LIVE-LOAD STRESSES The influence line for Ri is shown in Fig. 4a. The sum of the ordinate-load products within the shaded area rst equals the end reaction Ri, which at the same time is the end shear at Ri. From Fig. 4a, Ordinate-load products in \rst = " (ate " " " " [orb " brsc. By using formulas (4) and (5), this equation becomes R l = M 3 - l M l -W l = ^-W l . . (9) Any value of M or W may be read directly from Table 2 for the standard loadings given in Table 1. For example, in Fig. 4, if h = 10', k = 30', and w l of Cooper's #50 has advanced 14' beyond the left end of the span, we have from Table 2, At 1, 14' from wi, M 1 = 350.0* 1 W 1 = 62.50* At 2, 24' from Wi, M 2 = 1150.0 W z = 112.50 At 3, 54' from wi, M, = 5435.0 TF 3 = 177.50 The formula for R 2 is developed as for JBi, the method of writing the second member of the first equation bein^ abbreviated in a way readily understood. From the influ- ence line in Fig. 4b, and the formulas (4) and (5), R 2 = Ordinate-load products in (dvxe \ dvf -\- \fue) Or R 2 = W s - M t + ~M, = W, - M *~ Ml . (Qa) The sum of the reactions Ri and R 2 as given by (9) and (9a) equals TF 3 W\, or the sum of the loads on the bridge. From the influence line in Fig. 4c and formulas (5) or (7), the equation for bending moment may be written: M = Ordinate-load products in ( | gbh \ gak + \kz LIVE-LOAD STRESSES 15 Or M = l j-M, + |jf, - M, .... (10) Formula (10) readily follows, likewise, from the general formula (7), S = C.M, + <7 2 M 2 + C 3 M 3 = SCM. For example, in the case of the bending moment at point 2 in Fig. 4, C k ^ 1 ' L ~ L ~~ C ,.* c s - L - Whence M = r M, - M 2 + 4 M 3 . (lOa) Taking the derivative of M with respect to the advance dx of the loading toward the left or using formula (8) directly, the rate of variation of the bending moment is fF. + -TF, .... (11) . . , ... All positions for maximum M may be found by trying wheels at point 2 as directed by Rule 1 of Art. 3. In ap- plying this rule the simultaneous shifting of other wheels of the rigid loading from right to left of points 1 and 3 as a wheel is shifted from right to left of point 2, must be taken into account by substituting in formula (11) the corresponding changed values of Wi and W 3 . It is to be remembered, as stated in Art 3, that it is entirely unnec- essary to try wheels at points 1 and 3. From the influence line in Fig. 4d, the formula for the intermediate shear S follows by applying formulas (4) and (5) : S = Ordinate-load products in (| mfq mden \ ncq) 16 LIVE-LOAD STRESSES Or S = j-M 3 - TT. - \M, = M * ~ Mj - W> . (12) There is one more thing to be borne in mind in calcula- ting maximum bending moments in a girder bridge without panels: it is the rule for finding the section where the absolute maximum bending moment occurs. The rule is often spoken of as the "centre of gravity rule" and may be stated as follows: The bending moment under any given wheel becomes maxi- mum when the centre of the span bisects the distance from the wheel in question to the centre of gravity of the loading on the span. In the practical application of this rule, the procedure is first to find the wheel which gives maximum bending moment at the centre of the span and then to shift this wheel so that the bending moment beneath it becomes an absolute maximum according to the centre of gravity rule. For the usual standard loadings the maximum centre mo- ment closely approximates the absolute maximum bending moment for the spans greater than 70 feet. The proof of the centre of gravity rule follows. Refer to Fig. 5. Assume that it has been found by trial that the wheel w n gives the maximum centre moment. The general case where load has advanced beyond the span is taken. In order to get an absolute maximum bending moment under w n , this wheel must be shifted a certain distance from the centre. Let such position be distance y from Ri. The sum of the loads on the span is called P 2 and equals (W 3 Wi). The centre of gravity of the loads P 2 is distance x from R 2 . The sum of the loads on the span to the left of w n is called Pi, and their centre of gravity is at the fixed distance b from w n . Taking moments about R 2 , LIVE-LOAD STRESSES 17 Therefore, P r M = R,y - P,6 = ~ y - In this equation for M, the only variables are x and /y. Therefore, M will be a maximum when the product xy is maximum. Note, however, that the sum x + y = (L a) = constant. If two variables have a constant sum, their product is maximum when the two variables are equal. Therefore, M is maximum when x = y. But when x = y, the distance from w n to the centre of gravity of the loading is bisected FIG. 5. by the centre of the span. This proves the centre of gravity rule. In order to apply this rule, a general expression for x is needed. Since Ri = -y 2 - it follows that x = -5' Substitute the LJ r 2 value of Ri from formula (9), and the value (W s Wi) for P 2 . M 3 - x '' ! - LW i - W l (13) 18 LIVE-LOAD STRESSES In the special case where the loading has not advanced beyond the left end of the span, M i and Wi equal zero and x becomes - M 3 , x ~- : w, (13a) Problems relating to a girder bridge without panels will now be given to illustrate the application of the above formulas and the use of some of the tables following the text. Problem. Given a 40-foot deck-girder bridge consisting of one girder per rail. Use Cooper's E5Q loading. Find the maximum shear at the end, quarter point, and centre. Determine also the maximum bending moment at the quarter point and at the centre, and the absolute maximum bending moment. All values are to be given per rail. Solution. Table 5 following the text gives the position of Cooper's loadings for maximum end shear. This table is the result of the solution of end shears for a large num- ber of spans. As a general rule, however, it is safe to as- sume that w 2 of Cooper's and similar loadings will always give the maximum end or intermediate shear when placed immediately to the right of the given section, the live load being headed toward the left. The exceptions in Table 5 to this general rule are not of prime importance, for the actual value of the shear when w 2 is used is sufficiently close to the maximum even in the exceptional cases. There is no satisfactory criterion for determining the position of load- ing for maximum shear in girder bridges without panels, for it is as easy to calculate the actual values of the shears for the successive positions of loading as it is to apply any criterion. In the case of bending moment, however, time is saved by using the criterion. Maximum End Shear. Use formula (9), R, = ^~^ _ Wi. Place wheel 2 LIVE-LOAD STRESSES 19 of Cooper's #50 immediately to right of Ri. Take the values of moment and load sums for Cooper's E5Q from Table 2. Maximum end shear = : , -- 12.5 = 94.3*. Maximum Shear at Quarter Point. Use formula (12) with w 2 at quarter point. t S at Y point = 2838 4 7 5 ~ - 12.5 = 58.5*. Maximum Shear at Centre. Using formula (12) with w 2 at centre. at centre = - 12.5 = 27.5*. The values for the shears are given in Kips, or thou- sand of pounds. A comparison of the above shears with those in Table 7 shows agreement of results. Maximum Bending Moment at the One-Quarter Point. First compute successive pairs of values for-; for dif- ferent wheels, first placed to the right and then to the left of the quarter point. A change of sign from + to indi- cates a wheel that gives a maximum. Use formula (11), ' at J4 point. : : ^f = M (H2.5) + -M (o) - o = + No maximum. 20 LIVE-LOAD STEESSES ^ = M (H2.5) + M (0) - 12.5 = + w z at J4 point. ~ = V (145) + M (0) - 12.5 = + Maximum. = M (145) + M (0) - 37.5 = - waw at J4 point. M (145) + M (12.5) - 37.5 = + Maximum. ~ = K (161.25) + M (12.5) - 62.5 = - at J^ point. y (161.25) + % (12.5) - 62.5 = - No maximum. M (177.5) + M (37.5) - 87.5 = - Accordingly, compute the value of M by formula (10) for w z and w s at quarter point. M = l M s + jMi - M 2 (10) M for w 2 at quarter point, M = }/ (2838.75) + % (0) - 100 = 609.7 Kip feet. M for w 3 at quarter point, M = K (3563.75) + ^ (37.5) - 287.5 = 631.6 Kip feet. The latter value, 631.6, is the maximum bending mo- ment at the quarter point. A comparison of this value LIVE-LOAD STRESSES 21 with Table 11 shows agreement of results. Reference to Table 3 indicates that the correct wheel for maximum has been chosen. Maximum Bending Moment at the Centre. dM W 3 + W 1 TJ7 , in , -fa = 2 W *> ' Oa ^' M 3 + MI ,.. /v* \ i h M = M 2 , (Ha), when at centre, dM 128.75 dx 2 128.75 - 62.5 = + ^ at centre, dM 145 dM 145 No maximum. Maximum. at centre, 145 + 12.5 dx ' 2 161.25+ 12.5 dx - 87.5 = - No maximum. - 112.5 = - Therefore, maximum centre moment occurs with w 4 at centre. M - 283 ^' 75 - 600 = 819.37 Kip feet. Zj This value agrees with Table 11; and the position of loading, with Table 3. 22 LIVE-LOAD STRESSES Absolute Maximum Bending Moment. Shift w 4 according to centre of gravity rule, and then recompute the value of M under this wheel by formula (10). Note that new values for Zi, Z 2 , and M 3 must be determined. By formula (13a), when w* is at the centre, - _ M, _ 2838.75 _. ' W 3 ' 145 Therefore for absolute maximum bending moment under 104, shift loading to left - ^ - = 0'.21. The new values of Zi, Z 2 , and M 3 are h = 20.00 - 0.21 = 19.79 Z 2 = 20.00 + 0.21 = 20.21 M 3 = 2838.75 + .21(145) = 2869.2 The absolute maximum bending moment = M = j- M 3 -h T Mi M 2 = i^ (2869.2) + - 600 = 819.54 Kip feet. It appears, therefore, that the absolute maximum bend- ing moment is .17 Kip feet greater than the maximum cen- tre moment. The difference is not great in this particular case, as the required shift of the loading is comparatively small. The position of loading for absolute maximum bend- ing moment agrees with Table 4, and its value agrees with Table 7. ARTICLE V. PIER REACTION. IN Fig. 4e is given the influence line for the pier reac- tion R between two non-continuous beam spans li and Z 2 . From this influence line, the formulas (5) and (7) give R = Ordinate-load products in (| gbh \ gak + \ kzh) Or, MX Mi L L R - T + T - U M2 = Formula (14) may also be derived from formula (10) since the ordinates of the influence line for R bear the con- stant ratio 7-7 to the corresponding influence ordinates for LI li M, the position of the live load and the values of li and 1 2 remaining fixed. Therefore, R = ~ Substituting the value M = j- M 3 + j- Mi M 2 from formula (10) in formula (16), the result is again formula (14). For equal spans, 777 4- t> M 3 + MJ- 2M 2 li = Z 2 = I so that R = - j . (14a) The rate of change of R for a movement dx of the loading to the left is dR W, W l L L /Z, I ~dx = T + T " u* w * = E (~L W * + T w > ~ 23 24 LIVE-LOAD STRESSES For equal spans, h = 1 2 = I, so that dR W s + W, - 2W* dx = I (15a) In the last member of formula (15) the quantity within the parentheses is the same as the expression for -y in formula (11). It follows, therefore, that the same position of loading gives maximum R and maximum M for any given values of l\ and 2 - Problem. (a) Find the maximum pier reaction per rail between two simple beam spans li = 10 ft. and Z 2 = 30 ft. (b) Find the maximum pier reaction between two simple beam spans, each having a length of 20 feet. Use Cooper's #50 loading. Solution of Problem (a). Use formula (15) to find position of loading for maximum R. - (15) at pier. dR 40 /1 - - 62 - 5 ) - - 30 LIVE -LOAD STRESSES Note that the position of loading agrees with Table 3. For this position of loading formula (17) gives S a = i( (21895) + - 287.5) M 168.1*. For maximum shears in the intermediate panels, deter- mine the position of loading by formula (20a) and the shear by formula (19a). (20a) (19a) Panel 1-2. Try wheel 3 at panel point 2. = 27)(-6 < 306 - 25 ) - 37 ' 5 ) - + Maximum. f = 1(J (322.50) -62.5) = - Sb = ^o(l ( 1505L25 ) " 287 - 5 ) z ~- 11L0 *- Panel 2-3. Try wheel 3 at panel point 3. Maximum. & = ^Q (9345) - 287.5) = 63.5*. The above values for shears agree with the values given by Table 9. The wheel for maximum shear in panels of girder and truss bridges is given in Table 6. ARTICLE VII. THROUGH PRATT TRUSS. GENERAL FORMULAS FOR LIVE- LOAD STRESSES AND THEIR RATE OF VARIATION. ILLUSTRATIVE PROBLEMS. dS THE general formulas S = 2CM and -7 = 2CW may be used to write the equations for the live-load stresses in any member of a framed structure as soon as its influence Salient Points Ordinates Slopes Coefficients mkp h nv JL nv FIG. 7. line has been drawn and the ordinates at the salient points determined. In Figs. 7, 8, 9, and 10 are shown all the influence lines 31 32 LIVE-LOAD STRESSES needed in writing the formulas for the live-load stresses in a through Pratt truss with non-parallel or parallel chords. The influence ordinate at any salient point is the calcu- lated stress due to a one-pound load on the bridge at the panel point above this salient point. By easily discovered relations between similar triangles, the algebraic value of each stress, or influence ordinate, is expressed in terms that are most readily evaluated in any numerical problem. The derivation of any one formula for a live-load stress is typical. Refer to Fig. 7. The stress in the lower chord member S 5 is found by taking moments about C. The influence line for $ 5 is straight over each of the two inter- vals kp and mp. The ordinates at the salient points 1 and 4 are zero. The ordinate at salient point 3 must be found by placing a one-pound load at the lower panel point of the truss above this salient point and calculating the value of S 6 . For the unit load so placed, Reaction at A = = - np n By moments about C, Therefore, ~ (mp) = S, (v) = + ~ = Influence ordinate at 3. nv The slopes of the segments of this influence line follow. , mkp k Slope of ao = -- - -r- mp = -- nv nv r , . mkp m Slope of be = + - -r- kp = -\ -- nv nv The coefficients C for use in the general formula S = 2CM are now found. c 1 = o + - = +- nv nv LIVE-LOAD STRESSES 33 - - nv nv v nv nv Therefore, for the position of the live load advanced beyond the limits of the span, the general formula for However, in actual practice it is usually not necessary to advance the loading beyond the left end of the span in order to get a maximum value of S$. The usual formula will therefore not contain the term Mi, since this will be zero; thus, Inasmuch as the horizontal component of the stress *S 8 in an inclined top chord member or end post equals the stress $ 5 in a corresponding lower chord member, the stress $ 6 in any top chord member or end post may be found by SG = ~ Ss ........ (22) In Fig. 8 is shown the influence line for the stress $ 4 in any vertical post. The influence ordinates are deter- mined by taking moments about the intersection of the upper and lower chord members which are cut by the section. The algebraic values of these ordinates are trans- formed by use of easily discovered relations between sim- ilar triangles. The slopes and coefficients are then calcu- lated in the usual way. The influence line indicates that the live load should advance into but not beyond the panel p for a maximum compression, and for this reason MI and If 2 equal zero for the usual case. The numerical value of 34 LIVE-LOAD STRESSES the maximum compression S* in a vertical post is, therefore, The coefficients for the stress in any inclined web mem- ber are given by Fig. 9. The quantities for Si and >S 2 are Ordinates Slopes Coefficients . _ I b.L bL A. bl bLp\ a bL d bl . (>P\ _ IP FIG. 8. as shown, and the quantities for S 3 are of the same alge- braic form except that they are of opposite sign through- out. For the usual position of the live load advanced from the right into but not beyond the panel p for maximum stress, the moment sums Mi and M 2 equal zero, and the numerical values of the maximum tension Si and &> and of the maximum compression & are given by the following formula: (24) LIVE-LOAD STRESSES 35 In a special case where the loading must be advanced beyond the panel p until the tension in the inclined counter- web member $ 2 is balanced by the dead-load compression FIG. 9. in this same member, the value of M 2 is not zero, and the formula for $ 2 becomes Or, letting M c = - S). 36 LIVE-LOAD STRESSES Note that the coefficients of M 4 and M c in this formula are the same as the coefficients for M 4 and M 3 in formula (24) . The influence line for the counter-tension in a vertical post is shown in Fig. 10. For the usual case, the loading advances beyond the panel but not beyond the end of the span. Therefore Mi is equal to zero, so that FICJ. 10. (26) where K and M stand for the corresponding terms in the parentheses. In order that T be a maximum the live load must advance beyond the position for the maximum tension & until the tension as computed by formula (25) becomes equal to the dead-load compression in this same member. For this position of the live load, the value of T is then computed by using formula (26). It may be noted that LIVE-LOAD STRESSES 37 some specifications state that only % of the dead-load compression is to be counted as effective in counteracting the live-load tension in an inclined counter-web member. This specification has been observed in the problem to follow. A review of the preceding formulas shows that all the live-load stresses may be computed by formulas (21), (22), (23), and (24), except the counter-tension in a vertical post and the tension in a floor-beam hanger. Formula (25) makes it possible to find readily by trial the position of loading for maximum counter-tension in a vertical post, and formula (26) gives the value of this tension. The maximum ten- sion in the floor-beam hanger may be found by the use of formulas (14a) and (15a) for pier reaction between equal spans. If the chords of the Pratt truss are parallel, there will be no counter-tension in any vertical post. Formula (21) for the stress in a horizontal chord member and formula (22) for the stress in the inclined end post remain unchanged. Formulas (23) and (24) for web stresses are simplified be- cause a = b = depth of truss. The formulas, therefore, for the Pratt trusswith parallel chords are: Stress in horizontal chord members = Stress in inclined end post = S& = - $ 5 ....... (22) Stress in vertical post = 4 = j$* " ~ M ^ - - - ( 29 ) Stress in inclined web member = )*'-;& (30) One general formula will suffice for finding the position of loading for maximum chord and web stresses of a Pratt truss with either inclined or parallel chords. The formulas 38 LIVE-LOAD STRESSES (21), (23), (24), (29), and (30) for these stresses are of one general form S = (G) M, - (H) M, (27) where G and H are the corresponding coefficients of M 4 FIG. 11. and Ms in the preceding formulas. The rate of variation of S as the load advances is ^ = GWt - HW 3 = H (| Wt - W 3 ) . . (28) When any one of the above stresses is a maximum, the CC 1 \ jjWi W 3 J passes through zero as a wheel is shifted from right to left of the salient point 3 in Figs. 7, 8, or 9. The preceding formulas for the live-load stresses are summarized for convenient reference in Art. 11 preceding LIVE-LOAD STRESSES 39 the Tables. The important dimensions and quantities in Figs. 7, 8, and 9 are summarized in Fig. 11. If a uniform live load is used, the shaded areas in Fig. lla, b and c mul- tiplied by the intensity of the uniform load will give the maximum live-load stresses. The algebraic value of any one of these triangular areas is conveniently expressed as the base of the triangle times y% of the given algebraic ordi- nate. The lengths of the bases of the shaded areas in Figs, lla and b may be readily determined by one of the con- structions shown in Figs. 12a and 12b, which give the po- sition of the unit load for zero stress in the members indi- cated. The proofs that these constructions locate neutral points are not given, for they are generally known, and are proved in numerous texts on bridges. (See Marburg's " Framed Structures and Girders," Vol. I, page 392.) The application of the preceding formulas will now be made to the calculation of the live-load stresses in the two single track through Pratt trusses shown in Figs. 13 and 14. A convenient procedure is as follows: 1. Determine the lengths of all inclined members and write then- values on the truss outline. 2. Determine the values of the intercepts a as defined by Fig. 11 and write their values on the truss outline. 3. Write on the truss outline the distances of the sev- eral panel points from the right end of the span. 4. Write down the reciprocals of the span, panel length, and lengths of vertical members. 5. Make a form for tabulating calculations and list members in some convenient form as is done in Figs. 13 and 14. 6. Calculate the numerical values of the coefficients G and H for the several members by use of the formulas already derived. 7. Determine the position of the loading for maximum / C 1 \ stress by finding the position of loading causing (-77 TF 4 TF 3 J 40 LIVE-LOAD STRESSES to pass through zero, and for this position of loading select from Table 2 the corresponding values of M 4 and M 3 . At VARIOUS CONSTRUCTIONS USED TO FIND NEUTRAL POINTS IN PRATT TRUSSES. U, U 2 U,, U 4 U, the same time tabulate the length Z/i of loading causing maximum stress as this value is used in the impact formula LIVE-LOAD STRESSES 41 / = S' 300 Li + 300* 8. Calculate values of S = GM 4 HM S and combine with impact and dead-load stresses. When the dead- and live-load stresses are of opposite sign, the combination is usually not algebraic but according to the particular speci- fication that is used. a = 38' 2G.OO 26.08 A 1 2 a 1-4- 208 = .00480 1-^ 26 =.0385 Span 208' Live Ld. E J3 -f- 32 =.03125 -^36 .02778 FIG. 13. 6 7 A l-4-38=.0C32 >/r*%wv -rcru^^i M KJT OMj HMs T . 300 j DL Total Mem. w neci 4 HM \jrIYl4 Xijyi3 Mm Li+300 K EF 00373 .0385 3@3 33970 287 127 11 -116 143 .677 - 78 - 40 -234 ED 00481 .0442 3@2 46255 287 223 13 +210 169 .640 +134 + 83 +427 GH 00405 .0385 2@4 21531 100 87 4 - 83 112 .728 - 60 - 15 -158 GF 00500 .0450 3 @3 33970 287 170 13 +157 143 .677 +106 + 48 +311 IJ 00480 .0385 2 @5 12940 100 62 4 - 58 86 .777 - 45 + 7 IH 00580 .0466 3 @4 23375 287 136 13 +123 117 .719 + 88 + 21 +232 JK 00580 .0466 2 @5 12940 100 75 5 + 70 86 .777 + 54 - 21 ML 00777 .0493 2 @6 6550 100 51 5 + 46 60 .833 + 38 - 50 NO 01030 .0496 2 @7 2307 100 24 5 - 19 34 .898 - 17 + 83 No counter AC =AD 00390 .0312 4@1 63111 600 247 19 +228 200 .600 +137 +101 +466 BC -362 -217 -160 -739 AF 00695 .0278 7@2 59095 2694 410 75 +335 193 .608 +203 +154 +692 BE -339 -206 -156 -701 AH 00985 '0263 li@3 59661 7310 587 192 +395 194 'eo7 +239 +181 +815 BG -396 +240 -181 -817 BI 01315 '0263 13 4 50901 9585 670 252 -418 178 ^627 -262 -194 -874 CD 0385 .0770 4 @ 1 3725 600 144 46 + 98 44 .872 + 86 + 25 +209 Post at Mem. M4 Me S IB K M,, T Li 300 I D.L. Total Li+300 5 JK 22261 2390 + 16 __ -14.0020. 311340 +23 114 .725 +17 +3 + 43 6 ML 8865 687 +35|-34i 0021 1 5960 +13 71 .8 +10 +1 + 24 42 LIVE-LOAD STRESSES 9. Find positions of loading for maximum counter-ten- sions in posts and compute values by use of formulas (25) and (26). PROBLEM 1. Calculation of Live-load Stresses in a Pratt Truss with Inclined Chord. The complete data for this problem are given in Fig. 13. Items 1 to 5 of the above method of procedure need no explanation. The values of the coefficients G and //, the position of the loading for maximum stress, and the value of the maximum stress will be determined for several typical members; for example, vertical post, inclined web members, horizontal chords, end post, and inclined chords. Vertical Post EF. Formula S, = (^r) M 4 - (-) M 3 . . . . . . (23) Refer to Fig. 11 for definition of dimensions. G = - = |f (.00480) = .00373 DLi oO H = - = .0385 Try w s at panel point 3. Use Table 2. L = 143'. Therefore w 3 at 3 gives a maximum. S = GM, - HM 3 = .00373(33970) - .0385(287.5) = 126.7 -- 11.0 = 115.7" - 300 300 A77 Impact factor == -- =- == .677 Impact stress = .677 X 115.7 = 78.3 fc . LIVE-LOAD STRESSES 43 Inclined Web Member ED. Formula 8l = (^) M 4 - ()M, .... (24) Refer to Fig. 11 for definition of dimensions. H = ~ = ~ (.0385) = .0442 Try w s at panel point 2. Use Table 2. Za = 169'. nnj.si 37.5 -f- = ^^T (505.0)- or =or .U44^ }o c D^.O Therefore w 3 at 2 gives a- maximum. iS = GM* - HM S = .00481(46255) - .0442(287.5) - 223 - 13 = 210 fc . 300 Impact factor = -j = .640 Impact stress = .640 X 210 = 134*. Inclined Web Member ML. Formula S, = 4 - jtf, ... .(24) Refer to Fig. 9 or Fig. 11 for definition of dimensions. / 4fi 04 H = ^ = -^- ( - 0385) - - 0493 Try w 2 at panel point 6. Use Table 2. la = 60'. 00777 1^.5 + w( 19 )- 3 - = _ r Therefore w 2 at 6 gives a maximum. 44 LIVE-LOAD STRESSES S = GM, - HM, = .00777(6550) - .0493(100) = 51 -- 5 = 46*. 300 Impact factor = = .833 Impact stress = .833 X 46 = 38*. Lower Chord Member AC AD. Formula & == Q^ 4 - ()jf, ..... (21) Refer to Fig. 11 for definition of dimensions. syyi G = ~ = J (.03125) = .00390 H= = .0312 v Try w* at panel point 1. . Use Table 2. L, = 200'. Therefore io 4 at 1 gives a maximum. S = GM 4 - #Af, = .00390(63111) - .0312(600) = 247 -- 19 = 228*. Impact factor = = .600 Impact stress = .600 X 228 = 137*. End of Post BC. Formula S 6 = -^ S- ......... (22) S G = ^l 3 (228) = 362*, and impact = ^p (137) = 217*. Lower Chord Member AH. Formula S 5 = M 4 - M 3 . . . . (21) LIVE-LOAD STRESSES 45 Refer to Fig. 11 for definition of dimensions. /yyi G = ~ = | (.02632) = .00985 H = -i- = .0263 Try wu at panel point 3. Use Table 2. Li = 194'. ri 190 l^- Therefore MU at 3 gives a maximum. S = GM 4 - HM 3 = .00985(59661) - .0263(7310) = 587 -- 192 = 395*. Impact stress = S = .607 X 395 = 239*. Top Chord Member BG. Formula S G = S, .......... (22) S. = ^ (395) . 396 fc . 2fi AQ Impact = ^ (239) = 240". Counter-Tension in Post at Panel Point 5. Formulas S 2 = Stress JK = = tension in post. -)(- n ^-^)=K-M (26) Refer to Fig. 10 for definition of dimensions. The calculation of the dead-load compression in JK is 46 LIVE-LOAD STRESSES not given, but the value is 21 fc . Two-thirds of this com- pression, or 14 fc , will be considered effective in counterbal- ancing the live-load tension in JK. The live load must be advanced beyond the position of maximum live-load ten- sion in JK (i.e., w 2 at panel point 5) until &, or the stress in JK, equals 14 fe . This must be done by trial, S z being figured each time by formula (25). It is found that when 114' of loading has advanced upon the bridge, this condi- tion is approximately satisfied. For this position of loading M* = 22261 M c = M S - M 2 = (2565 - 175) = 2390 - Therefore, & = .00580(22261) - .0466(2390) = 16*. This value of S* = 16 fc balances % D = - 14 fc , nearly enough for practical purposes. Therefore, compute T for this position of the live load. K = = - 00203 M = $/8 (22261) - 2565 = 11340 T = .00203(11340) = 23* 300 Impact factor = T = .725 Impact stress for T = .725 X 23 = 17 fc . PROBLEM 2. Live-load Stresses in a Pratt Truss with Parallel Chords. The complete data for this problem are given in Fig. 14. Formulas (21), (29), and (30) give the values of the LIVE-LOAD STRESSES 47 coefficients G and H, which are identical for several mem- bers of any Pratt truss with parallel chords. The proce- dure for finding the positions of the loading and maximum stresses is exactly as in Problem 1. It should be noted that Stress FG = Stress EF X 37.54 HI = BC = (C GHX ACX 25 Live Load E 50 3 A -=^=.00667 ^ = ~j|f == ' 0400 Fig. 14. Secant Mem. G H Wheel M4 Ma S CD EF FG .0400 .00667 .0800 .0400 4@1 3 " 3 3564 13520 600 287 95 79 106 GH HI .00667 .0400 2 " 4 6170 100 37 50 JK DE BC .00894 . 00894 .0536 .0536 2 ' 5 3 ' 2 2179 21895 100 287 14 181 272 AC = AD AF = BE BG . 00595 .01190 . 01785 .0357 .0357 .0357 4 ' 1 7 ' 2 12 ' 3 33970 31375 34411 600 2694 8385 181 278 314 The stresses in all of the chord members may be checked by use of Table 8, and the stresses in the end post and web members may be checked by Table 9. The stress in CD agrees with the maximum pier reaction in Table 7. Table 3 may be used to find the position of loading for maximum chord stresses, and Table 6 gives position of loading for maximum web stresses. ARTICLE VIII. THREE-HINGED ARCH. APPLICATION OF THE GENERAL METHOD TO THE CALCULATION OF LIVE-LOAD STRESSES. THE general formulas -r- = 2CW and S = 2 CM may be used directly to find the position of loading and the FIG 15. value of the maximum live-load stress in any member of a framed structure as soon as the influence line for this member and the ordinates at all salient points have been 48 LIVE-LOAD STRESSES 49 determined. This method is applied to the calculation of maximum live-load stresses for the three-hinged arch shown in Fig. 15. Cooper's E40 loading is used. First are drawn the influence lines for the horizontal and vertical components of the reaction at the left hinge. The vertical component Vi is the same as for a simple span L. The horizontal component Hi equals the bending moment at the centre of the span L divided by the depth h. The influence-line ordinates for all members are now found by drawing five Maxwell diagrams, one of which is reproduced in Fig. 16. From the influence lines for Vi and Hi, the value of Vi is .8889 and Hi is .2187 for a one-pound load at Ui. The external loads acting on the left half of the arch are then as shown in Fig. 16a. The load line axbcya in Fig. 16b is drawn to a scale of 10" = 1 pound, and the Maxwell diagram completed in the usual way. The scaled TABLE A INFLUENCE-LINE ORDINATES FOR THREE-HINGED ARCH Members ORDINATES 1 Ib. at Ui 1 lb. at U 2 l.lb. at Us 1 lb. at U 4 1 lb. at U'4 E/ot/i= UiU t = U*U 3 = U 3 U* = - .403 - .417 - .378 .171 - .223 - .833 - .756 - 342 - .045 - .286 -1.135 513 4- .130 + .262 + .189 685 + .201 + .477 4- .757 -j- 548 LoLi = .295 - 590 - 885 -1 180 1 182 LxL 2 = LzL 3 - . + .221 4- 217 - .264 4- 434 - .740 408 -1.224 1 248 -1.302 1 484 L 3 L 4 - . 4- 164 4- 328 4- 491 1 086 1 674 L 4 L 5 - . 048 096 145 193 1 420 t/oLo - . 92 384 075 4- 234 _|_ 345 C/iLi = . 1 014 632 253 4- 129 4- 287 U'l 4- 022 955 490 043 + 165 U 3 L 3 = . 4- 075 4- 150 775 317 076 C/4^4 = . . + 114 4- 226 4- 342 545 364 C/oI/i = . + 800 4- 441 4- 085 270 400 UiL-. . 4- 019 4- 878 4- 350 180 398 U 2 L 3 - . . . - 044 088 4- 986 "4. 086 324 U 3 L 4 = 221 442 662 + 928 4- 224 f/4L 5 = H .. - .206 2187 - .412 4375 - .617 6562 - .823 8750 + .657 87^0 V .. 8889 7777 6666 5555 4444 6 14 29 44 58 63 50 LIVE-LOAD STRESSES values of these stresses are the influence ordinates for a one pound load at U\. In an exactly similar way the in- fluence ordinates for a unit load at U 2 , U s , U 4 , and U\ are determined. The influence lines are straight from U' Q to (a) FIG 16. 7' 4 . Table A gives the influence ordinates for all members and also for the horizontal and vertical components of the reaction at the left hinge. The angle 6 is the inclination of this reaction with the vertical. The calculation of the live-load stresses in any one mem- ber is typical. The member UJL is taken. The influence line for this member is drawn to scale in Fig. 15 by use of the influence ordinates from Table A. The salient points occur below panel points C7 3 , Ut, and U\. The distance LIVE-LOAD STRESSES 51 662 from U 3 to the neutral point equals AAO 4. oog Calculation of Slopes. Slope of df = jy - hk = km = mn = 68 - .662 - (.928) T .\JL\JU - .0758 + .0336 21 .928 - (.224) 21 .224 - 84 Calculation of Coefficients. C l = - (.0105) - .0105 C 2 = .0105 - ( -- .0758) - + .0863 C 8 = - .0758 - (.0336) - .1094 C 4 = .0336 - (.0027) = + .0309 C 5 = .0027 - = + .0027 The sum of these coefficients equals zero. This agrees with formula (6) of Art. 3. It should be remembered, as is pointed out in Art. 3, that the value of these coefficients may be measured graph- 2 59 ically. For example, in Fig. 15 the value of C 2 is -r- = .0863. 7O By use of the formula -,- = SCTF and Rule 1 of Art. 3, the position of loading for maximum tension in may now be determined. Try wheel 3 at t/ 4 with the load- ing advancing toward the left. Take the values of the load sums and moment sums for #40 from Table 2. 52 LIVE-LOAD STRESSES |~ = 2CW = - .1094(30) +.309(103) +.0027(302) = + .7 -.1094(50) + .309(103) +.0027(302) = - .7 Therefore w 3 at 17* gives a maximum tension in and its value is S = zCM = -.1094(230) + .309(1846)+.0027(19001)=83 fc . TOf By use of the formula - = ?CW and Rule 2 of Art. 3, the position of loading for maximum compression in is now determined. Try wheel 2 at U s with the loading advancing toward the right. Note that the signs of the coefficients remain unchanged. Take the values of the load sums and moment sums for #40 from Table 2. -.0105(192) + .0863(10) = -- 1.3 - = SCTF = -.0105(192) + .0863(30) = + 0.6 Therefore w 2 at U s gives a maximum negative stress, or compression, in U S L^ and its value is S = ?CM = -.0105(7092) + .0863(80) = - 67*. The above values of 83 fc and 67 fc for maximum tension and compression in C7 3 L 4 may be checked by use of formula S = qA z (2), the values of q being taken from Table 16. Tension U&L* by Equivalent Uniform Load. The area of the tension part of the influence line equals A z = 27.2 The influence line ohkm is not triangular, but a tri- angular influence line with intervals Zi = 10 ft. and 1 2 = 45 ft. approximates its shape closely enough for the selec- tion of an equivalent uniform load. For Zi = 10 r and Z 2 = 45', Table 16 gives 3.080 fc as the equivalent uniform load. LIVE-LOAD STRESSES 53 Therefore, S = qA z = (3.080) (27.2) = 84. This value checks very closely that obtained by the exact method. Compression U^L^ by Equivalent Uniform Load. Choose from Table 16 the equivalent uniform load for h = 10 ft. and Z 2 = 65 ft. From the influence line A z = 23.7. Therefore, S = qA z = (2.870) (23.7) = 68 fc . This checks closely the value obtained by the exact method. Calculation of other members of this arch and of some more complicated framed structures shows a close agree- ment between the two preceding methods. The latter method is the simpler when a table of equivalent uniform loads has been made, especially in the case of the more complex influence lines for members of swing bridges, two- hinged arches, arch ribs, etc. The method of calculating a table of equivalent uniform loads will be explained in the following article. ARTICLE IX. EQUIVALENT UNIFORM LOADS. AN equivalent uniform load is one which gives the same stress as does a loading which is not uniform. For any given standard loading, the equivalent uniform load is dif- ferent for stresses whose influence lines differ. Since the forms of influence lines are innumerable, a table of exact equivalent uniform loads for all stresses is impracticable. A table of equivalent uniform loads, however, for stresses whose influence lines are triangular may be used with little error in selecting equivalent uniform loads for stresses whose influence lines are not triangular. It is, therefore, sufficient for practical purposes to make tables of equivalent uniform loads for a series of triangular influence lines. It may be shown that the equivalent uniform load for any triangular influence line is dependent entirely upon the intervals Zi and 1 2) and is independent of the ordinate h at the apex of the influence line. Consider the triangular influence line in Fig. Ib to be for any stress S. Let the ordinate below C be any value h. If q equals the equivalent uniform load covering li and l z , S S = qA z , or q = j- (A) The area of this influence line is A z = \ (h + = \L (B) Furthermore, if the concentrated live loads have been placed so as to give the maximum pier reaction between two spans h and k, this same position of loading will give maximum S, if the influence line for S is a triangle with the 54 LIVE-LOAD STRESSES 55 same intervals li and Z 2 . Since the influence ordinates for S are related to the influence ordinates for R as h is to unity, S h R == 1.00 Or S = hR ..... . . . . (O Substituting the values of A z and S from equations (B) and (C) in equation (A), q = hR + L~ ...... (D) It appears, therefore, that q is independent of /t. From formula (16) of Art. 5, Li t-2 Substituting for R in equation (D), 2/2 2M (16) (3D The term M is the bending moment in the span L = li + 1 2 at the point where the intervals are h and Z 2 . Tables (10) to (18) inclusive have been calculated for the positions of the live load given by Table 3. The values of M were first found, then the values of R, and finally the values of the equivalent uniform loads. The three formulas that were used in succession are If - jJMt+jJfi- Jfi ..... (10) R = /y M .......... (16) tl t-2 2M 2R 56 LIVE-LOAD STRESSES An example of the use of equivalent uniform loads has already been given in Art. 8. The general formula S = qA z may be used in any case. For the special cases of bending moment in a beam and pier reaction between two simple spans, formula (31) gives <32> The quantities in the parentheses are the areas of the influence lines for M and R respectively. ARTICLE X. METHOD OF CALCULATING TABLE OF LOAD SUMS FOR ANY STANDARD LOADING. ILLUSTRATIVE EXAMPLE. THE definitions of moment sum and load sum are given at the beginning of Art. 2. It is at once evident that a table of load sums may be computed by adding the succes- sive loads. It may be shown that the table of moment sums may also be calculated by the process of addition. From formula (5a) of Art. 2, CnWn = dM a dx Or dM a = W a - dx. Expressed in words, the increase in the moment sum for an increase dx in the distance of the centre of moments from wheel 1 equals the load sum times dx. If the load sum is constant for an interval dx = 1 foot, as between con- centrated loads, the increase of the moment sum for dx = 1 foot equals the corresponding load sum. If the load sum is not constant, but uniformly increasing, as when the cen- tre of moments lies within the uniform load, the increase of the moment sum for dx = 1 foot equals the average value of the load sum for this one foot interval. The appli- cation of the foregoing principles is made clear by the fol- lowing example. Example. Give explicit directions for the calculation of a table of load sums and moment sums at intervals of 1 foot from IT to 400' for Cooper's #40 loading. Solution. Calculate the table of load sums by adding 57 58 LIVE-LOAD STRESSES the loads one by one, taking a sub-total for each addition. Thus, the following numbers are added,: 110 4 20's 4 13's 110 4 20's 4 13's 391 2's If the final total checks 284 + 391 X 2 = 866, the table of load sums is correct. Assume now that the table of load sums for E40 has been completed. The table of moment sums may now be found as directed below. The following numbers are to be added one by one, taking a sub-total for each addition: 8 10's 5 30's 5 50's 5 70's 9 90's 5 103's 6 116's 5 129's 8 142's 8 152's 5 172's 5 192's 5 212's 9 232's 5 245's 6 258's 5 271's 5 284's 1285 1287 1289 and all odd numbers up to 865. If the final total checks up 183,689, which is figured independently, the table of moment sums is correct. The preceding additions may be made most satisfac- torily on a recording adding machine. Table 2 was cal- culated in this way. It will be noted that the table of load sums serves as a table of differences for the table of moment sums. ARTICLE XI. SUMMARY OF FORMULAS. Art. 1. z = zwz . . '. . . .;; ... .. v ..-.- . . (i) Z = qA z ..... . . .,. . . (2) . Z = w2z ....... ....... (3) Z = z2w = zW . . . V . . ...... (4) Art. 2. Z = 2w n z n = C a 2w n x n = C a M a ..... (5) dZ d(C a M a ) C a dM a dx aa dx dx Art. 4. Girder Bridge without Panels. End reactions. Art. 3. ZC = ..... ../; . ., , ; : ! .... (6) S = 2CM . . /. . . . ....... (7) I ...... . R ^ =Ws _^-M, .;....'.;. (9a) Bending moment for unequal segments li and 1%. M = Klfa + -- Afi - M 2 , ; . . . . , (10) Wt-W, . ..... (11) 59 60 LIVE-LOAD STRESSES Bending moment at centre, h = Z 2 = ~^~ -. M 3 + Mi M = o ^/ 2 - dx 2 Shear at any section. s-*& Location of centre of gravity of loading on span. M 3 - Mi - LW l When Mi = 0, 5 = | . (13a) Ar. 5. Pier Reaction. For unequal spans Zi and Z 2 . cffi . Ej . Ej Aw _ J aic Z2 'i ZiZ2 Z: For equal spans Zi and k equal to Z. R = '^ g 1 " - 2 (14a) m = TF3 + ^- 2 ^ - - : - - (16a) Relation between R and M , R = rrM . (16) Art. 6. Girder Bridge with Panels. Shear in end panel; general case. = ^Mz+^M^Mz LIVE-LOAD STRESSES 01 f - Shear in intermediate panel; general case. : ' S ^M,_M, M,_M, ' L p p L dS b _W t W, W, TF, "^ == T" y y " T Shear in intermediate panel; usual case. . 7. Through Pratt Truss with Inclined Chord. Stress in hanger. Use formulas (14a) and (15a). Stress in any horizontal chord member; usual case. Compression in any inclined top chord member or end post; usual case. . (22) Compression in vertical post; usual case. (28) Stresses in inclined web members including counters; usual case. v ..- S t , &, S, = M 4 - Mt . . . (24) Stress in inclined counter; special case of loading ad- anced bevond panel. 62 LIVE-LOAD STRESSES / ta \ , t ( , f b , T \ ( ta S 2 = (-rriMi T-\Mz MZ) = (-rr \cbL' bp\ c / \coL Counter- tension in vertical post; usual case. Formulas (21), (23), and (24) are of the general form S = GM, - HM 3 . . '.,. , V . (27) where the coefficients G and H may be tabulated thus: Type of member G H Horizontal chord - Vertical post ry- Inclined web member. . .-7-7- 7 cbL bp The rate of variation of S in formula (27) is dS (28) 7 v r r 4 J.J. rr 3 J.x -rj r r 4 rr A I \***/ When S in formulas (21), (23), or (24) is a maximum CC* \ jjWi Wzj passes through zero. Through Pratt Truss Parallel Chords. Stress in hanger, use formulas (14a) and (15a) Stress in horizontal chord = S~ a = \ jM^ \-jM 3 , (21) " vertical post = & = (|)M 4 (-\M* (29) VJL// \ p / " inclined web = S, = M* - ~^i = (30) LIVE-LOAD STRESSES 63 Stress in end post = $ 6 = ~S b ........ (22) Formulas (21), (29), and (30) are of the general form S = G- M, - H-M t . . .... (27) and their rate of variation is . . . . -(28) G and H are the coefficients of M 4 and M s in equations (21), (29), and (30), respectively. When S in formulas (21), (29), or (30) is a maximum, / C 1 \ \JfWi W 3 } passes through zero. Art. 9. Equivalent Uniform Loads. ~~j ^O-Ly (32) INDEX TO TABLES NO. PAGE 1 Five standard loadings 66 2 Lengths, loads, load sums, and moment sums from 1 to 400 ft. for five standard loadings 67 3 Position of Cooper's loadings for maximum pier reaction between equal and unequal beam spans 88 4 Position of Cooper's loadings for absolute maximum bending mo- ment in beam spans 89 5 Position of Cooper's loadings for maximum end shear in beam spans 89 6 Position of Cooper's loadings for maximum panel shear in bridges with equal panels 90 7 Maximum moments, shears, and pier reactions for beam spans Cooper's #40 and #50 loadings 91 8 Maximum moments at panel points of truss bridges Cooper's #50 loading 94 9 Maximum shears in panels of truss bridges. Cooper's #50 loading 97 10 Maximum bending moments at 5-foot intervals of beams Cooper's #40 loading 100 11 Maximum bending moments at 5-foot intervals of beams Cooper's #50 loading 102 12 Maximum bending moments at 5-foot intervals of beams Cooper's #60 loading 104 13 Maximum pier reactions for equal and unequal beam spans Cooper's #40 loading 106 14 Maximum pier reactions for equal and unequal beam spans Cooper's #50 loading 108 15 Maximum pier reactions for equal and unequal beam spans Cooper's #60 loading 110 16 Equivalent uniform loads Cooper's #40 loading 112 17 Equivalent uniform loads Cooper's #50 loading 114 18 Equivalent uniform loads Cooper's #60 loading 116 19 Influence-line ordinates for bending moments at 5-foot intervals of beam spans 118 20 Reciprocals of influence-line ordinates for bending moments at 5-foot intervals of beam spans 120 21 Bending moments at 5-foot intervals of beam spans due to a unit uniform load . 122 65 LIVE-LOAD STRESSES TABLE 1 STANDARD LOADINGS Loads given are for one rail. COOPER'S E 40: COOPER'S E 50: 111 (jXJXJxt) cj> c|) cL cj> 4 fofcb6 ^cLo F 'ieis^sj COOPER'S E 60: SC50CS OOQS C50SC50S T-HT-IT--I -i CCCOCOCO rH rH -H i I ^o OO I 3000 I bs. per ft. 8H*-MJMW6H^H*W*6WWii COMMON STANDARD-1904-PACIF1C SYSTEM D. L. & W. R. R.: LIVE-LOAD STRESSES 67 TABLE 2 LOAD SUMS AND MOMENT SUMS FOR COOPER'S AND OTHER STANDARD LOADINGS NOTE. Load Sums and Moment Sums are given per rail in thousands of pounds and foot-pounds respectively. 68 LIVE-LOAD STRESSES COOPER'S #40. 0'-50' COOPER'S #40. 50'-100' Length Wheel Load Load Sums Moment Sums Length Wheel Load Load Sums Moment Sums w 1 10 10 o 50 3780 i W J. 10 51 3922 J. 2 20 52 4064 3 30 53 4206 4 40 54 4348 5 50 55 4490 6 t 60 56 w. 10 io 152 4632 7 70 57 4784 8 w. 2 20 '30 80 58 . 4936 9 110 59 5088 10 140 60 5240 11 170 61 5392 12 200 62 5544 13 w.'3 20 'so 230 63 5696 14 280 64 w. ii 20 172 5848 15 330 65 6020 16 380 66 6192 17 430 67 6364 1C w 4 20 70 480 68 6536 J.O 19 550 69 w. 12 20 i92 6708 20 620 70 6900 21 690 71 7092 22 760 72 7284 23 w.'5 20 '90 830 73 7476 24 920 74 w. 13 20 212 7668 25 1010 75 7880 26 .... 1100 76 8092 27 1190 77 8304 28 1280 78 8516 29 1370 79 w. 14 20 232 8728 30 1460 80 ;-_ 8960 31 1550 81 9192 32 w. 6 is 103 1640 82 9424 33 .... 1743 83 . . . 9656 34 .... 1846 84 9888 35 1949 85 * . 10120 36 2052 86 10352 37 w. 7 13 116 2155 87 10584 38 2271 88 w. 15 13 245 10816 39 2387 89 11061 40 2503 90 11306 41 2619 91 11551 42 2735 92 11796 43 w.8 13 129 2851 93 w. 16 13 258 12041 44 2980 94 12299 45 3109 95 12557 46 3238 96 . . . 12815 47 .... 3367 97 . . . . . 13073 48 Sv.9 is i42 3496 98 13331 49 3638 99 w. 17 is 27i 13589 50 3780 100 > 13860 LIVE-LOAD STRESSES 69 COOPER'S #40. 100'-150' COOPER'S #40. 150'-200' Length Wheel Load Load Sums Moment Sums Length Load Load Sums Moment Sums 100 13860 150 366 29689 101 14131 151 368 30056 102 103 14402 14673 152 153 370 372 30425 30796 104 105 W. 18 13 284 14944 15228 154 155 374 376 31169 31544 106 15512 156 378 31921 107 15796 157 380 32300 108 16080 158 382 32681 109 284 16364 159 384 33064 110 286 16649 160 386 33449 111 288 16936 161 388 33836 112 290 17225 162 390 34225 113 114 115 116 292 294 296 298 17516 17809 18104 18401 163 164 165 166 392 394 396 398 34616 35009 35404 35801 117 300 18700 167 400 36200 118 302 19001 168 402 36601 119 304 19304 169 404 37004 120 121 1 f_! 306 308 19609 19916 170 171 I JH 406 408 37409 37816 122 & 310 20225 172 OH 410 38225 123 a 312 20536 173 02 412 38636 124 125 314 316 20849 21164 174 175 1 o 414 416 39049 39464 126 318 21481 176 a 418 39881 127 320 21800 177 420 40300 128 f\T 322 22121 178 422 40721 129 130 131 132 :!:| 324 326 328 330 22444 22769 23096 23425 179 180 181 182 1077.0 187621.50 376 co" 1227.0 245221.50 327 II 1030.0 188700.00 377 II 1230.0 246450.00 328 T3 1083.0 189781.50 378 1233.0 247681.50 329 a 1086.0 190866.00 379 1 1236.0 248916.00 330 d 1089.0 191953.50 380 w 1239.0 250153.50 331 I 1092.0 193044.00 381 3 1242.0 251394.00 332 <2 1095.0 194137.50 382 1 1245.0 252637.50 333 '3 1098.0 195234.00 383 'd 1248.0 253884.00 334 ^ 1101.0 196333.50 384 tJ 1251.0 255133.50 335 1104.0 197436.00 385 1254.0 256386.00 336 1107.0 198541.50 386 1257.0 257641.50 337 1110.0 199650.00 387 1260.0 258900.00 338 1113.0 200761 . 50 388 1263.0 260161.50 339 1116.0 201876.00 389 1266.0 261426.00 340 1119.0 202993.50 390 1269.0 262693.50 341 1122.0 204114.00 391 1272.0 263964.00 342 1125.0 205237.50 392 1275.0 265237.50 343 1128.0 206364.00 393 1278.0 266514.00 344 1131.0 207493.50 394 1281.0 267793.50 345 1134.0 208626.00 395 1284.0 269076.00 346 1137.0 209761.50 396 1287.0 270361.50 347 1140.0 210900.00 397 1290.0 271650.00 348 1143.0 212041.50 398 1293.0 272941.50 349 1146.0 213186.00 399 1296.0 274236.00 350 1149.0 214333.50 400 1299.0 275533 . 50 80 LIVE-LOAD STRESSES COMMON STANDARD 0'-50' COMMON STANDARD 50'-100' Length Wheel Load Load Sums Moment Sums Length . Wheel Load Load Sums Moment Sums W 1 12 5 12.5 00.00 50 5120.00 1 12 50 51 5312 50 2 25 00 52 5505 00 3 37 50 53 5697.50 4 50 00 54 5890 . 00 5 62 50 55 6082.50 6 75 00 56 W. 10 12.5 205.0 6275.00 7 87 50 57 6480 00 8 W 2 2 7 5 400 100 00 58 6685 . 00 9 140.00 59 6890.00 10 180 00 60 7095.00 11 220 00 61 7300.00 12 260 00 62 7505.00 13 14 W. 3 27.5 67.5 300.00 367 . 50 63 64 W. 11 27.5 232.5 7710.00 7915.00 15 435.00 65 8147.50 16 502 50 66 8380.00 17 570 00 67 8612.50 18 19 w. 4 27.5 95.0 637.50 732.50 68 69 w. 12 27.5 260.0 8845.00 9077.50 20 827 50 70 9337 50 21 922 50 71 9597 . 50 22 1017.50 72 9857.50 23 w 5 27 5 122 5 1112 50 73 10117.50 24 1235.00 74 w. 13 27.5 287.5 10377.50 25 1357 50 75 10665.00 26 1480 00 76 10952.50 27 1602 50 77 11240.00 28 1725.00 78 11527.50 29 30 1847.50 1970 00 79 80 w. 14 27.5 315.0 11815.00 12130.00 31 2092 50 81 12445.00 32 w 6 17 5 1400 2215 00 82 12760 00 33 2355 00 83 13075 . 00 34 2495 00 84 13390.00 35 2635 . 00 85 13705.00 36 2775 00 86 14020.00 37 38 w. 7 17.5 157.5 2915.00 3072.50 87 88 w. 15 ii'.s 332.5 14335.00 14650.00 39 3230 00 89 14982.50 40 3387.50 90 15315.00 41 3545 00 91 15647.50 42 3702 . 50 92 15980.00 43 44 w. 8 17.5 175.0 3860.00 4035.00 93 94 w. 16 17.5 350.0 16312.50 16662.50 45 46 4210.00 4385 00 95 96 17012.50 17362.50 47 4560.00 97 17712.50 48 49 50 w. 9 17.5 192.5 4735.00 4927.50 5120 00 98 99 100 w'. 17 17^5 367.5 18062.50 18412.50 18780.00 LIVE-LOAD STRESSES 81 COMMON STANDARD 100'-150' COMMON STANDARD 150'-200' Length Wheel Load Load Sums Moment Sums Length Load Load Sums Moment Sums 100 18780.00 150 487.5 40061.25 101 19147.50 151 490.0 40550.00 102 19515.00 152 492.5 41041.25 103 19882.50 153 495.0 41535.00 104 W. 18 17.5 385^6 20250.00 154 497.5 42031.25 105 20635 . 00 155 500.0 42530.00 106 21020.00 156 502.5 43031.25 107 21405.00 157 505.0 43535.00 108 21790.00 158 507.5 44041.25 109 385 '. 6 22175.00 159 510.0 44550.00 110 387.5 22561.25 160 512.5 45061.25 111 390.0 22950.00 161 515.0 45575.00 112 392.5 23341.25 162 517.5 46091.25 113 395.0 23735.00 163 520.0 46610.00 114 397.5 24131.25 164 522.5 47131.25 115 400.0 24530.00 165 525.0 47655.00 116 a 402.5 24931.25 166 +j 527.5 48181.25 117 405.0 25335.00 167 M 530.0 48710.00 118 W 407.5 25741.25 168 532.5 49241.25 119 1 410.0 26150.00 169 & VI 535.0 49775.00 120 a 412.5 26561.25 170 1 537.5 50311.25 121 d H 415.0 26975.00 171 g 540.0 50850.00 122 1 417.5 27391.25 172 a 542.5 51391.25 123 M4 420.0 27810.00 173 o 545.0 51935.00 124 o 422.5 28231.25 174 1C 547.5 52481.25 125 <>f 425.0 28655.00 175 of 550.0 53030.00 126 427.5 29081.25 176 II 552.5 53581.25 127 430.0 29510.00 177 T3 555.0 54135.00 128 li 432.5 29941.25 178 O 557.5 54691.25 129 nS 435.0 30375.00 179 g 560.0 55250.00 130 . a 437.5 30811,25 180 562.5 55811.25 131 1 440.0 31250.00 181 '3 565.0 56375.00 132 c 442.5 31691.25 182 P 567.5 56941.25 133 P 445.0 32135.00 183 570.0 57510.00 134 447.5 32581.25 184 572.5 58081.25 135 450.0 33030.00 185 575.0 58655.00 136 452.5 33481.25 186 577.5 59231.25 137 455.0 33935.00 187 580.0 59810.00 138 457.5 34391.25 188 582.5 60391.25 139 460.0 34850.00 189 585.0 60975.00 140 462.5 35311.25 190 587.5 61561.25 141 465.0 35775.00 191 590.0 62150.00 142 467.5 36241.25 192 592.5 62741.25 143 470.0 36710.00 193 595.0 63335.00 144 472.5 37181.25 194 597.5 63931.25 145 475.0 37655.00 195 600.0 64530.00 146 477.5 38131.25 196 602.5 65131.25 147 480.0 38610.00 197 605.0 65735.00 148 482.5 39091.25 198 607.5 66341.25 149 485.0 39575.00 199 610.0 66950.00 150 487.5 40061.25 200 612.5 67561.25 82 LIVE-LOAD STRESSES COMMON STANDARD 200 '-250' COMMON STANDARD 250 '-300' Length Load Load Sums Moment Sums Length Load Load Sums Moment Sums 200 612.5 67561.25 250 737.5 101311.25 201 615.0 68175.00 251 740.0 102050.00 202 617.5 68791.25 252 742.5 102791.25 203 620.0 69410.00 253 745.0 103535.00 204 622.5 70031.25 254 747.5 104281.25 205 625.0 70655.00 255 750.0 105030.00 206 627.5 71281.25 256 752.5 105781.25 207 630.0 71910.00 257 755.0 106535.00 208 632.5 72541.25 258 757.5 107291.25 209 635.0 73175.00 259 760.0 108050.00 210 637.5 73811.25 260 762.5 108811.25 211 640.0 74450.00 261 765.0 109575.00 212 642.5 75091.25 262 767.5 110341.25 213 645.0 75735.00 263 770.0 111110.00 214 647.5 76381.25 264 772.5 111881.25 215 650.0 77030.00 265 775 '.0 112655.00 216 +3 652.5 77681.25 266 g 777.5 113431.25 217 1 655.0 78335.00 267 1 780.0 114210.00 218 (_, 657.5 78991.25 268 (H 782.5 114991.25 219 & 660.0 79650.00 269 I 785.0 115775.00 220 a 662.5 80311.25 270 3 787.5 116561.25 221 j 665.0 80975.00 271 790.0 117350.00 222 8< 667.5 81641.25 272 a 792.5 118141.25 223 Q 670.0 82310.00 273 795.0 118935.00 224 io 672.5 82981.25 274 g 797.5 119731.25 225 of 675.0 83655.00 275 c 1030.00 204710.00 318 8 907.5 157241.25 368 1032.50 205741.25 319 M Ld 910.0 158150.00 369 IM 1035.00 206775.00 320 ft 912.5 159061.25 370 1037.50 207811.25 321 915.0 159975.00 371 -8 1040.00 208850.00 322 917.5 160891.25 372 s 1042.50 209891.25 323 8, 920.0 161810.00 373 i 1045.00 210935.00 324 A 922.5 162731.25 374 M< 1047.50 211981.25 325 S 925.0 163655.00 375 o 1050.00 213030.00 326 ~ 927.5 164581.25 376 ^ 1052 . 50 214081.25 327 s ! 930.0 165510.00 377 c^ 1055.00 215135.00 328 -d 932.5 166441.25 378 T3 1057.50 216191.25 329 985.0 167375.00 379 J 1060.00 217250.00 330 a 937.5 168311.25 380 Ej 1062.50 218311.25 331 E o 940.0 169250.00 ,381 1065.00 219375.00 332 1 942.5 170191.25 382 1 1067.50 220441.25 333 1 945.0 171135.00 383 p ;5 1070.00 221510.00 334 947.5 172081.25 384 1072.50 222581.25 335 950.0 173030.00 385 1075 . 00 223655.00 336 952.5 173981.25 386 1077.50 224731.25 337 955.0 174935.00 387 1080.00 225810.00 338 957.5 175891.25 388 1082.50 226891.25 339 960.0 176850.00 389 1085.00 227975.00 340 962.5 177811.25 390 1087.50 229061.25 341 965.0 178775.00 391 1090.00 230150.00 342 967.5 179741.25 392 1092.50 231241.25 343 970.0 180710.00 393 1095.00 232335.00 344 972.5 181681.25 394 1097.50 233431.25 345 975.0 182655.00 395 1100.00 234530.00 346 977.5 183631.25 396 1102.50 235631.25 347 980.0 184610.00 397 1105.00 236735.00 34S 982.5 185591 .25 398 1107.50 237841.25 349 985.0 186575.00 399 1110.00 238950.00 350 987.5 187561.25 400 1112.50 240061.25 84 LIVE-LOAD STRESSES LACKAWANNA 0'-50' LACKAWANNA 50'-100' Length Wheel Load Load Sums Moment Sums Length Wheel Load Load Sums Moment Sums 1 W. 1 11 11.00 00.000 11 000 50 51 4744.000 4911 000 2 22.000 52 5078 000 3 33 000 53 5245 000 4 44 000 54 w 10 11 178 00 5412 000 5 55 000 55 5590 000 6 7 W. 2 25 36.66 66.000 77 000 56 57 5768.000 5946 000 8 113.000 58 6124 000 9 10 149.000 185.000 59 60 6302.000 6480 000 11 12 13 ' W.' 3 25 'ei'.oo 221.000 257.000 318 000 61 62 63 W. 11 25 203.00 6658.000 6861.000 7064 000 14 379 000 64 7267 000 15 440 000 65 7470 000 16 17 w. 4 25 86.66 501.000 562.000 66 67 w. 12 25 228.66 7673.000 7901 000 18 648.000 68 8129 000 19 734.000 69 8357 000 20 820.000 70 8585 000 21 22 23 ' w.' 5' 25 iii'.oo 906.000 992.000 1103 000 71 72 73 w. 13 25 253.00 8813.000 9066.000 9319 000 24 25 1214.000 1325 000 74 75 9572.000 9825 000 26 37 1436.000 1547.000 76 77 w. 14 25 278.00 10078.000 10356 000 28 1658.000 78 10634 000 29 1769 000 79 10912 000 30 1880.000 80 11190 000 31 32 w. 6 14 125.00 1991.000 2116 000 81 82 11468.000 11746 000 33 2241 000 83 12024 000 34 2366 000 84 12302 000 35 36 37 w. 7 i4 139'.00 2491.000 2616.000 2755 . 000 85 86 87 w. 15 14 292.00 12580.000 12872.000 13146 000 38 39 2894.000 3033.000 88 89 13456.000 13748 000 40 41 42 43 44 ' w.' 8' 14 153.00 3172.000 3311.000 3464.000 3617.000 3770 000 90 91 92 93 94 w. 16 14 306.00 14040.000 14346.000 14652.000 14958.000 15264 000 45 3923 000 95 w 17 14 320.00 15570 000 46 47 w. 9 14 167.00 4076.000 4243.000 96 97 15890.000 16210 000 48 4410 000 98 16530 000 49 4577 000 99 16850 000 50 4744.000 100 w. 18 14 334.00 17170.000 LIVE-LOAD STRESSES 85 LACKAWANNA 100'-150' LACKAWANNA 150'-200' Length Wheel Load Load Sums Moment Sums Length Load Load Sums Moment Sums 103 W. 18 14 334.00 17170.000 150 437.50 36250.500 101 17504.000 151 439.75 36689.125 102 17838.000 152 442.00 37130.000 103 18172 . 000 153 444.25 37573.125 104 334^00 18506.000 154 446.50 38018.500 105 336.25 18841 . 125 155 448.75 38466.125 106 338.50 19178.500 156 451.00 38916.000 107 340.75 19518.125 157 453.25 39368.125 108 343.00 19860.000 158 455.50 39822.500 109 345.25 20204.125 159 457.75 40279.125 110 347.50 20550.500 160 460.00 40738.000 111 349.75 20899.125 161 462.25 41199.125 112 352.00 21250.000 162 464.50 41662.500 113 354.25 21603 . 125 163 466.75 42128.125 114 356.50 21958.500 164 469.00 42596.000 115 358.75 22316.125 165 471.25 43066.125 116 361.00 22676.000 166 473.50 43538.500 117 t 363.25 23038.125 167 43 475.75 44013 . 125 118 a 365.50 23402.500 168 2 478.00 44490.000 119 & 367.75 23769.125 169 I 480.25 44969 . 125 120 02 370.00 24138.000 170 CQ 482.50 45450.500 121 T3 372.25 24509.125 171 TS a 484.75 45934.125 122 374.50 24882.500 172 487.00 46420.000 123 o R 376.75 25258.125 173 a 489.25 46908.125 124 o 379.00 25636.000 174 o 491.50 47398.500 125 t 53.7 58.4 74.13 13 0.13 58.4 63.2 75.37 13 1.37 63.2 70.00 74.07 13 0.07 .... NOTE. For spans greater than 70 feet, the maximum centre moment equals the absolute maximum bending moment with an error of less than one per cent. TABLE 5 POSITION OF COOPER'S LOADINGS FOR MAXIMUM END SHEAR IN GIRDER BRIDGES WITHOUT PANELS Span Direction Load Moves Position of Load Location of Maximum Shear O'to 23' 23 " 27 27 " 46 46 " 62 62 " 400 Right to left Right to left Right to left Right to left Right to left w z at left end Wn at right end Wz at left end w\\ at left end w-z at left end Left end Right end Left end Left end Left end 90 LIVE-LOAD STRESSES TABLE 6 POSITION OF COOPER'S LOADINGS FOR MAXIMUM SHEAR IN PANELS OF GIRDER AND TRUSS BRIDGES Number of Panels Panel PANEL LENGTH IN FEET 22 23 24 25 26 27 28 29 30 31 32 33 34 35 6 7 0-1 1-2 2-3 3-4 4-5 0-1 1-2 2-3 3^ 4-5 5-6 0-1 1-2 2-3 3-4 4-5 5-6 6-7 0-1 1-2 2-3 3^t 4-5 5-6 6-7 7-8 0-1 1-2 2-3 3 4 4-5 5-6 6-7 7-8 8-9 4 3 3 2 2 4 3 3 3 2 2 3 3 3 3 2 2 2 3 3 3 3 2 2 2 2 3 3 3 3 3 2 2 2 1 4 3 3 2 2 4 3 3 3 2 2 4 3 3 3 2 2 2 5 4 3 3 2 4 4 3 3 2 2 5 4 3 3 2 5 4 3 3 2 2 5 4 4 3 3 2 2 5 4 3 3 2 5 4 4 3 3 2 5 4 4 3 3 2 2 5 5 4 4 3 2 5 4 4 3 3 2 5 4 4 3 3 2 2 5 3 3 2 2 4 3 3 3 2 2 4 3 3 3 2 2 2 3 3 2 2 4 3 3 3 2 2 4 3 3 3 2 2 2 4 3 2 2 4 4 3 3 2 2 4 4 3 3 2 2 2 4 3 2 2 4 4 3 3 2 2 4 4 3 3 3 2 2 4 3 2 2 4 4 .3 3 2 2 4 3 2 2 4 4 3 3 2 2 4 3 2 2 4 4 3 3 2 2 4 3 3 2 4 4 3 3 2 2 8 9 10. .. . 4 3 3 3 2 2 4 3 3 3 2 2 4 3 3 3 2 2 4 4 3 3 2 2 4 4 3 3 2 2 3 3 3 3 2 2 2 4 3 3 3 3 2 2 2 1 3 3 3 3 2 2 2 4 3 3 3 3 2 2 2 1 3 3 3 3 2 2 2 4 3 3 3 3 2 2 2 1 4 3 3 3 2 2 2 4 4 3 3 3 2 2 2 1 4 3 3 3 2 2 2 4 4 3 3 3 2 2 2 1 4 3 3 3 2 2 2 4 4 3 3 3 3 2 2 1 3 3 3 2 2 2 4 4 3 3 3 3 2 2 1 4 3 3 2 2 2 4 4 4 3 3 3 2 2 2 4 3 3 3 2 2 4 4 4 3 3 3 2 2 2 4 3 3 3 2 2 4 4 4 3 3 3 2 2 2 4 3 3 3 2 2 4 4 4 3 3 3 2 2 2 4 3 3 3 2 2 5 4 4 3 3 3 2 2 2 4 3 3 3 2 2 5 4 4 4 3 3 2 2 2 NOTE. Place tabulated wheel at right end of corresponding panel with locomotive ad- vancing toward left. LIVE-LOAD STRESSES 91 TABLE 7 MAXIMUM MOMENTS, SHEARS, AND PIER REACTIONS FOR COOPER'S STANDARD LOADINGS (Figures for One Rail) Span 40 EoO Max. Moment Max. Shears Max. Pier React. Max. Moment Max. Shears Max. Pier React. End y* pt. Cent. End MPt. Cent. 10 56.3 65.7 80.0 95.0 110.0 125.0 140.0 155.0 170.0 186.6 206.3 226.0 245.7 265.4 2S5.2 305.0 324.8 344.6 365.5 388.0 410.5 432.9 455.4 477.9 500.6 523.0 548.6 574.3 600.0 625.6 655.6 684.6 713.6 742.6 771.6 800.6 829.8 858.6 887.6 918.8 950.9 983.1 1015.2 1047.4 30.0 32.7 35.0 36.9 38.6 40.0 42.5 44.7 46.7 48.4 50.0 51.4 52.7 53.9 55.4 56.8 58.1 59.2 60.4 61.6 63.0 64.4 65.7 66.9 68.1 69.2 70.6 71.9 73.1 74.3 75.4 76.8 78.4 79.4 80.6 81.7 82.8 83.8 85.0 86.1 87.2 88.4 89.3 90.5 20.0 20.9 21.7 22.3 23.6 25.0 26.3 27.4 28.3 29.2 30.0 31.4 32.7 33.9 35.0 38.0 38.9 37.8 38.6 39.3 40.0 40.7 41.3 42.0 42.8 43.5 44.1 44.8 45.4 46.0 48.8 47.5 48.2 48.9 49.5 50.1 50.7 51.4 52.1 52.8 53.5 54.1 54.8 55.4 10.0 10.9 11.7 12.3 12.9 13.3 13.7 13.8 13.9 14.0 14.0 14.5 15.0 15.4 15.8 16.2 16.5 16.9 17.1 17.4 17.7 18.2 18.8 19.2 19.7 20.1 20.6 21.0 21.3 21.7 22.0 22.3 22.6 22.9 23.2 23.4 23.7 23.9 24.2 24.5 24.9 25.2 25.5 25.8 40.0 43.7 46.7 49.2 52.2 54.7 56.9 58.8 60.7 62.9 65.6 68.0 70.2 72.2 74.0 75.7 77.7 80.2 82.3 84.4 86.3 88.5 91.0 93.3 95.5 97.5 99.6 101.5 103.7 105.9 108.0 110.0 112.1 114.3 116.5 118.6 120.7 122.7 124.8 126.8 128.7 131.0 133.3 135.6 70.4 82.1 100.0 118.8 137.5 156.3 175.0 193.8 212.5 233.3 257.9 282.5 307.1 331.8 356.5 381.3 406.0 430.8 456.9 485.0 513.0 541.1 569.3 597.4 625.8 653.8 685.8 717.9 750.0 783.3 819.5 855.8 892.0 928.3 964.5 1000.8 1037.3 1073.3 1109.5 1148.5 1188.6 1228.9 1269.0 1309.2 37.5 40.9 43.8 46.2 48.2 50.0 53.1 55.9 58.3 60.5 62.5 64.3 65.9 67.4 69.3 71.0 72.6 74.0 75.5 76.9 78.8 80.5 82.1 83.7 85.1 86.5 88.2 89.8 91.4 92.9 94.3 96.0 97.6 99.2 100.7 102.1 103.5 104.9 106.3 107.7 109.0 110.4 111.8 113.1 25.0 26.1 27.1 27.9 29.5 31.3 32.9 34.3 35.4 36.5 37.5 39.2 40.9 42.4 43.8 45.0 46.1 47.2 48.2 49.1 50.0 50.9 51.8 52.5 53.5 54.4 55.1 56.0 56.7 57.5 58.5 59.4 60.2 61.1 61.9 62.6 63.4 64.2 65.1 66.0 66.8 67.6 68.5 69.2 12.5 13.6 14.6 15.4 16.2 16.6 17.1 17.3 17.4 17.5 17.5 18.1 18.8 19.3 19.8 20.2 20.6 21.1 21.4 21.8 22.1 22.7 23.4 24.0 24.6 25.1 25.8 26.2 26.6 27.1 27.5 27.9 28.3 28.6 29.0 29.3 29.6 29.9 30.2 30.6 31.1 31.5 31.9 32.3 50.0 54.5 58.4 61.6 65.2 68.3 71.1 73.5 75.9 78.6 81.9 84.9 87.6 90.2 92.4 94.6 97.1 100.1 102.8 105.4 107.9 110.6 113.7 116.7 119.4 122.0 124.4 126.9 129.7 132.3 135.0 137.6 140.2 142.9 145.6 148.3 150.9 153.4 156.0 158.5 161.0 163.6 166.6 169.6 11 12 13 14 . ... 15 16 17 18 19 20 21 22 23 24 25 28 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 . 43 44 45 46 47 48 49 50 51. ... 52 53 92 LIVE-LOAD STRESSES TABLE 7. Continued MAXIMUM MOMENTS, SHEARS, AND PIER REACTIONS FOR COOPER'S STANDARD LOADINGS (Figures for One Rail) Span #40 50 Max. Moment Max. Shears Max. Pier React. Max. Moment Max. Shears Max. Pier React. End &Pt. Cent. End KPt. Cent. 54.. 55. . 1081.4 1116.9 1152.4 1187.9 1223.4 1261.0 1299.6 1338.3 1377.0 1415.6 1455.5 1497.5 1539.5 1581.5 1623.5 1665.5 1707.5 1749.3 1793.0 1833.9 1879.2 1925.8 1972.0 2019.1 2065.0 2112.3 2160.5 2207.7 2256.7 2306.5 2356.3 2406.9 2459.6 2510.6 2564.2 2615.9 2670.5 2723.0 2776.7 2831.5 2885.3 2939.5 2994.5 3049.0 91.5 92.6 93.7 94.8 95.9 97.0 98.0 99.2 100.1 101.3 102.6 103.8 105.0 106.4 107.8 109.2 110.5 111.8 113.3 114.8 116.3 117.7 119.1 120.4 121.7 123.0 124.2 125.6 126.9 128.2 129.5 130.7 132.1 133.4 134.7 136.0 137.2 138.5 139.8 141.1 142.4 143.6 144.8 146.2 56.1 56.8 57.5 58.2 58.8 59.5 60.1 60.7 61.3 61.8 62.4 63.0 63.6 64.2 64.8 65.4 65.9 66.5 67.0 67.5 68.0 68.6 69.2 69.9 70.5 71.1 71.7 72.3 73.0 73.7 74.4 75.1 75.8 76.5 77.1 77.9 78.7 79.5 80.3 81.0 81.7 82.5 83.3 84.2 26.1 26.4 26.6 26.9 27.2 27.5 27.9 28.2 28.5 28.8 29.1 29.4 29.7 30.0 30.2 30.5 30.7 31.1 31.4 31.7 32.0 32.3 32.6 32.9 33.2 33.4 33.7 34.0 34.4 34.7 35.0 35.3 35.6 35.9 36.2 36.5 36.7 37.0 37.3 37.5 37.8 38.0 38.3 38.5 138.0 140.3 142.7 145.4 148.1 150.6 153.2 155.7 158.2 160.4 162.6 165.2 167.8 170.1 172.5 174.8 177.1 179.3 181.5 183.7 186. C 188.2 190.4 192.5 194.7 196.8 198 ..9 200.9 203.0 205.0 206.9 208.9 210.8 212.8 214.7 216.7 218.6 220.6 222 5 224.4 226.3 228 . 1 230.0 231.8 1351.8 1396 . 1 1440.5 1484.9 1529.2 1576.2 1624.5 1672.9 1721.2 '1769.5 1819.4 1871.9 1924.4 1976.9 2029.4 2081.9 2134.4 2186.6 2241.2 2292.4 2349.0 2407.3 2465.0 2523.9 2581.2 2640.4 2700.6 2759.6 2820.9 2883 . 1 2945.4 3008.6 3074.5 3138.3 3205.3 3269.9 3338 . 1 3403 . 7 3470.9 3539 . 3 3606.6 3674 . 3 3743 . 1 3811.2 114.5 115.8 117.2 118.5 119.8 121.2 122.5 123.9 125.2 126.6 128.2 129.7 131.2 133.0 134.8 136.5 138.1 139.8 141.7 143.5 145.3 147.1 148.8 150.5 152.1 153.8 155.3 157.0 158.6 160.3 161.8 163.4 165.1 166.8 168.4 170.0 171.5 173.1 174.7 176.4 178.0 179.5 181.0 182.7 70.1 71.0 71.8 72.7 73.5 74.4 75.2 76.0 76.6 77.4 78.0 78,8 79.5 80.3 81.0 81.7 82.4 83.1 83.8 84.4 85.0 85.7 86.5 87.4 88.2 88.9 89.6 90.4 91.2 92.1 93.0 93.9 94.3 95.7 96.5 97.4 98.4 99.4 100.4 101.2 102.1 103.1 104.1 105.1 32.6 33.0 33.3 33.6 34.0 34.4 34.9 35.2 35.6 36.0 36.4 36.8 37.1 37.5 37.8 38.1 38.4 38.8 39.2 39.6 40.0 40.4 40.8 41.1 41.5 41.7 42.1 42.5 43.0 43.4 43.7 44.1 44.5 44.9 45.2 45.6 45.9 46.2 46.6 46.9 47.3 47.5 47.9 48.1 172.5 175.4 178.5 181.8 185.1 188.4 191.5 194.7 197.7 200.7 203.6 206.7 209.7 212.7 215.6 218.5 221.3 224.1 226.9 229.6 232.4 235.2 238.0 240.7 243.3 245.9 248.6 251.1 253.6 256.1 258.7 260.8 263.0 265.6 268.3 270.8 273.2 275.6 278.0 280.3 282.7 285.1 287.5 2F9.7 56 57 58 59 60 61. . 62. . . 63 64 65 66 67 68 69 70 71 72 73 74. . 75. . 76 77 78 79 80. . 81 82 83 84 85. 86. . 87.. 88 89 90 91 92 93 94. .. 95 96 97 LIVE-LOAD STRESSES 93 TABLE 7. Continued MAXIMUM MOMENTS, SHEARS AND PIER REACTIONS FOR COOPER'S STANDARD LOADINGS (Figures for One Rail) Span E40 750 Max. Moment Max. Shears Max. Pier React. Max. Moment Max. Shears Max. Pier React. End MPt. Cent. End HPt. Cent. 98 . ... 3106.5 3162.3 3219.9 3277.6 3335.9 3410.6 3475.2 3537.6 3600.3 3666.6 3745.3 3818.4 3886.8 3958.2 4026.9 4099.0 4172.0 4245.0 4318.8 4389.5 4463.8 4538.8 4614.1 4686.5 4762.7 4836.2 4917.4 4996.4 7062.3 9352.5 11873.0 17592.5 147.5 148.8 150.0 151.2 152.4 153.7 154.9 156.1 157.3 158.5 159.6 160.8 162.0 163.2 164.4 165.5 165.7 167.9 169.0 170.2 171.4 172.5 173.7 174.8 176.0 177.1 178.3 179.4 207.4 234.5 261.0 313.2 85.0 85.8 86.6 87.3 88.1 88.8 89.5 90.3 90.9 91.7 92.4 93.2 93.9 94.6 95.3 96.0 98.8 97.5 98.3 99.0 99.7 100.4 101.1 101.8 102.5 103.2 104.0 104.7 121.8 138.3 153.4 183.7 38.8 39.1 39.4 39.6 39.9 40.1 40.4 40.6 40.9 41.1 41.3 41.6 41.8 42.0 42.2 42.5 42.8 43.1 43.4 43.7 43.9 44.2 44.5 44.7 45.0 45.3 45.7 46.0 54.4 62.5 70.4 85.0 233.6 235.4 237.2 238.9 240.6 242.4 244.2 246.0 247.8 249.6 251.4 253.1 254.8 256.5 258.2 259.9 281.6 263.3 264.9 266.7 268.5 270.2 272.0 273.8 275.6 277.4 279.2 281.0 325.4 371.7 419.0 515.2 3883.1 3952.9 4024.9 4097.0 4169.9 4263.3 4344.0 4422.0 4500.4 4583.3 4681.6 4773.0 4858.5 4947.7 5033.6 5123.8 5215.0 5306.2 5398.5 5486.9 5579.7 5673.5 5767.6 5858.1 5953.4 6045.2 6146.7 6245.5 8827.9 11690.6 14841.2 21990.6 184.3 186.0 187.5 189.0 190.6 192.1 193.6 195.1 196.6 198.1 199.5 201.0 202.5 204.0 205.5 207.0 208.4 209.9 211.3 212.8 214.2 215.7 217.1 218.6 220.0 221.4 222.8 224.2 259.2 293.1 326.3 391.5 106.2 107.2 108.2 109.1 110.1 111.0 111.9 112.7 113.6 114.5 115.5 116.4 117.4 118.2 119.1 120.0 121.0 121.9 122.9 123.7 124.6 125.5 126.4 127.2 128.1 129.0 130.0 130.9 152.2 172.9 191.8 229.6 48.5 48.9 49.2 49.5 49.9 50.1 50.5 50.7 51.1 51.5 51.7 52.0 52.3 52.5 52.7 53.1 53.5 53.9 54.2 54.6 54.9 55.3 55.6 55.9 56.2 56.5 57.0 57.5 68.0 78.2 88.0 106.3 292.0 294.2 296.5 298.6 300.8 303.0 305.3 307.5 309.8 312.0 314.2 316.3 318.5 320.7 322.8 324.9 327.0 329.0 331.1 333.3 335.6 337.8 340.0 342.2 344.5 346.7 349.0 351.2 406.7 464.6 523.8 644.0 99 100 101 102 103 104 105 106 107 108 109 110 111 112 . 113 114 115 116 117 118 119 120 121 122 123 124 125 150 175 200 250 NOTES. Moments are given in thousand foot-pounds. Shears are given in thousand pounds. Pier reactions are given in thousand pounds and are for piers between two spans each equal to the tabulated span. 94 LIVE-LOAD STRESSES TABLE 8 MAXIMUM MOMENTS FOR TRUSS BRIDGES COOPER'S #50 FOR ONE RAIL Moments Given in Thousands of Foot-Pounds Panel Points h Panels in Truss 1? ll PANEL LENGTHS 8'0" 8' 6" 9'0" 9' 6" 10' 0" 10' 6" 11' 0" 11' 6" 12' 0" 12' 6" 13' 0" 13' 6" 3 1 325 359 392 425 464 503 541 580 619 661 707 755 4 1 2 433 569 483 625 533 683 582 747 632 819 688 892 743 964 799 1037 859 1110 918 1189 982 1269 1046 1352 5 1 2 540 790 599 877 662 964 728 1051 794 1149 861 1255 930 1361 1001 1468 1071 1574 1140 1675 1217 1792 1298 1910 6 1 2 3 641 1008 1109 710 1115 1221 784 1228 1351 859 1347 1484 937 1466 1618 1017 1587 1767 1100 1719 1925 1186 1857 2070 1280 1997 2240 1375 2135 2407 1485 2289 2581 1600 2451 2760 7 1 2 3 731 1215 1425 812 1344 1577 896 1477 1739 984 1615 1910 1080 1758 2086 1184 1904 2269 1293 2070 2465 1411 2252 2667 1530 2441 2879 1645 2642 3100 1775 2849 3332 1906 3050 3560 8 1 2 3 4 819 1402 1716 1819 915 1553 1899 2030 1021 1709 2100 2240 1133 1872 2311 2465 1254 2061 2529 2700 1375 2273 2752 2946 1501 2490 2991 3205 1631 2708 3241 3471 1776 2933 3498 3743 1900 3165 3775 4025 2047 3405 4078 4344 2200 3649 4383 4681 9 1 2 3 4 621 1583 1997 2208 1039 1764 2215 2459 1162 1960 2451 2719 1287 2179 2700 2997 1418 2405 2986 3291 1556 2642 3276 3592 1697 2888 3570 3899 1844 3139 3877 4226 1997 3400 4194 4588 2145 3670 4532 4970 2309 3946 4887 5370 2475 4224 5242 5770 l! !! PANEL LENGTHS 14' 0" 14' 6" 15' 0" 15' 6" 16' 0" 16' 6" 17' 0" 17' 6" 18' 0" 18' 6" 19' 0" 3 i 803 850 900 952 1008 1060 1115 1170 1228 1285 1347 4 i 2 1115 1441 1183 1529 1255 1624 1325 1721 1402 1820 1463 1924 1553 2030 1614 2134 1709 2240 1776 2349 1872 2465 5 . 1 2 1389 2047 1480 2177 1581 2310 1680 2440 1788 2581 1896 2725 2010 2881 2123 3030 2242 3190 2355 3350 2477 3518 6 1 2 3 1724 2616 2946 1840 2792 3138 1965 2986 3338 2090 3175 3539 2221 3372 3742 2352 3570 3953 2489 3775 4170 2626 3978 4422 2769 4194 4681 2910 4415 4948 3062 4650 5215 7 1 2 3 2047 3263 3802 2185 3485 4040 2332 3723 4310 2480 3958 4595 2634 4202 4898 2787 4450 5200 2945 4705 5509 3104 4958 5815 3268 5218 6135 3434 5480 6460 3605 5748 6800 8 1 2 3 4 2358 3900 4710 5034 2516 4165 5040 5398 2681 4436 5380 5768 2846 4710 5720 6147 3019 4994 6072 6516 3190 5280 6430 6915 3372 5576 6806 7331 3553 5873 7180 7740 3741 6180 7573 8163 3930 6487 7985 8595 4125 6805 8369 9043 9 1 2 3 4 2651 2828 4512 4804 5617* 5993 6187 6610 3012 5107 6390 7040 3196 5420 6790 7485 3389 5747 7204 7966 3583 6074 7620 6460 3785 6414 8054 8980 3987 6755 8496 9490 4198 7108 8959 10010 4410 7463 9415 10530 4629 7830 9892 11065 LIVE-LOAD STRESSES 95 TABLE 8. Continued MAXIMUM MOMENTS FOR TRUSS BRIDGES COOPER'S E50 FOR ONE RAIL Moments Given in Thousands of Foot-Pounds Panel Points h 4- 1 ll (5.2 3 1 1 PANEL LENGTHS 19' 6" 20' 0" 20' 6" 21' 0" 21' 6" 22' 0" 22' 6" 23 ' 0" 23' 6" 24' 0" 24' 6" 1404 1466 1527 1587 1653 1719 1788 1857 1927 1997 2066 4 1 2 1958 2581 2061 2700 2166 2821 2273 2946 2380 3074 2490 3205 2597 3338 2708 3471 2819 3607 2933 3743 3046 3883 5 1 2 2600 3685 2731 3943 2864 4144 3001 4347 3138 4555 3279 4767 3418 4978 3562 5193 3705 5415 3852 5640 3999 5865 6 1 2 3 3210 4885 5487 3362 5256 5746 3516 5501 6028 3678 5750 6321 3840 5998 6617 4008 6250 6921 4175 6501 7228 4349 6756 7538 4522 7011 7850 4700 7270 8166 4878 7525 8491 7 1 2 3 3778 6025 7140 3955 6326 7646 4130 6613 7990 4317 6914 8347 4505 7215 8710 4702 7530 9079 4897 7845 9448 5100 8173 9826 5303 8503 10207 5512 8842 10609 5721 9182 11017 8 1 2 3 4 4320 7125 8780 94,0 4525 7458 9234 9943 4727 7805 9330 10396 4939 8162 10070 10862 5150 8520 10515 11317 5373 8890 10993 11805 5592 9260 11475 12288 5829 9640 11976 12790 6061 10030 12472 13287 6300 10430 12981 13795 6540 10832 13490 14300 9 1 2 3 4 4850 8198 10372 11605 5379 8578 10880 12172 5308 8970 11375 12735 5545 9378 11900 13310 5780 9790 12425 13880 6030 10216 12978 14472 6280 10640 13535 15068 6542 11082 14118 15684 6804 11525 14705 16300 7074 11985 15308 16930 7344 12448 15910 17560 I 12 I.S ll 12 PANEL LENGTHS 25' 0" 25' 6" 26' 0" 26' 6" 27' 0" 27' 6" 28' 0" 28' 6" 29' 0" 29' 6" 30' 0" 3 1 2135 2215 2289 2370 2451 2534 2616 2700 2792 2889 2986 4 1 2 3165 4025 3282 4170 3405 4344 3526 4501 3649 4681 3774 4858 3900 5034 4031 5215 4165 5398 4300 5580 4436 5768 5 1 2 4150 6093 4301 6371 4456 6552 4611 6783 4770 7017 4929 7250 5092 7492 5255 7736 5422 7984 5589 8232 5760 8482 6 1 2 3 5061 7794 8821 5245 8088 9153 5433 8352 9490 5622 8654 9828 5816 8960 10170 6010 9268 10514 6208 9580 10862 6408 9897 11208 6612 10218 11565 6817 10547 11925 7026 10880 12296 7 1 2 3 5936 9530 11444 6151 9875 11870 6373 10236 12312 6595 10600 12752 6823 10980 13203 7051 11357 13653 7286 11742 14112 7521 12125 14571 7762 12520 15039 8003 12918 15507 8250 13330 15984 8 1 2 3 4 6787 11244 14010 14820 7035 11655 14528 15340 7289 12080 15063 15875 7540 12508 15605 16413 7806 12950 16163 16965 8069 13392 16718 17514 8338 13850 17285 18075 8608 14308 17852 18635 8887 14780 18431 19210 9165 15250 19010 19795 9450 15730 19600 20406 9 1 2 3 4 7622 12925 16528 18205 7900 13400 17145 18850 8188 13890 17778 19515 8477 14380 18414 20180 8774 14888 19070 20870 9070 15400 19730 21557 9376 15930 20405 22260 9686 16460 21080 22955 9996 17005 21770 23678 10310 17547 22461 24405 10633 18100 23168 25170 96 LIVE-LOAD STRESSES TABLE 8. Continued MAXIMUM MOMENTS FOR TRUSS BRIDGES COOPER'S "50 FOR ONE RAIL Moments Given in Thousands of Foot-Pounds Panel Points I Panels 1 in Truss ll && PANEL LENGTHS 30' 6" 31' 0" 31' 6" 32' 0" 32' 6" 33' 0" 33' 6" 34' 0" 34' 6" 35' 0" 35' 6" 3 1 3080 3175 3276 3372 3471 3570 3672 3775 3877 3978 4080 4 1 2 4573 5957 4710 6147 4852 6332 4994 6516 5137 6715 5280 6915 5428 7123 5576 7331 5725 7535 5873 7740 5923 7950 5 1 2 5937 8734 6113 8986 6295 9241 6477 9496 6678 9749 6849 10012 7039 10291 7228 10590 7423 10891 7617 11192 7814 11495 6 1 2 3 7238 11219 12668 7450 11558 13040 7671 11903 13418 7892 12248 13796 8120 12684 14180 8347 12979 14563 8581 13354 14952 8812 13729 15341 9050 14120 15745 9288 14510 16148 9628 14902 16654 7 1 2 3 8501 13748 16474 8752 14165 16964 9009 14590 17466 9266 15015 17968 9536 15460 18475 9806 15885 18981 10081 16358 19508 10355 16810 20015 10637 17284 20545 10919 17758 21024 11203 18234 21606 8 1 2 3 4 9740 16225 20206 21022 10030 16720 20812 21638 10326 17227 21432 22268 10622 1^33 22051 22898 10931 18252 22685 23549 11239 18770 23318 24200 11557 19311 23960 24860 11874 19852 24601 25531 12200 20407 25261 26216 12526 20961 25920 26901 12856 21518 26585 27590 9 1 2 3 4 10961 18672 23886 25943 11288 19244 24603 26715 11625 19832 25343 27498 11961 20419 26083 28281 12310 21019 26839 29096 1265C 21618 27595 29910 13018 22239 28365 30741 13378 22860 29135 31572 13747 23503 29923 32431 14116 24146 30710 33290 14490 24795 31500 34155 LIVE-LOAD STRESSES 97 TABLE 9 MAXIMUM SHEARS FOE TRUSS BRIDGES COOPER'S #50 FOE ONE RAIL Shears Given in Thousands of Pounds Panels d "3 PANEL LENGTHS l c 2 8' 0" 8' 6" 9' 0" 9' 6" 10' 0" 10' 6" 11' 0" 11' 6" 12' 0" 12' 6" 13' 0" 13' 6" 3 i 40.6 42.1 43.5 44.8 46.4 47.9 49.1 50.4 51.6 53.0 54.3 55.9 2 7.3 8.0 8.8 9.5 10.0 11.0 11.8 12.5 13.2 13.7 14.3 14.9 4 1 54.1 56.7 59il 61.3 63.1 65.5 67.4 69.4 71.6 73.6 75.5 77.6 2 23.5 25.4 27.4 28.6 30.0 31.3 32.4 33.4 34.4 35.6 36.7 37.7 8 2.4 3.1 3.9 4.5 5.0 5.9 6.5 7.2 7.9 8.4 8.9 9.4 5 1 675 70.4 73.6 76.6 79.4 82.3 84.5 87.1 89.2 91.4 93.6 96.4 2 38.8 41.0 43.0 44.9 46.7 48.7 50.3 51.9 53.8 55.5 57.1 58.7 3 16.3 18.0 19.5 20.8 22.0 23.1 24.0 25.0 25.9 26.9 27.8 28.7 6 1 80.1 83.5 86.9 90.1 93.6 96.9 100.1 103.1 106.7 110.5 114.3 118.7 2 52.7 55.3 57.9 60.5 62.9 65.5 67.8 70.1 72.1 74.2 76.3 78.1 8 30.2 33.5 34.0 35.6 37.4 39.0 40.8 41.9 43.4 44.9 46.3 47.7 4 11.5 13.0 14.4 15.6 16.6 17.8 18.8 19.4 20.2 21.1 21.9 22.6 7 1 91.1 94.6 99.2 103.4 108.0 112.8 1175 122.9 127.5 132.0 136.5 141.4 2 65.5 69.1 72.4 753 78.4 80.9 83.9 86.1 89.0 920 95.0 98.8 8 43.4 45.6 48.0 50.4 52.4 54.8 56.9 58.8 596 62.0 64.3 65.9 4 24.1 26.0 27.6 29.0 30.5 32.1 33.4 34.7 36.1 37.4 38.6 39.8 5 8.5 9.6 10.7 11.7 12.8 13.8 14.9 15.5 16.1 16.9 17.7 18.4 8 1 101.9 107.6 113.6 119.3 125.4 131.0 136.4 141.9 147.2 152.3 157.4 162.9 2 78.2 81.7 85.2 89.1 92.5 96.0 99.8 104.1 108.4 112.6 116.7 121.0 3 55.8 59.0 61.9 64.5 67.4 69.6 72.3 74.4 76.8 79.5 82.2 85.0 4 36.4 38.5 40.6 42.8 44.6 46.8 48.6 50.4 52.0 53.7 55.3 56.7 6 19.5 21.3 22.8 24.1 25.5 26.9 28.0 29.1 30.5 31.7 32.8 33.9 6 7.4 7.9 8.4 9.2 10.0 10.9 11.9 12.5 13.1 13.8 14.5 15.1 9 1 115.2 122.3 129.2 135.6 141.9 148.4 154.5 160.8 1664 172.0 177.6 183.5 2 89.0 93.6 98.3 103.3 108.3 113.6 118.6 123.4 128.2 132.9 137.5 142.5 3 68.1 71.4 74.5 77.6 81.2 84.3 87.8 91.6 95.4 99.2 102.9 106.4 4 48.2 51.1 53.8 56.5 58.5 60.8 63.1 65.1 67.4 69.8 72.2 74.8 5 31.0 32.9 34.9 36.9 38.5 40.5 42.3 43.8 45-3 46.8 48.3 49.6 6 16.0 17.5 19.1 20.3 21.5 22.7 23.9 25.0 26.2 27.3 28.3 29.3 a PANEL, LENGTHS fir* 1 .3 2 14' 0" 14' 6" 15' 0" 15' 6" 16' 0" 16' 6" 17' 0" 17' 6" 18' 0" 18' 6" 19' 0" 3 i 57.4 58.7 60.0 615 63.0 64.3 65.6 66.9 68.2 69.5 70.8 2 15.5 16.0 16.4 17.1 17.8 18.3 188 19.3 19.9 20.5 21.0 4 1 79.6 81.6 83.6 85.5 87.3 89.0 90.6 92.6 94.5 96.4 98.3 2 38.6 39.6 40.6 41.7 42.7 43.9 45.0 46.1 47.2 48.3 49.3 8 9.8 10.3 10.7 11.2 11.7 12.2 12.7 13.1 13.5 13.9 14.3 5 1 99.2 102.3 105.4 108.6 111.8 115.1 118.3 121.5 124.6 127.5 130.4 2 60.3 61.9 63.4 64.8 662 67.7 69.1 70.8 72.4 74.0 75.6 3 29.5 30.4 31.2 320 328 33.6 34.3 351 35.8 36.6 37.3 6 1 123.1 127.1 131.0 134.9 138.8 142.7 146.5 1502 153.8 157.5 161.1 2 79.8 82.2 84.6 86.9 90.1 93.0 95.8 98.5 101.1 103.6 106.1 8 49.1 50.4 51.7 52.9 54.0 55.3 56.5 57.6 58.6 59.7 60.7 4 23.3 24.1 24.8 25.6 26.3 27.0 27.6 28.3 28.9 29.6 30.2 7 1 146.2 150.9 155.5 160.1 164.6 169.0 173.3 177.5 181.6 185.7 189.7 2 102.6 106.1 109.6 113.0 116.4 119.7 123.1 126.4 129.6 132.8 135.9 8 67.4 69.3 71.1 73.1 75.0 77.4 79.7 82.1 84.4 86.6 88.8 4 41.0 42.2 43.4 44.4 45.4 46.5 47.5 48.5 49.4 50.4 51.3 5 19.0 19.7 203 21.0 21.6 22.2 22.8 23.4 24.0 24.6 25.1 8 1 168.4 173.6 178.8 183.8 188.7 193.6 198.4 203.1 207.8 212.5 217.1 2 125.3 129.5 133.7 1378 141.8 145.7 149.5 153.2 156.9 160.5 164.1 8 87.8 90.9 93.9 96.8 99.6 102.6 105.6 108.5 111.4 114.2 117.0 4 58.1 59.8 61.4 63.1 64.8 66.7 68.5 70.4 72.2 74.0 75.8 6 35.0 36.1 37.1 38.0 38.9 39.9 40.9 41.7 42.5 43.4 44.2 6 15.7 16.4 17.0 17.6 18.1 18.7 19.2 19.8 20.3 20.8 21.3 9 189.4 195.1 200.8 206.3 211.8 217.3 222.7 2280 233.2 238.4 243.6 2 147.4 152.1 156.8 1613 165.7 170.1 174.5 178.8 183.0 187.2 191.3 3 109.8 112.9 116.7 120.4 124.1 127.6 131.0 134.4 137.7 141.0 144.2 4 77.3 80.1 82.7 85.2 87.6 90.1 92.5 94.9 97.3 99.9 102.4 5 50.8 52.4 53.8 55.4 56.9 58.6 60.2 61.9 63.5 65.3 67.0 6 30.3 31.4 32.3 33.1 33.9 34.8 35.7 36.5 37.2 38.0 38.7 98 LIVE-LOAD STRESSES TABLE 9. Continued MAXIMUM SHEARS FOR TRUSS BRIDGES COOPER'S E5Q FOR ONE RAIL Shears Given in Thousands of Pounds 1 2 3 4.5.6,7.8.9. Panels I = 1 1 1 1 PANEL LENGTHS || 1 19' 6" 20' 0" 20' 6" 21' 0" 21' 6" 22' 0" 22' 6" 23' 0" 23' 6" 24' 0" 24' 6" 3 1 72.0 73.3 74.3 75.3 76.6 78.0 79.5 81.0 82.1 83.2 84.6 2 21.5 22.0 22.4 22.9 23.5 24.0 24.3 24.6 25.1 25.5 25.9 4 1 100.7 103.0 105.6 108.2 110.7 113.2 115.5 117.7 120.0 122.2 124.4 2 50.3 51.3 52.2 53.1 54.0 54.9 55.8 56.8 57.4 58.2 59.0 3 14.7 15.0 15.3 15.6 15.9 16.2 16.5 16.7 17.0 17.2 17.5 5 1 133.5 136.6 139.8 142.9 146.0 149.0 152.0 154.9 157.8 160.5 163.3 2 77.4 79.1 80.9 82.6 84.4 86.1 88.0 89.9 91.7 93.5 95.1 3 38.1 38.8 39.6 40.3- 40.9 41.6 42.3 42.9 43.7 44.3 45.0 6 1 164.6 168.1 171.7 175.2 178.8 182.3 185.8 189.2 192.6 195.9 199.2 2 108.6 111.0 113.6 116.0 118.5 120.8 123.2 125.4 127.9 130.1 132.4 3 62.1 63.5 65.1 66.6 68.2 69.6 71.3 72.9 74.5 75.9 77.4 4 30.8 31.4 32.1 32.8 33.4 34.0 34.5 35.0 35.5 36.0 36.6 7 1 193.9 197.8 201.7 205.5 209.6 213.7 217.8 221.8 225.8 229.7 233.6 2 139.0 142.0 145.0 147.9 150.9 153.7 156.1 159.3 162.1 164.8 167.6 3 91 93 1 95 4 97 5 99 6 101 6 103 8 105.8 107.9 109.8 111 8 4 52.4 53.4 54.5 55.5 56.7 57.8 59.3 60.6 62.1 63.4 64.7 5 25.7 26.3 26.9 27.4 28.0 28.5 29.0 29.4 29.9 30.3 30.8 8 1 221.7 226.3 230.8 235.2 239.8 244.3 248.9 253.4 258.0 262.5 267.1 2 167.7 171.3 174.8 178.2 181.7 185.0 188.4 191.7 195.1 198.3 201.7 3 119 8 122 5 125 1 127 6 130.5 132.8 135.4 137.8 140.3 142.7 145 2 4 77.8 79.8 81.7 83.6 85.5 87.3 89.2 91.0 92.8 94.5 96.3 5 45.2 46.1 47.1 48.0 49.0 49.4 51.0 52.1 53.1 54.1 55.3 6 21.9 22.4 22.9 23.4 23.9 24.4 24.9 25.3 25.7 26.0 26 5 9 1 248.8 253.9 259.0 264.0 269.2 274.2 279.4 284.5 289,7 294.8 299.9 2 195.4 199.5 203.5 207.5 211.5 215.5 219.4 223.3 227.2 231.0 234.9 3 147.4 150.6 163.8 156.9 160.0 163.0 166.0 169.0 172.0 175.0 177.9 4 104.9 107.3 109.7 112.0 114.3 116.6 118.9 121.1 123.4 125.5 127.8 5 68.6 70.1 71.7 73.3 74.9 76.4 78.0 79.5 81.2 82.8 84.3 6 39.6 40.4 41.3 42.1 43.0 43.9 44.9 45.8 46.7 47.6 48.6 I PANEL LENGTHS rj "3 Is 25' 0" 25' 6" 26' 0" 26' 6" 27' 0" 27' 6" 28' 0" 28' 6" 29' 0" 29' 6" 30' 0" 3 1 86.0 87.0 88.0 89.5 91.0 92.2 93.5 94.7 96.0 97.8 99.7 2 26.4 26.8 27.2 27.6 28.0 28.3 28.6 29.0 29.4 29.7 30.0 4 1 126.5 128.7 130.9 133.1 135.2 137.3 139.3 141.5 143.6 145.8 147.9 2 59.7 60.5 61.3 62.1 62.9 63.8 64.6 65.6 66.5 67.4 68.3 3 17.8 18.1 18.4 18.6 18.9 19.1 19.3 19.6 19.8 20.1 20.3 5 1 166.0 168.8 171.4 174.1 176.7 179.4 181.9 184.5 187.0 189.6 192.0 2 96.6 98.3 100.1 101.9 103.6 105.4 107.1 108.9 110.6 112.3 114.0 3 45.5 46.3 46.9 47.7 48.3 49.0 49.6 50.5 51.3 52.1 52.8 6 1 202.5 205.8 209.0 212.2 215.4 218.6 221.8 224.9 228.0 231.1 234.2 2 134.5 136.8 139.0 141.3 143.5 145.8 148.0 150.3 152.4 154.6 156.7 3 78.6 80.2 81.5 83.0 84.3 85.7 87.0 88.4 89.6 91.1 92.4 4 37.1 37.6 38.1 38.6 39.1 39.6 40.0 40.5 41.0 41.7 42.4 7 1 237.4 241.4 245.2 249.1 252.8 256.6 260.3 264.1 267.7 271.4 275.0 2 170.3 173.2 175.9 178.8 181.5 184.3 187.0 189.8 192.5 195.3 197.9 3 113.6 115.6 117.4 119.3 121.1 123.0 124.8 126.6 128.3 130.2 131.9 4 65.8 67.1 68.3 69.6 70.8 72.0 73.1 74.3 75.4 76.7 77.8 5 31.3 31.8 32.1 32.6 33.0 33.5 33.8 34.3 34.6 35.1 35.6 8 1 271.5 276.0 280.4 284.9 289.2 293.6 297.9 302.3 306.5 310.8 315.0 2 204.9 208.3 211.6 215.1 218.4 221.8 225.0 228 A 231.7 235.0 238.2 3 147.5 150.0 152.3 154.7 157.0 159.4 161.7 164.0 166.1 168.5 170.2 4 98.0 99.8 101.4 103.1 104.6 106.3 107.9 109.5 111.0 112.6 114.1 5 56.4 57.4 58.4 59.5 60.5 61.6 62.6 63.7 '64.8 65.9 66.9 6 26.9 27.3 27.6 28.0 28.4 28.8 29.1 29.5 29.9 30.4 30.8 9 1 304.9 310.0 315.0 320.1 325.0 330.0 334.9 339.9 344.7 349.7 354.5 2 238.8 242.8 246.7 250.6 254.5 258.5 262 .4 266.3 270.2 274.0 277.8 3 180.8 183.8 186.7 189.6 192.4 195.3 198.0 200.9 203.8 206.7 209 . ? 4 129.9 132.0 134.1 136.3 138 .4 140.5 142.5 144.6 146.6 148.6 150.fi 5 85.8 87.4 88.9 90.4 91.8 93.3 94.8 96.2 97.6 99.0 100.4 6 49.6 50.6 51.5 52 A 53.3 54.2 55.0 55.9 56.8 57.6 58.4 LIVE-LOAD STRESSES 99 TABLE 9. Continued MAXIMUM SHEARS FOR TRUSS BRIDGES COOPER'S #50 FOR ONE RAIL Shears Given in Thousands of Pounds Panels Panels in Truss 1 PANEL LENGTHS 30' 6" 31' 0" 31' 6" 32' 0" 32' 6" 33' 0" 33' 6" 34' 0" 34' 6" 35' 0" 35' 6" 3 1 101.1 102.6 104.6 106.6 108.1 109.6 111.5 113.4 114.8 116.2 117.6 2 30.4 30.8 31.2 31.5 31.8 32.2 32.5 32.8 33.1 33.4 33.7 4 1 149.9 152.0 154.0 156.1 158.0 160.0 161.9 163.8 165.8 167.9 169.8 2 69.1 70.0 71.7 73.3 74.4 75.4 76.4 77.4 78.4 79.4 80.5 3 20.6 20.9 21.1 21.3 21.6 22.0 22.2 22.5 22.7 23.0 23.3 5 1 194.6 197.1 199.8 202.4 205.0 207.5 210.1 212.6 215.1 217.6 220.2 2 115.6 117.3 118.9 120.4 122.0 123.5 125.0 126.5 128.0 129.5 131.0 3 53.6 54.3 55.1 55.9 56.7 57.4 58.3 59.1 60.0 60.8 61.7 6 1 237.3 240.3 243.5 246.6 249.8 252.9 256.0 259.1 262.3 265.4 268.5 2 158.8 160.9 163.0 165.1 167.2 169.3 171.4 173.4 175.4 177 .4 179.4 3 93.7 95.0 96.3 97.5 98.8 100.0 101.3 102.5 103.8 105.1 106.4 4 43.0 43.6 44.4 45.1 45.8 46.4 47.2 47.9 48.6 49.3 50.0 7 1 278.7 282.3 286.0 289.6 293.4 297.1 300.9 304.7 308.4 312.0 315.7 2 200.6 203.3 205.9 208.5 211.2 213.8 216.4 218.9 221.5 224.0 226.5 3 133.6 135.3 137.1 138.9 140.7 142.5 144.3 146.0 147.9 149.8 151.7 4 79.0 80.1 81.3 82.4 83.5 84.5 85.6 86.6 87.7 88.7 89.8 5 36.1 36.5 37.0 37.5 38.0 38.5 39.2 39.9 40.5 41.0 41.6 8 1 319.3 323.5 327.8 332.0 337.0 341.9 345.6 349.3 353.2 357.0 360.9 2 241.4 244.6 247.8 251.0 254.2 257.4 260.6 263.8 266.9 270.0 273.2 3 172.8 175.4 177.8 180.1 182.5 184.8 187.1 189.4 191.7 193.9 196.2 4 115.7 117.3 118.7 120.3 121.9 123.4 124.9 126.3 127.7 129.1 130.5 5 67.9 68.9 69.9 70.9 71.9 72.9 73.9 74.8 75.7 76.6 77.5 6 31.2 31.5 32.0 32.5 32.9 33.3 33.8 34.3 34.7 35.1 35.5 9 1 359.4 364.2 369.1 373.9 378.7 383.5 388.5 393.5 398.4 403.3 408.3 2 281.6 285.4 289.2 293.0 296.8 300.5 304.3 308.0 311.8 315.5 319.2 3 212.4 215.3 218.2 221.0 223.9 226.8 229.6 232.5 235.3 238.1 240.8 4 152.7 154.8 156.8 158.8 160.7 162.6 164.6 166.6 168.6 170.5 172 .5 5 101.8 103.1 104.5 105.9 107.3 108.6 110.0 111.4 112.7 114.0 115.4 6 59.4 60.3 61.2 62.0 62.9 63.8 64.7 65.5 66.3 67.1 67.8 100 LIVE-LOAD STRESSES TABLE 10 MAXIMUM BENDING MOMENTS IN GIRDER BRIDGES WITHOUT FLOOR-BEAMS, COOPER'S E0 LOADING Values in Thousands of Foot-Pounds per Rail SHORTER SEGMENT h 5 10 15 20 25 30 35 40 45 50 55 60 250.. 1534 3030 4514 5979 7411 8820 10203 11562 12916 14278 15628 16982 225.. 1404 2769 4122 5455 6758 8034 9288 10515 11743 12976 14198 15422 200. . 1273 2505 3727 4926 6098 7241 8364 9460 10560 11665 12759 13849 175.. 1139 2236 3326 4390 5430 6438 7430 8391 9364 10339 11306112266 160.. 1053 2073 3082 4063 5022 5950 6862 7742 8638 9535 10424 11300 150.. 1003 1962 2917 3843 4749 5620 6480 7304 8150 8994 9833 10664 140.. 947 1851 2750 3620 4471 5287 6093 6862 7658 8450 9236 10016 130.. 889 1738 2582 3394 4191 4951 5703 6417 7161 7901 8635 9363 120.. 834 1625 2410 3164 3906 4608 5307 5964 6658 7345 8028 8704 110.. 774 1509 2234 2930 3617 4260 4905 5514 6148 6782 7414 8038 M 100.. 714 1390 2055 2690 3320 3910 4494 5053 5650 6234 6813 7387 "-a 95.. 682 1329 1963 2566 3169 3730 4290 4864 5431 5991 6546 7096 1 90.. 650 1264 1866 2444 3016 3550 4114 4661 5202 5734 6263 6786 85.. 617 1200 1770 2314 2854 3365 3923 4442 4936 5458 5958 6449 bfi o 80.. 584 1134 1671 2186 2694 3200 3715 4205 4690 5171 5646 6117 CO 75.. 551 1070 1573 2054 2530 3008 3489 3964 4422 4874 5320 5761 3 70.. 516 1003 1474 1923 2366 2805 3254 3706 4132 4553 4967 5378 *G 65.. 482 931 1367 1792 2202 2602 3019 3437 3831 4221 4608 4993 2 60.. 453 864 1266 1649 2025 2389 2770 3155 3519 3884 4243 4597 55.. 425 805 1172 1518 1856 2195 2546 2884 3214 3514 3859 50. . 397 750 1091 1398 1713 2023 2336 2634 2928 3219 45. . 367 692 1005 1290 1567 1847 2136 2404 2669 40. . 335 635 918 1171 1419 1669 1921 2160 35. . 302 570 819 1050 1272 1490 1707 30.. 270 506 721 918 1109 1294 25. . 235 440 622 787 946 20. . 200 373 518 656 15. . 150 300 410 10. . 100 200 5.. 50 For h and Z 2 each > 142 ft. M LIVE-LOAD STRESSES TABLE 10. Continued MAXIMUM BENDING MOMENTS IN GIRDER BRIDGES WITHOUT FLOOR-BEAMS, COOPER'S #40 LOADING Values in Thousands of Foot-Pounds per Rail SHORTER SEGMENT l\ 250 225 200 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 65 70 75 80 85 90 95 100 110 120 130 140 18327 16639 14939 13224 12185 11487 10790 10088 9380 8666 7963 7642 7303 6943 6582 6197 5796 5374 19675 17862 16036 14205 13097 12354 11608 10857 10100 9338 8567 8182 7817 7428 7043 6629 6197 21062 19123 17172 15207 14018 13194 12395 11594 10786 9972 9150 8737 8321 7917 7500 7057 22421 20351 18269 16171 14906 14058 13206 12349 11486 10616 9738 9296 8851 8404 7954 23766 21569 19360 17134 15789 14887 13980 13069 12073 11226 10294 9824 9352 8876 25084 22757 20418 18017 16636 15681 14722 13756 12787 11812 10829 10334 9836 26364 23908 21440 18952 17450 16442 15430 14413 13421 11392 11348 10834 27660 25078 22482 19868 18289 17231 16169 15101 14026 12946 11857 30152 27315 24465 21597 19866 18706 17542 16372 15197 14014 32591 29502 26400 23278 21396 20151 18870 17600 16325 3503337455 3169133862 28231 &0255 2496326631 22930)24446 2156922986 2020321520 18834 1 For h and 1 2 each > 142 ft. M = h I, + 3800 j~ QJ DO O OOP Q o Influence Line for M 102' LIVE-LOAD STRESSES TABLE 11 MAXIMUM BENDING MOMENTS IN GIRDER BRIDGES WITHOUT FLOOR-BEAMS, COOPER'S, E50 LOADING Values in Thousands of Foot-Pounds per Rail SHORTER SEGMENT h 5 10 15 20 25 30 35 40 45 50 55 60 250. 1918 3788 5643 7474 9264 11025 12754 14452 16145 17848 19535 21228 225. 1755 3461 5153 6819 8447 10043 11610 13144 14679 16220 17748 19278 200. 1591 3131 4659 6158 7622 9052 10456 11825 13200 14581 15949 17311 175. 1424 2795 4158 5487 6787 8048 9288 10489 11705 12924 14132 15333 160. 1316 2591 3852 5079 6278 7437 8578 9677 10798 11919 13030 14125 150. 1254 2453 3646 4804 5936 7025 8100 9130 10187 11243 12291 13330 140. 1184 2314 3438 4525 5589 6609 7617 8578 9572 10562 11545 12520 130. 1114 2173 3227 4242 5239 6189 7129 8021 8951 9876 10794 11704 120. 1042 2031 3012 3955 4883 5760 6634 7455 8322 9181 10035 10880 110. 968 1886 2793 3662 4521 5325 6131 6892 7685 8478 9268 10048 100. 892 1737 2569 3362 4150 4887 5618 6316 7063 7793 8516 9234 +3 95. 853 1661 2454 3208 3961 4663 5363 6080 6789 7489 8183 8870 90. 812 1580 2333 3055 3770 4437 5143 5826 6502 7168 7829 8482 a 85. 771 1500 2213 2893 3568 4206 4904 5552 6170 6823 7448 8061 bC rtj 80. 730 1418 2089 2733 3368 4000 4644 5256 5862 6464 7058 7646 Qg 75. 689 1337 1966 2568 3163 3760 4361 4955 5528 6093 6650 7201 & 70. 645 1254 1843 2404 2958 3506 4068 4632 5165 5691 6209 6723 bD G 65. 602 1164 1709 2240 2753 3253 3774 4296 4789 5276 5760 6241 3 60. 566 1080 1582 2061 2531 2986 3463 3943 4399 4855 5304 5746 55. 531 1006 1465 1897 2320 2744 3182 3605 4017 4392 4824 50. 496 937 1364 1747 2141 2529 2920 3293 3660 4024 45. 459 865 1256 1613 1959 2309 2670 3005 3336 40. 419 794 1147 1464 1774 2086 2401 2700 . . . 35. 377 713 1024 1312 1590 1862 2134 30. 338 632 901 1148 1386 1617 25. 294 550 778 984 1182 20. 250 466 647 820 15. 187 375 513 10. 125 250 5. 62 For ^ and 1 2 each > 142 ft. M = 1.25 I, L + 4750 f LIVE-LOAD STRESSES 103 TABLE 11. Continued MAXIMUM BENDING MOMENTS IN GIRDER BRIDGES WITHOUT FLOOR-BEAMS, COOPER'S #50 LOADING Values in Thousands of Foot-Pounds per Rail SHORTER SEGMENT h 250 225 200 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 65 70 75 80 85 90 95 100 110 120 130 140 22909 20799 18674 16530 15231 14359 13488 12610 11725 10832 9954 9552 9129 8679 8228 7746 7237 6718 24594 22327 20045 17756 16371 15443 14510 13571 12625 11672 10709 10227 9771 9285 8804 8286 7746 26327 23904 21465 19009 17523 16492 15494 14492 13482 12465 11438 10921 10401 9896 9375 8821 28026 25439 22836 20214 18633 17573 16508 15436 14357 13270 12173 11620 11064 10505 9943 29707 26961 24200 21417 19736 18609 17475 16336 15091 14033 12867 12280 11690 11095 31355 28446 25522 22521 20795 19601 18402 17195 15984 14765 13536 12917 12295 32955 29885 26800 23690 21812 20553 19288 18016 16776 15490 14185 13543 34575 31347 28102 24835 22861 21539 20211 18876 17533 16182 14821 37690 34144 30581 26996 24832 23382 21927 20465 18996 17518 40739 36878 33000 29098 26745 25189 23588 22000 20406 43791 39614 35414 31204 28662 26961 25254 23542 46819 42327 37819 33289 30558 28732 26900 or 1, and I, each > 142 ft. M = 1.25 1, L -f 4750 -f Li O C ) O O O O o '///////////////A 142 ft. M = 1.5 + 5700^ LI LIVE-LOAD STRESSES 105 TABLE 12. Continued MAXIMUM BENDING MOMENTS IN GIRDER BRIDGES WITHOUT FLOOR-BEAMS, COOPER'S #60 LOADING Values in Thousands of Foot-pounds per Rail SHORTER SEGMENT h 65 70 75 80 85 90 95 100 110 120 130 140 250 225 200 175 160 150 40 130 120 110 100 95 90 85 80 75 70 65 27491 24959 22409 19836 18277 17231 16186 15132 14070 12998 11945 11462 10955 10415 9874 9295 8684 8062 29513 26792 24054 21307 19645 18532 17412 16285 15150 14006 12851 12272 11725 11142 10565 9943 9295 31592 28685 25758 22811 21028 19790 18593 17390 16178 14958 13726 13105 12481 11875 11250 10585 33631 30527 27403 24257 22360 21088 19810 18523 17228 15924 14608 13944 13277 12606 11932 35648 32353 29040 25700 23683 22331 20970 19603 18110 16840 15440 14736 14028 13314 37626 34135 30626 27025 24954 23521 22082 20634 19181 17718 16243 15500 14754 39546 35862 32160 28428 26174 24664 23146 21619 20131 18588 17022 16252 41490 37616 33722 29802 27433 25847 24253 22651 21040 19418 17785 45228 40973 36697 32395 29798 28058 26312 24558 22795 21022 48887 44254 39600 34918 32094 30227 28306 26400 24487 52549 47537 42497 37444 34394 32353 30305 28250 56183 50792 45383 39947 36670 34478 32280 1 For k and k each >142 ft. M = 1. + 5700^ Li OOOO noon Influence Line for M 106 LIVE-LOAD STRESSES TABLE 13 MAXIMUM PIER REACTIONS BETWEEN EQUAL AND UNEQUAL SPANS, COOPER'S #40 LOADING Values in Thousands of Pounds per Rail SHORTER SEGMENT h 5 10 15 20 25 30 35 40 45 50 56 250 314 314 315 318 3?!? 3?,6 329 332 336 338 342 346 225 287 287 290 294 298 301 304 306 309 312 317 321 200 175 160 150 140 130 261 234 218 207 196 185 261 234 218 207 196 185 263 236 220 210 198 187 268 241 225 214 203 192 271 244 228 218 206 196 275 248 232 222 210 278 251 236 225 214 203 281 254 238 229 218 ?,08 284 258 242 231 220 210 287 262 246 234 224 214 292 266 250 239 229 219 296 269 254 244 234 ?94 120 ... 174 174 176 181 184 189 19?, 196 198 ?,04 208 9 I 110 100 95 90 162 150 144 137 162 150 144 137 165 153 146 140 170 158 151 146 173 162 155 150 178 166 160 154 181 170 163 158 185 174 168 163 188 177 173 168 193 182 178 174 198 187 182 178 202 192 188 183 85 80 ... 131 124 131 124 134 127 139 133 142 137 148 142 152 146 158 153 163 158 168 163 174 168 178 174 1 75 70 65 118 110 104 118 110 104 122 114 107 126 120 112 130 124 118 135 128 122 140 134 126 146 139 133 152 146 139 158 150 144 162 156 149 167 162 155 60 98 98 101 106 110 115 119 125 131 137 142 148 55 50 45 ... . 93 87 82 93 87 82 95 90 85 99 94 90 103 98 93 108 102 98 113 108 102 118 114 109 125 118 114 130 124 118 134 129 141 40 75 75 79 84 88 92 98 102 108 35 69 69 74 78 82 87 92 98 30 63 63 67 72 77 82 86 25 57 57 6? 66 71 76 20 50 50 56 60 66 15 40 40 50 55 10 30 30 40 5.. 20 20 For h and k each > 142 f t. R = L + 3800 LIVE-LOAD STRESSES 107 TABLE 13 Continued MAXIMUM PIER REACTIONS BETWEEN EQUAL AND UNEQUAL SPANS, COOPER'S #40 LOADING Values in Thousands of Pounds per Rail SHORTER SEGMENT h 60 65 70 75 80 85 90 95 100 110 120 130 140 ~s I -r 250 225 200 175 160 150 140 130 1?0 350 326 300 274 258 248 238 229 218 356 330 305 279 264 254 242 233 222 359 334 309 284 269 259 249 239 228 365 340 314 290 274 264 253 243 233 370 345 320 294 280 269 259 250 239 374 350 324 300 284 274 264 254 242 379 354 329 303 289 278 270 258 248 382 358 333 308 293 282 273 262 253 387 362 337 312 297 287 277 267 ?57 395 370 345 319 305 295 284 274 265 402 377 352 327 312 302 292 282 ?,7?, 410 385 359 334 320 310 299 290 417 392 367 342 328 318 308 1 110 100 95 PO 207 197 192 188 212 202 198 194 218 208 203 198 223 214 208 203 230 219 214 209 234 224 219 214 238 229 223 218 243 233 229 247 238 255 . . . 85 183 189 194 198 204 209 80 178 184 188 194 199 75 173 178 183 188 70 166 171 178 65 160 165 60 153 For Zi and k each >142 ft. R = L + p o O O O O O O O 108 LIVE-LOAD STRESSES TABLE 14 MAXIMUM PIER REACTIONS BETWEEN EQUAL AND UNEQUAL SPANS, COOPER'S #50 LOADING Values in Thousands of Pounds per Rail SHORTER SEGMENT h 5 10 15 20 25 30 35 40 45 50 55 250 225 200 175 160 150 392 359 326 293 273 259 392 359 326 293 273 259 394 362 329 295 275 262 398 367 335 301 281 267 403 372 339 305 285 272 407 376 344 310 290 277 411 380 347 314 295 281 415 383 351 318 298 286 420 386 355 323 302 289 423 390 359 327 307 293 428 396 365 332 313 299 432 401 370 336 318 305 140 245 245 248 254 258 263 268 273 275 280 286 293 130 231 231 234 240 245 251 ?M 260 262 268 274 280 ~$ 1 120 110 100 95. 90 85 80 217 202 187 180 171 164 755 217 202 187 180 171 164 155 220 206 191 183 175 168 159 226 212 197 189 182 174 166 230 216 202 194 187 178 171 236 222 208 200 192 185 177 240 226 212 204 197 190 183 245 231 218 210 204 198 191 248 235 221 216 210 204 197 255 241 227 222 218 210 204 260 247 234 228 223 217 210 266 253 240 235 229 223 217 ^ 75 147 147 152 158 163 169 175 183 190 197 203 209 I 70 65 60 55 50 45 40 138 130 123 116 109 102 94 138 130 123 116 109 102 94 143 134 126 119 112 106 99 150 140 132 124 118 112 105 155 147 137 129 122 116 110 160 152 144 135 128 122 115 167 158: 149 141 135 128 1?,?, 174 166 156 148 142 136 128 182 174 164 156 148 142 135 188 180 171 162 155 148 195 186 178 168 161 202 194 185 176 35 30 86 79 86 79 92 84 98 90 103 96 109 102 115 108 122 25 71 71 77 83 89 95 20 63 63 70 75 82 15 50 50 62 69 10 QQ QQ 50 5 25 25 . . For li and k each >142 ft. R = 1.25 L + 4750 LIVE-LOAD STRESSES 109 TABLE 14. Continued MAXIMUM PIER REACTIONS BETWEEN EQUAL AND UNEQUAL SPANS, COOPER'S #50 LOADING Values in Thousands of Pounds per Rail SHORTER SEGMENT l\ 60 65 70 75 80 85 90 95 100 110 120 130 140 250... 437 445 449 456 463 468 474 478 484 494 502 512 521 225... 407 413 418 425 431 437 442 448 452 462 471 481 490 200... 375 381 386 393 400 405 411 416 421 431 440 449 459 175... 343 349 355 362 368 375 379 385 390 399 409 418 427 160... 323 330 336 343 350 355 361 366 371 381 390 400 410 150... 310 317 324 330 336 343 348 353 359 369 378 387 397 140... 298 303 311 316 324 330 337 341 346 355 365 374 385 130... 286 291 299 304 312 317 323 328 334 343 352 362 120... 272 278 285 291 299 303 310 316 321 331 340 110... 259 265 273 279 287 292 298 304 309 319 100... 246 253 260 267 274 280 286 291 296 95... 240 247 254 260 267 274 279 286 90. 235 242 248 254 261 268 273 85... 229 236 242 248 255 261 80... 223 230 235 242 249 75... 216 222 229 235 70... 208 214 222 65. . . 200 206 60... 191 For Zi and k each > 142 ft. R = 1.25 L -f 4750 O O O O O O O O O 110 LIVE-LOAD STRESSES TABLE 15 MAXIMUM PIER REACTIONS BETWEEN EQUAL AND UNEQUAL SPANS, COOPER'S #60 LOADING Values in Thousands of Pounds per Rail SHORTER SEGMENT h 5 10 15 20 25 30 35 40 45 50 55 250. . 470 470 473 478 484 488 493 498 504 508 514 518 225 431 431 434 440 446 451 456 460 463 468 475 481 200 175 391 352 391 352 395 354 402 361 407 366 413 372 417 377 421 38? 426 388 431 39? 438 398 444 403 160 . 328 328 330 337 342 348 354 358 36? 368 376 38? 150 140 130 311 294 ?77 311 294 ?,77 314 298 ?,81 320 305 288 326 310 294 332 316 301 337 322 305 343 328 31? 347 330 314 352 336 3?,?, 359 343 3?,9 366 352 336 120 ?60 ?60 ?64 ?71 ?76 283 ?88 ?94 ?98 306 31?, 319 Segment Z 2 110 100 95 90 85 80 75 242 224 216 205 197 186 176 242 224 216 205 197 186 176 247 229 220 210 202 191 18'? 254 236 227 218 209 199 190 259 242 233 224 214 205 196 266 250 240 230 222 212 203 271 254 245 236 228 220 210 277 262 252 245 238 229 220 282 265 259 252 245 236 ??8 289 272 266 262 252 245 ?,36 296 281 274 268 260 252 ?44 304 288 282 275 268 260 ?51 1 70 65 166 156 166 156 172 161 180 168 186 176 192 182 200 190 209 199 218 ?09 226 ?16 234 ??3 242 ?33 9 60 148 148 151 158 164 173 179 187 197 ?05 ?14 ??? H-l 55 50 45 139 131 122 139 131 122 143 134 127 149 142 134 155 146 139 162 154 146 169 162 154 178 170 163 187 178 170 194 186 178 202 193 211 40 113 113 119 126 132 138 146 154 16? 35 103 103 110 118 124 131 138 146 30 25 95 85 95 85 101 92 108 100 115 107 122 114 130 20 76 76 84 90 98 15 60 60 74 83 10.. 5 46 30 46 30 60 For Zj and Z 2 each >142 ft. R 1 Z T i 57GO 1.5 L -f -- LIVE-LOAD STRESSES 111 TABLE 15. Continued MAXIMUM PIER REACTIONS BETWEEN EQUAL AND UNEQUAL SPANS, COOPER'S #60 LOADING Values in Thousands of Pounds per Rail SHORTER SEGMENT Zi 60 65 70 75 80 85 90 95 100 110 120 130 140 250... 225... 200... 175... 160... 150... 140... 130... 120 524 488 450 412 388 372 358 343 326 534 496 457 419 396 380 364 349 334 539 502 463 426 403 389 373 359 342 547 510 472 434 412 396 379 365 349 556 517 480 442 420 403 389 374 359 562 524 486 450 426 412 396 380 364 569 530 493 455 433 418 404 388 372 574 538 499 462 439 424 409 394 379 581 542 505 468 445 431 415 401 385 593 554 517 479 457 443 426 412 397 602 565 528 491 468 454 438 422 408 614 577 539 502 480 464 449 434 625 588 551 512 492 476 462 110... 100... 95 311 295 288 318 304 296 328 312 305 335 320 312 344 329 320 350 336 329 358 343 335 365 349 343 371 356 383 90 282 290 298 305 313 322 328 85 275 283 290 298 306 313 80 268 276 282 290 299 75. 259 266 275 282 70... 65... 60.. 250 240 229 257 247 266 . . . ... For h and Z 2 each >142 ft. R = 1.5 L + 5700 112 LIVE-LOAD STRESSES TABLE 16 EQUIVALENT UNIFORM LOADS FOR COOPER'S #40 LOADING Values in Pounds per Lineal Foot per Rail SHORTER SEGMENT l\ Longer Segment lz o 5 10 15 20 25 30 35 40 45 50 55 2270 2300 2320 2340 2370 2380 2400 2420 2430 2460 2480 2500 2540 2550 2570 2580 2580 2580 2580 2550 250 2500 2550 2610 2680 2730 2760 2800 2850 2900 2940 3000 3020 3050 3080 3110 3140 3160 3190 3270 3370 3490 3630 3770 3960 4200 4540 5000 5336 6000 8000 2450 2500 2540 2610 2630 2670 2700 2740 2770 2810 2850 2880 2890 2920 2920 2940 2940 2960 3020 3090 3180 3260 3350 3450 3610 3770 4000 4000 4000 4000 2430 2460 2500 2550 2590 2620 2650 2670 2710 2740 2780 2800 2810 2820 2840 2860 2870 2870 2880 2930 3000 3080 3180 3260 3380 3520 3730 4000 4000 2410 2450 2490 2540 2570 2590 2620 2650 2680 2710 2740 2760 2770 2780 2790 2800 2810 2810 2820 2840 2910 2980 3060 3120 3200 3320 3450 3650 2380 2430 2460 2510 2540 2570 2580 2610 2640 2660 2690 2700 2720 2730 2740 2740 2750 2760 2750 2760 2800 2870 2930 3010 3060 3150 3280 2370 2400 2440 2490 2510 2540 2560 2580 2610 2630 2660 2670 2680 2700 2710 2700 2700 2700 2700 2700 2740 2780 2840 2900 2960 3020 2350 2380 2420 2460 2480 2500 2520 2540 2560 2580 2610 2620 2630 2640 2670 2670 2670 2670 2660 2660 2700 2740 2780 2840 2880 2330 2360 2390 2420 2450 2460 2490 2510 2530 2550 2570 2580 2620 2640 2660 2660 2660 2660 2640 2650 2670 2710 2740 2790 2310 2340 2370 2400 2420 2430 2450 2470 2490 2500 2530 2560 2590 2620 2620 2640 2650 2650 2630 2620 2630 2670 2700 2300 2320 2350 2380 2400 2420 2430 2450 2460 2490 2510 2540 2570 2580 2610 2620 2620 2620 2610 2600 2600 2640 2290 2310 2340 2360 2380 2400 2420 2430 2450 2460 2500 2520 2550 2570 2580 2600 2600 2600 2590 2560 2580 225 . ... 200 175 160 150 140 130 120 110 . 100 95 90 85 80 75 70 65 . . 60 55 50 45 40 35 30 25 20. . 15 10 5 . i For h and 1 2 each > 142 ft. q 1000 LIVE-LOAD STRESSES 113 TABLE 16. Continued EQUIVALENT UNIFORM LOADS FOR COOPER'S .#40 LOADING Values in Pounds per Lineal Foot per Rail SHORTER SEGMENT h 60 65 70 75 80 85 90 95 100 110 120 130 140 250. . 225 200 175 160 150 140 130 2260 2290 2310 2340 2350 2370 2380 2400 2420 2440 2460 2500 2510 2530 2550 2560 2560 2560 2550 2260 2280 2300 2320 2340 2350 2380 2390 2410 2420 2460 2480 2500 2510 2540 2540 2540 2540 2250 2270 2290 2320 2340 2360 2370 2390 2410 2420 2450 2460 2480 2500 2520 2530 2530 2250 2270 2290 2320 2340 2350 2360 2380 2400 2420 2440 2460 2460 2490 2500 2510 2240 2260 2280 2310 2330 2340 2360 2380 2400 2420 2440 2450 2460 2470 2490 2230 2260 2280 2300 2320 2340 2350 2370 2370 2400 2420 2440 2450 2460 2220 2250 2270 2290 2310 2330 2340 2350 2370 2390 2410 2420 2430 2220 2240 2260 2280 2300 2300 2320 2340 2350 2380 2390 2400 2210 2220 2250 2270 2280 2300 2310 2330 2340 2350 2380 2200 2220 2230 2240 2260 2270 2280 2290 2300 2320 2180 2180 2200 2210 2230 2240 2250 2260 2270 2160 2170 2180 2200 2210 2220 2220 2230 2140 2150 2160 2180 2180 2190 2200 120 110 100 ... 95 90 85 80 75 70 65 60. . For ?! and 1 2 each > 142 ft. q (2.0 + ^1000 liL I Pounds nor Lineal Foot per Rail. 114 LIVE-LOAD STRESSES TABLE 17 EQUIVALENT UNIFORM LOADS FOR COOPER'S #50 LOADING Values in Pounds per Lineal Foot per Rail SHORTER SEGMENT h -s 1 J 5 10 15 20 25 30 35 40 45 50 55 2840 2870 2900 2930 2960 2980 3000 3020 3040 3065 3095 3130 3165 3185 3210 3225 3225 3220 3215 3190 250.. 225 3130 3190 3265 3350 3410 3455 3505 3560 3620 3680 3750 3780 3810 3850 3885 3920 3945 3990 4085 4215 4360 4540 4715 4945 5255 5680 6250 6670 7500 10000 3060 3120 3180 3260 3290 3340 3380 3420 3460 3510 3560 3600 3610 3650 3650 3670 3680 3700 3780 3860 3970 4080 4190 4310 4510 4710 5000 5000 5000 5000 3040 3080 3130 3190 3240 3270 3305 3340 3385 3430 3470 3500 3510 3530 3545 3565 3585 3580 3595 3660 3750 3850 3975 4080 4215 4400 4660 5000 5000 3010 3060 3110 3170 3210 3240 3275 3310 3350 3385 3425 3445 3455 3470 3480 3495 3510 3505 3515 3550 3635 3720 3825 3900 4000 4150 4315 4560 2980 3040 3080 3140 3170 3210 3230 3260 3295 3330 3360 3375 3395 3405 3415 3425 3435 3445 3435 3450 3495 3585 3660 3760 3825 3935 4100 2960 3000 3050 3110 3140 3170 3195 3225 3255 3285 3320 3340 3350 3370 3385 3380 3380 3375 3375 3380 3425 3480 3550 3630 3695 3780 2940 2980 3020 3070 3100 3130 3150 3175 3200 3225 3260 3275 3290 3300 3335 3340 3340 3335 3315 3325 3370 3420 3475 3545 3595 2910 2950 2990 3030 3060 3080 3110 3135 3160 3185 3210 3225 3265 3295 3315 3325 3320 3325 3300 3305 3335 3390 3430 3485 2890 2920 2960 3000 3020 3040 3064 3085 3106 3133 3158 3200 3237 3266 3284 3303 3308 3305 3286 3277 3293 3339 3375 2870 2900 2940 2970 3000 3020 3040 3060 3080 3105 3140 3175 3210 3225 3255 3275 3280 3270 3260 3245 3250 3295 2860 2890 2920 2950 2980 3000 3018 3039 3060 3083 3117 3153 3186 3210 3232 3250 3252 3246 3237 3194 3219 200 175 . 160. . . . 150 140 130 120 110 100 95.... 90. ... 85 80 75 70 65 60.. 55.... 50 45 40 35 30 25.. 20...... 15 10 5 For h and k each >142 ft. LIVE-LOAD STRESSES 115 TABLE 17. Continued EQUIVALENT UNIFORM LOADS FOR COOPER'S E5Q LOADING Values in Pounds per Lineal Foot per Rail SHORTER SEGMENT h 60 I 65 70 75 80 250 2830 2820 2810J2810 2800J2790 2780 2770 2760J2750 2720 2700 2680 225 2860 2850 284012840 2830 2820 2810 1 2800 ! 27802770 i 2730>2710'2690 200 S2890 287012860 2860 2850 2850 2840 ; 2820 ! 28 10 2790 2750 12720 12700 2920 2900:2900 2900i2890j2880 2860 2850,2840 2800|2760 2750! 2720 2940i2930|2920 2920 2910 2900 2890 2870|2850 2820 2790 2760J2730 2960 2940|2950 2940 2930 2920 2910|2880 2870 2840 2800 27701 2740 2980 2965 2960 2950i2950 2940 2920 2900 2890 2850|2810 2775 2750 3000 2985 2985 2975'2970 2955 2940 2920 \ 2905 2860 2820i2785 3020 3005 3005 2995 s 2995 2960 2960 2940 2920 2880 2835 304513030 3030 3020^3015 3000 2985 2965 2940 2895 3080 3065 3060 3050| 3045 1 3030 301012985 2965 3115 3095 3075 3065^3060 3050 3020 ! 3001 3140 3120 3100 3080 3075 3060 3035 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 60. . 90 95 100 110 120 130 140 3160 3140 3120 31053090 3070 31853165314531253110 3200318031553140 320031803160 3200:3180 3190 . . For h and I, each >142 ft. q = /2.5 +-f \ h 1000 Pounds per Lineal Foot per Rail. 116 LIVE-LOAD STRESSES TABLE 18 EQUIVALENT UNIFORM LOADS FOR COOPER'S #60 LOADING Values in Pounds per Lineal Foot per Rail SHORTER SEGMENT h Longer Segment k 5 10 15 20 25 30 35 40 45 ;o 55 250.. 225 200 175 160 150 140 :.... 130 3760 3830 3920 4020 4090 4150 4210 4270 4340 4420 4500 4540 4570 4620 4660 4700 4730 4790 4900 5060 5230 5450 5660 5930 6310 6820 7500 8000 9000 12000 3670 3740 3820 3910 3950 4010 4060 4110 4150 4210 4270 4320 4330 4380 4380 4400 4420 4440 4540 4630 4760 4900 5030 5-170 5410 5650 6000 6000 6000 6000 3650 3700 3760 3830 3890 3920 3970 4010 4070 4120 4160 4200 4210 4240 4260 4280 4310 4300 4320 4390 4500 4620 4780 4900 5060 5280 5590 6000 6000 3610 3670 3730 3800 3850 3890 3940 3970 4020 4070 4120 4140 4150 4160 4180 4200 4210 4210 4220 4260 4370 4460 4600 4680 4800 4980 5180 5470 3580 3650 3700 3770 3800 3850 3880 3910 3960 4000 4030 4060 4080 4080 4100 4120 4130 4140 4130 4140 4200 4310 4390 4510 4600 4730 4920 3550 3600 3660 3730 3770 3800 3840 3850 3910 3950 3980 4010 4020 4040 4070 4060 4060 4060 4060 4060 4120 4180 4260 4360 4440 4540 3530 3580 3620 3680 3720 3760 3780 3820 3840 3880 3910 3940 3950 3960 4010 4010 4010 4010 3980 4000 4040 4100 4180 4260 4320 3490 3540 3590 3640 3670 3700 3730 3770 3790 3830 3850 3880 3920 3960 3980 4000 3980 4000 3960 3970 4010 4070 4120 4190 3470 3500 3550 3600 3620 3650 3680 3710 3730 3760 3790 3840 3890 3920 3940 3960 3970 3970 3950 3940 3950 4010 4060 3440 3480 3530 3560 3600 3620 3650 3670 3700 3760 3770 3820 3850 3880 3910 3940 3940 3920 3910 3900 3900 3960 3430 3470 3500 3540 3580 3600 3630 3650 3670 3700 3740 3780 3830 3850 3880 3900 3900 3900 3890 3840 3860 3410 3440 3480 3520 3550 3580 3600 3620 3650 3680 3720 3760 3800 3830 3850 3870 3870 3860 3860 3830 120 110 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 For J, and k each >142 ft. q = (s.O + -}~ * liL 1000 LIVE-LOAD STRESSES 117 TABLE 18. Continued EQUIVALENT UNIFORM LOADS FOR COOPER'S #60 LOADING Values in Pounds per Lineal Foot per Rail SHORTER SEGMENT h 60 65 70 75 80 85 90 95 100 110 120 130 140 250 3400 3430 3470 3500 3530 3550 3580 3600 3620 3650 3700 3740 3770 3790 3830 3840 3840 3840 3830 3380 3420 3440 3480 3520 3530 3560 3590 3610 3640 3680 3720 3740 3770 3800 3820 3820 3820 3370 3410 3430 3480 3500 3540 3550 3580 3600 3640 3670 3690 3720 3740 3770 3780 3790 3370 3410 3430 3480 3500 3530 3540 3570 3590 3630 3660 3680 3700 3730 3750 3770 3360 3400 3420 3470 3490 3520 3540 3560 3590 3620 3650 3670 3690 3710 3730 3350 3380 3420 3460 3480 3500 3530 3550 3550 3600 3640 3660 3670 3680 3340 3370 3410 3430 3470 3490 3530 3550 3550 3590 3610 3620 3650 3320 3360 3380 3420 3440 3460 3480 3500 3530 3560 3590 3600 3310 3340 3370 3410 3420 3440 3470 3490 3500 3530 3560 3300 3320 3350 3360 3380 3410 3420 3430 3460 3480 3260 3280 3300 3310 3350 3360 3370 3380 3410 3240 3250 3260 3300 3310 3320 3340 3350 3220 3230 3240 3260 3280 3290 3300 225 200 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 60 For ?! and I* each > 142 ft. q = (s.O + 1000 Pounds per "Lineal Foot per Rail 118 LIVE-LOAD STRESSES TABLE 19 INFLUENCE-LINE ORDINATES FOR M FOR GIRDER BRIDGES WITHOUT FLOOR- BEAMS Values of ^ SHORTER SEGMENT h Longer begment Iz 5 10 15 20 25 30 35 40 45 50 55 60 250.. 225 . . 4.90 4.90 4.88 4.85 4.85 4.83 4.83 4.81 4.80 4.78 4.76 4.75 4.74 4.72 4.71 4.69 4.67 4.64 4.62 4.58 4.55 4.50 4.44 4.37 4.29 4.17 4.00 3.75 3.33 2.50 9.62 9.62 9.52 9.43 9.43 9.35 9.34 9.29 9.23 9.17 9.09 9.05 9.00 8.94 8.89 8.83 8.75 8.67 8.58 8.46 8.33 8.18 8.00 7.78 7.50 7.14 6.67 6.00 5.00 14.14 14.06 13.97 13.83 13.70 13.64 13.55 13.44 13.33 13.19 13.05 12.95 12.85 12.76 12.63 12.50 12.35 12.20 12.00 11.79 11.53 11.25 10.91 10.50 10.00 9.38 8.58 7.50 18.5 18.4 18.2 17.9 17.8 17.6 17.5 17.3 17.2 16.9 16.7 16.5 16.4 16.2 16.0 15.8 15.6 15.3 15.0 14.7 14.3 13.9 13.3 12.7 12.0 11.1 10.0 22.7 22.5 22.2 21.9 21.6 21.5 21.2 21.0 20.7 20.4 20.0 19.8 19.6 19.3 19.0 18.8 18.4 18.0 17.6 17.2 16.7 16.1 15.4 14.6 13.6 12.5 26.7 26.5 26.1 25.6 25.3 25.0 24.7 24.4 24.0 23.6 23.1 22.8 22.5 22.2 21.8 21.5 21.0 20.5 20.0 19.4 18.8 18.0 17.2 16.2 15.0 30.7 30.3 20.9 29.2 28.7 28.4 28.0 27.6 27.1 26.6 25.9 25.6 25.2 24.8 24.3 23.9 23.4 22.7 22.1 21.4 20.6 19.7 18.7 17.5 34.5 33.9 33.3 32.6 32.0 31.6 31.1 30.6 30.0 29.3 28.6 28.1 27.7 27.2 26.7 26.1 25.5 24.8 24.0 23.2 22.2 21.2 20.0 38.2 37.6 36.8 35.8 35.2 34.7 34.1 33.4 32.7 31.9 31.1 30.6 30.0 29.4 28.8 28.1 27.4 26.6 25.8 24.8 23.7 22.5 41.7 41.0 40.0 38.9 38.0 37.6 36.8 36.1 35.3 34.4 33.3 32.8 32.2 31.5 30.8 30.0 29.2 28.3 27.3 26.2 25.0 45.2 44.2 43.1 42.0 41.0 40.3 39.5 38.6 37.7 36.6 35.5 34.8 34.1 33.4 32.6 31.8 30.8 29.8 28.7 27.5 48.3 47.4 46.1 44.6 43.7 42.9 42.0 41.0 40.0 38.7 37.5 36.7 36.1 35.2 34.3 33.3 32.4 31.2 30.0 200 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 . 25 20 15 10 5...... LIVE-LOAD STRESSES 119 TABLE 19. Continued INFLUENCE-LINE ORDINATES FOR M FOR GIRDER BRIDGES WITHOUT FLOOR- BEAMS Values of ^- 2 LI SHORTER SEGMENT h 65 70 75 80 85 90 95 100 110 120 130 140 250 . 51.5 50.5 49.0 47.2 46.1 45.2 44.4 43.3 42.2 40.8 39.4 38.6 37.7 36.8 35.8 34.8 33 8 54.6 53.2 51.8 50.0 48.5 47.6 46.7 45.5 44.3 42.7 41.2 40.3 39.4 38.3 137.3 i36.2 35 57.5 56.2 54.6 52.4 51.0 50.0 49.0 47.6 46.3 44.6 42.9 42.0 41.0 39.8 38.7 37.5 60.6 58.8 57.1 54.9 53.2 52.1 51.0 49.5 48.1 46.3 44.4 43.5 42.4 41.2 40.0 63.3 61.7 59.5 57.1 55.6 54.3 52.9 51.6 49.8 48.1 46.1 44.8 43.7 42.5 66.2 64.1 62.1 59.5 57.5 56.2 54.6 53.2 51.5 49.5 47.4 46.3 45.0 .... 69.0 66.7 64.5 61.7 59.5 58.1 56.5 55.0 53.2 51.0 48.8 47.5 71.4 69.4 66.8 63.7 61.7 59.9 58.5 56.5 54.6 52.4 50.0 76.3 73.5 70.9 67.6 64.9 63.3 61.7 59.5 57.5 55.0 81.3 78.1 75.2 71.4 68.5 66.7 64.9 62.5 60.0 85.5 82.0 78.7 74.6 71.4 69.4 67.6 65.0 89.3 86.2 82.0 78.1 74.6 72.5 70.0 225 200 175 160 150 140 . . . 130 . 120 110 100 95 90 85 80 75 70. . . 65 32 5 1 1.00* Influence Line for M 120 LIVE-LOAD STRESSES TABLE 20 RECIPROCALS OF INFLUENCE-LINE ORDINATES FOR M FOR GIRDER BRIDGES WITHOUT FLOOR-BEAMS Values Of ry lit, SHORTER SEGMENT h 5 10 15 20 25 30 35 40 45 50 55 60 250 .204 .104 .0707 .0540 .0440 .0374 .0326 .0290 .0262 .0240 .0221 .0207 225 .204 .104 .0711 .0544 .0444 .0378 .0330 .0295 .0266 .0244 .0226 .0211 200 .205 .105 .0716 .0550 .0450 .0383 .0335 .0300 .0272 .0250 .0232 .0217 175 .206 .106 .0723 .0558 .0457 .0390 .0342 .0307 .0279 .0257 .0238 .0224 160 .206 .106 .0730 .0562 .0462 .0396 .0348 .0313 .0284 .0263 .0244 .0229 150 .207 .107 .0733 .0567 .0466 .0400 .0352 .0317 .0288 .0266 .0248 .0233 140 .207 .107 .0738 .0571 .0472 .0405 .0357 .0321 .0293 .0271 .0253 .0238 130 .208 .108 .0744 .0577 .0477 .0410 .0363 .0327 .0299 .0277 .0259 .0244 120 .208 .108 .0750 .0583 .0483 .0417 .0369 .0333 .0306 .0283 .0265 .0250 -4? 110 .209 .109 .0758 .0591 .0491 .0424 .0376 .0341 .0314 .0291 .0273 .0258 ** d 100 .210 .110 .0766 .0600 .0500 .0433 .0386 .0350 .0322 .0300 .0282 .0267 V r* 95 .211 .111 .0772 .0605 .0505 .0438 .0391 .0355 .0327 .0305 .0287 .0272 a 90 .211 .111 .0778 .0611 .0511 .0444 .0397 .0361 .0333 .0311 .0293 .0277 1 85 .212 .112 .0784 .0618 .0517 .0451 .0403 .0368 .0340 .0318 .0299 .0284 S3 80 .213 .113 .0792 .0625 .0525 .0458 .0411 .0375 .0347 .0325 .0307 .0292 bC 75 .213 .113 .0800 .0633 .0533 .0466 .0419 .0383 .0356 .0333 .0315 .0300 70 .214 .114 .0810 .0643 .0543 .0476 .0428 .0393 .0365 .0343 .0325 .0309 J 65 .215 .115 .0820 .0654 .0554 .0487 .0440 .0404 .0376 .0353 .0336 .0321 60 .217 .117 .0833 .0666 .0567 .0500 .0452 .0417 .0388 .0366 .0348 .0333 55 .218 .118 .0848 .0682 .0582 .0515 .0467 .0432 .0404 .0382 .0364 50 .220 .120 .0867 .0700 .0600 .0533 .0486 .0450 .0422 .0400 45 .222 .122 .0889 .0722 .0622 .0555 .0508 .0472 .0444 40 .225 .125 .0917 .0750 .0650 .0583 .0536 .0500 35 .229 .129 .0952 .0786 .0686 .0619 .0571 30 .233 .133 .1000 .0833 .0733 .0666 25 .240 .140 .1066 .0900 .0800 20 .250 .150 .1166 .1000 15 .267 .167 .1333 10 .300 .200 5 .400 LIVE-LOAD STRESSES 121 TABLE 20. Continued RECIPROCALS OF INFLUENCE-LINE ORDINATES FOR M FOR GIRDER BRIDGES WITHOUT FLOOR-BEAMS Values of SHORTER SEGMENT 65 70 75 80 85 90 95 100 110 120 130 140 250 225 200 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 .0194 .0198 .0204 .0212 .0217 .0221 .0225 .0231 .0237 .0245 .0254 .0259 .0265 .0272 .0279 .0287 .0296 .0307 .0183 .0188 .0193 .0200 .0206 .0210 .0214 .0220 .0226 .0234 .0243 .0248 .0254 .0261 .0268 .0276 .0286 .0174 .0178 .0183 .0191 .0196 .0200 .0204 .0210 .0216 .0224 .0233 .0238 .0244 .0251 .0258 .0286 .0165 .0170 .0175 .0182 .0188 .0192 .0196 .0202 .0208 .0216 .0225 .0230 .0236 .0243 .0250 .0158 .0162 .0168 .0175 .0180 .0184 .0189 .0194 .0201 .0208 .0217 .0223 .0229 .0235 .0151 .0156 .0161 .0168 .0174 .0178 .0183 .0188 .0194 .0202 .0211 .0216 0222 .0145 .0150 .0155 .0162 .0168 .0172 .0177 .0182 .0188 .0196 .0205 .0211 .0140 .0144 .0150 .0157 .0162 .0167 .0171 .0177 .0183 .0191 .0200 .0131 .0136 .0141 .0148 .0154 .0158 .0162 .0168 .0174 .0182 .0123 .0128 .0133 .0140 .0146 .0150 .0154 .0160 .0167 .0117 .0122 .0127 .0134 .0140 .0144 .0148 .0154 .0112 .0116 .0122 .0128 .0134 .0138 .0143 Influence Line for M 122 LIVE-LOAD STRESSES TABLE 21 BENDING MOMENTS IN BEAMS DUE TO UNIFORM LOAD OP 1 POUND PER LINEAL FOOT Values in Foot-pounds Values equal -^ = Area of Influence Line for M SHORTER SEGMENT h 5 10 15 20 25 30 35 40 4b 50 55 60 250 625 1250 1875 2500 3125 3750 4375 5000 5625 6250 6875 7500 225 562.5 1125 1687.5 2250 2812.5 3375 3937.5 4500 5062.5 5625 6187.5 6750 200 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 175 437.5 875 1312.5 1750 2187.5 2625 3062.5 3500 3937.5 4375 4812.5 5250 160 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 150 375 750 1125 1500 1875 2250 2625 3000 3375 3750 4125 4500 140 350 700 1050 1400 1750 2100 2450 2800 3150 3500 3850 4200 130 325 650 975 1300 1625 1950 2275 2600 2925 3250 3575 3900 120 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 ^ 110 275 550 825 1100 1375 1650 1925 2200 2475 2750 3025 3300 +a 100 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 g 95 237.5 475 712.5 950 1187.5 1425 1662.5 1900 2137.5 2375 2612.5 2850 g 90 225 450 675 900 1125 1350 1575 1800 2025 2250 2475 2700 o 3 85 212.5 425 637.5 850 1062.5 1275 1487.5 1700 1912.5 2125 2337.5 2550 r/j 80 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 I 75 187.5 375 562.5 750 937.5 1125 1312.5 1500 1687.5 1875 2062.5 2250 a 70 175 350 525 700 875 1050 1225 1400 1575 1750 1925 2100 2 65 162.5 325 487.5 650 812.5 975 1137.5 1300 1462.5 1625 1787.5 1950 60 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800 55 137.5 275 412.5 550 687.5 825 962.5 1100 1237.5 1375 1512.5 . . . . 50 125 250 375 500 625 750 875 1000 1125 1250 45 112.5 225 337.5 450 562.5 675 787.5 900 1012.5 40 100 200 300 400 500 600 700 800 35 87.5 175 262.5 350 437.5 525 612.5 30 75.0 150 225 300 375 450 25 62.5 125 187.5 250 312.5 20 50.0 100 150 200 15 37.5 75 112.5 10 25.0 50 5 12.5 LIVE-LOAD STRESSES 123 TABLE 21. Continued BENDING MOMENTS IN BEAMS DUE TO UNIFORM LOAD OF 1 POUND PER LINEAL FOOT Values in Foot-pounds Values equal ^ = Area of Influence Line for M SHORTER SEGMENT h 65 70 75 80 85 90 95 100 110 120 130 140 250 225 200 175 160 150 140 130 120 110 100 95 90 85 80 75 70 65 8125 7312.5 6500 5687.5 5200 4875 4550 4225 3900 3575 3250 3087.5 2925 2762.5 2600 2437.5 2275 2112.5 8750 7875 7000 6125 5600 5250 4900 4550 4200 3850 3500 3325 3150 2975 2800 2625 2450 9375 8437.5 7500 6562.5 6000 5625 5250 4875 4500 4125 3750 3562.5 3375 3187.5 3000 2812.5 10000 9000 8000 7000 6400 6000 5600 5200 4800 4400 4000 3800 3600 3400 3200 10625. 9562.5 8500 7437.5 6800 6375 5950 5525 5100 4675 4250 4037.5 3825 3612.5 11250 10125 9000 7875 7200 6750 6300 5850 5400 4950 4500 4275 4050 11875 10687.5 9500 8312.5 7600 7125 6650 6175 5700 5225 4750 4512.5 12500 11250 10000 8750 8000 7500 7000 6500 6000 5500 5000 13750 12375 11000 9625 8800 8250 7700 7150 6600 6050 15000 13500 12000 10500 9600 9000 8400 7800 7200 16250 14625 13000 11375 10400 9750 9100 8450 17500 15750 14000 12225 11200 10500 9800 1 Pound per Lineal Foot THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $I.OO ON THE SEVENTH DAY OVERDUE. MAR 161934 LD 21-100m-7,'33 337856 - UNIVERSITY OF CALIFORNIA LIBRARY