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Civil Engineering Formulas 2nd Edition PDF Book

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Summary of Contents

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    CIVILENGINEERINGFORMULAS

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    ABOUT THE AUTHORTyler G. Hicks, P.E., is a consulting engineer and a successful engi-neering book author. He has worked in plant design and operationin a variety of industries, taught at several engineering schools, andlectured both in the United States and abroad. Mr. Hicks holds abachelor’s d...

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    CIVILENGINEERINGFORMULASTyler G. Hicks, P.E.International Engineering AssociatesMember: American Society of Mechanical EngineersUnited States Naval InstituteSecond EditionNew YorkChicagoSan FranciscoLisbonLondonMadridMexico CityMilanNew DelhiSan JuanSeoulSingaporeSydneyToronto

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    Copyright © 2010, 2002 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permittedunder the United States Copyright Act of 1976, no part of this publication may be reproduced or distrib-uted in any form or by any means, or stored in a database or retrieval system, without the pri...

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    CONTENTS 12,Preface 12,xi 14,Acknowledgments 14,xiii 16,How to Use This 16,Book 16,xv 20,Chapter 1. 20,Conversion Factors for 20,Civil 20,Engineering Practice 20,1 30,Chapter 2. 30,Beam Formulas 30,11 30,Continuous Beams 30,/ 30,11 65,Ultimate Strength of 65,Continuous Beams 65,/ 65,46 71...

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    124,Chapter 4. 124,Piles and Piling Formulas 124,105 124,Allowable Loads on Piles 124,/ 124,105 124,Laterally Loaded Vertical Piles 124,/ 124,105 126,Toe Capacity Load 126,/ 126,107 126,Groups of Piles 126,/ 126,107 128,Foundation-Stability Analysis 128,/ 128,109 131,Axial-Load Capacity...

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    176,Chapter 6. 176,Timber Engineering Formulas 176,157 176,Grading of Lumber 176,/ 176,157 176,Size of Lumber 176,/ 176,157 178,Bearing 178,/ 178,159 178,Beams 178,/ 178,159 179,Columns 179,/ 179,160 180,Combined Bending and Axial Load 180,/ 180,161 180,Compression at Angle to Grain ...

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    226,Chapter 9. 226,Building and Structures 226,Formulas 226,207 226,Load-and-Resistance Factor Design 226,for Shear in Buildings 226,/ 226,207 227,Allowable-Stress Design for Building 227,Columns 227,/ 227,208 228,Load-and-Resistance Factor Design for Building 228,Columns 228,/ 228,209...

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    294,Chapter 11. 294,Highway and Road Formulas 294,275 294,Circular Curves 294,/ 294,275 296,Parabolic Curves 296,/ 296,277 297,Highway Curves and Driver Safety 297,/ 297,278 298,Highway Alignments 298,/ 298,279 300,Structural Numbers for Flexible 300,Pavements 300,/ 300,281 303,Transi...

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    380,Chapter 13. 380,Stormwater, Sewage, Sanitary 380,Wastewater, and Environmental Protection 380,361 380,Determining Storm Water Flow 380,/ 380,361 380,Flow Velocity in Straight Sewers 380,/ 380,361 383,Design of a Complete-Mix 383,Activated Sludge Reactor 383,/ 383,364 387,Design of a C...

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    PREFACEThe second edition of this handy book presents some 2,500 formulas and calcu-lation guides for civil engineers to help them in the design office, in the field,and on a variety of construction jobs, anywhere in the world. These formulasand guides are also useful to design drafters, structur...

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    When using any of the formulas in this book that may come from an indus-try or regulatory code, the user is cautioned to consult the latest version of thecode. Formulas may be changed from one edition of code to the next. In a workof this magnitude it is difficult to include the latest formulas f...

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    ACKNOWLEDGMENTSMany engineers, professional societies, industry associations, and governmen-tal agencies helped the author find and assemble the thousands of formulas pre-sented in this book. Hence, the author wishes to acknowledge this help andassistance.The author’s principal helper, advisor,...

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    the author consulted several hundred civil engineering reference and textbooksdealing with the topics in the current book. The author is grateful to the writersof all the publications cited here for the insight they gave him to civil engi-neering formulas. A number of these works are also cited i...

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    HOW TO USE THIS BOOKThe formulas presented in this book are intended for use by civil engineersin every aspect of their professional work—design, evaluation, construction,repair, etc.To find a suitable formula for the situation you face, start by consulting theindex. Every effort has been made ...

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    CIVILENGINEERINGFORMULAS

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    CHAPTER 11CONVERSION FACTORSFOR CIVIL ENGINEERINGPRACTICECivil engineers throughout the world accept both the United States CustomarySystem(USCS) and the System International(SI) units of measure for bothapplied and theoretical calculations. However, the SI units are much morewidely used than tho...

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    2CHAPTER ONETABLE 1.1Commonly Used USCS and SI Units*Conversion factor (multiply USCS unit by this factor to USCS unitSI unitSI symbolobtain SI unit)Square footSquare meterm20.0929Cubic footCubic meterm30.2831Pound per KilopascalkPa6.894square inchPound forceNewtonN4.448Foot pound Newton meterN m...

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    CONVERSION FACTORS FOR CIVIL ENGINEERING PRACTICE3TABLE 1.3Factors for Conversion to SI Units of MeasurementTo convert fromToMultiply byAcre foot, acre ftCubic meter, m31.233489E03AcreSquare meter, m24.046873E03Angstrom, ÅMeter, m1.000000* E10Atmosphere, atmPascal, Pa1.013250* E05(standard)Atmos...

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    4CHAPTER ONETABLE 1.3Factors for Conversion to SI Units of Measurement(Continued)To convert fromToMultiply byCentimeter, cm, ofPascal, Pa9.80638E01water (4°C)ChainMeter, m2.011684E01Circular milSquare meter, m25.067075E10DaySecond, s8.640000* E04Day (sidereal)Second, s8.616409E04Degree (angle)Ra...

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    CONVERSION FACTORS FOR CIVIL ENGINEERING PRACTICE5TABLE 1.3Factors for Conversion to SI Units of Measurement(Continued)To convert fromToMultiply byFoot per secondMeter per second3.048000† E01squared, ft/s2squared, m/s2Footcandle, fcLux, lx1.076391E01Footlambert, fLCandela per square3.426259E00m...

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    6CHAPTER ONETABLE 1.3Factors for Conversion to SI Units of Measurement(Continued)To convert fromToMultiply byInch of mercury, in HgPascal, Pa3.38638E03(32°F) (pressure)Inch of mercury, in HgPascal, Pa3.37685E03(60°F) (pressure)Inch of water, in Pascal, Pa2.4884E02H2O (60°F) (pressure)Square in...

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    CONVERSION FACTORS FOR CIVIL ENGINEERING PRACTICE7TABLE 1.3Factors for Conversion to SI Units of Measurement(Continued)To convert fromToMultiply byMicroinch, inMeter, m2.540000† E08Micron, mMeter, m1.000000† E06Mil, miMeter, m2.540000† E05Mile, mi (international)Meter, m1.609344† E03Mile,...

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    8CHAPTER ONETABLE 1.3Factors for Conversion to SI Units of Measurement(Continued)To convert fromToMultiply byPerm inch, perm inKilogram per pascal1.45929E12(23°C)second meter,kg/(Pa s m)Pint, pt (U.S. dry)Cubic meter, m35.506105E04Pint, pt (U.S. liquid)Cubic meter, m34.731765E04Poise, P (absolut...

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    CONVERSION FACTORS FOR CIVIL ENGINEERING PRACTICE9TABLE 1.3Factors for Conversion to SI Units of Measurement(Continued)To convert fromToMultiply byQuart, qt (U.S. dry)Cubic meter, m31.101221E03Quart, qt (U.S. liquid)Cubic meter, m39.463529E04RodMeter, m5.029210E00Second (angle)Radian, rad4.848137...

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    CHAPTER 2BEAMFORMULASIn analyzing beams of various types, the geometric properties of a variety ofcross-sectional areas are used. Figure 2.1 gives equations for computing area A,moment of inertia I, section modulus or the ratio SI/c,where cdistancefrom the neutral axis to the outermost fiber of ...

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    A = bdbdb2 + d2c1 = d/2S1 =c113 c32bbRectangleTriangle1dd2333c1c2bHalf ParabolaParaboladc11122c211212b1d2c2 = dc3 =bd26l1 = bd336S1 = bd224c1 = 2d3A =bd23l2 = bd312r1 = d18r1 =d12l1 = bd312l2 = bd33l3 =S3 =b3d3b2d26 (b2 + d2)b2 + d26r3 =bd6 (b2 + d2)l1 =bd3bd8175l3 = 16105l2 = b3d30A =bd23l1 =bd3...

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    FIGURE 2.1Geometric properties of sections.SectionEquilateral polygonA = Arear = Rad inscribed circlen = No. of sidesa = Length of sideAxis as in preceding sectionof octagonR = Rad circumscribed circleMoment of inertiab1b122cbhI =(6R2− a2)6R2− a22412r2+ a24812b2+ 12bb1 + 2b126 (2b+ b1)R...

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    b2b2b2b2B2B2hHHHhhchchhHHHBbBbbbBBBBH3 + bh312BH3 – bh312BH3 + bh36HBH3 – bh36H =IIc= =I=BH3 + bh312 (BH + bh)BH3 – bh312 (BH – bh)IcSectionMoment of inertiaSection modulusRadius of gyration14

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    FIGURE 2.1(Continued)b2HHHHhhh1bc2aaaa2a2bBBBbBc1d1B12ddddb2b2c2c1c2c1hdI(Bd + bd1) + a(h + h1)r=c2 = II – c1I= (Bc13 – bh3 + ac23)c1 =I[Bd + a(II – d)]ali2 + bd2ali + bdI= (Bc13 – B1h3 + bc23 – b1h13)c1 =aH2 + B1d2 + b1d1 (2H – d1)aH + B1d + b1d112131212SectionMoment of inertia and ...

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    Rrrddm = (D + d)(D4 – d4)s = (D – d)DdR4 + r22I= π64I=r2=πd464πr44= A4(R4 – r4)= π4A4(R2 + r2)== 0.05 (D4 – d4) (approx)D4 – d4Dπ32D2 + d22R4 – r4R==π4whenis very small= 0.8dm2s (approx)0.05d4 (approx)=r=πd332πr34= A41212Ic0.1d3 (approx)==Ic=d4r2sdm=SectionMoment of ...

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    FIGURE 2.1(Continued)17c2c2c1c1rrr1RI = r 4I = 0.1098 (R4 – r 4)= 0.3tr13 (approx)whenis very small0.283R2r2 (R – r)R+ rπ889π_= 0.1098r4–tr1Ic2Ic1c1= 0.1908r3= 0.2587r3= 0.4244rc1 =c2 = R – c143πR2 + Rr + r 2R + r9π2 – 646πr = 0.264r2Iπ (R2 – r2)= 0.31r1 (approx)ta1aattb1bb(appr...

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    18B4B4B2bbdB2HhhB(approx)I = l12B2I =++t4t233π16Ic2+ hπB4πISectionMoment of inertia and section modulusRadius of gyration+ B2hh3h= H –πB316πBh22+ 2b (h – d)d24d4+ b(h3 – d3)+ b3(h – d)I= l6hIc= 2IH + t3π16d4+ b(h3 + d3)+ b3(h – d)tb2h2h1HBCorrugated sheet iron,parabolically curved...

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    FIGURE 2.1(Continued)Approximate values of least radius of gyration rDDDDDDCrossDBDDDDCarnegiecolumnPhoenixcolumnr =r =0.3636D0.295DD/4.58BD/2.6 (B + D)D/4.74D/4.74D/5D/3.54D/6Z-barI-beamT-beamChannelAngleUnequal legsAngleEqual legsDeckbeam19

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    FIGURE 2.1(Continued)DdmdsdmabsdoDOuter fiberShaft sectionShaft torque formulas and location ofmaximum shear stress in the shaftShaft sectionShaft angle-of-twist section formulasAngle-of-twistθ =D, do, di, dm, ds, s, a, b = shaft sectional dimensions, in (mm)N = modulus of rigidity, psi (MPa)L =...

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    CASE 2. Beam Supported Both Ends—Concentrated Load at Any PointR = WbLWaLR1 =V (max) = R when a < b and R1 when a > bAt point of load:At x: when x = a(a + 2b)+ 3 and a > bD (max) = Wab (a + 2b) 3a (a + 2b)+ 27 EILAt x: when x < aAt x: when x > aAt x: when x < aL WDRR1MxVa...

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    CASE 3. Beam Supported Both Ends—Two Unequal Concentrated Loads, Unequally DistributedR = 1LW (L – a) + W1bM = aLW (L – a) + W1bM1 = bLWa + W1 (L – b)M= WaLbxL(L – x) + W1R1 = 1LWa + W1 (L – b)V (max) = Maximum reactionAt point of load W:At point of load W1:At x: when x > a or <...

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    23FIGURE 2.2(Continued)CASE 4. Beam Supported Both Ends—Three Unequal Concentrated Loads, Unequally DistributedR =Wb + W1b1 + W2b2LR1 =V (max) = Maximum reactionAt x: when x > a and < a1At x: when x = aAt x: when x = a1M = RaM1 = Ra1 – W (a1 – a)At x: when x = a2M2 = Ra2 – W (a2 ...

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    DR1At center:At supports:At x:At x:RVVWxLMM1M (max) =(L2 – 2Lx + x2)WL24M1 (max) = WL12L26M=–+ Lx – x2W2LAt center:At x:D (max) =1384WL3EID =Wx224 EILR = R1 = V (max) =W2V =–W2WxLCASE 5. Beam Fixed Both Ends—Continuous Load, Uniformly Distributed24

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    FIGURE 2.2(Continued)25CASE 6. Beam Fixed Both Ends—Concentrated Load at Any Pointb2(3a + b)L3V (max) = Rwhen a < b= R1 when a > bAt x: when x =when x < aAt x: when x < aAt support R:At support R1:At x:At point of load:R1 = WR = WM1= – WV = Ra2(3b + a)L3ab2L2and a> b2 aL3a + b...

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    R1LoadingAt fixed end:At free end:At x:At x:BendingShearDVWxLMM (max) =(x4 – 4L3x + 3L4)WL2M = Wx22LAt x:D (max) =WL38EID =W24EILR1 = V (max) = WV =WxLCASE 7. Beam Fixed at One End (Cantilever)—Continuous Load, Uniformly Distributed26

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    FIGURE 2.2(Continued)27DR1VWxabLMCASE 8. Beam Fixed at One End (Cantilever)—Concentrated Load at Any PointR1 = V (max) = WM (max) = WbM = W(x = a)At x: when x > aAt fixed end:At free end:At x: when x < aAt x: when x > aAt x: when x > aV = WV = 0WL36EI33aLD(max) =2 – aL+W6EI–3aL...

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    At x: when x < aAt x: when x > aAt x: when x > aAt x: when x = a = 0.414LAt x: when x < aAt x: when x > aAt x: when x < aAt point of load:At fixed end:V = RV = R – WR1 = W3aL2 – a32L316EIR = W3b2L – b32L33RL2x – Rx3 –R1(2L3 – 3L2x + x3) –3b2L – b32L3axWLDR1M1MVbR...

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    FIGURE 2.2(Continued)29DAt fixed end:At x:At x: when x =At x: when x = 0.4215LAt x:RR1VWxLMM1M (max) =WL9128M1 (max) =M=xL – WxLWL183812L38At x:D (max) = 0.0054WL3EID =Wx48EILR1 = V (max) =58 WR = 38 WV =W–38WxLCASE 10. Beam Fixed at One End, Supported at Other—Continuous Load, Uniformly D...

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    At x: when x < aAt x1: when x1 < LAt x1: when x1 < L V = R – w(a + x1)At x2: when x2 < b V = w(b – x2)At x: when x < a V = w (a – x)At x2: when x2 < bAt x1: when x1 =At R:At R1:axx1x2WLR1M1M1MbR– awa212R2w= W= load per unit of lengthWa+ L + b– aRwM(max) = RV(max)...

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    FIGURE 2.2(Continued)31DDVVR1At x1: when x1 < LAt x: when x < aAt x: when x < aRW2xaaLx1M (max) =(a – x)Wa2MM =W2At center:At free ends:D =Wa2(3L + 2a)12EID =WaL216EIR = R1 = V (max) =W2V =W2CASE 12. Beam Overhanging Both Supports, Symmetrically Placed—Two Equal Concentrated Loads a...

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    LLoadx(l – x)wL2x(l – 2x2 + x3)– xwLShearMomentElastic curve(a)wxLR=RRwL2R= wL2wL28wL324EI5wL4384EIwL424EIL2121212x<12cLLLoada+(2c + b)(x′ < a)(x″ < c)ShearMoment(b)bwacR=R2R1wb2L(2c + b)R2 = wb2LR1wR12wR1 – w(x – a)w2x(a < x < a+ b)(x –a)2–x′x″R1x′R1xR1(a +...

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    33PPPPLLoad(1 – 2k)LPxLx(3k – 3k2 – x2)k(3x′ – 3x′2 – k2)k(3 – 4k2)ShearMomentElastic curve(e)xL(x < k)k(1 – k)PL22EIPL36EIPL324EIPkLPL36EIL2L′′2kLkLR = PR = Px′(k < x′ < (1 – k))PPRPPPPPPPPaLaLaLaLaLaL(For n aneven number)(For n an even number)LoadmaLShearMom...

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    PPLoadShearMomentElastic curve(g)PR1 = L′LL′xPL′x′L′xLPx′L′PL′LPR2 =R2R1L + L′L2(L + L′) – L′x′ (1 + x′)L3(L + L′)PL′23EIx(1 – x2)PL′L26EIPL′2 (1 – x′)6EIdmax = PL′L2 3EI9PPLLoadShearMomentElastic curve(h)R = PLPPLxLPxLR(2 – 3x + x3)PL33EIPL33EILL...

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    35LoadShearMomentElastic curve(j)R = wLwLxwLxLLwwL22wL22(3 – 4x + x4)x2wL424EIwL48EIwL212LoadShearMomentElastic curve(k)R1 =R2R1wL′wL′x′x′L′xLLL′wwL′22LR2 =wL′2LwL′22L3(4L + 3L′)x(1 – x2)x′2wL′324EIdmax =wL′2L2wL′2L212EIwL′2x12wL′2123EI18(2L + L′)LoadShearMom...

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    ShearMoment(m)0.5774lR1R1 = V1R2 = V2maxMmax∆max∆xat x = = 0.5774lVxMxR2V1lxWMmaxV2= W3W 3= 2W3==W3– Wx2l2(l2 – x2)Wl2EI== 0.01304(3x4 – 10l2x2 + 7l4)Wx180EIl2=Wx3l2= 0.1283Wl2Wlat x = l1 –= 0.5193l8153936FIGURE 2.3Elastic-curve equations for prismatic beams. (m) Simple beam—load ...

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    BEAM FORMULAS37ShearMoment(n)R= VMmax (at center)∆max (at center)∆xRVRxWlVMmax=(l2 – 4x2)W2l2= W2= Wl6(5l2 – 4x2)2Wx480EIl2=l2l2VxMxwhen x < l2= Wx–when x < l22x23l212= Wl360EIShearMoment(o)R1 = V1max(2l – a)MmaxR2R1R1WV2awaxlV1Mmax=wx22R2 = V2R1 – wx= wa22lwa2l==wx24EIl=V (wh...

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    38CHAPTER TWOShearMoment(p)R= V= P= Pl= Px=Mmax (at fixed end)∆max (at free end)∆xMxRPVxlMmaxPl33EI= P6EI(2l3 – 3l2x + x3)ShearMoment(q)R= VMmax (at center and ends)∆max (at center)∆xRPVxlVRMmaxMmaxMmax= P2=(4x – l)P8=(3l – 4x)Pl3192EI= Px248EI= Pl8l4l2l2Mx when x <l2FIGURE 2.3El...

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    PPRLLRALLRLRLRR(a)(b)(c)MLMLMLMRMRMRkLVRVLkLkLk(l – k)PL++=k(l – k)PLBCFIGURE 2.4Any span of a continuous beam (a) can be treated as a simple beam, as shown in(b) and (c). In (c), the moment diagram is decomposed into basic components.39

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    40CHAPTER TWOR1d1y11d1 = y11R1 + y12R2 + y13R3d2 = y21R1 + y22R2 + y23R3d3 = y31R1 + y32R2 + y33R3y21y31d2d3R0L1L2WL3L4R2R3R4(a)(b)(c)y13111y23y33y12y22y32(e)(d)FIGURE 2.5Reactions of continuous beam (a) found by makingthe beam statically determinate. (b) Deflections computed with inte-rior supp...

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    BEAM FORMULAS41kLkLkLkLkL–k(l – k)2 WLk(l – k)WLk(l – k)WLk2(l – k)WLWLLL(a)(b)(c)(d)wLWL12WL12L2WLW = wLL4L4L4L418WL18WL18WL16WL548WL548W1312k(l – k)WL12kWL12W2W2W13W13–––LcFIGURE 2.6Fixed-end moments for a prismatic beam. (a) For a concentrated load.(b) For a uniform load. (c)...

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    42CHAPTER TWO00.05 0.10 0.15m =MFWLWaL0.2001.00.10.90.20.8G 2 = 0.05G2 = 0G2 = 0G 2 = 0.15G 2 = 0.20G 2 = 0.250.30.7a0.40.6Use upper line for MRFUse lower line for MLF0.60.40.70.30.80.20.90.11.000.5G2 = 0S3 = 0S3 = 0.05 for mR(–0.05for mL)S3 = –0.05 for mR(0.05for mL)G 2 = 0.10G2 = 0.10FIGURE...

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    BEAM FORMULAS43W = PW = wyLW =W = (n + 1)PnPyLxLxLyLwyLxLwX = 0x =yn1 + nG2 = 0G2 =S3 = 0S3 = 0Case 1Case 3Case 4Case 2Px =y12y2n(1 + n)2G2 =y2112S3 = –x =y13G2 =y2118y31135S3 =y3n(n – 1)(1 + n)3wyL2–––––––FIGURE 2.8Characteristics of loadings.

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    44CHAPTER TWOW =wyL2yLxLyL yL yL yL yL yLaLPPPPPPPyLxL–xLLdxxLyLy0wCase 6G2 =y2 =n2– 112n+ 1n– 1LÚ wx′dx′Wy0W= LÚ f(x′)dx′a212.S3 = 0Case 5S3 = 0Case 7Case 8x =W = nPyyL12x =x =y0LÚ wx2dxWG2 =y0LÚ wx3dxWS3 =yn– 1212yL12G2 =y2124S3 = –x =x =y14G2 =y2380y31160W =13wyLw = ky2...

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    BEAM FORMULAS45S3Pn/W. These values are given in Fig. 2.8 for some common typesof loading.Formulas for moments due to deflection of a fixed-end beam are given inFig. 2.9. To use the modified moment distribution method for a fixed-end beamsuch as that in Fig. 2.9, we must first know the fixe...

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    46CHAPTER TWOreciprocal deflections, we obtain the end moments of the deflected beam inFig. 2.9 as(2.1)(2.2)In a similar manner the fixed-end moment for a beam with one end hinged andthe supports at different levels can be found from (2.3)where Kis the actual stiffness for the end of the beam ...

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    BEAM FORMULAS47The mechanism method can be used to analyze rigid frames of constant sec-tion with fixed bases, as in Fig. 2.11. Using this method with the vertical load atmidspan equal to 1.5 times the lateral load, the ultimate load for the frame is4.8MP /Llaterally and 7.2MP /Lvertically at mi...

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    L2LBBAEELDADBPCCCP(a)(b) Beammechanism(c) Framemechanism(d) Combinationmechanism(e)(f)(g)AAADDDBPBBEEECCCADBEC1.5P1.5P1.5PL2L2L2θL2θL2θL2θL2θθθθθθθθθθθ2θ2θL2L2L2FIGURE 2.11Ultimate-load possibilities for a rigid frame of constant section with fixed bases.48

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    BEAM FORMULAS49or if(2.6)A plastic hinge forms at this point when the moment equals kMP. For equilibrium,leading to(2.7)When the value of MP previously computed is substituted,from which k0.523. The ultimate load is(2.8)In any continuous beam, the bending moment at any section is equal to thebend...

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    TABLE 2.1Uniformly Loaded Continuous Beams over Equal Spans(Uniform load per unit lengthw; length of each spanl)Shear on each sideDistance to point Notationof support. Lleft,of max moment,Distance toNumberofRright. reaction atMomentMax.measured to point of inflection,ofsupportany support is LRov...

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    BEAM FORMULAS51on each side of the supports. Note that the shear is of the opposite sign on eitherside of the supports and that the sum of the two shears is equal to the reaction.Figure 2.13 shows the relation between the moment and shear diagrams for auniformly loaded continuous beam of four equ...

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    52CHAPTER TWOMaxwell’s theorem states that if unit loads rest upon a beam at two points, Aand B, the deflection at Adue to the unit load at Bequals the deflection at Bdueto the unit load at A.Castigliano’s TheoremThis theorem states that the deflection of the point of application of an ext...

  • Page 72

    BEAM FORMULAS53Shear at the end of a beam necessitates modification of the forms deter-mined earlier. The area required to resist shear is P/Sv in a cantilever and R/Sv ina simple beam. Dashed extensions in Figs. 2.15 and 2.16 show the changes nec-essary to enable these cantilevers to resist she...

  • Page 73

    Beam1. Fixed at One End, Load P Concentrated at Other End12Cross sectionBABPPAllbbxxyy2=xyyyhh6PbSshhRectangle:width (b) con-stant, depth(y) variableDeflection at A:Elevation:1, top, straightline; bottom,parabola. 2,complete pa-rabolaPlan:rectangleFormulasElevationand planh=f=6PlbSs8PbElh3h2h254

  • Page 74

    BBzPAAPllxbbzxyby=xbyyyhhhh6Ph2Ss23= k (const)hRectangle:width (y) var-iable, depth(h) constantRectangle:width (z) var-iable, depth(y) variablePlan:cubic parabolaElevation:cubic parabolaPlan:triangleDeflection at A:Elevation:rectangleb= 6Plh2Ssf= 6PbEy3=x6PkSslhh=z= kyb= kh6PlbSs332 3bzyFIGURE 2....

  • Page 75

    BPBAxAllxydy3=xbydyyhh32PπSsCircle:diam (y)variableRectangle:width (b) con-stant, depth(y) variablePlan:rectangleElevation:trianglePlan:cubic parabolaElevation:cubic parabolaf= 6 PbElhd=32PlπSs3y= x3PblSh=3PlbSs323dBeam1. Fixed at One End, Load P Concentrated at Other End (Continued)Cross secti...

  • Page 76

    FIGURE 2.17(Continued)BBAAxxllyyyzhbby=bbzyhh3Px2lSsh2Rectangle:width (z) var-iable, depth(h) constantRectangle:width (z) var-iable, depth(y) variable,Elevation:semicubicparabolaPlan:two paraboliccurves withvertices at freeendPlan:semicubicparabolab= 3PlSsh2y3=3Px2kSslz= kyb= khf= 3PbElhh=3PlkSs3...

  • Page 77

    BAPCxxllyydy3=x2xdbyhhy16PπlSsRectangle:width (b) con-stant, depth(y) variableElevation:two parabolas,vertices atpoints of supportPlan:semicubicparabolaPlan:rectangled=16PlπSs3y=3PSsbh=3Pl2bSsd3l2Beam2. Fixed at One End, Load P Uniformly Distributed Over l3. Supported at Both Ends, Load P Conce...

  • Page 78

    FIGURE 2.17(Continued)PCCABbPxxPby=xxbyxy3PSsh2Rectangle:width (y) vari-able, depth(h) constantRectangle:width (b) con-stant, depth(y or y1)variableElevation:two parabolas,vertices atpoints of supportPlan:two triangles,vertices atpoints of supportPlan:rectangleb= 3Pl2Ssh2y2=6P(l – p)blSs6P(l ...

  • Page 79

    OCPxABbRectangle:width (b) con-stant, depth(y) variablePlan:rectangle3Pl4bSsh=f=yyhhl2Elevation:ellipseDeflection at O:l2h2= 1+y23Pl4bSsx2164= 316PbEl22lh34. Supported at Both Ends, Load P Uniformly Distributed Over lPl3EIOxABbRectangle:width (b) con-stant, depth(y) variableMajor axis = lMinor ax...

  • Page 80

    FIGURE 2.17(Continued)xRectangle:width (y)variable, depth(h) constantPlan:two parabolaswith vertices atcenter of spany=ybybhl2Elevation:rectanglex –3PSsh2b= 3Pl4Ssh2x2lBeamCross sectionFormulasElevationand plan4. Supported at Both Ends, Load P Uniformly Distributed Over l61

  • Page 81

    TABLE 2.2Approximate Safe Loads in Pounds (kgf) on Steel Beams* (Percoyd Iron Works)(Beams supported at both ends; allowable fiber stress for steel, 16,000 lb/in2 (1.127 kgf/cm2) (basis of table) for iron, reduce values given in tableby one-eighth)Greatest safe load, lb†Deflection, in†Shape...

  • Page 82

    TABLE 2.3Coefficients for Correcting Values in Table 2.2 for Various Methods of Support and of Loading*Max relativedeflection underMax relativemax relative safeConditions of loadingsafe loadloadBeam supported at endsLoad uniformly distributed over span1.01.0Load concentrated at center of span0....

  • Page 83

    64CHAPTER TWOFigure 2.19 shows the condition when two equal loads are equally distant onopposite sides of the center of the beam. The moment is then equal under thetwo loads.If two moving loads are of unequal weight, the condition for maximummoment is the maximum moment occurring under the heavie...

  • Page 84

    BEAM FORMULAS65FIGURE 2.20Two moving loads of unequal weight.P2PR1R2l2a6aMomentShearl2When several wheel loads constituting a system are on a beam or beams, theseveral wheels must be examined in turn to determine which causes the greatestmoment. The position for the greatest moment that can occur...

  • Page 85

    66CHAPTER TWORis the radius of the centroidal axis; Z is a cross-section propertydefinedby(2.19)Analyticalexpressions for Zof certain sections are given in Table 2.4. Zcanalso be found by graphicalintegration methods (see any advanced strengthbook). The neutral surfaceshifts toward the center of ...

  • Page 86

    TABLE 2.4Analytical Expressions for ZSectionExpressionA2 [(tb)C1bC2]Z1RAb ln RC2RC2(tb) ln RC1RC1 and AtC1(bt) C3bC2Z1RAt ln (RC1)(bt) ln (RC0)b ln (RC2)Z12RrRrB Rr21Z1RhlnRCRChRSectionCrRbbC1C1C3C2RC2Rttr67

  • Page 87

    68CHAPTER TWOand(2.23)at(2.24)By Castigliano,(2.25)Eccentrically Curved BeamsThese beams (Fig. 2.23) are bounded by arcs having different centers of cur-vature. In addition, it is possible for either radius to be the larger one. Theone in which the section depth shortens as the central section is...

  • Page 88

    BEAM FORMULAS69TABLE 2.5Stress Factors for Inner Boundary at Central Section (see Fig. 2.23)1.For the arch-type beams(a)(b)(c)In the case of larger section ratios use the equivalent beam solution2.For the crescent I-type beams(a)(b)(c)3.For the crescent II-type beams(a)(b)(c) K1.081hRoRi0.0270 i...

  • Page 89

    70CHAPTER TWOTABLE 2.6Crescent-Beam Position Stress Factors (see Fig. 2.23)*Angle ,kdegree Inner Outer1010.055 H/h10.03 H/h2010.164 H/h10.10 H/h3010.365 H/h10.25 H/h4010.567 H/h10.467 H/h5010.733 H/h6011.123 H/h7011.70 H/h8012.383 H/h901 3.933 H/h*Note: All formulas are valid for 0 H/h0.325. Form...

  • Page 90

    BEAM FORMULAS71where Lunbraced length of the memberEmodulus of elasticityIymoment of inertial about minor axisGshear modulus of elasticityJtorsional constantThe critical moment is proportional to both the lateral bending stiffness EIy/Land the torsional stiffness of the member GJ/L.For the case o...

  • Page 91

    72CHAPTER TWOmoment, Mcr may be obtained by multiplying Mcr given by the previous equa-tions by an amplification factor(2.28)where(2.29)and Mmaxabsolute value of maximum moment in the unbraced beam segmentMAabsolute value of moment at quarter point of the unbraced beamsegmentMBabsolute value of ...

  • Page 92

    BEAM FORMULAS73minimum value of MPdis zero. The deflection dfor axial compression andbending can be closely approximated by(2.32)where d0deflection for the transverse loading alone, in (mm); and Pccriti-cal buckling load 2EI / L2, lb (N).UNSYMMETRICAL BENDINGWhen a beam is subjected to loads th...

  • Page 93

    74CHAPTER TWOwhere Across-sectional area, in2 (mm2)cdistance from neutral axis to outermost fiber, in (mm)Imoment of inertia of cross section about neutral axis, in4 (mm4)rradius of gyration, in (mm)Figure 2.1 gives values of the radius of gyration for several cross sections.If there is to be no...

  • Page 94

    Type of Support Fundamental ModeLSecond ModeThird ModeFourth ModeCantilever20.01253500.31556.0FixedFixed-hinged0.1122240.02815020.01250.774L0.5L0.356L0.644L6840.132L0.05030.01800.00920.5L0.359L0.359L 0.278L0.278L1,1330.0056LLSimplewL4/EI =EI/wL4 =wL4/EIωωω=EI/wL4TTTωT=wL4/EI =EI/wL4 =wL4/EI =...

  • Page 95

    76CHAPTER TWOmultiply it by . To get T, divide the appropriate constant by .In these equations, natural frequency, rad/sWbeam weight, lb per linear ft (kg per linear m)Lbeam length, ft (m)Emodulus of elasticity, lb/in2 (MPa)Imoment of inertia of beam cross section, in4 (mm4)Tnatural period, sTo d...

  • Page 96

    TABLE 2.7Polar Moment of Intertia and Maximum Torsional ShearBEAM FORMULAS77Strain Energy in TorsionFor a member subjected to torsion(2.40)(2.41)UJG 22LUT 2L2JG2r2R2raabaPolar moment ofinertia JMaximum shear*vmaxat periphery2Tπr3at midpointof each sideat midpointof longer sides– 0.211 – R= 2...

  • Page 97

    78CHAPTER TWOPab2L2aPPwk/twk/twk/twk/taLaLwk/tPPPPPPL–2aL/4L/4L/4L/5 L/5 L/5L/3PPPL/3L/2L/2L/3L/5 L/5L/4L/2aaPPbPa2bL2PL8PL82PL92PL5wL2125wL219211wL2192wk/twk/twk/tL/2L/2L/2L/2L/2L/2L/2L/2L/4L/4L/23wL2160wL2307wL2960wk/twk/twk/t5wL296wL2325wL219223wL2960wL230wL220wa2(6 – 8a + 3a2)L212wa3(4 ...

  • Page 98

    BEAM FORMULAS79where Ttorqueangle of twistLlength over which the deformation takes placeJpolar moment of inertiaGshear modulus of elasticityStrain Energy in BendingFor a member subjected to pure bending (constant moment)(2.42)(2.43)where Mbending momentangle through which one end of beam rotates ...

  • Page 99

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  • Page 100

    CHAPTER 3 COLUMNFORMULAS GENERAL CONSIDERATIONS Columns are structural members subjected to direct compression. All columnscan be grouped into the following three classes: 1. Compression blocksare so short (with a slenderness ratio — that is, unsup-ported length divided by the least radius of g...

  • Page 101

    TABLE 3.1Strength of Round-Ended Columns According to Euler’s Formula*Low-Medium-WroughtcarboncarbonMaterial†Cast ironironsteelsteelUltimate compressive strength, lb/in2107,00053,40062,60089,000Allowable compressive stress, lb/in27,10015,40017,00020,000(maximum)Modulus of elasticity14,200,000...

  • Page 102

    COLUMN FORMULAS83ShortSCompression blocksLongEuler columnCritical L/rParabolic typeStraight linetypeL/rFIGURE 3.1L/r plot for columns.general, based on the assumption that the permissible stress must be reducedbelow that which could be permitted were it due to compression only. The mannerin which...

  • Page 103

    84CHAPTER THREEIf, as in certain classes of masonry construction, the material cannot withstandtensile stressand, thus, no tension can occur, the center of moments (Fig. 3.3) istaken at the center of stress. For a rectangular section, Pacts at distance kfrom thenearest edge. Length under compress...

  • Page 104

    COLUMN FORMULAS85PPkeebPAPeyIPeyIPeyIPeyIPAPAFIGURE 3.2Load plot for columns.FIGURE 3.3Load plot for columns.Smaxr1rSAeZZSmaxFIGURE 3.4Circular columnload plot.Pkebc. of g.OSM3k

  • Page 105

    TABLE 3.3Values of the Ratio z/r0.00.50.60.70.80.91.00.252.000.250.301.820.300.351.661.891.980.350.401.511.751.841.930.400.451.371.611.711.811.900.450.501.231.461.561.661.781.892.000.500.551.101.291.391.501.621.741.870.550.600.971.121.211.321.451.581.710.600.650.840.941.021.131.251.401.540.650.70...

  • Page 106

    TABLE 3.4Values of the Ratio Smax/Savg0.00.50.60.70.80.91.00.001.001.001.001.001.001.001.000.000.051.201.161.151.131.121.111.100.050.101.401.321.291.271.241.221.200.100.151.601.481.441.401.371.331.300.150.201.801.641.591.541.491.441.400.200.252.001.801.731.671.611.551.500.250.302.231.961.881.811....

  • Page 107

    88CHAPTER THREEThe kernis the area around the center of gravity of a cross section withinwhich any load applied produces stress of only one sign throughout the entirecross section. Outside the kern, a load produces stresses of different sign.Figure 3.5 shows kerns (shaded) for various sections.Fo...

  • Page 108

    COLUMN FORMULAS89Nomenclature for formulas Eqs. (3.4) through (3.12):Qallowable load, lbPultimate load, lbAsection area of column, sq inLlength of column, inrleast radium of gyration of column section, inSuultimate strength, psiSyyield point or yield strength of material, psiEmodulus of elasticit...

  • Page 109

    90CHAPTER THREE(3.11)(3.12)COLUMN BASE PLATE DESIGNBase plates are usually used to distribute column loads over a large enough areaof supporting concrete construction that the design bearing strength of the con-crete is not exceeded. The factored load, Pu, is considered to be uniformly dis-tribut...

  • Page 110

    COLUMN FORMULAS91The thickness of the base plate tp, in (mm), is the largest of the values givenby the equations that follow(3.17)(3.18)(3.19)where mprojection of base plate beyond the flange and parallel to the web,in (mm)(N 0.95d)/2nprojection of base plate beyond the edges of the flange and ...

  • Page 111

    92CHAPTER THREEusually defined as shown in Fig. 3.6. (If the base plate is small, the area of thebase plate inside the column profile should be treated as a beam.) Yield-lineanalysis shows that an equivalent cantilever dimension can be defined as, and the required base plate thickness tp can be ...

  • Page 112

    be at least 4 percent of the total composite area. The concrete should be rein-forced with longitudinal load-carrying bars, continuous at framed levels, andlateral ties and other longitudinal bars to restrain the concrete; all should haveat least in (38.1 mm) of clear concrete cover. The cross-se...

  • Page 113

    94CHAPTER THREEThe portion of the required strength of axially loaded encased compositecolumns resisted by concrete should be developed by direct bearing at connec-tions or shear connectors can be used to transfer into the concrete the loadapplied directly to the steel column. For direct bearing,...

  • Page 114

    COLUMN FORMULAS95where Emodulus of elasticity of the column material, psi (Mpa)Acolumn cross-sectional area, in2 (mm2)rradius of gyration of the column, in (mm)Figure 3.8 shows some ideal end conditions for slender columns and corre-sponding critical buckling loads. Elastic critical buckling load...

  • Page 115

    96CHAPTER THREEwhere Gshear modulus of elasticity, psi (MPa)Jtorsional constantAcross-sectional area, in2 (mm2)Ippolar moment of inertiaIxIy, in4 (mm4)If the section possesses a significant amount of warping rigidity, the axial buck-ling load is increased to(3.28)where Cw is the warping constant...

  • Page 116

    COLUMN FORMULAS97Restraining bulk-head partiallyfixedPPPPPPPPc = 1.00c = 1.25 to 1.50c = 2.00c = 4.00FIGURE 3.9Values of c, column-end fixity, for determining the critical L/ ratio of dif-ferent loading conditions.ULTIMATE STRENGTH DESIGN CONCRETE COLUMNSAt ultimate strength Pu, kip (N), columns...

  • Page 117

    TABLE 3.5Material Constants for Common Aluminum AlloysAverageValuesFcyFceType JohnsonMaterialpsiMPapsiMPaKknequation14S–T434,000234.439,800274.40.3853.001.0Straight line24S–T3 and T440,000275.848,000330.90.3853.001.0Straight line61S–T635,000241.341,100283.40.3853.001.0Straight line14S–T65...

  • Page 118

    COLUMN FORMULAS99The axial-load capacity Pu kip (N), of short, rectangular members subject toaxial load and bending may be determined from(3.34)(3.35)whereeccentricity, in (mm), of axial load at end of member with respect tocentroid of tensile reinforcement, calculated by conventional methodsof f...

  • Page 119

    100CHAPTER THREEwhere eb is the eccentricity, in (mm), of the axial load with respect to the plasticcentroid and is the distance, in (mm), from plastic centroid to centroid of ten-sion reinforcement.When Pu is less than Pb or the eccentricity, e, is greater than eb, tension gov-erns. In that case...

  • Page 120

    COLUMN FORMULAS101When Pu is greater than Pb, or eis less than eb, compression governs. In that case,the ultimate strength is approximately(3.41)(3.42)where Mu is the moment capacity under combined axial load and bending, inkip (kNm) and Po is the axial-load capacity, kip (N), of member when conc...

  • Page 121

    102CHAPTER THREEWhen tension controls,(3.47)When compression governs,(3.48)Slender Columns When the slenderness of a column has to be taken into account, the eccentricityshould be determined from eMc/Pu, where Mc is the magnified moment.DESIGN OF AXIALLY LOADED STEEL COLUMNS*Design of columns th...

  • Page 122

    COLUMN FORMULAS103with 0.85. For c1.5Fcr0.658 c2 Fy(3.50)forc1.5(3.51)wherec (KL/r )Fyminimum specified yield stress of steel, ksiAggross area of member, in2Eelastic modulus of the steel 29,000 ksiFor ASD, the allowable compression stress depends on whether bucklingwill be elastic or inelastic, ...

  • Page 123

    104CHAPTER THREEIn the United States, land-based wind turbines (also called onshore turbines)have been the most popular type because there is sufficient land area for, singleor multiple, wind turbine installations. In Europe, land scarcity led to offshorewind farms where the wind strength and de...

  • Page 124

    CHAPTER 4105PILES ANDPILING FORMULASALLOWABLE LOADS ON PILESA dynamic formula extensively used in the United States to determine theallowable static load on a pile is the Engineering Newsformula. For piles dri-ven by a drop hammer, the allowable load is(4.1)For piles driven by a steam hammer, the...

  • Page 125

    106CHAPTER FOURThe lateral-load versus pile-head deflection relationship is developed fromcharted nondimensional solutions of Reese and Matlock. The solution assumesthe soil modulus Kto increase linearly with depth z; that is, Knh z, where nhcoefficient of horizontal subgrade reaction. A characte...

  • Page 126

    PILES AND PILING FORMULAS107of the pile head on yand Mcan be evaluated by substituting the value of Mtfrom the preceding equation into the earlier yand Mequations. Note that, forthe fixed-head case,(4.7)TOE CAPACITY LOADFor piles installed in cohesive soils, the ultimate tip load may be computed ...

  • Page 127

    108CHAPTER FOURgroup. The efficiency factor Eg is defined as the ratio of the ultimate groupcapacity to the sum of the ultimate capacity of each pile in the group. Eg is conventionally evaluated as the sum of the ultimate peripheral frictionresistance and end-bearing capacities of a block of soil...

  • Page 128

    PILES AND PILING FORMULAS109thickness of the consolidating soil layers penetrated by the piles, and theirundrained shear strength, respectively. Such forces as Qgd could only beapproached for the case of piles driven to rock through heavily surcharged,highly compressible subsoils.Design of rock s...

  • Page 129

    110CHAPTER FOURThe net bearing capacityper unit area, qu, of a long footing is convention-ally expressed as(4.14)wheref1.0 for strip footings and 1.3 for circular and square footingscuundrained shear strength of soilvoeffective vertical shear stress in soil at level of bottom offootingf0.5 for st...

  • Page 130

    PILES AND PILING FORMULAS111where Bwidth of rectangular footingLlength of rectangular footingeeccentricity of loadingFor the other case Fig. 4.2(c), the soil pressure ranges from 0 to a maximum of(4.17)qm2P3L(B/2e)TABLE 4.2Shape Corrections for Bearing-Capacity Factors of ShallowFoundations*Corre...

  • Page 131

    112CHAPTER FOURFor square or rectangular footings subject to overturning about two principalaxes and for unsymmetrical footings, the loading eccentricities e1 and e2 aredetermined about the two principal axes. For the case where the full bearingarea of the footings is engaged, qm is given in term...

  • Page 132

    PILES AND PILING FORMULAS113These include elastic, semiempirical elastic, and load-transfer solutions forsingle shafts drilled in cohesive or cohesionless soils.Resistance to tensile and lateral loads by straight-shaft drilled shafts shouldbe evaluated as described for pile foundations. For relat...

  • Page 133

    114CHAPTER FOURdegree. For piles less than 50 ft (15.2 m) long, Kis more likely to be in therange of 1.0 to 2.0, but can be greater than 3.0 for tapered piles.Empirical procedures have also been used to evaluate from in situtests,such as cone penetration, standard penetration, and relative densit...

  • Page 134

    CHAPTER 5115CONCRETEFORMULASREINFORCED CONCRETEWhen working with reinforced concrete and when designing reinforced con-crete structures, the American Concrete Institute(ACI) Building Code Require-ments for Reinforced Concrete, latest edition, is widely used. Future referencesto this document are ...

  • Page 135

    116CHAPTER FIVEnon-air-entrained concrete and from 0.54 to 0.35 for air-entrained concrete.These values are for a specified 28-day compressive strength in lb/in2 orMPa, of 2500 lb/in2 (17 MPa) to 5000 lb/in2 (34 MPa). Again, refer to the ACICodebefore making any design or construction decisions.M...

  • Page 136

    CONCRETE FORMULAS117TENSILE STRENGTH OF CONCRETEThe tensile strength of concrete is used in combined-stress design. In normal-weight, normal-density concrete the tensile strength can be found from(5.5)REINFORCING STEELAmerican Society for Testing and Materials(ASTM) specifications cover ren-forci...

  • Page 137

    118CHAPTER FIVENegative moment at face of all supports for(a) slabs with spans not exceeding 10 ft (3 m)and (b) beams and girders where the ratio of sum of column stiffness to beam stiffness exceeds 8 at each end of the spanNegative moment at interior faces of exterior supports, for members built...

  • Page 138

    CONCRETE FORMULAS119where bwidth of beam [equals 12 in (304.8 mm) for slab], in (mm)deffective depth of beam, measured from compressive face of beamto centroid of tensile reinforcing (Fig. 5.1), in (mm)Mbending moment, lb . in (k . Nm)fccompressive stress in extreme fiber of concrete, lb/in2 (MPa...

  • Page 139

    120CHAPTER FIVEFor a balanced design, one in which both the concrete and the steel arestressed to the maximum allowable stress, the following formulas may be used: (5.7)Values of K, k, j, and pfor commonly used stresses are given in Table 5.2.T-Beams with Tensile Reinforcing OnlyWhen a concrete s...

  • Page 140

    CONCRETE FORMULAS121Checking Stresses in BeamsBeams designed using the preceding approximateformulas should be checked to ensure that the actual stresses do not exceed theallowable, and that the reinforcing is not excessive. This can be accomplished bydetermining the moment of inertia of the beam...

  • Page 141

    122CHAPTER FIVEwhere fc, fsc, fsactual unit stresses in extreme fiber of concrete, in compressivereinforcing steel, and in tensile reinforcing steel, respectively,lb/in2 (MPa)cc, csc, csdistances from neutral axis to face of concrete, to compressivereinforcing steel, and to tensile reinforcing st...

  • Page 142

    CONCRETE FORMULAS123where ubond stress on surface of bar, lb/in2 (MPa)Vtotal shear, lb (N)deffective depth of beam, in (mm)0sum of perimeters of tensile reinforcing bars, in (mm)For preliminary design, the ratio jmay be assumed to be 7/8. Bond stressesmay not exceed the values shown in Table 5.3....

  • Page 143

    TABLE 5.3Allowable Bond Stresses*Horizontal bars with more than 12 in (30.5 mm) of concretecast below the bar†Other bars†Tension bars with sizes and or 350, whichever is lessor 500, whichever is lessdeformations conforming toASTM A305Tension bars with sizes and deformations conforming toASTM ...

  • Page 144

    CONCRETE FORMULAS125earlier for spirally reinforced columns. The ratio pg for a tied column should not beless than 0.01 or more than 0.08. Longitudinal reinforcing should consist of at leastfour bars; minimum size is No. 5.Long ColumnsAllowable column loads where compression governs designmust be...

  • Page 145

    126CHAPTER FIVEtdiameter of column or overall depth of column, in (mm)ddistance from extreme compression fiber to centroid of tension rein-forcement, in (mm)fvyield point of reinforcement, lb/in2 (MPa)Design of columns controlled by compression is based on the followingequation, except that the a...

  • Page 146

    CONCRETE FORMULAS127where Mz and My are bending moments about the xand yaxes, and M0x and M0yare the values of M0 for bending about these axes.PROPERTIES IN THE HARDENED STATEStrengthis a property of concrete that nearly always is of concern. Usually, it isdetermined by the ultimate strength of a...

  • Page 147

    128CHAPTER FIVETENSION DEVELOPMENT LENGTHSFor bars and deformed wire in tension, basic development length is defined bythe equations that follow. For No. 11 and smaller bars,(5.30)where Abarea of bar, in2 (mm2)fyyield strength of bar steel, lb/in2 (MPa)28-day compressive strength of concrete, lb/...

  • Page 148

    CONCRETE FORMULAS129positive and negative moment should be proportioned for crack control so thatspecific limits are satisfied by(5.35)where fscalculated stress, ksi (MPa), in reinforcement at service loadsdcthickness of concrete cover, in (mm), measured from extreme ten-sion surface to center of...

  • Page 149

    130CHAPTER FIVEDEFLECTION COMPUTATIONS AND CRITERIA FORCONCRETE BEAMSThe assumptions of working-stress theory may also be used for computingdeflections under service loads; that is, elastic-theory deflection formulas may beused for reinforced-concrete beams. In these formulas, the effective momen...

  • Page 150

    CONCRETE FORMULAS131Equating the compression and tension at the critical section yields(5.43)The criterion for compression failure is that the maximum strain in the con-crete equals 0.003 in/in (0.076 mm/mm). In that case,(5.44)where fssteel stress, ksi (MPa)Esmodulus of elasticity of steel29,000...

  • Page 151

    132CHAPTER FIVEThe shear strength Vc carried by the concrete alone should not exceedwhere bw is the width of the beam web and d, the depth of the centroidof reinforcement. (As an alternative, the maximum value for Vc may be taken as(5.49)wherewAs/bwd and Vu and Mu are the shear and bending moment...

  • Page 152

    CONCRETE FORMULAS133Development of Tensile ReinforcementAt least one-third of the positive-moment reinforcement in simple beams and one-fourth of the positive-moment reinforcement in continuous beams should extendalong the same face of the member into the support, in both cases, at least 6 in(152...

  • Page 153

    134CHAPTER FIVEkddistance from extreme compression surface to neutral axis, in (mm)ddistance from extreme compression to centroid of reinforcement, in(mm)When the steel ratio As /bd, where Asarea of tension reinforcement,in2 (mm2); and bbeam width, in (mm), is known, kcan be computed from(5.57)Wh...

  • Page 154

    CONCRETE FORMULAS135The excess shear vvc should not exceed 4.4in sections with web rein-forcement. Stirrups and bent bars should be capable of resisting the excessshear VVvc bd.The area required in the legs of a vertical stirrup, in2 (mm2), is (5.63)where sspacing of stirrups, in (mm); and fvallo...

  • Page 155

    136CHAPTER FIVEwhere adepth of equivalent rectangular compressive stress distribution bwidth of beam, in (mm)ddistance from extreme compression surface to centroid of tensilesteel, in (mm)distance from extreme compression surface to centroid of compres-sive steel, in (mm)Asarea of tensile steel, ...

  • Page 156

    CONCRETE FORMULAS137for the effects of creep and nonlinearity of the stress–strain diagram for con-crete. However, should not exceed the allowable tensile stress for the steel.Because total compressive force equals total tensile force on a section,(5.71)where Ctotal compression on beam cross se...

  • Page 157

    138CHAPTER FIVEwhere Mbending momentMsmoment-resisting capacity of compressive steelM1moment-resisting capacity of concreteULTIMATE-STRENGTH DESIGN OF I- AND T-BEAMSWhen the neutral axis lies in the flange, the member may be designed as a rec-tangular beam, with effective width band depth d. For ...

  • Page 158

    CONCRETE FORMULAS139an I- or T-beam may be designed by the following formulas, which ignore thecompression in the stem, as is customary: (5.82)where kddistance from extreme compression surface to neutral axis, in (mm)ddistance from extreme compression surface to centroid of tensilesteel, in (mm)f...

  • Page 159

    140CHAPTER FIVEULTIMATE-STRENGTH DESIGN FOR TORSIONWhen the ultimate torsion Tu is less than the value calculated from the Tu equa-tion that follows, the area Av of shear reinforcement should be at least(5.90)However, when the ultimate torsion exceeds Tu calculated from the Tu equa-tion that foll...

  • Page 160

    CONCRETE FORMULAS141The spacing of closed stirrups, however, should not exceed (x1y1)/4 or 12 in(304.8 mm). Torsion reinforcement should be provided over at least a distanceof dbbeyond the point where it is theoretically required, where bis thebeam width.At least one longitudinal bar should be pl...

  • Page 161

    142CHAPTER FIVEFLAT-SLAB CONSTRUCTIONSlabs supported directly on columns, without beams or girders, are classified asflat slabs. Generally, the columns flare out at the top in capitals (Fig. 5.3).However, only the portion of the inverted truncated cone thus formed that liesinside a 90° vertex an...

  • Page 162

    SlabDrop panelL ColumnsA4HalfcolumnstripColumnstripMiddlestripDrop panelCapitalColumn(a)(b)A4A4AA2B –BA4A4cL ColumnscL ColumnscL ColumnsCcA2FIGURE 5.3Concrete flat slab: (a) Vertical section through drop panel and column at a support. (b) Plan view indicatesdivision of slab into column and midd...

  • Page 163

    144CHAPTER FIVEIn both methods, a flat slab is considered to consist of strips parallel to columnlines in two perpendicular directions. In each direction, a column stripspansbetween columns and has a width of one-fourth the shorter of the two perpendic-ular spans on each side of the column center...

  • Page 164

    CONCRETE FORMULAS1451.The sum of the flexural stiffnesses of the columns above and below the slabKc should be such that (5.102)where Kcflexural stiffness of columnEccIcEccmodulus of elasticity of column concreteIcmoment of inertia about centroidal axis of gross section of columnKsEcsIsKbEcbIbminm...

  • Page 165

    146CHAPTER FIVEwhere boperimeter of critical section and cratio of long side to short sideof critical section.However, if shear reinforcement is provided, the allowable shear may beincreased a maximum of 50 percent if shear reinforcement consisting of bars isused and increased a maximum of 75 per...

  • Page 166

    CONCRETE FORMULAS147computed from(5.107)where Vu is the design shear, kip (kN), at the section; fy is the reinforcementyield strength, but not more than 60 ksi (413.7 MPa); and , the coefficient offriction, is 1.4 for monolithic concrete, 1.0 for concrete placed against hardenedconcrete, and 0.7 ...

  • Page 167

    148CHAPTER FIVEwhere M1smaller of two end moments on column as determined by con-ventional elastic frame analysis, with positive sign if column is bent insingle curvature and negative sign if column is bent in double curvature; andM2absolute value of larger of the two end moments on column as det...

  • Page 168

    CONCRETE FORMULAS149where Vutotal design shear forcecapacity reduction factor0.85d0.8lwhoverall thickness of walllwhorizontal length of wallThe shear Vc carried by the concrete depends on whether Nu, the design axi-al load, lb (N), normal to the wall horizontal cross section and occurring simul-t...

  • Page 169

    150CHAPTER FIVEwhere Abbearing area of anchor plate, and maximum area of portionof anchorage surface geometrically similar to and concentric with area ofanchor plate.A more refined analysis may be applied in the design of the end-anchorageregions of prestressed members to develop the ultimate str...

  • Page 170

    CONCRETE FORMULAS151falls outside the middle third, the pressure vanishes under a zone around theheel, and pressure at the toe is much larger than for the other cases.The variables in the five formulas in Fig. 5.4 areP1 and P2the pressure, lb/ft2 (MPa), at the locations shownLand adimensions, ft ...

  • Page 171

    152CHAPTER FIVEFIGURE 5.4(Continued)(a) Resultant in middle third(b) Resultant at edge middle third(c) Resultant outside middle thirdRvP2RWhen a =,L/3aRvRL/3aLP1P1P1 = (4L – 6a) RvL2P2 = (6a – 2L) RvL2L2P1 = P2 =P2 = OP2RvLP1 = 2RvLP = 2Rv3aRvRL/3a3aP(b)

  • Page 172

    CONCRETE FORMULAS153In usual design work on retaining walls the sum of the righting momentsand the sum of the overturing moments about the toe are found. It is assumedby designers that if the retaining wall is overturned, it will overturn about thetoe of the retaining wall. Designers then apply a...

  • Page 173

    154CHAPTER FIVEShear unit stress on a horizontal section of a counterfort may be computedfrom vcV1/bd, where bis the thickness of the counterfort and dis the hori-zontal distance from face of wall to main steel,(5.120)where Vshear on sectionMbending moment at sectionangle earth face of counterfor...

  • Page 174

    CONCRETE FORMULAS155WALL FOOTINGSThe spread footing under a wall (Fig. 5.7) distributes the wall load horizontallyto preclude excessive settlement.The footing acts as a cantilever on opposite sides of the wall under down-ward wall loads and upward soil pressure. For footings supporting concretewa...

  • Page 175

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  • Page 176

    CHAPTER 6157TIMBERENGINEERINGFORMULASGRADING OF LUMBERStress-grade lumber consists of three classifications:1. Beams and stringers. Lumber of rectangular cross section, 5 in (127 mm) ormore thick and 8 in (203 mm) or more wide, graded with respect to its strengthin bending when loaded on the narr...

  • Page 177

    TABLE 6.1Properties of Sections for Standard Lumber Sizes[Dressed (S4S) sizes, moment of inertia, and section modulus are given with respect to xx axis, with dimensions b and h, as shown on sketch]StandardMoment ofSectionNominaldressed sizeArea ofinertiamodulusBoard feetsizeS4Ssectionper linearbh...

  • Page 178

    TIMBER ENGINEERING FORMULAS159BEARINGThe allowable unit stresses given for compression perpendicular to the grainapply to bearings of any length at the ends of beams and to all bearings 6 in(152.4 mm) or more in length at other locations. When calculating the requiredbearing area at the ends of b...

  • Page 179

    160CHAPTER SIXV1modified total end shear, lb (N)Wtotal uniformly distributed load, lb (N)xdistance from reaction to concentrated load in (mm)For simple beams, the span should be taken as the distance from face to face ofsupports plus one-half the required length of bearing at each end; and for co...

  • Page 180

    TIMBER ENGINEERING FORMULAS161For columns of circular cross section, the formula becomes(6.8)The allowable unit stress, P/A, may not exceed the allowable compressive stress,c. The ratio, l/d, must not exceed 50. Values of P/A are subject to the duration ofloading adjustment given previously.Nomen...

  • Page 181

    162CHAPTER SIXwhere callowable unit stress at angle to grain, lb/in2 (MPa)callowable unit stress parallel to grain, lb/in2 (MPa)callowable unit stress perpendicular to grain, lb/in2 (MPa)angle between direction of load and direction of grainRECOMMENDATIONS OF THE FOREST PRODUCTSLABORATORYThe Wood...

  • Page 182

    TIMBER ENGINEERING FORMULAS163dsmaller dimension of rectangular sectionEmodulus of elasticityfcallowable compressive stress parallel to grain in short column of givenspeciesfallowable compressive stress parallel to grain in given columnCOMPRESSION ON OBLIQUE PLANEConsider that a timber member sus...

  • Page 183

    164CHAPTER SIXADJUSTMENT FACTORS FOR DESIGN VALUESDesign values obtained by the methods described earlier should be multiplied byadjustment factors based on conditions of use, geometry, and stability. Theadjustments are cumulative, unless specifically indicated in the following.The adjusted desig...

  • Page 184

    TIMBER ENGINEERING FORMULAS165For shear, the adjusted design value FV is computed from(6.21)where FV is the design value for shear and CH is the shear stress factor 1—pemitted for FV parallel to the grain for sawn lumber members.For compression perpendicular to the grain, the adjusted design va...

  • Page 185

    166CHAPTER SIXvalues for southern pine are based on the adjustment equation given in Ameri-can Society for Testing and Materials(ASTM) D1990. This equation, based onin-grade test data, accounts for differences in Fb, Ft, and Fc related to width and inFb and Ft related to length (test span).For vi...

  • Page 186

    TIMBER ENGINEERING FORMULAS167modification for duration of load. If these values are exceeded, mechanical rein-forcement sufficient to resist all radial tensile stresses is required.When Mis in the direction tending to increase curvature (decrease theradius), the stress is compressive across the ...

  • Page 187

    168CHAPTER SIXKcE0.3 for visually graded lumber and machine-evaluated lumber0.418 for products with a coefficient of variation less than 0.11c0.80 for solid-sawn lumber0.85 for round timber piles0.90 for glued-laminated timberFor a compression member braced in all directions throughout its length...

  • Page 188

    TIMBER ENGINEERING FORMULAS169The beam stability factor CL may be calculated from(6.34)wheredesign value for bending multiplied by all applicable adjustmentfactors, except Cfu, CV, and CLFbE0.438 for visually graded lumber and machine-evaluated lumber0.609 for products with a coefficient of varia...

  • Page 189

    170CHAPTER SIXThe loads apply where the nail or spike penetrates into the member, receiving itspoint at least 10 diameters for Group I species, 11 diameters for Group IIspecies, 13 diameters for Group III species, and 14 diameters for Group IVspecies. Allowable loads for lesser penetrations are d...

  • Page 190

    TIMBER ENGINEERING FORMULAS171The allowable lateral load for wood screws driven into end grain is two-thirdsthat given for side grain.ADJUSTMENT OF DESIGN VALUES FOR CONNECTIONSWITH FASTENERSNominal design values for connections or wood members with fasteners shouldbe multiplied by applicable adj...

  • Page 191

    172CHAPTER SIXFor wood screws,(6.44)(6.45)where Ceg is the end-grain factor.For lag screws,(6.46)(6.47)For metal plate connectors,(6.48)For drift bolts and drift pins,(6.49)(6.50)For spike grids,(6.51)ROOF SLOPE TO PREVENT PONDINGRoof beams should have a continuous upward slope equivalent to in/f...

  • Page 192

    TIMBER ENGINEERING FORMULAS173BENDING AND AXIAL TENSIONMembers subjected to combined bending and axial tension should be propor-tioned to satisfy the interaction equations(6.53)and(6.54)where fttensile stress due to axial tension acting alonefbbending stress due to bending moment alonedesign valu...

  • Page 193

    174CHAPTER SIXFor either uniaxial or biaxial bending, fc should not exceed(6.56)where Eis the modulus of elasticity multiplied by adjustment factors. Also, forbiaxial bending, fc should not exceed(6.57)and fb1 should not be more than(6.58)where d1 is the width of the wide face and d2 is the width...

  • Page 194

    TIMBER ENGINEERING FORMULAS175(6.60)(6.61)The ultimate unit load is:(6.62)For lumber of various types, the following formulas can be used, whereSallowable compressive stress; and dleast dimension of the lumbercross section.(6.63)(6.64)(6.65)(6.66)For general structural use under continuously dry ...

  • Page 195

    176CHAPTER SIXFor solid circular columns with ends flat in general structural use under con-tinuously dry conditions: Q/A is same as for square column of equal area. For tapered round column,dand A are taken as for section distant 1/3 Lfrom smaller end. Q/A at smallend must not exceed S.Note: Whe...

  • Page 196

    CHAPTER 7177SURVEYINGFORMULASUNITS OF MEASUREMENTUnits of measurement used in past and present surveys areFor construction work: feet, inches, fractions of inches (m, mm)For most surveys: feet, tenths, hundredths, thousandths (m, mm)For National Geodetic Survey(NGS) control surveys: meters, 0.1, ...

  • Page 197

    178CHAPTER SEVEN1 rad57 17 44.8 or about 57.301 grad (grade)circlequadrant100 centesimal min104 cen-tesimals (French)1 milcircle0.056251 military pace (milpace) ft (0.762 m)THEORY OF ERRORSWhen a number of surveying measurements of the same quantity have beenmade, they must be analyzed on the bas...

  • Page 198

    SURVEYING FORMULAS179where E1, E2, E3 . . . are probable errors of the separate measurements.Error of the mean is (7.4)where Esspecified error of a single measurement.Probable error of the mean is (7.5)MEASUREMENT OF DISTANCE WITH TAPESReasonable precisions for different methods of measuring dist...

  • Page 199

    180CHAPTER SEVENFor ordinary taping, a tape accurate to 0.01 ft (0.00305 m) should be used. Thetension of the tape should be about 15 lb (66.7 N). The temperature should bedetermined within 10°F (5.56°C); and the slope of the ground, within 2 percent;and the proper corrections, applied. The cor...

  • Page 200

    SURVEYING FORMULAS181hand calculator operation. For slopes of 10 percent or less, the correction to beapplied to Lfor a difference din elevation between tape ends, or for a horizon-tal offset dbetween tape ends, may be computed from(7.12)For a slope greater than 10 percent, Cs may be determined f...

  • Page 201

    182CHAPTER SEVENwhere M, F, and Kare distances in miles, thousands of feet, and kilometers,respectively, from the point of tangency to the earth.Refractioncauses light rays that pass through the earth’s atmosphere tobend toward the earth’s surface. For horizontal sights, the average angular d...

  • Page 202

    SURVEYING FORMULAS183STADIA SURVEYINGIn stadia surveying, a transit having horizontal stadia crosshairs above andbelow the central horizontal crosshair is used. The difference in the rod read-ings at the stadia crosshairs is termed the rod intercept. The intercept maybe converted to the horizonta...

  • Page 203

    184CHAPTER SEVENwhere Rintercept on rod between two sighting wires, ft (m)ffocal length of telescope, ft (m) (constant for specific instrument)idistance between stadia wires, ft (m)Cfc(7.28)cdistance from center of spindle to center of objective lens, ft (m)Cis called the stadia constant, althoug...

  • Page 204

    CHAPTER 8185SOIL ANDEARTHWORKFORMULASPHYSICAL PROPERTIES OF SOILSBasic soil properties and parameters can be subdivided into physical, index, andengineering categories. Physical soil properties include density, particle size anddistribution, specific gravity, and water content.The water content w...

  • Page 205

    186CHAPTER EIGHTINDEX PARAMETERS FOR SOILSIndex parameters of cohesive soils include liquid limit, plastic limit, shrinkagelimit, and activity. Such parameters are useful for classifying cohesive soils andproviding correlations with engineering soil properties.The liquid limitof cohesive soils re...

  • Page 206

    SOIL AND EARTHWORK FORMULAS187(8.6)(8.7)(8.8)Unit weights are generally expressed in pound per cubic foot or gram per cubiccentimeter. Representative values of unit weights for a soil with a specific grav-ity of 2.73 and a void ratio of 0.80 are(8.9)(8.10)(8.11)The symbols used in the three prece...

  • Page 207

    188CHAPTER EIGHTINTERNAL FRICTION AND COHESIONThe angle of internal frictionfor a soil is expressed by(8.12)whereangle of internal frictiontancoefficient of internal frictionnormal force on given plane in cohesionless soil massshearing force on same plane when sliding on plane isimpendingFor medi...

  • Page 208

    SOIL AND EARTHWORK FORMULAS189The Westergaard equation applies to an elastic material laterally reinforcedwith horizontal sheets of negligible thickness and infinite rigidity, which preventthe mass from undergoing lateral strain. The vertical stress at a point in themass, assuming a Poisson’s r...

  • Page 209

    190CHAPTER EIGHTLATERAL PRESSURE OF COHESIONLESS SOILSFor walls that retain cohesionless soils and are free to move an appreciableamount, the total thrust from the soil is(8.15)When the surface behind the wall is level, the thrust is(8.16)where(8.17)The thrust is applied at a point H/3 above the ...

  • Page 210

    SOIL AND EARTHWORK FORMULAS191LATERAL PRESSURE OF COHESIVE SOILSFor walls that retain cohesive soils and are free to move a considerable amountover a long period of time, the total thrust from the soil (assuming a level sur-face) is(8.21)or, because highly cohesive soils generally have small angl...

  • Page 211

    192CHAPTER EIGHTSTABILITY OF SLOPESCohesionless SoilsA slope in a cohesionless soil without seepage of water is stable if(8.25)With seepage of water parallel to the slope, and assuming the soil to be satu-rated, an infinite slope in a cohesionless soil is stable if(8.26)whereislope of ground surf...

  • Page 212

    SOIL AND EARTHWORK FORMULAS193(8.30)where quultimate bearing capacity of soil, lb/ft2 (kg/m2)ccohesion, lb/ft2 (kg/m2)angle of internal friction, degreedryunit weight of dry soil, lb/ft3 (kg/m3)bwidth of footing, ft (m)ddepth of footing below surface, ft (m)Kpcoefficient of passive pressure e2.71...

  • Page 213

    194CHAPTER EIGHTCharacteristics of the soil are computed from(8.32)(8.33)(8.34)(8.35)(8.36)Maximum density is found by plotting a density–moisture curve.Load-Bearing TestOne of the earliest methods for evaluating the in situdeformability ofcoarse-grained soils is the small-scale load-bearing te...

  • Page 214

    SOIL AND EARTHWORK FORMULAS195California Bearing RatioThe California bearing ratio(CBR) is often used as a measure of the quality ofstrength of a soil that underlies a pavement, for determining the thickness of thepavement, its base, and other layers.(8.39)where Fforce per unit area required to p...

  • Page 215

    196CHAPTER EIGHTCompaction production can be computed from(8.41)where Wwidth of roller, ft (m)Sroller speed, mi/h (km/h)Llift thickness, in (mm)Fratio of pay yd3 (m3) to loose yd3 (m3)Eefficiency factor (allows for time losses, such as those due to turns):0.90, excellent; 0.80, average; 0.75, poo...

  • Page 216

    SOIL AND EARTHWORK FORMULAS197Additional power is required to overcome rolling resistance on a slope.Grade resistance also is proportional to weight:(8.44)where Ggrade resistance, lb (N)Rggrade-resistance factor 20 lb/ton (86.3 N/t) 1% lb/lb (N/N)spercent grade, positive for uphill motion, negati...

  • Page 217

    198CHAPTER EIGHTThe load, or amount of material a machine carries, can be determined byweighing or estimating the volume. Payload estimating involves determinationof the bank cubic yards (cubic meters) being carried, whereas the excavatedmaterial expands when loaded into the machine. For determin...

  • Page 218

    SOIL AND EARTHWORK FORMULAS199Vibrations caused by blasting are propagated with a velocity V, ft/s (m/s);frequency f, Hz; and wavelength L, ft (m), related by(8.55)Velocity v, in/s (mm/s), of the particles disturbed by the vibrations depends onthe amplitude of the vibrations A, in (mm):(8.56)If t...

  • Page 219

    200CHAPTER EIGHTBlasting OperationsBlasting operations are used in many civil engineering projects—for basementsin new buildings, bridge footings, canal excavation, dam construction, and so on.Here are a number of key formulas used in blasting operations of many types.*Borehole DiameterGenerall...

  • Page 220

    SOIL AND EARTHWORK FORMULAS201determining the burden. The length of the burden relative to the depth of the cuthas a significant effect on fragmentation.Langefors’ FormulaLangefors suggested that the burden determination wasbased on more factors, including diameter of hole, weight strength of e...

  • Page 221

    202CHAPTER EIGHTwhere Bburden, ftLlength of borehole, ftIf the length-to-burden ratio is greater than 4, then the spacing is twice theburden. Therefore, if the L/B is 5 the spacing is determined by S= 2B. Forexample, if the burden were equal to 7 ft (2.1 m), the spacing would beagain showing the ...

  • Page 222

    SOIL AND EARTHWORK FORMULAS203Air ConcussionAir concussion, or air blast, is a pressure wave traveling through the air; it isgenerally not a problem in construction blasting. The type of damage created byair concussion is broken windows. However, it must be realized that a properlyset window glas...

  • Page 223

    204CHAPTER EIGHTVibration Control in Blasting*Vibrations caused by blasting are propagated with a velocity V, ft/s; frequencyf, Hz; and wavelength L, ft, related by(8.69)Velocity v, in/s, of the particles disturbed by the vibrations depends on theamplitude of the vibrations A, in(8.70)If the velo...

  • Page 224

    SOIL AND EARTHWORK FORMULAS205The acceleration a, in/s2, of the particles is given by(8.72)For a charge exploded on the ground surface, the overpressure P, psi, may becomputed from(8.73)where Wmaximum weight of explosives, lb per delayDdistance, ft, from explosion to exposureThe sound pressure le...

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  • Page 226

    CHAPTER 9207BUILDING ANDSTRUCTURESFORMULASLOAD-AND-RESISTANCE FACTOR DESIGN FOR SHEAR IN BUILDINGSBased on the American Institute of Steel Construction(AISC) specifications forload-and-resistance factor design(LRFD) for buildings, the shear capacity Vu,kip (kN4.448kip), of flexural members may be...

  • Page 227

    208CHAPTER NINEALLOWABLE-STRESS DESIGN FOR BUILDING COLUMNSThe AISC specification for allowable-stress design(ASD) for buildings pro-vides two formulas for computing allowable compressive stress Fa, ksi(MPa), for main members. The formula to use depends on the relationshipof the largest effective...

  • Page 228

    BUILDING AND STRUCTURES FORMULAS209LOAD-AND-RESISTANCE FACTOR DESIGN FOR BUILDING COLUMNSPlastic analysis of prismatic compression members in buildings is permitted ifdoes not exceed 800 and Fu65 ksi (448 MPa). For axially loaded members with b/tr, the maximum load Pu, ksi (MPa6.894ksi), may beco...

  • Page 229

    210CHAPTER NINEsections. Fy is the minimum specified yield strength of the steel, ksi (MPa).Table 9.2 lists values of Fb for two grades of steel.The allowable extreme-fiber stress of 0.60Fy applies to laterally supported,unsymmetrical members, except channels, and to noncompact box sections.Compr...

  • Page 230

    BUILDING AND STRUCTURES FORMULAS211When, however, the compression flange is solid and nearly rectangular incross section, and its area is not less than that of the tension flange, the allow-able stress may be taken as (9.12)When Eq. (9.12) applies (except for channels), Fb should be taken as the ...

  • Page 231

    212CHAPTER NINEFor I-shaped beams, symmetrical about both the major and the minor axis orsymmetrical about the minor axis but with the compression flange larger thanthe tension flange, including hybrid girders, loaded in the plane of the web: (9.17)where Fycminimum yield stress of compression fla...

  • Page 232

    BUILDING AND STRUCTURES FORMULAS213where Fywspecified minimum yield stress of web, ksi (MPa)Frcompressive residual stress in flange10 ksi (68.9 MPa) for rolled shapes, 16.5 ksi (113.6 MPa), forwelded sectionsFLsmaller of FyfFr or FywFyfspecified minimum yield stress of flange, ksi (MPa)X1X2(4Cw /...

  • Page 233

    214CHAPTER NINEFor shapes to which Eq. (9.24) applies,(9.25)For solid rectangular bars and symmetrical box sections,(9.26)For determination of the flexural strength of noncompact plate girders andother shapes not covered by the preceding requirements, see the AISC manualon LRFD.ALLOWABLE-STRESS D...

  • Page 234

    BUILDING AND STRUCTURES FORMULAS215plane AAin Fig. 9.2, exceed Fv, that is, when Fis larger than dctwFv, where dcis the depth and tw is the web thickness of the member resisting F. The shearmay be calculated from(9.30)where Vsshear on the sectionM1M1LM1GM1Lmoment due to the gravity load on the le...

  • Page 235

    216CHAPTER NINEassume normal forces and shears to be uniformly distributed over the shellthickness and bending stresses to be linearly distributed.Then, normal stresses can be computed from equations of the form:(9.31)where zdistance from middle surfacetshell thicknessMxunit bending moment about ...

  • Page 236

    BUILDING AND STRUCTURES FORMULAS217With Nestablished, usually rounded to full inches (millimeters), the mini-mum width of plate B, in (mm), may be calculated by dividing A1 by Nandthen rounded off to full inches (millimeters), so that BNA1. Actual bearingpressure fp, ksi (MPa), under the plate th...

  • Page 237

    218CHAPTER NINETo minimize material requirements, the plate projections should be nearlyequal. For this purpose, the plate length N, in (mm) (in the direction of d), maybe taken as(9.39)The width B, in (mm), of the plate then may be calculated by dividing A1 by N.Both Band Nmay be selected in ful...

  • Page 238

    BUILDING AND STRUCTURES FORMULAS219PLATE GIRDERS IN BUILDINGSFor greatest resistance to bending, as much of a plate girder cross section aspracticable should be concentrated in the flanges, at the greatest distance fromthe neutral axis. This might require, however, a web so thin that the girderwo...

  • Page 239

    220CHAPTER NINEwhere Awweb area, in2 (mm2)Afarea of compression flange, in2 (mm2)0.6Fyw /Fb1.0Fywminimum specified yield stress, ksi, (MPa), of web steelIn a hybrid girder, where the flange steel has a higher yield strength than theweb, the preceding equation protects against excessive yielding o...

  • Page 240

    BUILDING AND STRUCTURES FORMULAS221Units used in these equations are those commonly applied in United StatesCustomary System(USCS) and the System International(SI) measurements,that is, kip (kN), lb/in2 (MPa), ft (m), and in (mm).Where shear walls contain openings, such as those for doors, corrid...

  • Page 241

    222CHAPTER NINEWhen fa /Fa0.15, the following equation may be used instead of the preced-ing two:(9.54)In the preceding equations, subscripts xand yindicate the axis of bending aboutwhich the stress occurs, andFaaxial stress that would be permitted if axial force alone existed, ksi (MPa)Fbcompres...

  • Page 242

    BUILDING AND STRUCTURES FORMULAS223To prevent web crippling, the AISC specification requires that bearing stiff-eners be provided on webs where concentrated loads occur when the compres-sive force exceeds R, kip (kN), computed from the following:For a concentrated load applied at a distance from ...

  • Page 243

    224CHAPTER NINEStiffeners are required if the concentrated load exceeds R, kip (kN), computedfrom(9.60)where hclear distance, in (mm), between flanges, and rwf is less than 2.3when the loaded flange is restrained against rotation. If the loaded flange is notrestrained and rwf is less than 1.7,(9....

  • Page 244

    BUILDING AND STRUCTURES FORMULAS225and a pair of stiffeners should be provided opposite the tension flange when thethickness of the column flange tf is less than(9.64)Stiffeners required by the preceding equations should comply with the follow-ing additional criteria:1.The width of each stiffener...

  • Page 245

    226CHAPTER NINEcompletely braced laterally, the allowable stress in the flanges is 0.66Fy, whereFy is the yield strength, ksi (MPa), of the steel. Assuming the steel to carry thefull dead load and the composite section to carry the live load, the maximum unitstress, ksi (MPa), in the steel is(9.6...

  • Page 246

    BUILDING AND STRUCTURES FORMULAS227least 1 in (25.4 mm) of concrete cover in all directions; and unless studs arelocated directly over the web, stud diameters may not exceed 2.5 times thebeam-flange thickness.With heavy concentrated loads, the uniform spacing of shear connectorsmay not be suffici...

  • Page 247

    228CHAPTER NINEIn continuous composite construction, longitudinal reinforcing steel may beconsidered to act compositely with the steel beam in negative-moment regions.In this case, the total horizontal shear, kip (kN), between an interior support andeach adjacent point of contraflexure should be ...

  • Page 248

    BUILDING AND STRUCTURES FORMULAS229For structural design computations, flat compression elements of cold-formed structural members can be divided into two kinds—stiffened elementsand unstiffened elements. Stiffened compression elementsare flat compressionelements; that is, plane compression fla...

  • Page 249

    230CHAPTER NINEbut not less than 4.8t. A simple lip should not be used as an edge stiffener forany element having a flat-width ratio greater than 60.For safe-load determination, that is, in computing effective area and sectionmodulus:(9.77)where fcomputed unit stress in psi in the element based u...

  • Page 250

    BUILDING AND STRUCTURES FORMULAS231For safe-load determinations, that is, in computing effective area and sectionmodulus(9.83)where funit stress in psi in the element computed on basis of reduced section,psi (MPa)wwidth of the element, in (mm)tthickness, in (mm)bt8,040f12,010(w/t) fFIGURE 9.7Typi...

  • Page 251

    232CHAPTER NINEwttttt(a)(b)(c)12b12b12b212b112b112b112b112b112b112b112b112b12bw12b12bw12b12bw12b12bw1w1w1w2w2w2tttt(e)(f)Columns effective area for computing column factor Qa(g)12b212b212b212b212b2w1(d)Beams top flange in compression12b12bwwFIGURE 9.8Effective width of stiffened compression eleme...

  • Page 252

    BUILDING AND STRUCTURES FORMULAS233For deflection determinations, that is, in computing moment of inertia to beused in deflection calculations or in other calculations involving stiffness,(9.84)with f, w, and tthe same as for Eq. (9.83).Web Stresses in Light-gage MembersThe shear on the gross web...

  • Page 253

    234CHAPTER NINE(9.87)NOTE: In solving this formula, B should not be assigned any value greaterthan h.This formula applies only where the distance xis greater than 1.5h. Other-wise, Eq. (9.86) governs.(9.88)Pmax(3)20,000t2 7.40.93BBthtthBBPmax (3)Pmax (3)t = web thickness (each web sheet), mmh = c...

  • Page 254

    BUILDING AND STRUCTURES FORMULAS235(9.89)Values of Pmax (4) apply only where the distance xis greater than 1.5h. Other-wise use Pmax (3).The effective length of bearing Bfor substitution in the above formulasshould not be taken as greater than h.The column-design formulas recommended by American ...

  • Page 255

    236CHAPTER NINE(When the effective-width treatment of stiffened elements is used, Qcan bedefined as the ratio between effective area and total area of the cross section,and Ain the quantity P/A can mean total area of section.)Since fcr can never be greater than fy, the value of Qas a form factor ...

  • Page 256

    BUILDING AND STRUCTURES FORMULAS237Members subject to both axial compression and bending stresses shall be pro-portioned to meet the requirements of both of the following formulas, as applicable:(9.96)and at braced points only(9.97)where Famaximum axial unit stress in compression permitted by thi...

  • Page 257

    238CHAPTER NINEHowever, the wall material and its attachments to the studs must comply withthe following:1.Wall sheathing is attached to both faces or flanges of the studs.2.The spacing in inches of attachments of wall material to each face or flangeof the stud does not exceed ain either:(9.99)(9...

  • Page 258

    BUILDING AND STRUCTURES FORMULAS239For continuous attachment, use nequal to four times the attached lengthalong each faying surface.Bolting Light-gage MembersBolting is employed as a common means ofmaking field connections in light-gage steel construction. The AISI Specificationfor the Design of ...

  • Page 259

    240CHAPTER NINEBeamsThe optimal section modulus for an elastically designed I-shaped beamresults when the area of both flanges equals half the total cross-sectional area ofthe member. Assume now two members made of steels having different yieldpoints and designed to carry the same bending moment,...

  • Page 260

    BUILDING AND STRUCTURES FORMULAS241(9.112)where C carbon content, %Mnmanganese content, %Crchromium content, %Momolybdenum, %Vvanadium, %Ninickel content, %Cucopper, %To prevent underbead cracking, the weld must be cooled at an acceptable rate.A higher carbon equivalent requires a longer cooling ...

  • Page 261

    242CHAPTER NINEWith pinned supports and a uniform load on the horizontal member, Fig. 9.11:(9.117)(9.118)With pinned supports and a uniform load on the vertical member, Fig. 9.12:(9.119)(9.120)V12WL1H2(L1L2)L3H218W5L31(I3/I1)2L1L2L34L21L3L31(I3/I1)L32(I3/I2)L21L3L22L3L1L2L3V112WL3H(L1L2)L3HWL32(L...

  • Page 262

    BUILDING AND STRUCTURES FORMULAS243With fixed supports and a concentrated load on the horizontal member,Fig. 9.13:(9.121)(9.122)(9.123)With fixed supports and and a concentrated load on one vertical member,Fig. 9.14:(9.124)(9.125)(9.126)With fixed supports and a uniform load on the horizontal mem...

  • Page 263

    244CHAPTER NINEWith fixed supports and a uniform load on the vertical member, Fig. 9.16:(9.130)(9.131)(9.132)ROOF LIVE LOADS*Some building codes specify that design of flat, curved, or pitched roofs shouldtake into account the effects of occupancy and rain loads and be designed forminimum live lo...

  • Page 264

    BUILDING AND STRUCTURES FORMULAS245Snow LoadsDetermination of designing snow loads for roofs is often based on the maxi-mum ground snow load in 50-year mean recurrence period (2% probability ofbeing exceeded in any year). This load or data for computing it from anextreme-value statistical analysi...

  • Page 265

    246CHAPTER NINEwhere Kzvelocity exposure coefficient evaluated at height zKzttopographic factorKdwind directionality factorIimportance factorVbasic wind speed corresponding to a 3-s gust speed at 33 ft abovethe ground in exposure CVelocity pressures due to wind to be used in building design vary ...

  • Page 266

    BUILDING AND STRUCTURES FORMULAS247example, ASCE 7-95 stipulates that the total lateral force, or base shear, V(kips) acting in the direction of each of the principal axes of the main structuralsystem should be computed from(9.139)where Csseismic response coefficientWtotal dead load and applicabl...

  • Page 267

    248CHAPTER NINEFor horizontal shear distribution, the seismic design story shear in any story,Vx, is determined by the following:(9.145)where Fithe portion of the seismic base shear induced at level i. The seismicdesign story shear is to be distributed to the various elements of the force-resisti...

  • Page 268

    CHAPTER 10249BRIDGE ANDSUSPENSION-CABLE FORMULASSHEAR STRENGTH DESIGN FOR BRIDGESBased on the American Association of State Highway and Transportation Offi-cials(AASHTO) specifications for load-factor design(LFD), the shear capacity,kip (kN), may be computed from(10.1)for flexural members with un...

  • Page 269

    250CHAPTER TENALLOWABLE-STRESS DESIGN FOR BRIDGE COLUMNSIn the AASHTO bridge-design specifications, allowable stresses in concentri-cally loaded columns are determined from the following equations:When Kl/r is less than Cc,(10.3)When Kl/r is equal to or greater than Cc,(10.4)See Table 10.1.LOAD-A...

  • Page 270

    BRIDGE AND SUSPENSION-CABLE FORMULAS251where Fcrbuckling stress, ksi (MPa)Fyyield strength of the steel, ksi (MPa)Keffective-length factor in plane of bucklingLclength of member between supports, in (mm)rradius of gyration in plane of buckling, in (mm)Emodulus of elasticity of the steel, ksi (MPa...

  • Page 271

    252CHAPTER TENEmodulus of elasticity of material, psimfactor of safety(L/r)critical slenderness ratio(10.11)Suggested Sy32,000; suggested m1.7(10.12)or(10.13)(10.14)(10.15)When the column ends are pinned, the following formulas apply:(10.16)(10.17)(10.18)(10.19)For structural silicon steel with S...

  • Page 272

    BRIDGE AND SUSPENSION-CABLE FORMULAS253(10.21)(10.22)(10.23)When the column ends are pinned,(10.24)(10.25)(10.26)For structural nickel steel with Sy55,000 psi, using structural shapes, orbeing fabricated with column ends being riveted,(10.27)(10.28)(10.29)(10.30)When the column ends are pinned,(1...

  • Page 273

    254CHAPTER TENALLOWABLE-STRESS DESIGN FOR BRIDGE BEAMSAASHTO gives the allowable unit (tensile) stress in bending as Fb0.55Fy. Thesame stress is permitted for compression when the compression flange is support-ed laterally for its full length by embedment in concrete or by other means.When the co...

  • Page 274

    BRIDGE AND SUSPENSION-CABLE FORMULAS255where M1smaller beam end moment; and M2larger beam end moment.The algebraic sign of M1/M2 is positive for double-curvature bending and nega-tive for single-curvature bending.STIFFENERS ON BRIDGE GIRDERSThe minimum moment of inertia, in4 (mm4), of a transvers...

  • Page 275

    256CHAPTER TENThickness of stiffener, in (mm), should be at least where bis the stiff-ener width, in (mm); and fb is the flange compressive bending stress, ksi (MPa).The bending stress in the stiffener should not exceed that allowable for the material.HYBRID BRIDGE GIRDERSThese may have flanges w...

  • Page 276

    TABLE 10.3Design Criteria for Symmetrical Flexural Sections for Load-Factor Design of BridgesMaximumFlange minimumWeb minimum Maximum Type ofbending strength Mu,thickness tf,thickness tu,unbraced length lb,sectionin kip (mm kN)in (mm)in (mm)in (mm)Compact*FyZBracedFySnoncompact†Unbraced________...

  • Page 277

    258CHAPTER TENBEARING ON MILLED SURFACESFor highway design, AASHTO limits the allowable bearing stress on milledstiffeners and other steel parts in contact to Fp0.80Fu. Allowable bearingstresses on pins are given in Table 10.4.The allowable bearing stress for expansion rollers and rockers used in...

  • Page 278

    BRIDGE AND SUSPENSION-CABLE FORMULAS259COMPOSITE CONSTRUCTION IN HIGHWAY BRIDGESShear connectors between a steel girder and a concrete slab in compositeconstruction in a highway bridge should be capable of resisting both hori-zontal and vertical movement between the concrete and steel. Maximumspa...

  • Page 279

    260CHAPTER TENBending StressesIn composite beams in bridges, stresses depend on whether or not the members areshored; they are determined as for beams in buildings (see “Composite Construc-tion” in Chap. 9,“Building and Structures Formulas”), except that the stresses in thesteel may not e...

  • Page 280

    BRIDGE AND SUSPENSION-CABLE FORMULAS261where wchannel length, in (mm), in transverse direction on girder flange; andBcyclic variable4.0 for 100,000 cycles, 3.0 for 500,000 cycles, 2.4for 2 million cycles, and 2.1 for over 2 million cycles.For welded studs (with height/diameter ratio H/d4):(10.47)...

  • Page 281

    262CHAPTER TENAt points of maximum positive moments, Pis the smaller of P1 and P2,computed from(10.49)(10.50)where Aceffective concrete area, in2 (mm2)fc28-day compressive strength of concrete, ksi (MPa)Astotal area of steel section, in2 (mm2)Fysteel yield strength, ksi (MPa)The number of connect...

  • Page 282

    BRIDGE AND SUSPENSION-CABLE FORMULAS263for flexural members with unstiffened webs with h/tw150 or for girders withstiffened webs with a/h exceeding 3 and 67,600(h/tw)2:or stiffeners are not requiredFor girders with transverse stiffeners and a/h less than 3 and 67,600(h/tw)2, theallowable shear st...

  • Page 283

    TABLE 10.6Maximum Width/Thickness Ratios b/tafor Compression Elements for Highway BridgesbLoad-and-resistance-factor designcDescription of elementCompactNoncompactdFlange projection of rolled orfabricated I-shaped beamsWebs in flexural compression150Allowable-stress design ffa0.44FyDescription of...

  • Page 284

    In bracing and other1211secondary membersPlates supported on two edges or3227webs of box shapesgSolid cover plates supported on4034two edges or solid webshPerforated cover plates supported4841on two edges for box shapesa bwidth of element or projection; tthickness. The point of support is the inn...

  • Page 285

    266CHAPTER TENhorizontal, the cable assumes the form of a parabolic arc. Then, the ten-sion at midspan is(10.57)where Hmidspan tension, kip (N)wload on a unit horizontal distance, klf (kN/m)Lspan, ft (m)dsag, ft (m)The tension at the supports of the cable is given by(10.58)where Ttension at suppo...

  • Page 286

    BRIDGE AND SUSPENSION-CABLE FORMULAS267(10.60)Then(10.61)(10.62)Span length then is L2c, with the previous same symbols.GENERAL RELATIONS FOR SUSPENSION CABLESCatenaryFor any simple cable (Fig. 10.3) with a load of qo per unit length of cable, kip/ft(N/m), the catenary length s, ft (m), measured ...

  • Page 287

    268CHAPTER TENThe distance from the low point Cto the left support is(10.65)where fLvertical distance from Cto L, ft (m). The distance from Cto the rightsupport Ris(10.66)where fRvertical distance from Cto R.Given the sags of a catenary fL and fR under a distributed vertical load qo, thehorizonta...

  • Page 288

    BRIDGE AND SUSPENSION-CABLE FORMULAS269where lspan, or horizontal distance between supports Land Rab; hvertical distance between supports.The distance from the low point Cto the right support Ris(10.71)Supports at Different LevelsThe horizontal component of cable tension Hmay be computed from(10....

  • Page 289

    270CHAPTER TENLength of parabolic arc LCis(10.79)(10.80)Supports at Same LevelIn this case, fLfRf, h0, and abl/2. The horizontal component ofcable tension Hmay be computed from(10.81)The vertical components of the reactions at the supports are(10.82)Maximum tension occurs at the supports and equa...

  • Page 290

    BRIDGE AND SUSPENSION-CABLE FORMULAS271Elastic elongation of a parabolic cable is approximately(10.88)where Across-sectional area of cableEmodulus of elasticity of cable steelHhorizontal component of tension in cableIf the corresponding change in sag is small, so that the effect on His negligible...

  • Page 291

    272CHAPTER TENIf elastic elongation and can be ignored,(10.95)Thus, for a load uniformly distributed horizontally wL,(10.96)and the increase in the horizontal component of tension due to live load is(10.97)(10.98)CABLE SYSTEMSThe cable that is concave downward (Fig. 10.4) usually is considered th...

  • Page 292

    BRIDGE AND SUSPENSION-CABLE FORMULAS273where ninteger, 1 for fundamental mode of vibration, 2 for second mode, . . .lspan of cable, ft (m)wload on cable, kip/ft (kN/m)gacceleration due to gravity 32.2 ft/s2Tcable tension, kip (N)The spreaders of a cable truss impose the condition that under a giv...

  • Page 293

    274CHAPTER TENThe drain should have a large enough area to handle the maximum runoffanticipated. Rainwater runoff from a bridge should be led a specially designedand built, or existing, sewerage system. Where a new sewerage system is beingdesigned, the bridge must be considered as an additional w...

  • Page 294

    CHAPTER 11275HIGHWAY AND ROAD FORMULASCIRCULAR CURVES Circular curves are the most common type of horizontal curve used to con-nect intersecting tangent (or straight) sections of highways or railroads. Inmost countries, two methods of defining circular curves are in use: the first,in general use ...

  • Page 295

    276CHAPTER ELEVENEquations of Circular Curves(11.1)(11.2)(11.3)(11.4)MR vers 12 IR (1cos 12 I) exactER exsec 12 IR (sec 12 I1) exactTR tan 12 I exact50sin 12 D exact for chord definitionR5,729.578D exact for arc definition, approxi-mate for chord definitionPIITPTCRII2TPCEMRI2FIGURE 11.1...

  • Page 296

    HIGHWAY AND ROAD FORMULAS277(11.5)(11.6)(11.7)(11.8)(11.9)(11.10)(11.11)(11.12)PARABOLIC CURVESParabolic curves are used to connect sections of highways or railroads of differ-ing gradient. The use of a parabolic curve provides a gradual change in directionalong the curve. The terms and symbols g...

  • Page 297

    278CHAPTER ELEVENPVTpoint of vertical tangency, end of curveG1grade at beginning of curve, ft/ft (m/m)G2grade at end of curve, ft/ft (m/m)Llength of curve, ft (m)Rrate of change of grade, ft/ft2 (m/m2)Velevation of PVI, ft (m)E0elevation of PVC, ft (m)Etelevation of PVT, ft (m)xdistance of any po...

  • Page 298

    HIGHWAY AND ROAD FORMULAS279Superelevated roadway cross sections are typically employed on curves ofrural highways and urban freeways. Superelevation is rarely used on local streetsin residential, commercial, or industrial areas.HIGHWAY ALIGNMENTSGeometric design of a highway is concerned with ho...

  • Page 299

    280CHAPTER ELEVENFor crest vertical curves, AASHTO defines the minimum length Lmin, ft (m), ofcrest vertical curves based on a required sight distance S, ft (m), as that given by(11.18)When eye height is 3.5 ft (1.07 m) and object height is 0.5 ft (0.152 m):(11.19)Also, for crest vertical curves:...

  • Page 300

    HIGHWAY AND ROAD FORMULAS281STRUCTURAL NUMBERS FOR FLEXIBLE PAVEMENTSThe design of a flexible pavement or surface treatment expected to carry morethan 50,000 repetitions of equivalent single 18-kip axle load (SAI) requiresidentification of a structural number SN that is used as a measure of the a...

  • Page 301

    282CHAPTER ELEVENFigure 11.8 shows the use of curves in at-grade four-leg intersections ofhighways. Figure 11.9 shows the use of curves in at-grade T (three-leg)intersections. Figure 11.10 shows street space and maneuvering space usedfor various parking positions.Gradeseparation(a)(b)(c)(d)(e)(f)...

  • Page 302

    HIGHWAY AND ROAD FORMULAS283(a)(b)(c)FIGURE 11.8Highway turning lanes: (a) Unchannelized; (b) channelized; (c) flared.(a)(b)(c)(d)FIGURE 11.9Highway turning lanes: (a) Unchannelized; (b) intersec-tion with a right-turn lane: (c) intersection with a single-turning roadway;(d) channelized intersect...

  • Page 303

    284CHAPTER ELEVENTRANSITION (SPIRAL) CURVESOn starting around a horizontal circular curve, a vehicle and its contents areimmediately subjected to centrifugal forces. The faster the vehicle enters thecircle and the sharper the curvature is, the greater the influence on vehicles anddrivers of the c...

  • Page 304

    HIGHWAY AND ROAD FORMULAS285DESIGNING HIGHWAY CULVERTSA highway culvert is a pipelike drainage facility that allows water to flowunder the road without impeding traffic. Corrugated and spiral steel pipe arepopular for culverts because they can be installed quickly, have long life, arelow in cost,...

  • Page 305

    286CHAPTER ELEVENConduit deflectionis given by the Iowa formula. This formula gives the rel-ative influence on the deflection of the pipe strength and the passive side pres-sure resisting horizontal movement of the pipe wall, or(11.29)wherexhorizontal deflection of pipe, in (mm)D1deflection lag f...

  • Page 306

    HIGHWAY AND ROAD FORMULAS287than one pipe diameter, the total load TL is assumed to act on the pipe, andTLPv; that is,(11.30)When the height of cover is equal to, or greater than, one pipe diameter,(11.31)where Pvdesign pressure, kip/ft2 (MPa/m2)Kload factorDLdead load, kip/ft2 (MPa/m2)LLlive loa...

  • Page 307

    288CHAPTER ELEVENcarried by the steel. The ring compression is an axial load acting tangentiallyto the conduit wall (Fig. 11.12). For conventional structures in which the toparc approaches a semicircle, it is convenient to substitute half the span for thewall radius. Then,(11.32)Allowable Wall St...

  • Page 308

    HIGHWAY AND ROAD FORMULAS289A safety factor of 2 is applied to the ultimate wall stress to obtain the designstress Fc, ksi (MPa):(11.36)Wall ThicknessRequired wall area A, in2/ft (mm2/m), of width, is computed from the calculatedcompression Cin the pipe wall and the allowable stress Fc:(11.37)Fro...

  • Page 309

    290CHAPTER ELEVENCheck Bolted SeamsStandard factory-made pipe seams are satisfactory for all designs within themaximum allowable wall stress of 16.5 ksi (113.8 MPa). Seams bolted in theshop or field, however, continue to be evaluated on the basis of test values foruncurved, unsupported columns. A...

  • Page 310

    CHAPTER 12291HYDRAULICS AND WATERWORKS FORMULASTo simplify using the formulas in this chapter, Table 12.1 presents symbols,nomenclature, and United States Customary System(USCS) and System Inter-national(SI) units found in each expression.CAPILLARY ACTIONCapillarityis due to both the cohesive for...

  • Page 311

    292TABLE 12.1Symbols, Terminology, Dimensions, and Units Used in Water Engineering SymbolTerminologyDimensionsUSCS unitsSI units AAreaL2ft2mm2CChezy roughness coefficientL1/2/Tft5/sm0.5/sC1Hazen–Williams roughness coefficientL0.37/Tft0.37/sm0.37/sdDepthLftm dcCritical depthLftm DDiameterLftm EM...

  • Page 312

    293wSpecific weightF/L3lb/ft3kg/m3yDepth in open channel, distance from solid boundaryLftm ZHeight above datumLftm Size of roughnessLftm ViscosityFT/L2lb s/ftkg s/mKinematic viscosityL2/Tft2/sm2/sDensityFT2/L4lb s2/ft4kg s2/m4Surface tensionF/Llb/ftkg/m Shear stressF/L2lb/in2MPaSymbols for dimens...

  • Page 313

    294CHAPTER TWELVEratio of the tangential shearing stresses between flow layers to the rate of changeof velocity with depth:(12.2)whereshearing stress, lb/ft2 (N/m2)Vvelocity, ft/s (m/s) ydepth, ft (m) Viscosity decreases as temperature increases but may be assumed independentof changes in pressur...

  • Page 314

    HYDRAULICS AND WATERWORKS FORMULAS295PRESSURE ON SUBMERGED CURVED SURFACESThe hydrostatic pressure on a submerged curved surface (Fig. 12.2) is given by(12.3)where Ptotal pressure force on the surface PHforce due to pressure horizontallyPVforce due to pressure vertically P2P 2HP 2VAC(a)(b)EPV = P...

  • Page 315

    296CHAPTER TWELVEFUNDAMENTALS OF FLUID FLOW For fluid energy, the law of conservation of energy is represented by theBernoulli equation:(12.4)where Z1elevation, ft (m), at any point 1 of flowing fluid above an arbitrary datum Z2elevation, ft (m), at downstream point in fluid above same datum p1pr...

  • Page 316

    HYDRAULICS AND WATERWORKS FORMULAS297section, however, varies with velocity. Usually, Zp/w at the midpoint andthe average velocity at a section are assumed when the Bernoulli equation isapplied to flow across the section or when total head is to be determined. Aver-age velocity, ft/s (m/s)Q/A, wh...

  • Page 317

    298CHAPTER TWELVESimilar equations may be written for the Yand Zdirections. The impulse–momentum equation often is used in conjunction with the Bernoulli equation butmay be used separately. SIMILITUDE FOR PHYSICAL MODELSA physical model is a system whose operation can be used to predict thechar...

  • Page 318

    HYDRAULICS AND WATERWORKS FORMULAS299where FFroude number (dimensionless) Vvelocity of fluid, ft/s (m/s) Llinear dimension (characteristic, such as depth or diameter), ft (m) gacceleration due to gravity, 32.2 ft/s2 (9.81 m/s2)For hydraulic structures, such as spillways and weirs, where there is ...

  • Page 319

    300CHAPTER TWELVEFor true models, Sr1, RrLr. Hence,(12.15)In models of rivers and channels, it is necessary for the flow to be turbulent. TheU.S. Waterways Experiment Station has determined that flow is turbulent if(12.16)where Vmean velocity, ft/s (m/s) Rhydraulic radius, ft (m) kinematic viscos...

  • Page 320

    HYDRAULICS AND WATERWORKS FORMULAS301where Vfluid velocity, ft/s (m/s) Dpipe diameter, ft (m)density of fluid, lb s2/ft4 (kg s2/m4) (specific weight divided by g,32.2 ft/s2)viscosity of fluid lb s/ft2 (kg s/m2)/kinematic viscosity, ft2/s (m2/s)For a Reynolds number less than 2000, flow is laminar...

  • Page 321

    302CHAPTER TWELVEDarcy–Weisbach FormulaOne of the most widely used equations for pipe flow, the Darcy–Weisbach for-mula satisfies the condition described in the preceding section and is valid forlaminar or turbulent flow in all fluids:(12.20)where hfhead loss due to friction, ft (m)ffriction ...

  • Page 322

    HYDRAULICS AND WATERWORKS FORMULAS303where Vvelocity, ft/s (m/s)Ccoefficient, dependent on surface roughness of conduitSslope of energy grade line or head loss due to friction, ft/ft (m/m) ofconduitRhydraulic radius, ft (m)Hydraulic radiusof a conduit is the cross-sectional area of the fluid in i...

  • Page 323

    304CHAPTER TWELVEFor pipes flowing full:(12.29)(12.30)(12.31)(12.32)where Vvelocity, ft/s (m/s)C1coefficient, dependent on surface roughness (given in engineeringhandbooks)Rhydraulic radius, ft (m)Shead loss due to friction, ft/ft (m/m) of pipe Ddiameter of pipe, ft (m)Llength of pipe, ft (m)Qdis...

  • Page 324

    HYDRAULICS AND WATERWORKS FORMULAS305(12.33)(12.34)where pDpressure at D; and wunit weight of liquid. Comparison of the Darcy-Weisbach, Manning, and Hazen-Williams Formulas*Because the Darcy-Weisbach, Manning, and Hazen-Williams equations are allused frequently in practice, it is important to kno...

  • Page 325

    306CHAPTER TWELVEAll three expressions are approximately of the form(12.40)where Kconstant dependent on pipe roughness.If these expressions are equated and simplified, the following relationshipbetween f, n, and Cis obtained:(SI units)(12.41)(U.S. customary units)(12.42)When one of the coefficien...

  • Page 326

    HYDRAULICS AND WATERWORKS FORMULAS307A special application of these two preceding formulas is the discharge from apipe into a reservoir. The water in the reservoir has no velocity, so a full velocityhead is lost.Gradual EnlargementsThe equation for the head loss due to a gradual conical enlargeme...

  • Page 327

    308CHAPTER TWELVEFLOW THROUGH ORIFICESAn orifice is an opening with a closed perimeter through which water flows. Ori-fices may have any shape, although they are usually round, square, or rectangular. Orifice Discharge into Free Air Discharge through a sharp-edged orifice may be calculated from(1...

  • Page 328

    HYDRAULICS AND WATERWORKS FORMULAS309(12.53)where hLlosses in head, ft (m), between 1 and 2.By assuming V10, setting h1h2h, and using a coefficient of dis-charge Cto account for losses, the following formula is obtained:(12.54)QCa2g hV2B2g h1h2V 212ghL1hCc2XYFIGURE 12.8Fluid jet takes a parabolic...

  • Page 329

    310CHAPTER TWELVEValues of Cfor submerged orifices do not differ greatly from those for nonsub-merged orifices.Discharge under Falling Head The flow from a reservoir or vessel when the inflow is less than the outflowrepresents a condition of falling head. The time required for a certain quantityo...

  • Page 330

    HYDRAULICS AND WATERWORKS FORMULAS311The Ycoordinate is(12.60)where Vavgaverage velocity over period of time t. The equation for the pathof the jet:(12.61)ORIFICE DISCHARGE INTO DIVERGING CONICAL TUBES This type of tube can greatly increase the flow through an orifice by reducing thepressure at t...

  • Page 331

    312CHAPTER TWELVEWATER HAMMERWater hammer is a change in pressure, either above or below the normal pres-sure, caused by a variation of the flow rate in a pipe.The equation for the velocity of a wave in a pipe is(12.63)where Uvelocity of pressure wave along pipe, ft/s (m/s) Emodulus of elasticity...

  • Page 332

    HYDRAULICS AND WATERWORKS FORMULAS313where pinternal pressure, lb/in2 (MPa)Doutside diameter of pipe, in (mm) Fforce acting on each cut of edge of pipe, lb (N) Hence, the stress, lb/in2 (MPa) on the pipe material is(12.65)where Aarea of cut edge of pipe, ft2 (m2); and tthickness of pipe wall,in (...

  • Page 333

    F2m =Section 1Section 2Directionof flowF2 = P2 A2P1 = P1 A1V1, P1, A1V2, P2, A2P2F1mXP1ααθθF2mYYXRV2QwgResultantforceF1m =V1QwgFIGURE 12.12Forces produced by flow at a pipe bend and change in diameter. 314

  • Page 334

    HYDRAULICS AND WATERWORKS FORMULAS315A1area before size change in pipe, ft2 (m2)A2area after size change in pipe, ft2 (m2)F2mforce due to momentum of water in section 2V2Qw/gF1mforce due to momentum of water in section 1V1Qw/gP2pressure of water in section 2 times area of section 2 p2A2P1pressure...

  • Page 335

    316CHAPTER TWELVEFrom the Bernoulli equation for the entrance and exit, and the Manningequation for friction loss, the following equation is obtained:(12.69)Solution for the velocity of flow yields(12.70)where Helevation difference between headwater and tailwater, ft (m) Vvelocity in culvert, ft/...

  • Page 336

    HYDRAULICS AND WATERWORKS FORMULAS317Entrance Submerged or Unsubmerged but Free Exit.For these conditions,depending on the head, the flow can be either pressure or open channel (Fig. 12.14). The discharge for the open-channel condition is obtained by writing theBernoulli equation for a point just...

  • Page 337

    318CHAPTER TWELVETo solve the preceding head equation, it is necessary to try different valuesof dn and corresponding values of Runtil a value is found that satisfies theequation.OPEN-CHANNEL FLOW Free surface flow, or open-channel flow, includes all cases of flow in which theliquid surface is op...

  • Page 338

    HYDRAULICS AND WATERWORKS FORMULAS319whereis an empirical coefficient that represents the degree of turbulence.Experimental data indicate that may vary from about 1.03 to 1.36 for pris-matic channels. It is, however, normally taken as 1.00 for practical hydraulic workand is evaluated only for pre...

  • Page 339

    320CHAPTER TWELVEwhere Aarea of flow, ft2 (m2)Rhydraulic radius, ft (m) Qamount of flow or discharge, ft3/s (m3/s)nManning’s roughness coefficient Sslope of energy grade line or loss of head, ft (m), due to friction perlinear ft (m), of channel AR2/3is referred to as a section factor.Critical D...

  • Page 340

    HYDRAULICS AND WATERWORKS FORMULAS321(12.84)Because the discharge QVA, this equation may be written:(12.85)where Aarea of flow, ft2 (m2); and Qquantity of flow, ft3/s (m3/s).HYDRAULIC JUMPThis is an abrupt increase in depth of rapidly flowing water (Fig. 12.16).Flow at the jump changes from a sup...

  • Page 341

    322CHAPTER TWELVEThe pressure force Fdeveloped in hydraulic jump is (12.86)where d1depth before jump, ft (m) d2depth after jump, ft (m) wunit weight of water, lb/ft3 (kg/m3)The rate of change of momentum at the jump per foot width of channel equals(12.87)where Mmass of water, lb s2/ft (kg s2/m)V1...

  • Page 342

    HYDRAULICS AND WATERWORKS FORMULAS323NONUNIFORM FLOW IN OPEN CHANNELS Symbols used in this section are Vvelocity of flow in the open channel, ft/s(m/s); Dccritical depth, ft (m); gacceleration due to gravity, ft/s2 (m/s2);Qflow rate, ft3/s (m3/s); qflow rate per unit width, ft3/ft (m3/m); and Hmm...

  • Page 343

    765Ld243Steady jumpUndularjumpWeakjumpOscillatingjumpWavySurfaceturbulence onlyBestperformanceAcceptableperformanceRollerLd1d2V1Strong jumpExpensive stilling basin andrough surface conditions20191817161514131211109876543210F1 = V1/√gd1FIGURE 12.18Length of hydraulic jump in a horizontal channel...

  • Page 344

    HYDRAULICS AND WATERWORKS FORMULAS325Nonuniform flow occurs in open channels with gradual or sudden changesin the cross-sectional area of the fluid stream. The terms gradually varied flowand rapidly varied floware used to describe these two types of nonuniformflow. Equations are given next for fl...

  • Page 345

    326CHAPTER TWELVEand the critical depth is(12.99)Then the discharge per foot (meter) of width is given by(12.100)With g32.16, Eq. (12.100) becomes(12.101)Triangular Channels In a triangular channel (Fig. 12.20), the maximum depth Dc and the mean depthDm equalDc. Then, (12.102)and(12.103)As shown ...

  • Page 346

    HYDRAULICS AND WATERWORKS FORMULAS327or(12.107)With g32.16,(12.108)Parabolic ChannelsThese channels can be conveniently defined in terms of the top width Tand thedepth Dc. Then the area aDcT and the mean depthDm.Then (Fig. 12.21), (12.109)and(12.110)Further,(12.111)With g32.16,(12.112)and(12.113)...

  • Page 347

    328CHAPTER TWELVETrapezoidal ChannelsFigure 12.22 shows a trapezoidal channel having a depth of Dc and a bottomwidth b. The slope of the sides, horizontal divided by vertical, is z. Expressingthe mean depth Dm in terms of channel dimensions, the relations for criticaldepth Dc and average velocity...

  • Page 348

    HYDRAULICS AND WATERWORKS FORMULAS329(12.122)(12.123)Flow quantity is then given by(12.124)WEIRSA weir is a barrier in an open channel over which water flows. The edge or sur-face over which the water flows is called the crest.The overflowing sheet ofwater is the nappe.If the nappe discharges int...

  • Page 349

    330CHAPTER TWELVESharp-crested weirs are useful only as a means of measuring flowing water.In contrast, weirs not sharp crested are commonly incorporated into hydraulicstructures as control or regulation devices, with measurement of flow as theirsecondary function.FLOW OVER WEIRSRectangular WeirT...

  • Page 350

    HYDRAULICS AND WATERWORKS FORMULAS331Trapezoidal (Cipolletti) WeirThe Cipolletti weir, extensively used for irrigation work, is trapezoidal inshape. The sides slope outward from the crest at an inclination of 1:4 (horizontal:vertical). The discharge is(12.126)where b, h, and Qare as defined earli...

  • Page 351

    332CHAPTER TWELVEPREDICTION OF SEDIMENT-DELIVERY RATETwo methods of approach are available for predicting the rate of sediment accu-mulation in a reservoir; both involve predicting the rate of sediment delivery.One approach depends on historical records of the silting rate for existingreservoirs ...

  • Page 352

    HYDRAULICS AND WATERWORKS FORMULAS333eaactual vapor pressure, in (mm), of mercury, in air based on monthlymean air temperature and relative humidity at nearby stations forsmall bodies of shallow water or based on information obtainedabout 30 ft (9.14 m) above the water surface for large bodies of...

  • Page 353

    334CHAPTER TWELVEwhere Kand bare dependent on the storm frequency and region of the UnitedStates (Fig. 12.26 and Table 12.4).The Steel formula gives the average maximum precipitation rates for dura-tions up to 2 h.GROUNDWATERGroundwater is subsurface water in porous strata within a zone of satura...

  • Page 354

    HYDRAULICS AND WATERWORKS FORMULAS335Permeabilityindicates the ease with which water moves through a soil anddetermines whether a groundwater formation is an aquifer or aquiclude.The rate of movement of groundwater is given by Darcy’s law:(12.134)where Qflow rate, gal/day (m3/day)Khydraulic con...

  • Page 355

    336CHAPTER TWELVEECONOMICAL SIZING OF DISTRIBUTION PIPINGAn equation for the most economical pipe diameter for a distribution system forwater is(12.138)where Dpipe diameter, ft (m)fDarcy–Weisbach friction factorbvalue of power, $/hp per year ($/kW per year)Qaaverage discharge, ft3/s (m3/s)Sallo...

  • Page 356

    HYDRAULICS AND WATERWORKS FORMULAS337where Qflow rate, ft3/s (m3/s)c empirical discharge coefficient dependent on throat velocity anddiameterd1diameter of main section, ft (m)d2diameter of throat, ft (m)h1pressure in main section, ft (m) of waterh2pressure in throat section, ft (m) of waterHYDROE...

  • Page 357

    338CHAPTER TWELVEPUMPS AND PUMPING SYSTEMSCivil engineers use centrifugal and other rotating pumps for a variety oftasks—water supply, irrigation, sewage treatment, fire-fighting systems, shipcanals—and numerous other functions. This section of Chap. 12 presents per-tinent formulas for applyi...

  • Page 358

    FIGURE 12.28Definition sketch for the head on a pump. (Metcalf & Eddy—Wastewater Engineering: Collection and Pumping ofWastewater, McGraw-Hill.)hdhsHohfd +Σhmdhfs +ΣhmsHtHsVs2MotorEntrance lossSuction intakeEnergy grade lineDatumPumpdischargePumpHydraulic grade lineHstat2gVd22gVd22g339

  • Page 359

    340CHAPTER TWELVEFriction Head.The head of water that must be supplied to overcome thefrictional loss caused by the flow of fluid through the piping system is the frictionhead. The frictional head loss in the suction (hfs) and discharge (hfd) pipingsystem may be computed with the Darcy-Weisbach o...

  • Page 360

    HYDRAULICS AND WATERWORKS FORMULAS341whereHttotal dynamic head, m (ft)Hd (HS)discharge (suction) head measured at discharge (suction) nozzleof pump referenced to the centerline of the pump impeller, m (ft)Vd (Vs)velocity in discharge (suction) nozzle, m/s (ft/s)gacceleration due to gravity, 9.81 ...

  • Page 361

    342CHAPTER TWELVEfunction of the design. Information on the design is furnished by the pump manu-facturer in a series of curves for a given pump. Pump efficiency Ep—the ratio ofthe useful output power of the pump to the input power to the pump—is given by(SI units)(12.151)(U.S. customary unit...

  • Page 362

    HYDRAULICS AND WATERWORKS FORMULAS343The operating points at which similar flow patterns occur are called corre-sponding points;the Eqs. (12.153) to (12.155) apply only to correspondingpoints. However, every point on a pump head-capacity curve corresponds to apoint on the head-capacity curve of a...

  • Page 363

    344CHAPTER TWELVEthe vapor pressure of the water, expressed in meters (feet). Cavitation occurswhen the NPSHA is less than the NPSHR. The NPSHA is found by adding theterm to the right-hand side of Eq. (12.148) or to the energy(Bernoulli’s) equation applied to the suction side of the pump. Thus,...

  • Page 364

    HYDRAULICS AND WATERWORKS FORMULAS345Qdischarge, cfs (cu m/s)Q1discharge at 1 ft headcfs (cu m/s)tthickness, in, or time, secucircumferential velocity of a point on runner, fps (m/s)Vabsolute velocity of water, fps (m/s)Vutangential component of absolute velocity VcosVrcomponent of Vin radial pla...

  • Page 365

    346CHAPTER TWELVEA convenient way of determining the highest practicable speed is by therelation of the specific speed to the head. For Francis turbines this isUSCS(12.169) SIFor propeller-type turbines, USCS(12.170) SIAffinity Laws.The relationships between head, discharge, speed, horsepower...

  • Page 366

    HYDRAULICS AND WATERWORKS FORMULAS347for(12.176)for(12.177)whereVand Hare the penstock velocity, ft/s (m/s);and head, ft (m), prior to closure; Lis the penstock length, ft (m); and Tis thetime of gate closure. For full load rejection, Tmay be taken as 85 percent of thetotal gate traversing time t...

  • Page 367

    348CHAPTER TWELVEDAMSA structure that bars or detains the flow of water in an open channel or water-course. Dams are constructed for several principal purposes. Diversion damsdivert water from a stream; navigation damsraise the level of a stream toincrease the depth for navigation purposes; power...

  • Page 368

    HYDRAULICS AND WATERWORKS FORMULAS349Utotal uplift forceHalgebraic summation of all active horizontal forcesValgebraic summation of all active vertical forcesMalgebraic summation of all momentszvertical normal stress1first principal stress2second principal stresszyhorizontal and vertical shearing...

  • Page 369

    350CHAPTER TWELVEThe resistance of a gravity dam to sliding is primarily dependent upon thedevelopment of sufficient shearing strength. The factor of safety due to com-bined shearing and sliding resistance may be expressed by the formula.(12.183)In practice, this resistance is attained in part by...

  • Page 370

    HYDRAULICS AND WATERWORKS FORMULAS351usually considered as gravity dams, although some parts of the loads may becarried by arch action. Many early arch dams were built of rubble, ashlar, orcyclopean masonry. However, practically all arch dams constructed duringrecent years have been built of conc...

  • Page 371

    352CHAPTER TWELVERock Movements.Considerations of rock movements and their effects on thesection of arch dams may be based on approximate formulas.* If the ends of thearch elements are vertical, and the bases of the cantilever elements, horizontal,rock rotations and deflections of elements with p...

  • Page 372

    HYDRAULICS AND WATERWORKS FORMULAS353In the preceding equations, Mand Vare the arch and cantilever momentsand shears, Hthe arch thrust, Mt the cantilever twisting moment, Er the elasticmodulus of the rock, tthe radial thickness of the element, and K1, K2, K3, K4,and K5 constants depending on Pois...

  • Page 373

    354CHAPTER TWELVEwherexintensity of normal stress on horizontal planeNtotal vertical load on section (masonry water)Asectional area of baseMmomentNeeeccentricity (distance from point of application to center of gravityof section)Ydistance from center of gravity to most remote fiberImoment of iner...

  • Page 374

    HYDRAULICS AND WATERWORKS FORMULAS355where Qdischarge, cfsHhead, ftLlength, ftC1a coefficient that depends on the character of the materialSubstituting for Q, the value Av [Eq. 12.200] becomes(12.201)For each class of foundation material, homogeneity being assumed, there is adefinite maximum velo...

  • Page 375

    356CHAPTER TWELVEand the effective stress in the soil(12.207)but 1/Li and iN1/N2, therefore(12.208)Were the soil structure of such a nature and the reservoir head Hgreatenough to create a large amount of percolation or high seepage pressure, theflow Qwould exceed the limits for a safe foundation ...

  • Page 376

    HYDRAULICS AND WATERWORKS FORMULAS357The coefficient of permeability may be determined in the laboratory by testson distributed or remolded samples of the soil.Quantity of Seepage.The quantity of seepage can be computed directlyfrom a flow net. Or, in certain instances, from charts or equations w...

  • Page 377

    358CHAPTER TWELVECaseII. When I< 2U,(12.213)Figure 12.33 gives 10 formulas for seepage through the embankment of anearth dam with an impervious foundation.Slope Protection.The slopes of earth embankments must be protected againsterosion by wave action, rainwash, frost, and wind action. Protect...

  • Page 378

    HYDRAULICS AND WATERWORKS FORMULAS359economically by provided against the erosion which would occur if the dam wereovertopped. A major cause of earth-dam failures has been overtopping; therefore,care must be exercised that freeboard and spillway capacity are ample to prevent it.Wave Action.The he...

  • Page 379

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  • Page 380

    CHAPTER 13361STORMWATER, SEWAGE,SANITARY WASTEWATER,AND ENVIRONMENTALPROTECTION*DETERMINING STORM WATER FLOWSewers must handle storm water flows during, and after, rain storms, and snow-and ice-melting events. The rational formula for peak storm-water runoff isQCIA(13.1)where Qpeak runoff, ft3/sA...

  • Page 381

    362CHAPTER THIRTEENwhere ncoefficient dependent on roughness of conduit surfaceRhydraulic radius, ft area, ft2, of fluid divided by wetted perimeter, ftSenergy loss, ft/ft of conduit length; approximately the slope of theconduit invert for uniform flowC1.486 (conversion factor to account for chan...

  • Page 382

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION363When a grate inlet is used to collect the storm water, an opening of 18 in, ormore, is recommended. With a flow depth of up to 4.8 in, the inlet capacity is(13.7)where Pperimeter, ft, of grate opening over which water may flow, ignoringthe ba...

  • Page 383

    364CHAPTER THIRTEENwhere llength of weir, fthdepth of flow over weir at down-stream end, ftSiphon spillways are better adapted to handling large storm water and sewerflows. The area for the siphon throat, A sq ft, is:(13.10)where Qdischarge, ft3/sccoefficient of discharge, which varies from 0.6 t...

  • Page 384

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION365where Vrreactor volume (Mgal) (m3)cmean cell residence time, or the average time that the sludge remainsin the reactor (sludge age). For a complete-mix activated sludgeprocess,c ranges from 5 to 15 days. The design of the reactor isbased on c...

  • Page 385

    366CHAPTER THIRTEENIt is recommended that aeration equipment be designed with a factor ofsafety of at least 2.The food to microorganism ratio is defined asF:MSO Xa(13.17)where F:M is the food to microorganism ratio in d–1.F:M is simply a ratio of the “food” or BOD5 of the incoming waste, to...

  • Page 386

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION367Xmixed liquor suspended solids (MLSS)Qrreturn activated sludge pumping rate (Mgd)Xrconcentration of sludge in the return line (mg/L). When lacking sitespecific operational data, a value commonly assumed is 8000 mg/L.Qeeffluent flow rate (Mgd)...

  • Page 387

    368CHAPTER THIRTEENThe ratio of RAS pumping rate to influent flow rate, or recirculation ratio ( ),may now be calculated:(13.24)Recirculation ratio can vary from 0.25 to 1.50 depending upon the type ofactivated sludge process used. Common design practice is to size the RASpumps so that they are c...

  • Page 388

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION369SIZING A POLYMER DILUTION/FEED SYSTEMDepending on the quality of settled secondary effluent, organic polymer addi-tion is often used to enhance the performance of tertiary effluent filters in adirect filtration process: see Design of a Rapid ...

  • Page 389

    370CHAPTER THIRTEENThe sludge feed rate for the dewatering facility is(13.35)And the sludge weight flow rate is(13.36)Wsweight flow rate of sludge feed, lb/h (kg/h)Vvolume flow rate of sludge feed, gal/min (L/s)s.g.specific gravity of sludge% solids percent solids expressed as a decimaldensity of...

  • Page 390

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION371The wet cake discharge (lb/h) is(13.39)The volume of wet cake, assuming a cake density of 60 lb/ft3 is(13.40)The percent reduction in sludge volume is then(13.41)The centrifugal acceleration force (G), defined as multiples of gravity, is afun...

  • Page 391

    372CHAPTER THIRTEENThe NRC equations for trickling filter performance are empirical equations,which are primarily applicable to single and multistage rock systems with recir-culation.The overall efficiency of the two-stage trickling filter is(13.45)Also, overall efficiency E1E2(1 – E1); and E1E...

  • Page 392

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION373where Rrecirculation ratio Qr /QQrrecirculation flowQwastewater flowThe BOD5 loading for the first stage filter is calculated usingW(influent BOD5, mg/L)(wastewater flow, Mgd)\(8.34 lb/Mgal/mg/L)(13.49)The BOD5 loading for the second stage tr...

  • Page 393

    374CHAPTER THIRTEENThe power input per volume of liquid is generally used as a rough measureof mixing effectiveness, based on the reasoning that more input power createsgreater turbulence, and greater turbulence leads to better mixing. The followingequation is used to calculate the required power...

  • Page 394

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION375The weight of oxygen, (lb2/day) required to destroy the VSS is approximately(13.58)Then, the volume of air required at standard conditions of 14.7 lb/sq in and68 F with 23.2 percent oxygen by weight and a density of 0.075 lb/ft3 is:(13.59)The...

  • Page 395

    376CHAPTER THIRTEENAdjustment for depth. The treatability constant is then adjusted from thestandard depth of 20 ft (6.1 m) to the example filter depth of 25 ft (7.6 m) usingthe following equation:(13.62)where k30/25treatability constant at 30°C (86°F) and 25 ft (7.6 m) filter depth(as example ...

  • Page 396

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION377achieve uniform growth and sloughing, higher periodic dosing rates arerequired. The required dosing rate in inches per pass of distributor arm may beapproximated using the following:Dosing rate (organic loading, lb/103 ft3 d)(0.12)(13.67)Typi...

  • Page 397

    378CHAPTER THIRTEENIn the anaerobic digestion process, the organic material is converted biologi-cally, under anaerobic conditions, to a variety of end products includingmethane (CH4) and carbon dioxide (CO2). The process is carried out in an air-tight reactor. Sludge, introduced continuously or ...

  • Page 398

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION379where Pxvolatile solids produced, lb/d (kg/d)Yyield coefficient (lb VSS/lb BODL)kdendogenous coefficient (d–1)cmean cell residence time (d)The volume of methane gas produced at standard conditions (32°F and 1 atm)(0°C and 101.3 kPa) is ca...

  • Page 399

    380CHAPTER THIRTEENThe average daily consumption of chlorine isCl2 lb/d(average dosage, mg/L) (Mgd)(8.34)(13.77)A typical chlorination flow diagram is shown in Fig. 13.8. This is a com-pound loop system, which means the chlorine dosage is controlled through sig-nals received from both effluent fl...

  • Page 400

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION381To size the main and lateral sewers, use the Manning formula and the appro-priate conveyance factorfrom Table 13.2.When the conveyance factor Cf is used, the Manning formula becomes(13.80)where Qflow rate through the pipe, ft3/s; Cfconveyance...

  • Page 401

    382CHAPTER THIRTEENInfiltration is usually expressed in gallons per day per mile of sewer. Withvery careful construction, infiltration can be kept down to 5000 gal/(day mi)[0.14 L/(km s)] of pipe even when the groundwater level is above the pipe.With poor construction, porous soil, and high groun...

  • Page 402

    STORMWATER, WASTEWATER, AND ENVIRONMENTAL PROTECTION383DESIGN OF AN AERATED GRIT CHAMBERGrit removal in a wastewater treatment facility prevents unnecessary abrasionand wear of mechanical equipment such as pumps and scrappers, and grit depo-sition in pipelines and channels. Grit chambers are desi...

  • Page 403

    384CHAPTER THIRTEENLength of grit chamber (ft) (volume)/[(width, ft) (depth, ft)] (13.86)Length-width ratios range from 3:1 to 5:1.The air supply requirement for an aerated grit chamber ranges from 2.0 to5.0 ft3/min/ft of chamber length (0.185 to 0.46 m3/min m).(13.87)Grit quantities must be esti...

  • Page 404

    INDEX

  • Page 405

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  • Page 406

    387Activated sludge reactor, design of, 383,364Adjustment factors for lumber, 188,169Aerated grit chamber, design of, 402,383Aerobic digester, design 393,of, 374Allowable bond stresses in concrete 143,beams, 124Allowable loads for 124,piles, 105Allowable stress design for buildings, 227,208A...

  • Page 407

    388INDEXBridge and suspension-cable formulas(Cont.):bearing on milled 277,surfaces, 258bridge 277,fasteners, 258cable 291,systems, 272–273composite construction in highway 278,bridges, 259bending 279,stresses, 260effective width of 278,slabs, 259shear 279,range, 260span/depth 278,ratios,...

  • Page 408

    INDEX389Cable 291,systems, 272–273California bearing 214,ratio, 195Cantilever retaining walls, 172,153–155Centrifugal 361,pumps, 342–343Chezy 321,formula, 302Chlorination system, design 398,of 379Circular 347,channels, 328Circular 294,curves, 275Circular settling tank, design of, 3...

  • Page 409

    390INDEXConversion table, 21,typical, 2Crack 147,control, 128Culverts, highway, designing, 304,285–290Darcy–Weisbach 321,formula, 302Design 137,methods, 118–127 137,beams, 118–123 142,columns, 123–127Direct design method for flat-plateconcrete 163,slabs, 144Distance measurements in...

  • Page 410

    INDEX391Hydraulic and waterworks formulas, 310,291–359capillary 310,action, 291computing rainfall 352,intensity, 333 334,culverts, 315entrance and exit submerged, 334,315on subcritical 335,slopes, 316 367,dams, 348–359 367,arch, 348, 369,350 372,buttress, 353–354 367,diversion, 348 367...

  • Page 411

    392INDEXLangefors’ 220,formula, 201Laterally loaded vertical piles, 124,105–107Load-and-resistance factor design, 226,207for bridge 275,beams, 256for bridge 269,columns, 250for building 230,beams, 211for building 228,columns, 209for building 226,shear, 207Load-bearing 167,walls, 148Lum...

  • Page 412

    INDEX393Pumps and pumping systems (Cont.):efficiency and power 360,input, 341friction 359,head, 340 357,head, 338static discharge 357,head, 338static 357,head, 338static suction 357,head, 338velocity 359,head, 340Rainfall intensity, 352,computing, 333Steel 352,formula, 333Rainwater accumu...

  • Page 413

    394INDEXSoils and earthwork formulas (Cont.):forces on retaining walls, 208,189index 205,parameters, 186internal friction and cohesion, 207,188lateral 208,pressures, 189load-bearing 213,test, 194 214,permeability, 195physical properties of soils, 204,185Rankine 208,theory, 189scraper 216,...

  • Page 414

    INDEX395Venturi meter flow computations, 355,336Vertical control in 201,surveying, 182Vibration, natural circular and periods 93,of, 74 310,Viscosity, 291Wall 174,footings, 155Walls, load 167,bearing, 148Water flow for 354,firefighting, 335Water 331,hammer, 312in 365,penstocks, 346 348,We...