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CIVIL
ENGINEERING
FORMULAS

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CIVIL
ENGINEERING
FORMULAS
Tyler G. Hicks, P.E.
International Engineering Associates
Member: American Society of Mechanical Engineers
United States Naval Institute

McGRAW-HILL
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CONTENTS
9

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Preface xiii
Acknowledgments xv
How to Use This Book xvii

Chapter 1. Conversion Factors for Civil
Engineering Practice

Chapter 2. Beam Formulas

1

15

Continuous Beams / 16
Ultimate Strength of Continuous Beams / 53
Beams of Uniform Strength / 63
Safe Loads for Beams of Various Types / 64
Rolling and Moving Loads / 79
Curved Beams / 82
Elastic Lateral Buckling of Beams / 88
Combined Axial and Bending Loads / 92
Unsymmetrical Bending / 93
Eccentric Loading / 94
Natural Circular Frequencies and Natural Periods
of Vibration of Prismatic Beams / 96

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CONTENTS

Chapter 3. Column Formulas

99

General Considerations / 100
Short Columns / 102
Eccentric Loads on Columns / 102
Column Base Plate Design / 111

American Institute of Steel Construction Allowable-Stress
Design Approach / 113
Composite Columns / 115
Elastic Flexural Buckling of Columns / 118
Allowable Design Loads for Aluminum Columns / 121
Ultimate-Strength Design of Concrete Columns / 124

Chapter 4. Piles and Piling Formulas

W
U
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131

Allowable Loads on Piles / 132
Laterally Loaded Vertical Piles / 133
Toe Capacity Load / 134
Groups of Piles / 136
Foundation-Stability Analysis / 139
Axial-Load Capacity of Single Piles / 143
Shaft Settlement / 144
Shaft Resistance to Cohesionless Soil / 145

Chapter 5. Concrete Formulas

147

Reinforced Concrete / 148
Water/Cementitious Materials Ratio / 148

Job Mix Concrete Volume / 149
Modulus of Elasticity of Concrete / 150
Tensile Strength of Concrete / 151
Reinforcing Steel / 151
Continuous Beams and One-Way Slabs / 151
Design Methods for Beams, Columns, and Other Members / 153
Properties in the Hardened State / 167

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CONTENTS

Compression at Angle to Grain / 220
Recommendations of the Forest Products Laboratory / 221
Compression on Oblique Plane / 223
Adjustments Factors for Design Values / 224
Fasteners for Wood / 233
Adjustment of Design Values for Connections with
Fasteners / 236
Roof Slope to Prevent Ponding / 238
Bending and Axial Tension / 239
Bending and Axial Compression / 240

Chapter 7. Surveying Formulas

C
F
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V

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243

Units of Measurement / 244
Theory of Errors / 245
Measurement of Distance with Tapes / 247
Vertical Control / 253
Stadia Surveying / 253
Photogrammetry / 255

Chapter 8. Soil and Earthwork Formulas

257

Physical Properties of Soils / 258
Index Parameters for Soils / 259
Relationship of Weights and Volumes in Soils / 261
Internal Friction and Cohesion / 263
Vertical Pressures in Soils / 264
Lateral Pressures in Soils, Forces on Retaining Walls / 265
Lateral Pressure of Cohesionless Soils / 266
Lateral Pressure of Cohesive Soils / 267
Water Pressure / 268
Lateral Pressure from Surcharge / 268
Stability of Slopes / 269
Bearing Capacity of Soils / 270
Settlement under Foundations / 271
Soil Compaction Tests / 272

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ix

CONTENTS

Compaction Equipment / 275
Formulas for Earthmoving / 276
Scraper Production / 278
Vibration Control in Blasting / 280

Chapter 9. Building and Structures Formulas

3

7

283

Load-and-Resistance Factor Design for Shear in Buildings / 284
Allowable-Stress Design for Building Columns / 285
Load-and-Resistance Factor Design for Building Columns / 287
Allowable-Stress Design for Building Beams / 287
Load-and-Resistance Factor Design for Building Beams / 290
Allowable-Stress Design for Shear in Buildings / 295
Stresses in Thin Shells / 297
Bearing Plates / 298

Column Base Plates / 300
Bearing on Milled Surfaces / 301
Plate Girders in Buildings / 302
Load Distribution to Bents and Shear Walls / 304
Combined Axial Compression or Tension and Bending / 306
Webs under Concentrated Loads / 308
Design of Stiffeners under Loads / 311
Fasteners for Buildings / 312
Composite Construction / 313
Number of Connectors Required for Building Construction / 316
Ponding Considerations in Buildings / 318

Chapter 10. Bridge and Suspension-Cable
Formulas

321

Shear Strength Design for Bridges / 322
Allowable-Stress Design for Bridge Columns / 323
Load-and-Resistance Factor Design for Bridge Columns / 324
Allowable-Stress Design for Bridge Beams / 325
Stiffeners on Bridge Girders / 327
Hybrid Bridge Girders / 329

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CONTENTS

Load-Factor Design for Bridge Beams / 330
Bearing on Milled Surfaces / 332
Bridge Fasteners / 333
Composite Construction in Highway Bridges / 333
Number of Connectors in Bridges / 337
Allowable-Stress Design for Shear in Bridges / 339
Maximum Width/Thickness Ratios for Compression
Elements for Highway Bridges / 341
Suspension Cables / 341
General Relations for Suspension Cables / 345
Cable Systems / 353

Chapter 11. Highway and Road Formulas

355

Circular Curves / 356
Parabolic Curves / 359
Highway Curves and Driver Safety / 361
Highway Alignments / 362
Structural Numbers for Flexible Pavements / 365
Transition (Spiral) Curves / 370
Designing Highway Culverts / 371
American Iron and Steel Institute (AISI) Design
Procedure / 374

Chapter 12. Hydraulics and Waterworks
Formulas

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381

Capillary Action / 382
Viscosity / 386
Pressure on Submerged Curved Surfaces / 387
Fundamentals of Fluid Flow / 388
Similitude for Physical Models / 392
Fluid Flow in Pipes / 395

Pressure (Head) Changes Caused by Pipe Size Change / 403
Flow through Orifices / 406

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xi

CONTENTS

5

Fluid Jets / 409
Orifice Discharge into Diverging Conical Tubes / 410
Water Hammer / 412
Pipe Stresses Perpendicular to the Longitudinal Axis / 412
Temperature Expansion of Pipe / 414
Forces Due to Pipe Bends / 414
Culverts / 417
Open-Channel Flow / 420
Manning’s Equation for Open Channels / 424
Hydraulic Jump / 425
Nonuniform Flow in Open Channels / 429
Weirs / 436
Flow Over Weirs / 438
Prediction of Sediment-Delivery Rate / 440
Evaporation and Transpiration / 442

Method for Determining Runoff for Minor
Hydraulic Structures / 443
Computing Rainfall Intensity / 443
Groundwater / 446
Water Flow for Firefighting / 446
Flow from Wells / 447
Economical Sizing of Distribution Piping / 448
Venturi Meter Flow Computation / 448
Hydroelectric Power Generation / 449

Index

451

1

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PREFACE

This handy book presents more than 2000 needed formulas
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 formulas are also useful to design drafters,
structural engineers, bridge engineers, foundation builders,
field engineers, professional-engineer license examination
candidates, concrete specialists, timber-structure builders,
and students in a variety of civil engineering pursuits.
The book presents formulas needed in 12 different specialized branches of civil engineering—beams and girders,
columns, piles and piling, concrete structures, timber engineering, surveying, soils and earthwork, building structures, bridges, suspension cables, highways and roads, and
hydraulics and open-channel flow. Key formulas are presented for each of these topics. Each formula is explained
so the engineer, drafter, or designer knows how, where, and
when to use the formula in professional work. Formula
units are given in both the United States Customary System
(USCS) and System International (SI). Hence, the text is
usable throughout the world. To assist the civil engineer

using this material in worldwide engineering practice, a comprehensive tabulation of conversion factors is presented in
Chapter 1.
In assembling this collection of formulas, the author
was guided by experts who recommended the areas of
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PREFACE

greatest need for a handy book of practical and applied civil
engineering formulas.
Sources for the formulas presented here include the various regulatory and industry groups in the field of civil engineering, authors of recognized books on important topics in
the field, drafters, researchers in the field of civil engineering, and a number of design engineers who work daily in

the field of civil engineering. These sources are cited in the
Acknowledgments.
When using any of the formulas in this book that
may come from an industry or regulatory code, the user
is cautioned to consult the latest version of the code.
Formulas may be changed from one edition of a code to
the next. In a work of this magnitude it is difficult to
include the latest formulas from the numerous constantly changing codes. Hence, the formulas given here are
those current at the time of publication of this book.
In a work this large it is possible that errors may occur.
Hence, the author will be grateful to any user of the book
who detects an error and calls it to the author’s attention.
Just write the author in care of the publisher. The error will
be corrected in the next printing.
In addition, if a user believes that one or more important
formulas have been left out, the author will be happy to
consider them for inclusion in the next edition of the book.
Again, just write him in care of the publisher.
Tyler G. Hicks, P.E.

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ACKNOWLEDGMENTS

Many engineers, professional societies, industry associations, and governmental agencies helped the author find and
assemble the thousands of formulas presented in this book.
Hence, the author wishes to acknowledge this help and
assistance.
The author’s principal helper, advisor, and contributor
was the late Frederick S. Merritt, P.E., Consulting Engineer.

For many years Fred and the author were editors on companion magazines at The McGraw-Hill Companies. Fred
was an editor on Engineering-News Record, whereas the
author was an editor on Power magazine. Both lived on
Long Island and traveled on the same railroad to and from
New York City, spending many hours together discussing
engineering, publishing, and book authorship.
When the author was approached by the publisher to prepare this book, he turned to Fred Merritt for advice and help.
Fred delivered, preparing many of the formulas in this book
and giving the author access to many more in Fred’s extensive files and published materials. The author is most grateful to Fred for his extensive help, advice, and guidance.
Further, the author thanks the many engineering societies, industry associations, and governmental agencies whose
work is referred to in this publication. These organizations
provide the framework for safe design of numerous structures of many different types.
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ACKNOWLEDGMENTS

The author also thanks Larry Hager, Senior Editor, Professional Group, The McGraw-Hill Companies, for his
excellent guidance and patience during the long preparation
of the manuscript for this book. Finally, the author thanks
his wife, Mary Shanley Hicks, a publishing professional,
who always most willingly offered help and advice when
needed.
Specific publications consulted during the preparation of
this text include: American Association of State Highway
and Transportation Officials (AASHTO) “Standard Specifications for Highway Bridges”; American Concrete Institute
(ACI) “Building Code Requirements for Reinforced Concrete”; American Institute of Steel Construction (AISC)
“Manual of Steel Construction,” “Code of Standard Practice,” and “Load and Resistance Factor Design Specifications for Structural Steel Buildings”; American Railway
Engineering Association (AREA) “Manual for Railway
Engineering”; American Society of Civil Engineers
(ASCE) “Ground Water Management”; American Water
Works Association (AWWA) “Water Quality and Treatment.” In addition, the author consulted several hundred
civil engineering reference and textbooks dealing with the
topics in the current book. The author is grateful to the
writers of all the publications cited here for the insight they
gave him to civil engineering formulas. A number of these
works are also cited in the text of this book.

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HOW TO USE
THIS BOOK

The formulas presented in this book are intended for use by
civil engineers in every aspect of their professional work—
design, evaluation, construction, repair, etc.
To find a suitable formula for the situation you face,

start by consulting the index. Every effort has been made to
present a comprehensive listing of all formulas in the book.
Once you find the formula you seek, read any accompanying text giving background information about the formula.
Then when you understand the formula and its applications,
insert the numerical values for the variables in the formula.
Solve the formula and use the results for the task at hand.
Where a formula may come from a regulatory code,
or where a code exists for the particular work being
done, be certain to check the latest edition of the applicable code to see that the given formula agrees with the
code formula. If it does not agree, be certain to use the
latest code formula available. Remember, as a design
engineer you are responsible for the structures you plan,
design, and build. Using the latest edition of any governing code is the only sensible way to produce a safe and
dependable design that you will be proud to be associated with. Further, you will sleep more peacefully!
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CHAPTER 1

CONVERSION
FACTORS FOR
CIVIL
ENGINEERING
PRACTICE

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2

CHAPTER ONE

Civil engineers throughout the world accept both the
United States Customary System (USCS) and the System
International (SI) units of measure for both applied and
theoretical calculations. However, the SI units are much
more widely used than those of the USCS. Hence, both the
USCS and the SI units are presented for essentially every
formula in this book. Thus, the user of the book can apply
the formulas with confidence anywhere in the world.
To permit even wider use of this text, this chapter contains the conversion factors needed to switch from one system to the other. For engineers unfamiliar with either
system of units, the author suggests the following steps for
becoming acquainted with the unknown system:
1. Prepare a list of measurements commonly used in your
daily work.
2. Insert, opposite each known unit, the unit from the other
system. Table 1.1 shows such a list of USCS units with
corresponding SI units and symbols prepared by a civil
engineer who normally uses the USCS. The SI units
shown in Table 1.1 were obtained from Table 1.3 by the
engineer.
3. Find, from a table of conversion factors, such as Table 1.3,
the value used to convert from USCS to SI units. Insert
each appropriate value in Table 1.2 from Table 1.3.
4. Apply the conversion values wherever necessary for the
formulas in this book.
5. Recognize—here and now—that the most difficult

aspect of becoming familiar with a new system of measurement is becoming comfortable with the names and
magnitudes of the units. Numerical conversion is simple,
once you have set up your own conversion table.

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CONVERSION FACTORS

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TABLE 1.1 Commonly Used USCS and SI Units†
Conversion factor
(multiply USCS unit
by this factor to
SI symbol
obtain SI unit)

USCS unit

SI unit

square foot
cubic foot
pound per
square inch
pound force
foot pound
torque
kip foot
gallon per
minute
kip per square
inch

square meter
cubic meter

m2
m3

0.0929
0.2831

kilopascal
newton

kPa
Nu

6.894
4.448

newton meter
kilonewton meter

Nиm
kNиm

1.356
1.355

liter per second

L/s

0.06309

megapascal

MPa

6.89

†

This table is abbreviated. For a typical engineering practice, an actual table
would be many times this length.

Be careful, when using formulas containing a numerical
constant, to convert the constant to that for the system you
are using. You can, however, use the formula for the USCS
units (when the formula is given in those units) and then
convert the final result to the SI equivalent using Table 1.3.
For the few formulas given in SI units, the reverse procedure should be used.

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4

CHAPTER ONE

TABLE 1.2 Typical Conversion Table†
To convert from

To

square foot
foot per second
squared
cubic foot
pound per cubic
inch
gallon per minute
pound per square
inch
pound force
kip per square foot
acre foot per day

square meter
meter per second

squared
cubic meter
kilogram per cubic
meter
liter per second

acre
cubic foot per
second

kilopascal
newton
pascal
cubic meter per
second
square meter
cubic meter per
second

T
Multiply by‡
9.290304

E Ϫ 02

3.048
2.831685

E Ϫ 01
E Ϫ 02

2.767990
6.309

E ϩ 04
E Ϫ 02

6.894757
4.448222
4.788026

a
a
a
a
a

b
b
E ϩ 04
E Ϫ 02

1.427641
4.046873

E ϩ 03

2.831685

E Ϫ 02

b
B
B

†

This table contains only selected values. See the U.S. Department of the
Interior Metric Manual, or National Bureau of Standards, The International
System of Units (SI), both available from the U.S. Government Printing
Office (GPO), for far more comprehensive listings of conversion factors.
‡
The E indicates an exponent, as in scientific notation, followed by a positive
or negative number, representing the power of 10 by which the given conversion factor is to be multiplied before use. Thus, for the square foot conversion factor, 9.290304 ϫ 1/100 ϭ 0.09290304, the factor to be used to
convert square feet to square meters. For a positive exponent, as in converting acres to square meters, multiply by 4.046873 ϫ 1000 ϭ 4046.8.
Where a conversion factor cannot be found, simply use the dimensional
substitution. Thus, to convert pounds per cubic inch to kilograms per cubic
meter, find 1 lb ϭ 0.4535924 kg and 1 in3 ϭ 0.00001638706 m3. Then,
1 lb/in3 ϭ 0.4535924 kg/0.00001638706 m3 ϭ 27,680.01, or 2.768 E ϩ 4.

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5

CONVERSION FACTORS

TABLE 1.3 Factors for Conversion to SI Units of Measurement

2
1
2
4
2

4
2
3
2

e

To convert from

To

acre foot, acre ft
acre
angstrom, Å
atmosphere, atm
(standard)
atmosphere, atm
(technical
ϭ 1 kgf/cm2)
bar
barrel (for petroleum,
42 gal)
board foot, board ft
British thermal unit,
Btu, (mean)
British thermal unit,
Btu (International
Table)иin/(h)(ft2)
(°F) (k, thermal
conductivity)
British thermal unit,
Btu (International
Table)/h
British thermal unit,
Btu (International
Table)/(h)(ft2)(°F)
(C, thermal

conductance)
British thermal unit,
Btu (International
Table)/lb

cubic meter, m3
square meter, m2
meter, m
pascal, Pa

1.233489
4.046873
1.000000*
1.013250*

Multiply by

pascal, Pa

9.806650* E ϩ 04

pascal, Pa
cubic meter, m2

1.000000* E ϩ 05
1.589873 E Ϫ 01

cubic meter, m3
joule, J

2.359737
1.05587

E Ϫ 03
E ϩ 03

watt per meter
kelvin, W/(mиK)

1.442279

E Ϫ 01

watt, W

2.930711

E Ϫ 01

watt per square
meter kelvin,
W/(m2иK)

5.678263

E ϩ 00

joule per kilogram,
J/kg

2.326000* E ϩ 03

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E ϩ 03
E ϩ 03
E Ϫ 10
E ϩ 05

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6

CHAPTER ONE

TABLE 1.3 Factors for Conversion to SI Units of Measurement
(Continued)
To convert from
British thermal unit,

Btu (International
Table)/(lb)(°F)
(c, heat capacity)
British thermal unit,
cubic foot, Btu
(International
Table)/ft3
bushel (U.S.)
calorie (mean)
candela per square
inch, cd/in2
centimeter, cm, of
mercury (0°C)
centimeter, cm, of
water (4°C)
chain
circular mil
day
day (sidereal)
degree (angle)
degree Celsius
degree Fahrenheit
degree Fahrenheit
degree Rankine
(°F)(h)(ft2)/Btu
(International
Table) (R, thermal
resistance)

To

T
(

Multiply by

joule per kilogram
kelvin, J/(kgиK)

4.186800* E ϩ 03

(

joule per cubic
meter, J/m3

3.725895

E ϩ 04

cubic meter, m3
joule, J
candela per square
meter, cd/m2
pascal, Pa

3.523907
4.19002
1.550003

E Ϫ 02
E ϩ 00
E ϩ 03

d
f
f
f
f

1.33322

E ϩ 03

pascal, Pa

9.80638

E ϩ 01

meter, m
square meter, m2

2.011684 E ϩ 01
5.067075 E Ϫ 10
8.640000* E ϩ 04
8.616409 E ϩ 04
1.745329 E Ϫ 02
TK ϭ tC ϩ 273.15
tC ϭ (tF Ϫ 32)/1.8

TK ϭ (tF ϩ 459.67)/1.8
TK ϭ TR /1.8
1.761102 E Ϫ 01

second, s
second, s
radian, rad
kelvin, K
degree Celsius, °C
kelvin, K
kelvin, K
kelvin square
meter per watt,
Kиm2/W

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s
s

s
c
c
c
f

f
f

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7

CONVERSION FACTORS

TABLE 1.3 Factors for Conversion to SI Units of Measurement
(Continued)
To convert from
2

8

To

Multiply by

kelvin meter per
watt, Kиm/W

fathom
foot, ft
foot, ft (U.S. survey)
foot, ft, of water
(39.2°F) (pressure)
square foot, ft2
square foot per hour,
ft2/h (thermal
diffusivity)
square foot per
second, ft2/s
cubic foot, ft3 (volume
or section modulus)
cubic foot per minute,
ft3/min
cubic foot per second,
ft3/s
foot to the fourth
power, ft4 (area
moment of inertia)
foot per minute,
ft/min
foot per second,
ft/s

meter, m
meter, m
meter, m
pascal, Pa

1.828804
3.048000†
3.048006
2.98898

square meter, m2
square meter per
second, m2/s

9.290304† E Ϫ 02
2.580640† E Ϫ 05

square meter per
second, m2/s
cubic meter, m3

9.290304† E Ϫ 02
2.831685

E Ϫ 02

cubic meter per
second, m3/s
cubic meter per
second, m3/s
meter to the fourth
power, m4

4.719474

E Ϫ 04

2.831685

E Ϫ 02

8.630975

E Ϫ 03

meter per second,
m/s
meter per second,
m/s

5.080000† E Ϫ 03

newton, N

6.933471

E ϩ 00

(°F)(h)(ft )/(Btu
(International
Table)иin) (thermal
resistivity)
dyne, dyn

1.000000† E Ϫ 05

E ϩ 00
E Ϫ 01
E Ϫ 01
E ϩ 03

3.048000† E Ϫ 01

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