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FE Reference Handbook 10.0.1
Copyright ©2020 by NCEES®. All rights reserved.
All NCEES material is copyrighted under the laws of the United States. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted in any form or by any means without the prior written permission of NCEES. Requests for permissions
should be addressed in writing to permissions@ncees.org.
ISBN 978-1-947801-11-0
Printed in the United States of America
Second printing May 2020
Edition 10.0.1
Preface
About the Handbook
The Fundamentals of Engineering (FE) exam is computer-based, and the FE Reference Handbook is the only resource
material you may use during the exam. Reviewing it before exam day will help you become familiar with the charts,
formulas, tables, and other reference information provided. You won't be allowed to bring your personal copy of the
Handbook into the exam room. Instead, the computer-based exam will include a PDF version of the Handbook for your use.
No printed copies of the Handbook will be allowed in the exam room.
The PDF version of the FE Reference Handbook that you use on exam day will be very similar to the printed version. Pages
not needed to solve exam questions—such as the cover, introductory material, index, and exam specifications—will not be
included in the PDF version. In addition, NCEES will periodically revise and update the Handbook, and each FE exam will
be administered using the updated version.
The FE Reference Handbook does not contain all the information required to answer every question on the exam. Basic
theories, conversions, formulas, and definitions examinees are expected to know have not been included. Special material
required for the solution of a particular exam question will be included in the question itself.
Updates on exam content and procedures
NCEES.org is our home on the web. Visit us there for updates on everything exam-related, including specifications, exam-
day policies, scoring, and practice tests. A PDF version of the FE Reference Handbook similar to the one you will use on
exam day is also available there.
Errata
To report errata in this book, send your correction using our chat feature or your account on NCEES.org. We will also post
errata on the website. Examinees are not penalized for any errors in the Handbook that affect an exam question.
iii
CONTENTS
Units and Conversion Factors ..........................................................................................................1
Ethics and Professional Practice ......................................................................................................4
Safety .............................................................................................................................................13
Mathematics ...................................................................................................................................34
Engineering Probability and Statistics ...........................................................................................63
Chemistry and Biology ..................................................................................................................85
Materials Science/Structure of Matter ...........................................................................................94
Statics ...........................................................................................................................................107
Dynamics .....................................................................................................................................114
Mechanics of Materials ................................................................................................................130
Thermodynamics ..........................................................................................................................143
Fluid Mechanics ...........................................................................................................................177
Heat Transfer ................................................................................................................................204
Instrumentation, Measurement, and Control ...............................................................................220
Engineering Economics ...............................................................................................................230
Chemical Engineering ..................................................................................................................238
Civil Engineering .........................................................................................................................259
Environmental Engineering .........................................................................................................310
Electrical and Computer Engineering ..........................................................................................355
Industrial and Systems Engineering .............................................................................................417
Mechanical Engineering ..............................................................................................................431
Index ............................................................................................................................................461
FE Exam Specifications ...............................................................................................................471
v
Units and Conversion Factors
Distinguishing pound-force from pound-mass
The FE exam and this handbook use both the metric system of units and the U.S. Customary System (USCS). In the USCS sys-
tem of units, both force and mass are called pounds. Therefore, one must distinguish the pound-force (lbf) from the pound-mass
(lbm).
The pound-force is that force which accelerates one pound-mass at 32.174 ft/sec2. Thus, 1 lbf = 32.174 lbm-ft/sec2. The expression
32.174 lbm-ft/(lbf-sec2) is designated as g and is used to resolve expressions involving both mass and force expressed as pounds.
c
For instance, in writing Newton's second law, the equation would be written as F = ma/g , where F is in lbf, m in lbm, and a is in ft/
c
sec2.
Similar expressions exist for other quantities. Kinetic Energy, KE = mv2/2g , with KE in (ft-lbf); Potential Energy,
c
PE = mgh/g , with PE in (ft-lbf); Fluid Pressure, p = ρgh/g , with p in (lbf/ft2); Specific Weight, SW = ρg/g , in (lbf/ft3); Shear
c c c
Stress, τ = (µ/g )(dv/dy), with shear stress in (lbf/ft2). In all these examples, g should be regarded as a force unit conversion
c c
factor. It is frequently not written explicitly in engineering equations. However, its use is required to produce a consistent set of
units.
Note that the force unit conversion factor g [lbm-ft/(lbf-sec2)] should not be confused with the local acceleration of gravity g,
c
which has different units (m/s2 or ft/sec2) and may be either its standard value (9.807 m/s2 or 32.174 ft/sec2) or some other local
value.
If the problem is presented in USCS units, it may be necessary to use the constant g in the equation to have a consistent set
c
of units.
Constants and conversion factors provided are approximate, with sufficient accuracy to solve exam questions.
METRIC PREFIXES
COMMONLY USED EQUIVALENTS
Multiple Prefix Symbol
10–18 atto a
10–15 femto f 1 gallon of water weighs 8.34 lbf
10–12 pico p 1 cubic foot of water weighs 62.4 lbf
10–9 nano n 1 cubic inch of mercury weighs 0.491 lbf
10–6 micro µ
The mass of 1 cubic meter of water is 1,000 kilograms
10–3 milli m
1 mg/L is 8.34 × 10–6 lbf/gal
10–2 centi c
10–1 deci d
101 deka da
TEMPERATURE CONVERSIONS
102 hecto h
103 kilo k
106 mega M ºF = 1.8 (ºC) + 32
109 giga G ºC = (ºF – 32)/1.8
1012 tera T ºR = ºF + 459.69
1015 peta P K = ºC + 273.15
1018 exa E
1
Units and Conversion Factors
Significant Figures
Significant figures of numbers in math operations will determine the accuracy of the result. General rules for significant digits
are:
Rule 1: Non-zero digits are always significant.
Rule 2: Any zeros between two significant digits are significant.
Rule 3: All zeros in the decimal portion are significant.
Rule 4 (Addition and Subtraction): The number used in the calculation with the least number of significant digits after the deci-
mal point dictates the number of significant figures after the decimal point. The number with the most significant figures to the
left of the decimal point dictates the number of significant digits to the left of decimal point.
Rule 5 (Multiplication and Division): The result of the operation has the same number of significant digits as the input
number with the least number of significant digits.
Rule 6: In the solution of engineering problems, it is customary to retain 3–4 significant digits in the final result.
Ideal Gas Constants
The universal gas constant, designated as R in the table below, relates pressure, volume, temperature, and number of moles of an ideal
gas. When that universal constant, R, is divided by the molecular weight of the gas, the result, often designated as R, has units of energy
per degree per unit mass [kJ/(kg·K) or ft-lbf/(lbm-ºR)] and becomes characteristic of the particular gas. Some disciplines, notably chemi-
cal engineering, often use the symbol R to refer to the universal gas constant R.
Fundamental Constants
Quantity Symbol Value Units
electron charge e 1.6022 × 10−19 C (coulombs)
Faraday constant F 96,485 coulombs/(mol)
gas constant metric R 8,314 J/(kmol·K)
gas constant metric R 8.314 kPa·m3/(kmol·K)
gas constant USCS R 1,545 ft-lbf/(lb mole-ºR)
R 0.08206 L·atm/(mole·K)
gravitation−Newtonian constant G 6.673 × 10–11 m3/(kg·s2)
gravitation−Newtonian constant G 6.673 × 10–11 N·m2/kg2
gravity acceleration (standard) metric g 9.807 m/s2
gravity acceleration (standard) USCS g 32.174 ft/sec2
molar volume (ideal gas), T = 273.15 K, p = 101.3 kPa V 22,414 L/kmol
m
speed of light (exact) c 299,792,458 m/s
Stefan-Boltzmann constant σ 5.67 × 10–8 W/(m2·K4)
2
Units and Conversion Factors
Multiply By To Obtain Multiply By To Obtain
acre 43,560 square feet (ft2) joule (J) 9.478 × 10–4 Btu
ampere-hr (A-hr) 3,600 coulomb (C) J 0.7376 ft-lbf
ångström (Å) 1 × 10–10 meter (m) J 1 newton•m (N•m)
atmosphere (atm) 76.0 cm, mercury (Hg) J/s 1 watt (W)
atm, std 29.92 in., mercury (Hg)
atm, std 14.70 lbf/in2 abs (psia) kilogram (kg) 2.205 pound-mass (lbm)
atm, std 33.90 ft, water kgf 9.8066 newton (N)
atm, std 1.013 × 105 pascal (Pa) kilometer (km) 3,281 feet (ft)
km/hr 0.621 mph
bar 1 × 105 Pa kilopascal (kPa) 0.145 lbf/in2 (psi)
bar 0.987 atm kilowatt (kW) 1.341 horsepower (hp)
barrels–oil 42 gallons–oil kW 3,413 Btu/hr
Btu 1,055 joule (J) kW 737.6 (ft-lbf )/sec
Btu 2.928 × 10–4 kilowatt-hr (kWh) kW-hour (kWh) 3,413 Btu
Btu 778 ft-lbf kWh 1.341 hp-hr
Btu/hr 3.930 × 10–4 horsepower (hp) kWh 3.6 × 106 joule (J)
Btu/hr 0.293 watt (W) kip (K) 1,000 lbf
Btu/hr 0.216 ft-lbf/sec K 4,448 newton (N)
calorie (g-cal) 3.968 × 10–3 Btu liter (L) 61.02 in3
cal 1.560 × 10–6 hp-hr L 0.264 gal (U.S. Liq)
cal 4.184 joule (J) L 10–3 m3
cal/sec 4.184 watt (W) L/second (L/s) 2.119 ft3/min (cfm)
centimeter (cm) 3.281 × 10–2 foot (ft) L/s 15.85 gal (U.S.)/min (gpm)
cm 0.394 inch (in)
centipoise (cP) 0.001 pascal•sec (Pa•s) meter (m) 3.281 feet (ft)
centipoise (cP) 1 g/(m•s) m 1.094 yard
centipoise (cP) 2.419 lbm/hr-ft m/second (m/s) 196.8 feet/min (ft/min)
centistoke (cSt) 1 × 10–6 m2/sec (m2/s) mile (statute) 5,280 feet (ft)
cubic feet/second (cfs) 0.646317 million gallons/day (MGD) mile (statute) 1.609 kilometer (km)
cubic foot (ft3) 7.481 gallon mile/hour (mph) 88.0 ft/min (fpm)
cubic meters (m3) 1,000 liters mph 1.609 km/h
mm of Hg 1.316 × 10–3 atm
electronvolt (eV) 1.602 × 10–19 joule (J) mm of HO 9.678 × 10–5 atm
2
foot (ft) 30.48 cm newton (N) 0.225 lbf
ft 0.3048 meter (m) newton (N) 1 kg•m/s2
ft of HO 0.4332 psi N•m 0.7376 ft-lbf
2
ft-pound (ft-lbf) 1.285 × 10–3 Btu N•m 1 joule (J)
ft-lbf 3.766 × 10–7 kilowatt-hr (kWh)
ft-lbf 0.324 calorie (g-cal) pascal (Pa) 9.869 × 10–6 atmosphere (atm)
ft-lbf 1.356 joule (J) Pa 1 newton/m2 (N/m2)
ft-lbf/sec 1.818 × 10–3 horsepower (hp) Pa•sec (Pa•s) 10 poise (P)
pound (lbm, avdp) 0.454 kilogram (kg)
gallon (U.S. Liq) 3.785 liter (L) lbf 4.448 N
gallon (U.S. Liq) 0.134 ft3 lbf-ft 1.356 N•m
gallons of water 8.3453 pounds of water lbf/in2 (psi) 0.068 atm
gamma (γ, Γ) 1 × 10–9 tesla (T) psi 2.307 ft of HO
2
gauss 1 × 10–4 T psi 2.036 in. of Hg
gram (g) 2.205 × 10–3 pound (lbm) psi 6,895 Pa
hectare 1 × 104 square meters (m2) radian (rad) 180/π degree
hectare 2.47104 acres
horsepower (hp) 42.4 Btu/min slug 32.174 pound-mass (lbm)
hp 745.7 watt (W) stokes 1 × 10–4 m2/s
hp 33,000 (ft-lbf)/min
hp 550 (ft-lbf)/sec tesla 1,000 weber/m2
hp-hr 2,545 Btu therm 1 × 105 Btu
hp-hr 1.98 × 106 ft-lbf ton (metric) 1,000 kilogram (kg)
hp-hr 2.68 × 106 joule (J) ton (short) 2,000 pound-force (lbf)
hp-hr 0.746 kWh
watt (W) 3.413 Btu/hr
inch (in.) 2.540 centimeter (cm) W 1.341 × 10–3 horsepower (hp)
in. of Hg 0.0334 atm W 1 joule/s (J/s)
in. of Hg 13.60 in. of HO weber/m2 (Wb/m2) 10,000 gauss
2
in. of HO 0.0361 lbf/in2 (psi)
2
in. of HO 0.002458 atm
2
3
Ethics and Professional Practice
Code of Ethics
Engineering is considered to be a "profession" rather than an "occupation" because of several important characteristics
shared with other recognized learned professions, law, medicine, and theology: special knowledge, special privileges, and
special responsibilities. Professions are based on a large knowledge base requiring extensive training. Professional skills
are important to the well-being of society. Professions are self-regulating, in that they control the training and evaluation
processes that admit new persons to the field. Professionals have autonomy in the workplace; they are expected to utilize
their independent judgment in carrying out their professional responsibilities. Finally, professions are regulated by ethical
standards. (Harris, C.E., M.S. Pritchard, & M.J. Rabins, Engineering Ethics: Concepts and Cases, Wadsworth Publishing
company, pages 27–28, 1995.)
The expertise possessed by engineers is vitally important to societal welfare. In order to serve society effectively, engineers
must maintain a high level of technical competence. However, a high level of technical expertise without adherence to
ethical guidelines is as much a threat to public welfare as is professional incompetence. Therefore, engineers must also be
guided by ethical principles.
The ethical principles governing the engineering profession are embodied in codes of ethics. Such codes have been adopted
by state boards of registration, professional engineering societies, and even by some private industries. An example of one
such code is the NCEES Rules of Professional Conduct, found in Section 240 of the Model Rules and presented here. As
part of his/her responsibility to the public, an engineer is responsible for knowing and abiding by the code. Additional rules
of conduct are also included in the Model Rules.
The three major sections of the Model Rules address (1) Licensee's Obligation to the Public, (2) Licensee's Obligation to
Employers and Clients, and (3) Licensee's Obligation to Other Licensees. The principles amplified in these sections are
important guides to appropriate behavior of professional engineers.
Application of the code in many situations is not controversial. However, there may be situations in which applying
the code may raise more difficult issues. In particular, there may be circumstances in which terminology in the code is
not clearly defined, or in which two sections of the code may be in conflict. For example, what constitutes "valuable
consideration" or "adequate" knowledge may be interpreted differently by qualified professionals. These types of questions
are called conceptual issues, in which definitions of terms may be in dispute. In other situations, factual issues may also
affect ethical dilemmas. Many decisions regarding engineering design may be based upon interpretation of disputed or
incomplete information. In addition, tradeoffs revolving around competing issues of risk vs. benefit, or safety vs. economics
may require judgments that are not fully addressed simply by application of the code.
No code can give immediate and mechanical answers to all ethical and professional problems that an engineer may face.
Creative problem solving is often called for in ethics, just as it is in other areas of engineering.
Model Rules, Section 240.15 Rules of Professional Conduct
To safeguard the health, safety, and welfare of the public and to maintain integrity and high standards of skill and practice in
the engineering and surveying professions, the rules of professional conduct provided in this section shall be binding upon
every licensee and on all firms authorized to offer or perform engineering or surveying services in this jurisdiction.
A. Licensee’s Obligation to the Public
1. Licensees shall be cognizant that their first and foremost responsibility is to safeguard the health, safety, and
welfare of the public when performing services for clients and employers.
2. Licensees shall sign and seal only those plans, surveys, and other documents that conform to accepted engineering
and surveying standards and that safeguard the health, safety, and welfare of the public.
3. Licensees shall notify their employer or client and such other authority as may be appropriate when their
professional judgment is overruled when the health, safety, or welfare of the public is endangered.
4. Licensees shall, to the best of their knowledge, include all relevant and pertinent information in an objective and
truthful manner within all professional documents, statements, and testimony.
5. Licensees shall express a professional opinion publicly only when it is founded upon an adequate knowledge of the
facts and a competent evaluation of the subject matter.
4
Ethics and Professional Practice
6. Licensees shall issue no statements, criticisms, or arguments on engineering and surveying matters that are inspired
or paid for by interested parties, unless they explicitly identify the interested parties on whose behalf they are
speaking and reveal any interest they have in the matters.
7. Licensees shall not partner, practice, or offer to practice with any person or firm that they know is engaged in
fraudulent or dishonest business or professional practices.
8. Licensees who have knowledge or reason to believe that any person or firm has violated any rules or laws applying
to the practice of engineering or surveying shall report it to the board, may report it to appropriate legal authorities,
and shall cooperate with the board and those authorities as requested.
9. Licensees shall not knowingly provide false or incomplete information regarding an applicant in obtaining
licensure.
10. Licensees shall comply with the licensing laws and rules governing their professional practice in each of the
jurisdictions in which they practice.
B. Licensee’s Obligation to Employer and Clients
1. Licensees shall undertake assignments only when qualified by education or experience in the specific technical
fields of engineering or surveying involved.
2. Licensees shall not affix their signatures or seals to any plans or documents dealing with subject matter in which
they lack competence, nor to any such plan or document not prepared under their responsible charge.
3. Licensees may accept assignments and assume responsibility for coordination of an entire project if each technical
segment is signed and sealed by the licensee responsible for preparation of that technical segment.
4. Licensees shall not reveal facts, data, or information obtained in a professional capacity without the prior consent
of the client, employer, or public body on which they serve except as authorized or required by law or rules.
5. Licensees shall not solicit or accept gratuities, directly or indirectly, from contractors, their agents, or other parties
in connection with work for employers or clients.
6. Licensees shall disclose to their employers or clients all known or potential conflicts of interest or other
circumstances that could influence or appear to influence their judgment or the quality of their professional service
or engagement.
7. Licensees shall not accept compensation, financial or otherwise, from more than one party for services pertaining
to the same project, unless the circumstances are fully disclosed and agreed to in writing by all interested parties.
8. Licensees shall not solicit or accept a professional contract from a governmental body on which a principal or
officer of their organization serves as a member. Conversely, licensees serving as members, advisors, or employees
of a government body or department, who are the principals or employees of a private concern, shall not participate
in decisions with respect to professional services offered or provided by said concern to the governmental body that
they serve.
9. Licensees shall not use confidential information received in the course of their assignments as a means of making
personal profit without the consent of the party from whom the information was obtained.
C. Licensee’s Obligation to Other Licensees
1. Licensees shall not falsify or permit misrepresentation of their, or their associates’, academic or professional
qualifications. They shall not misrepresent or exaggerate their degree of responsibility in prior assignments nor the
complexity of said assignments. Presentations incidental to the solicitation of employment or business shall not
misrepresent pertinent facts concerning employers, employees, associates, joint ventures, or past accomplishments.
2. Licensees shall not offer, give, solicit, or receive, either directly or indirectly, any commission, or gift, or other
valuable consideration in order to secure work, and shall not make any political contribution with the intent to
influence the award of a contract by public authority.
3. Licensees shall not injure or attempt to injure, maliciously or falsely, directly or indirectly, the professional
reputation, prospects, practice, or employment of other licensees, nor indiscriminately criticize other licensees’
work.
4. Licensees shall make a reasonable effort to inform another licensee whose work is believed to contain a material
discrepancy, error, or omission that may impact the health, safety, or welfare of the public, unless such reporting is
legally prohibited.
5
Ethics and Professional Practice
Model Law, Section 110.20 Definitions
A. Engineer
1. Engineer—The term "Engineer," within the intent of this Act, shall mean an individual who is qualified to practice
engineering by reason of engineering education, training, and experience in the application of engineering
principles and the interpretation of engineering data.
2. Professional Engineer—The term "Professional Engineer," as used in this Act, shall mean an individual who has
been duly licensed as a professional engineer by the board. The board may designate a professional engineer, on the
basis of education, experience, and examination, as being licensed in a specific discipline or branch of engineering
signifying the area in which the engineer has demonstrated competence.
3. Professional Engineer, Retired—The term "Professional Engineer, Retired," as used in this Act, shall mean an
individual who has been duly licensed as a professional engineer by the board and who chooses to relinquish or not
to renew a license and who applies to and is approved by the board to be granted the use of the title "Professional
Engineer, Retired."
4. Engineer Intern—The term "Engineer Intern," as used in this Act, shall mean an individual who has been duly
certified as an engineer intern by the board.
5. Practice of Engineering—The term "Practice of Engineering," as used in this Act, shall mean any service or
creative work requiring engineering education, training, and experience in the application of engineering principles
and the interpretation of engineering data to engineering activities that potentially impact the health, safety, and
welfare of the public.
The services may include, but not be limited to, providing planning, studies, designs, design coordination,
drawings, specifications, and other technical submissions; teaching engineering design courses; performing
surveying that is incidental to the practice of engineering; and reviewing construction or other design products for
the purposes of monitoring compliance with drawings and specifications related to engineered works.
Surveying incidental to the practice of engineering excludes the surveying of real property for the establishment
of land boundaries, rights of way, easements, and the dependent or independent surveys or resurveys of the public
land survey system.
An individual shall be construed to practice engineering, within the meaning and intent of this Act, if he or she
does any of the following:
a. Practices any discipline of the profession of engineering or holds himself or herself out as able and entitled to
practice any discipline of engineering
b. Represents himself or herself to be a professional engineer by verbal claim, sign, advertisement, letterhead, or
card or in any other way
c. Through the use of some other title, implies that he or she is a professional engineer under this Act
6. Inactive Status—Licensees who are not engaged in engineering practice that requires licensure in this jurisdiction
may be granted inactive status. No licensee granted inactive status may practice or offer to practice engineering in
this jurisdiction unless otherwise exempted in this Act.
B. Professional Surveyor (Professional Land Surveyor, Professional Surveyor and Mapper, Geomatics Professional, or
equivalent term); See Model Law.
C. Board—The term "Board," as used in this Act, shall mean the jurisdiction board of licensure for professional engineers
and professional surveyors, hereinafter provided by this Act.
D. Jurisdiction—The term "Jurisdiction," as used in this Act, shall mean a state, the District of Columbia, or any territory,
commonwealth, or possession of the United States that issues licenses to practice and regulates the practice of
engineering and/or surveying within its legal boundaries.
E. Responsible Charge—The term "Responsible Charge," as used in this Act, shall mean direct control and personal
supervision of engineering or surveying work, as the case may be.
F. Rules of Professional Conduct—The term "Rules of Professional Conduct," as used in this Act, shall mean those rules
of professional conduct, if any, promulgated by the board as authorized by this Act.
G. Firm—The term "Firm," as used in this Act, shall mean any form of business or entity other than an individual
operating as a sole proprietorship under his or her own name.
H. Managing Agent—The term "Managing Agent," as used in this Act, shall mean an individual who is licensed under this
Act and who has been designated pursuant to Section 160.20 of this Act by the firm.
I. Rules—The term "Rules," as used in this Act, shall mean those rules and regulations adopted pursuant to Section
120.60 A, Board Powers, of this Act.
6
