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ES 230 Strengths Fall 2015

ES 230 – STRENGTH OF MATERIALSThis course will provide students with the opportunity to learn how to apply the fundamentals of strength of materials as related to concepts of stress, strain, bending, torsion, stability, and deflection of deformable bodies to applications concerning bars, shafts, beams, columns, and pressure vessels. Course material focuses on the fundamentals of applying equilibrium, deformation compatibility, and force-deformation relationships in solving problems. Students are required to demonstrate proficiency in calculating stresses and strains in bars and beams subjected to axial and shear forces and torsional/bending moments, and also combinations of these loadings; analyzing external and internal forces and moments; calculating deflections of beams; calculating angles of twist of shafts; determining support reactions for indeterminate bars, shafts, and beams; and performing engineering design of bars, shafts, and beams.Prerequisite: Equilibrium forces and moments and analysis of trusses, beams, and frames. Met by passing Statics.

Professor: Anne Raich, Ph.D. 322 AEC [email protected] x5590 http://sites.lafayette.edu/raicha

Lecture: Tues. 11:00-12:15 & Thurs. 11:00-12:15 in 315 AECRecitation: Tues. 7:00–7:50 pm or 8:00–8:50 pm in 223B AECOffice Hours: Mon. 1:15 – 2:30 pm; Tues. 1:15 – 2:30 pm; Tues. 4:00 – 5:30 pm; & Wed. 11:00 am – 1:00 pm

Other times, if the door to my office is open, feel free to stop by and ask questions

Textbook: Hibbeler, R.C., Mechanics of Materials. 9th ed. or equivalent

Exam I: Thursday, Oct. 8th, 2015 (Evening Exam – 7:00 to 9:00 pm) Exam II: Thursday, Nov. 12th, 2015 (Evening Exam – 7:00 to 9:00 pm) Final Exam: T.B.A.Notes: For all exams you will be able to bring one page of handwritten notes (one-side only). The use of other notes or books during the exam will not be permitted. The final exam is comprehensive.

Grade Distribution: Exam I: 20% Homework: 15% Exam II: 20% Combined Stress Project: 5% Final Exam: 25% Recitation Work: 15%

Final Grading Scale: A 92; 92 > A- 90; 90 > B+ 87; 87 > B 82; 82 > B- 80; 80 > C 70; 70 > D 60; F < 60

Expected Workload:This course requires what is called “old-fashioned gumption”, as will many other engineering courses you encounter in your studies. There is a lot of homework assigned in the course. The main purpose of the homework is to help you learn the fundamentals, and there is typically a strong correlation between effort on the homework and exam scores. You should expect to spend 8-12 hours each week solving homework problems, including reworking problems that you have trouble with (recommended because often these types of problems appear on the exams). Engineers are known for their intellectual curiosity and their willingness to work and these skills allow engineers to rise to leadership positions.

Learning Objectives:Specific learning objectives are defined for each class. Learning objectives are tailored to let the student know what skills and knowledge they are expected to learn. Therefore, the learning objectives directly define the skills and knowledge on which the student will be tested in the exams. Full understanding of a learning objective is accomplished through in-class and out-of-class work (i.e. through notes, discussion, demonstrations, reading assignments, recitation problems and homework).

Attendance and Participation:Regular and on-time class attendance is required, although no grade is assigned for attendance. Class participation is expected and provides an opportunity to ask your questions, which serve the class at large in learning the course material more thoroughly. You are responsible for all material covered in class, even if absent.

Academic Integrity Statement: “Students are expected to be honorable, ethical, and mature in every regard”Just as ethical conduct is an essential part of the engineering profession, academic integrity is essential to ensure a fair and positive learning environment. No form of scholastic misconduct will be tolerated. Academic dishonesty includes cheating, fabrication, falsification, plagiarism, copying homework from other students (even if you have received their permission) or from a solution, etc. It is the student’s responsibility to comply with the Lafayette College Student Handbook (http://studentlife.lafayette.edu/resources/) and to be familiar with the Principles of Intellectual Honesty (http://fye.lafayette.edu/academics/). Violations will be handled in accordance with the Procedural Standards in Disciplinary Proceedings outlined in the Student Handbook.

Success is achieved by developing our strengths, not by eliminating our weaknesses

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“I give you two examinations, one in trigonometry and one in honesty. I hope you pass them both, but if you must fail one, let it be trigonometry for there are many good people in this world today who cannot pass an examination in trigonometry, but there are no good people in the world who cannot pass an examination in honesty” -Vanderbilt University past-Chancellor Madison SarrattClass Date Tentative Topic (Subject to Change) Reading for Class (Fund.Prob)

1 T 9/1 Introduction/Internal Forces by Statics/Normal Stress 1.1, 1.2, 1.3, 1.4 ( F1.1-F1.6)R1 T 9/1 Review of Statics 1.1 – 1.42 R 9/3 Shear Stress/Bolted Connections/Bearing Stress 1.5 (F1.7-F1.12)3 T 9/8 Factor of Safety/Oblique Planes/General Stress State 1.6, 1.7 (F1.13-F1.24)

R2 T 9/8 Design & Optimization4 R 9/10 Strain & Deformation 2.1, 2.2 (F2.1-F2.5)5 T 9/15 Material Properties 3.1-3.4&3.6-3.8 (F3.1-F3.16)

R3 T 9/15 Visualization and Stress/Strain6 R 9/17 Axial I - Member Analysis 4.1, 4.2, 4.3 (F4.1-F4.6)7 T 9/22 Axial II - Indeterminate Members 4.4

R4 T 9/22 Axial Deformations and Superposition8 R 9/24 Axial III – Thermal Stresses and Stress Concentrations 4.6, 4.7 9 T 9/29 Torsion I - Shear Stress & Angle of Twist 5.1, 5.2, 5.3, 5.4 (F5.1-F5.14)

R5 T 9/29 Torsion of Non-Circular Cross-Sections & Stress Conc. 4.7, 5.610 R 10/1 Torsion II - Indeterminate Members 5.4, 5.5, 5.6 11 T 10/6 Flexure I – Graphical V and M Diagrams/M of I 6.1, 6.2, A.1, A.2 (F6.1-F6.8)R6 T 10/6 Practice Graphical Methods and Stress Visualization12 R 10/8 Flexure II – Flexural Equation/Bending Stresses 6.3, 6.4 (F6.9-F6.13)

R 10/8 EXAM I (Evening Exam 7:00 pm – 9:00 pm) Classes 1 - 10T 10/13 Fall Break

13 R 10/15 Flexure III– Bending Stresses/Combined Bending/Axial 6.3, 6.4, 6.5 (F6.14-F6.15)14 T 10/20 Flexure IV – Composite Beams 6.6, 6.7 R7 T 10/20 Intro to FEA Modeling – Axial Force ANSYS Tutorial 115 R 10/22 Shear Stress Equation/Shear Stress Distribution 7.1, 7.2 (F7.1-F7.5)16 T 10/27 Shear Flow/Shear Connectors(Glued/Nailed/Bolted) 7.3, 7.4 (F7.6-F7.9)R8 T 10/27 FEA Modeling & Analysis – Bending ANSYS Tutorial 217 R 10/29 Combined Loading & Pressure Vessels 8.1, 8.2, 11.1,11.2 (F8.1-F8.8)18 T 11/3 Combined Loading & Beam Design 8.2 (F.11.1-F11.6)R9 T 11/3 FEA Theory - Elements & Meshes ANSYS Tutorial 319 R 11/5 Stress Transformation - Equations 9.1, 9.2, 9.3 (F9.1-F9.6)20 T 11/10 Stress Transformation – Mohr’s Circle 9.4, 9.5 (F9.7-F9.12)

R10 T 11/10 FEA Modeling & Analysis – Focus Principal Stresses ANSYS Tutorial 421 R 11/12 Strain Transformation – Equations/General Hooke’s Law 10.1, 10.2

R 11/12 EXAM II (Evening Exam 7:00 pm – 9:00 pm) Classes 1 - 1822 T 11/17 Failure Theories/Strain Gages 10.5, 10.6, 10.7

R11 T 11/17 FEA Modeling & Analysis – Focus Stress Concentrations ANSYS Tutorial 5 & 4.723 R 11/19 Beam Deflection I - Integration 12.1, 12.2, 12.3 (F12.1-F12.6)24 T 11/24 Beam Deflection II - Superposition 12.5

R12 T 11/24 FEA Modeling & Analysis – Composite Sections ANSYS Tutorial 6R 11/26 Thanksgiving Break

25 T 12/1 Indeterminate Beams 12.6, 12.7R13 T 12/1 FEA Modeling & Analysis – Combined Loads ANSYS Tutorial 726 R 12/3 Column Analysis – Buckling and Bracing 13.1, 13.2, 13.3 (F13.1-F13.6)27 T 12/8 FEA Modeling & Analysis – Pressure Vessels ANSYS Tutorial 8 & 8.1

R14 T 12/8 FEA Modeling & Analysis – Modeling Issues ANSYS Tutorial 928 R 12/10 FEA Modeling & Analysis – Focus on Failure ANSYS Tutorial 10

“Problems worthy of attack prove their worth by fighting back” – Paul Erdos


FINAL: TBA Classes 1 - 26

ABET Outcomes: This course focuses on ABET program outcome (a.), which involves applying math and science principles daily to perform analysis of systems by applying concepts of equilibrium, kinematics, and compatibility to axial, torsional, bending, and combined loading problems. Other outcomes addressed include (j.) and (k.), which involve discussing contemporary structural and mechanical systems, including bridges, buildings, and machines, in order to highlight their impact on the design profession and society, and providing opportunities to gain proficiency in the use of engineering tools and programs through completion of the combined stress project and the acquisition of basic ANSYS finite element modeling and analysis skills.

Students with Disabilities: The college policy regarding student disabilities can be found at this website: http://attic.lafayette.edu/disability-services and will be adopted in this course. All documentation should be forwarded to Rebecca Brenner, Disability Services Coordinator, Scott Hall 312. Documentation can be emailed to [email protected] or faxed to (610) 330-3065.

Homework Policy: Homework is generally assigned every class meeting and is collected at the beginning of next class meeting,

unless otherwise specified. Late homework will be accepted only by making a prior arrangement with the instructor either during office hours or by email, subject to Lafayette College Dean’s Excuse Policy outlined in section 7.3.2 of the Faculty Handbook under the heading Class Attendance

This class emphasizes developing skills in problem solving. All homework should be submitted on non-spiral-edged, grid-background paper using one side only in a professional manner, which includes neatness and organization. The following should be provided for each homework problem:o Provide a brief written description of the problem being solved and the result requested.o Provide straightedge-sketches of system, including straightedge-sketches of all necessary FBDs.o Clearly identify the analysis steps and calculations made and check your units as you work. o Provide a summary of your final answers with correct units labeled and box them.o Check your answers to make sure they seem reasonable for reality.Neatness and clarity are important – messy, unorganized problem solutions will be given a grade of zero.

Unless otherwise stated, all homework in this class is expected to be individual work. Copying the work of others, including homework, is in violation of the College’s Principles of Intellectual Honesty, which can be accessed at http://fye.lafayette.edu/academics/. You may discuss the homework assignments with other students. All work submitted, however, must be your own and it is your responsibility to properly acknowledge the source of ideas and facts received from others, including other students. If you work or consult with someone else you must identify them by name at the top of your homework as collaborators. Copying is not collaboration. A student who commits academic dishonesty is subject to a range of penalties, including suspension or expulsion.

Discussions about re-grading of homework/exams are not conducted in person or by email. If you would like to request re-grading, attach a signed statement to your work that details where you feel you lost points and submit it to the professor within one week after the homework/exam has been returned.

HW extension request email and response email:

Why the Emphasis on Clear Calculations on Your Homework? (PCA, Ken Hover)When you work for an engineering firm your calculations are not your personal property but are the legal property of that firm (including what you may consider to be scratch notes or rough drafts). When another engineer needs to make a modification to your work, your ‘calcs’ will be his or her starting point. If you get promoted, transferred to another office, take a leave, or win the lottery, another engineer may have to pick up where you left off, and if your ‘calcs’ are not clear, that other engineer may have to start all over, meaning that all of your time (and salary) were wasted. Should your firm ever need to defend the correctness of your

“Adopting the right attitude can convert a negative stress into a positive one” – Hans Selye


work in a court of law, your calculations are likely to be subpoenaed. If the calculations are sloppy or unprofessional, the judge and jury are likely to assume (mistakenly, perhaps) that you are likewise a sloppy and unprofessional engineer. If your calculations contain an error, the opposition’s expert witness WILL catch it and soon everyone will know about it.

Recitation Sessions: Recitation sessions will be held once a week. You are required to attend the Recitation session at the time assigned to you on Tuesday evening in 223B AEC. During the 1st half of the semester, the recitation sessions focus on building problem solving skills by asking you

to apply concepts learned in class to more challenging problems. Students will work individually or in small groups to complete a recitation problem set during the session. You will be able to ask questions and get help during each session. At the end of each recitation session, the problem sets will be collected and graded for completeness. No homework will be assigned during the 1st half of the semester in the recitation sessions.

During the 2nd half of the semester, the recitation sessions focus on using the finite element analysis (FEA) program ANSYS to model and analyze structures subjected to axial, shear, bending, torsional and pressure loads. o To prepare for each ANSYS Recitation session you will be required to complete an assigned ANSYS

Tutorial before attending the corresponding ANSYS Recitation session. You will be asked to complete the self-guided tutorial problems before class. The self-guided tutorial will be available no later than the Thursday before our Tuesday night recitation sessions. There is no grade assigned for completing the assigned ANSYS Tutorials. However, if you do not complete the self-guided ANSYS tutorials you will have quite a bit of difficulty completing any of the ANSYS Homework assignments.

o During each ANSYS Recitation session you will either focus on completing the assigned ANSYS Tutorial problems (if there are errors or other issues with your outside-of-class prep work) or on working through the assigned ANSYS Homework problems during the recitation session. During each ANSYS Recitation session you will be able to ask questions and to get help from the instructor and other students.

o The assigned ANSYS Homework will be due at the start of the following Tuesday night ANSYS recitation session and will be submitted in hard copy and/or file format as defined in each homework assignment. The ANSYS Homework assignments will be graded for correctness and completeness.

o In many cases you will be able to complete the assigned ANSYS Homework during the recitation session. If you do not complete the assigned ANSYS Homework then you will need to work on the Homework as well as work through the new ANSYS Tutorial before the next ANSYS recitation class the following Tuesday.

o There will be at least one quiz held during the ANSYS Recitation sessions. You will be notified beforehand about the quiz so that you will be able to prepare for them as needed. Any quizzes will be submitted in file format and graded for correctness and completeness.

Combined Stress Project Information : The objective of the project is to allow students to take their knowledge gained from working the directed

homework problems and apply it to solve a more open-ended engineering problem involving the design of a system that is subjected to combined loading and therefore subjected to combined stresses. In addition, the project will involve the creation of an Excel Spreadsheet to simplify the design effort in evaluating the effect of different loading configurations.

Students will be given a longer time period (3 weeks) to work on the project during the last month of the course. For the project, students can choose to work individually or in teams of two students.

More detailed information concerning the project analysis, design, and reporting requirements will be presented and discussed in class later in the semester and made available on the course website at that time.

Availability of Help from People and Example Problems: Drop by during Office Hours or arrange to meet the Instructor at other times during the week to get help with

concepts or problems you do not understand. Peer Tutoring sessions will be held twice a week at regularly schedule times and rooms, which will be finalized

during the first two weeks of the semester. The Peer Tutor is Ethan Ossolinski, who is a CEE junior. You should feel free to attend as many Peer Tutoring sessions as you like. In the Peer Tutoring sessions you will have the opportunity to ask questions about course concepts and to ask for help with solving class and homework problems. These are not supplemental instruction sessions; these are drop-in tutoring sessions.

The Attic also will help find Individual Tutors for students upon request. Please contact the Attic if you would like additional to receive individual tutoring help during the semester.

Work out the Preliminary Problems and Fundamental Problems in the textbook and use the partial solutions and answers provided in the back of the textbook to check your work and the problems solving process you are using. Ask the Peer Tutor or Instructor for help with any problems you do not understand.

Cover-up and work out the Example Problems in each chapter of the textbook. Ask the Peer Tutor or Instructor for help with any problems you do not understand.

Availability of Help from Software:



Students can complete the homework and the project by hand using a calculator, a spreadsheet program like Excel, Matlab, Mathematica, or any other useful programs, including ANSYS in the second half of the semester. As engineers you will continually be looking for new tools to learn and apply that make your life easier. Therefore, you are encouraged to try to use some of the programs, which are available on computers in AEC.

Availability of Help from Useful Web-based Tutorial Sites: There are only a few fundamental concepts in Strengths; but many, many, many problems to apply them to. The more problems you think about solving or trying to solve the better you will become at recognizing how to solve newly presented problems. These software apps provide additional problems to work on and to review: MecMovies: http://web.mst.edu/~mecmovie MDSolids: www.mdsolids.comMechanics eCourse: http://ecourses.ou.edu/cgi-bin/ebook.cgi?topic=me

Availability of Help from the Web for ANSYSWe will be using ANSYS Workbench 15.0. This is Release 15.0 in the life of ANSYS Software, which started in 1971. You will be able to find help on the Web, especially on YouTube and other video forums, but be sure to search for ‘ANSYS Workbench’ or ‘ANSYS 15.0’ or ‘ANSYS Workbench 15.0’ to make sure you are not looking at sites that are using older releases of ANSYS. Anything older than Release 14.0 looks different enough to be confusing.

Federal Credit Hour Policy: The student work in this course is in full compliance with the federal definition of a four credit hour course. Please see the Registrar’s Office webpage (http://registrar.lafayette.edu/files/2013/04/Federal-Credit-Hour-Policy-Web-Statement.doc) for the full policy and practice statement.

Engineers Have Attitude:In addition to focusing on picking up basic knowledge & developing the technical skills required to solve engineering problems, such as problem solving, critical thinking, teamwork, & communication, there is another goal of all engineering courses. This is to promote an “engineering attitude”. The characteristics of having a strong engineering attitude are accepting mistakes, having common sense, patience, ethics, high standards, confidence, persistence, curiosity, flexibility, & understanding that there is not always a single right answer. Students with an engineering attitude possess a well-founded confidence in their ability to solve both routine & novel technical problems.

ES 230 Class Specific Learning Outcomes: Introduction/Internal Forces by Statics/Normal Stress 1. Review how to calculate external reaction forces (support reactions) in trusses, beams, and frames using equations of equilibrium.2. Review how to calculate internal forces (Axial force - N, Shear force - V, and Bending moment - M) in members by taking a section

cut from a truss, beam, or frame using equations of equilibrium.3. Calculate the average normal stress in a bar or cable subjected to a tensile or compressive axial force.4. Determine the minimum required cross-sectional area of a bar or cable subjected to tensile or compressive axial force given the

maximum allowable stress of the material (σall).Shear Stress/ Bolted Connections/Bearing Stress 1. Identify the forces and areas involved in calculating normal stress, shear stress, and bearing stress.2. Identify three types of problems that can be solved: find ave, find min Areq’d, or find max P that can be carried.3. Calculate average shear stress acting on a transverse area in connections and bolts due to shear forces (V).4. Calculate the average bearing stress in connection elements or other bearing situations due to normal forces (P).Factor of Safety/Oblique Planes/General Stress State 1. Explain why a factor of safety is used in engineering design. 2. Determine allowable stress (normal or shear) of a material given its ultimate stress or yield stress and a specific factor of safety.3. Select the required member area, bolt diameter, or plate thickness given the ultimate average normal or shear stress, factor of safety,

and load that must be carried.4. Calculate the average normal and shear forces that act on an oblique plane in a tension or compression member.5. Identify the six general normal and shear stress components that are used to characterize the state of stress at a point. Strain & Deformation1. Identify the difference between displacement (rigid body) and deformation. 2. Describe how normal strain is defined in terms of change in length of a line segment.3. Describe how shear strain is defined in terms of change of angles between line segments (originally at 90°).4. Calculate the average normal strain or average shear strain at a point in the body given information concerning the deformed

geometry of the system or body (assuming small deformations occur).Material Properties 1. Identify the yield stress, proportional limits and ultimate stress of a material from its stress stain curve obtained from a tension test.

Understand the following mechanical properties: elasticity, resilience, and toughness. 2. Describe how to calculate the modulus of elasticity, E, of a material.3. Understand the general form of the stress-strain diagram of ductile and brittle materials.4. Apply Hooke’s Law to solve for unknown stress or strains in axial loaded member linear elastic problems.5. Apply Poisson’s ratio, , in solving problems that require determining lateral strains.6. Investigate whether a material can have a negative Poisson’s ratio.Axial I – Member Analysis


Engineering problems are under-defined, there are many solutions, good, bad and indifferent. The art is to arrive at a good solution. This is a creative activity, involving imagination, intuition and deliberate choice. - Ove Arup

http://ecourses.ou.edu/cgi-bin/ebook.cgi?topic=me

http://www.mdsolids.com/

http://web.mst.edu/~mecmovie


1. Describe the local effects that the specific loading and support configuration has the stress near the ends of an axial loaded bar. Discuss the assumption we make in claiming there is uniform normal stress across the cross section (aka. S.t Venant’s Theorem).

2. Show how to obtain the equation = PL/AE. Discuss how to find if the area, A, or load, P, is a function of x.3. Apply force-deformation, stress-strain, and Poisson’s ratio relationships to determine forces, stresses, strains, or deformations in

axially loaded members. 4. Understand the importance of the principle of superposition in the analysis of axial loaded bars in linear, elastic problems.Axial II - Indeterminate Members1. Show and/or Describe the difference between determinate and indeterminate axial bar systems (bars in series, bars in parallel,

multiple–bar beam supports)2. Apply concepts of statics and deformation compatibility to solve for reaction forces and internal member forces and stresses in

statically indeterminate axial member problems, including bars in series, bars in parallel, and multiple-bar beam supports.3. Apply concepts of statics and deformation compatibility to determine the equations required to solve for reaction forces and internal

member forces and stresses in statically indeterminate axial member problems, including “gap” problems and “giving-in” problems.Axial III - Thermal Stresses and Stress Concentrations1. Understand the effect of stress concentrations due to holes or changes in width or radius of axial loaded bars.2. Handle thermal stresses and strains that may develop in both determinate and indeterminate members 3. Apply concepts of statics and deformation compatibility to solve for reaction forces and internal member forces and stresses in

statically indeterminate axial member problems, including bars in series, bars in parallel, and multiple-bar beam supports.Torsion I – Shear Stresses & Angle of Twist1. Explain the assumptions that are made in solving for the Torsional Shear Stress Formula for circular shafts and tubes. 2. Explain the how to use deformation and stress-strain relationships along with the cross-sectional property of polar moment of inertia

to obtain the Torsional Shear Stress Formula for circular shafts and tubes.3. Apply concepts of statics and the Torsional Shear Stress Formula to determine the stresses at points along shafts and bars in 2-D and

3-D in statically determinate problems (where statics is enough to find the internal torques)4. Draw the state of shear stress on a stress element that summarizes the shear stresses obtained at a point along a shaft or bar.5. Describe the parameters that control the angle of twist and state the angle of twist formula for circular shafts.6. Apply statics and the angle of twist formula to calculate the angle of twist in a circular shaft having constant torque in each segment. Torsion II –Indeterminate Members1. Explain the how a torsionally-loaded bar or tube may fail if the material is ductile or brittle (weak in tension, compression, or shear)2. Describe the parameters that control the angle of twist and state the angle of twist formula for circular shafts. 3. Apply statics and the angle of twist formula to calculate the angle of twist in a circular shaft in which the torque is constant in each

segment and in which the torque is a function of position, x, over the length of the shaft.4. Apply statics, compatibility of deformations (angles of twist) and the angle of twist formula to calculate forces in indeterminate

torsionally-loaded shafts, in which shafts are placed in series or in parallel.Flexure I – Graphical V and M Diagrams/Moment of Inertia Calculations1. Compute the internal shear force, normal force, and bending moment at any point in a member by taking a section cut at that point. 2. Write out and describe the differential and integral relationships between w(x), V(x), and M(x).3. Write equations for moment and shear force, M(x) and V(x), as a function of location x along the beam. Watch out for

discontinuities wherever the applied load changes or at support locations.4. Graph shear force and moment diagrams that show how the shear force and moment vary across the beam using Graphical Methods

and/or equations for moment and shear force.5. Calculate the moment of inertia of beam cross-sections consisting of common geometrical shapes.Flexure II – Flexural Equation/Bending Stresses1. Determine equations that state the relationships between normal strain, normal stress, and curvature occurring in beams subjected to

bending moments.2. Calculate the moment of inertia for a beam cross-section consisting of common geometrical shapes.3. Calculate the normal stress, x, or normal strain, x, at any point on a beam cross-section, including determining the location and

magnitude of maximum normal stress or strain (max or mac)4. Draw the linear strain and stress distributions over the cross-section of a beam subjected to bending considering a wide variety of

cross-sectional shapes (square, rectangular, circular, tube, triangle, I-beam, T-beam, channel, etc).5. Calculate the curvature of a beam given the applied moment and cross-section or the maximum stress. Flexure III - Bending Stresses/Combined Bending/Axial1. Calculate normal stresses due to eccentric loadings by superimposing the uniform normal stress induced by the centric, axial load

and the linear normal stress distribution induced by the pure bending moment. 2. Calculate the normal stresses on cross-sections that are subjected to arbitrary moments that create moments about both axes of

bending of a member and normal, axial forces using superposition of stresses. Flexure IV - Composite Beams/Reinforced Concrete Beams/Beam Design 1. Explain why composite beam cross-sections must be transformed into equivalent single material cross-sections before calculating the

moment of inertia of the transformed section.2. Explain the difference between bending stress distributions in non-composite (homogeneous) and composite (made up of two or

more different materials) cross-sections.3. Transform a composite beam cross-section and compute the moment of inertia of the transformed cross-section. 4. Calculate the bending stresses acting on the transformed section, x, and also the actual bending stresses that act on each material in

the composite section, 1 , 2 , etc.Shear Stress Equation/Shear Stress Distribution1. Explain why complimentary longitudinal and transverse shear stress components exist in beams that are subjected to bending.2. Explain how to calculate Q (1st moment of area) in order to calculate the shear stress as a specific depth in a beam cross-section. 3. Calculate the shear stresses acting at various locations over a beam cross-section and sketch the distribution of shear stresses over the

cross-section for rectangular, circular, I-beam, T-beam, etc. cross-sections.


Learning Strengths as an engineer is similar to getting a driver’s license as a teenager.It is essential, it allows you to head out on your own, and it helps you get a “date to the big dance” (i.e. job/grad school)


Shear Flow/Shear Connectors Design1. Explain the difference between shear stress, , and shear flow, q. 2. Explain how to calculate the shear flow at a specific depth in a beam cross-section. 3. Calculate the shear flow and determine either the required spacing of connectors given the connector strength or determine the

connector strength required given the spacing or number of connectors.4. Determine cross-section dimensions based on the allowable bending stress, all, and the allowable shear stress, all, of the material.Thin-Walled Pressure Vessels1. Explain how the stress equations for cylindrical and spherical pressure vessels are obtained from FBDs and equilibrium.2. Calculate the stresses (hoop and/or axial stress) in thin-walled pressure vessels.3. Calculate the stresses or forces in radial band or connectors that are used to construct pressure vessels.4. Using Hooke’s Law and definition of axial strain determine strain or elongation of radial bands.Combined Loading1. Identify and calculate the stresses that are created by axial force, shear force, torsional moment, and bending moments on a member

cross-section and at any point in the member along its length. 2. Use the principle superposition to determine the resultant normal and shear stress acting at a specific point designated anywhere in a

member cross-section along the member length.3. Represent the results on an element of material located at a specific point or show the results as a distribution of stress acting over

the member’s cross-sectional area, either in 3D or 2D. 4. Determine the equivalent force/moment system for any eccentrically-applied load (load not applied through the centroid of cross-

section) either in 2-D or 3-D problems. Stress Transformation – Equations1. Determine normal and shear stresses acting on any plane cut at any angle through any plane stress element.2. Calculate the magnitude and direction of the principal stresses of a given plane stress state using equations.3. Calculate the magnitude and direction of the max in-plane shear stress and avg normal stress of a given plane stress state.Stress Transformation – Mohr’s Circle1. Sketch Mohr’s Circle given a state of plane stress consisting of applied normal stresses (x, y) and shear stress ( xy )2. Determine the magnitude and direction of the principal stresses of a given plane stress state using Mohr’s Circle.3. Determine the magnitude and direction of the maximum in-plane shear stress and average normal stress of a given plane stress state

using Mohr’s Circle. 4. Determine the normal and shear stresses acting on any plane cut at any angle through any plane stress element using Mohr’s Circle

and/or Stress Transformation equations.Strain Gages/Strain Transformation Equations/Generalized Hooke’s Law 1. Determine the magnitude and direction of the principal strains of a given strain state using strain-transformation equations.2. Determine the magnitude and direction of the maximum in-plane shear stress and average normal stress of a given strain state using

strain-transformation equations and/or Mohr’s circle.3. Determine the normal and shear strains acting on any plane cut at any angle through any strain element using strain-transformation

equations and/or Mohr’s circle.4. Sketch the geometry of a deformed element under a given or calculated state of strain.5. Calculate strains given strain measurements obtained from a strain gage rosette.6. Calculate stresses from strains for a given material point using generalized Hooke’s Law Equations.Failure Theories1. Determine whether failure has occurred in a brittle material by applying the Rankine Principal Stress Criteria.2. Determine whether failure has occurred in a ductile material by applying the Tresca Maximum Shear Stress Criteria.3. Determine whether failure has occurred in a ductile material by applying the von Mises Maximum Distortional Energy Criteria 4. Calculate the absolute maximum shear stress (which is greater than or equal to the maximum in-plane shear stress), given the

principal stresses. Consider two cases: both principal stresses have the same sign or both have different signsBeam Deflections I – Integration and Discontinuity Functions1. Define the basic equations for the elastic curve of a loaded beam in terms of M, E, I, and curvature. 2. Define the sequence of equations obtained by integration that lead from the moment equation to the slope equation to deflection

equation for a loaded beam.3. Determine the slope and deflection equations for beam with a continuous load effect over its entire length by writing the moment

equation, integrating to get the slope & deflection equations, and applying two boundary conditions (beam support conditions).4. Determine the slope and deflection equations for beam loaded with discontinuities in load effect over its length by writing the

moment equation for each continuous beam segment, integrating to get the slope and deflection equations for each segment, and applying as many boundary conditions as necessary, including continuity equations between segments (slope equal on either side and deflection equal on either side) to solve for the constants of integration.

5. Write out one moment equation for an entire beam using discontinuity functions to handle changes in loading. 6. Perform integration to get the slope and deflection equations for the entire beam having load and/or support discontinuities and

applying two boundary conditions to solve for the constants of integration. 7. Determine using the principle of superposition and tabulated deflection and slope equations from beam tables the deflection or slope

at various locations along a beam having several types and variations in loading along its lengthBeam Deflections II – Superposition/Deflections and Indeterminate Beams with One DOI1. Determine using the principle of superposition and tabulated deflection and slope equations from beam tables the values of

deflection or slope at specific points along a beam with a known loading pattern. 2. Determine using the principle of superposition and tabulated deflection and slope equations from beam tables the values of

unknown support reactions in indeterminate beams having one or two degrees of indeterminancy (DOI). 3. Determine using statics the remaining unknown support reactions and draw the indeterminate beam shear and moment diagrams.4. Determine using the principle of superposition and tabulated deflection and slope equations from beam tables the values of

unknown support reactions in indeterminate beams having one DOI and a yielding support (spring or tension rod).


“Nothing is particularly hard if you divide it into small jobs” – Anon


Column Analysis - Buckling and Bracing1. Describe buckling behavior of a slender column and how this differs from yielding behavior of a short column.2. Discuss how the column length, cross-sectional properties, end support conditions, and bracing locations influences the critical

buckling load a column can carry (L, A, I, K).3. Derive Euler’s formula for a pinned-pinned column using statics of equilibrium of the deformed column geometry.4. Calculate a column’s load carrying capacity, Pcr, given cross-sectional properties, end support conditions, and bracing locations.


We were put on this earth to make things – W.H. Auden

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