CIEG 212


CIEG 212: Solid Mechanics
Spring Semester 2008

Instructors:
Victor N. Kaliakin
e-mail

James N. Scarborough
e-mail

CONTENT OF LECTURES

Lecture #1 (02-12-08):
  • Discussion of Class Syllabus.
  • Introduction to Solid Mechanics (associated reading: Section 1-1 in the textbook).
  • Review of Statics (associated reading: Sections 1-2 to 1-3 and 1-5 in the textbook).
  • Today's lecture material (available until 02-19-08).
Lecture #2 (02-14-08):
Lecture #3 (02-19-08):
  • General idea of the concept of stress (intensity of force) (associated reading: Section 2-1 in the textbook).
  • Average normal stress in an axially loaded member (associated reading: Section 2-2 in the textbook). Example involving a two-story column.
  • Avearge shear stress (associated reading: Section 2-3 in the textbook). Example involving three glued 2 x 4 timber boards.
  • NOTE: answers to selected homework problems in the textbook (Riley et al.) are available on-line at http://higheredbcs.wiley.com/legacy/college/riley/047170511X/select_ans/select_ans.pdf
Lecture #4 (02-21-08):
  • Office hours:
    • Mondays: 10:00 to noon and 2:00 to 3:00 p.m. in Room 314 P.S. DuPont Hall (with Mike Cann (e-mail).
    • Wednesdays: 10:00 to noon and 2:00 to 3:00 p.m. in Room 360F P.S. DuPont Hall (Dr. Kaliakin).
    • Or by appointment.
  • Other examples of average shear stress.
    • Example of a single shear bolted connection.
    • Example of a double shear bolted connection.
  • Calculation of avearge normal & shear stress on an inclined plane in an axially loaded member (associated reading: Section 2-6 in the textbook).
  • Interactive examples related to stress concepts from University of Michigan (see http://www.engin.umich.edu/students/ELRC/me211/stress.html).
  • Homework Assignment #2 assigned, due 02-28-08.

Lecture #5 (02-26-08):
  • Concept of stress; state of stress at a point (associated reading: Section 2-7 in the textbook).
  • State of plane stress (associated reading: Section 2-8 in the textbook).
  • General equations of plane stress transformation (associated reading: Section 2-9 in the textbook).
Lecture #6 (02-28-08):
  • QUIZ #1
    • Shall be administered during the last 20 minutes of the class period on March 6, 2008.
    • Will be CLOSED BOOK & CLOSED NOTES.
    • Will cover two-dimensional problems in Statics, emphacizing: a) Computation of reactions, b) Computation of internal forces and moments, and c) Proper treatment of distributed forces.
  • General equations of plane stress transformation (associated reading: Section 2-9 in the textbook).
  • Begin discussion of principal stresses under conditions of plane stress (associated reading: Section 2-10 in the textbook).
  • Solution to Homework Problem Set #1
  • Homework Assignment #3 assigned, due 03-06-08.
Lecture #7 (03-04-08):
  • General equations of plane stress transformation (continued) (associated reading: Section 2-9 in the textbook).
  • Determination of principal stresses (associated reading: Section 2-10 in the textbook).
  • Principal stresses and maximum in-plane shear stress (associated reading: Section 2-10 in the textbook).
Lecture #8 (03-06-08):
Lecture #9 (03-11-08):
  • Strain (continued) (associated reading: Sections 3-1 to 3-5 in the textbook).
    • Normal strains; shearing strains.
    • State of strain at a point.
    • Special case of plane strain.
    • Today's lecture material (available until 03-13-08).
    • Simple example of uniaxial state of stress.
    • Example Problem 3-3 in text.
    • Computation of shearing strain.

Lecture #10 (03-13-08):
  • QUIZ #2
    • Shall be administered during the last 20 minutes of the class period on March 20, 2008.
    • Will be CLOSED BOOK & CLOSED NOTES.
    • Will cover stress (see appropriate sections in Chapter 2 of the textbook).
  • 75-Minute Exam
    • Shall be administered on March 27, 2008.
    • Will cover stress, strains and mechanical properties of materials (solids).
    • Will be CLOSED BOOK & CLOSED NOTES.
    • One page of formulas will be allowed.
  • Solution to Quiz #1
  • Mechanical properties of materials (associated reading: Chapter 4 in the textbook).
    • The stress-strain diagram (associated reading: Section 4-2 in the textbook).
    • Hooke's Law (associated reading: Section 4-2-4 in the textbook).
    • Poisson's ratio (associated reading: Section 4-2-12 in the textbook).
    • The shear stress-shear strain response (associated reading: Section 4-2-4 in the textbook).
  • Solution to Homework Problem Set #3
  • Homework Assignment #5 assigned, due 03-20-08.
Lecture #11 (03-18-08):
  • Mechanical properties of materials (continued).
    • Two examples involving Hooke's Law and Poisson's ratio.
  • Mechanical properties of materials (continued).
    • Review of linearity & elasticity.
    • Typical uniaxial stress-strain curve.
  • Lecture material for the last three lectures (available until 03-25-08).
Lecture #12 (03-20-08):
  • NOTE: 75-minute exam #1 will be administered on March 27, 2008 (9:30 to 10:45 a.m.) in Room 100 Kirkbride Hall.
    • The exam will cover pertinent sections from Chapters 1, 2, 3 and 4. This includes:
      • Basic concepts from Statics: determination of reactions, internal forces, equivalent loads associated with distributed applied loadings, etc.
      • Average normal and shear stresses and strains.
      • Transformation of stress (plane stress conditions) & strain (plane strain conditions).
      • Mechanical properties of materials: Elastic modulus, Poisson's ratio, Modulus of rigidity; stress vs. strain curves, etc.
    • The exam will not cover material from Chapter 5 (deformation of axially loaded members).
    • The exam will be closed book & closed notes.
    • One page of formulas will be allowed.
    • Bring paper (stapler will be provided) and a calculator.
  • General equations of mechanics
    • Equilibrium.
    • Geometric compatibility.
    • Stress-strain (constitutive) relations.
  • Deformation of axially loaded members (associated reading: Sections 5-1 and 5-2 in the textbook).
  • Quiz #2 administered during final the 20 minutes of class period.
  • Solution to Homework Problem Set #4
Lecture #13 (03-25-08):
  • Deformation of axially loaded members (associated reading: Sections 5-1 and 5-2 in the textbook) (continued).
    • Example of a statically determinate problem (associated reading: Section 5-2 in the textbook).
    • Example of a statically indeterminate problem (associated reading: Section 5-4 in the textbook).
  • Solution to Quiz #2
  • Solution to Homework Problem Set #5
Lecture #14 (03-27-08):
  • 75-minute examination.
Lecture #15 (04-08-08):
  • Results of first 75-minute examination:
    • Range: 100 to 59.0 out of a possible 100 points.
    • Mean = 82.6/100
    • Solution.
  • Professor Scarborough lecturing.
  • Torsion of circular cross-sections (associated reading: Chapter 6 in the textbook).
  • Introduction
    • Why do we need to study torsion?
    • Terms used – torque, rotational moment, twisting moment or couple.
    • Torque will develop a shearing stress in a structural member.
  • How torque is used in a shaft with gears.
    • Applied torque and resultant torque in shaft.
    • Assumptions for analysis limit shapes to solid circular members and to hollow circular members.
  • Example problem – (similar to book example 6.1)
    • Determine the resisting torque in a shaft with a force applied to several gears.
    • Determine the torque in individual shaft sections
  • Homework Assignment #6 assigned, due 04-15-08.
Lecture #16 (04-10-08):
  • Torsional Shear Strain (associated reading: Section 6.2 in the textbook).
    • Twisting deformation of a rod under torque.
    • Angle of Twist, deformation angle over length equals shear strain
    • Shear strain = radius *angle of twist/ length of section
    • Shear strain at any distance from center =( shear strain at surface divided by radius)*distance from center.
  • Torsional Shear Stress (associated reading: Section 6.3 in the textbook).
    • Using Hooke’s Law for materials in the elastic range, we get: Shear stress at any distance from center = (shear stress at surface/radius)*distance from center.
    • Polar second moment of area definition and use.
      • J=pi*radius^4/2 for solid shafts
      • J=pi*(outer radius^4-inner radius^4)/2 for hollow shafts
    • Torque equation T = shear stress*J/radius -we can find shear stress for a known diameter or radius for an allowable shear stress.
    • Angle of Twist = TL/GJ ; if multiple diameters on shaft then use summation of each section to find overall angle of twist.
    • Continuation of example problem, find the angles of twist for various sections and the shear stresses for them. Table of limiting angles of twist for a couple of situations was put on board.
  • Homework problems assigned, due 04-15-08.
Lecture #17 (04-15-08):
  • Power Transmission (associated reading: Section 6.6 in the textbook).
    • Often need to size a shaft for a motor of a given horsepower or find the largest horsepower a shaft can handle.
    • Definition of hp = 550 ft-lbs/sec.
    • Power is equal to Torque times angular velocity w.
    • Conversion of shaft rpm’s to angular velocity 2*pi*rpm/60 (rpm/60=f).
    • So P = 2*pi*f*T.
    • Example done in class to find shaft diameter for a given hp.
  • Bending of Beams
    • Flexural Loading in Beams (associated reading: Section 7.1 in the textbook).
      • Classes of beams – Simple, Overhanging, Continuous, Cantilever.
      • Beam may have more unknowns than the three equations of equilibrium can solve. For example, 3 supports under a single beam. Need strength of materials equations also.
      • Resisting moment and Shear and their relation to the normal and shear stress at any point on the cross section of a beam.
    • Flexural Strain (associated reading: Section 7.2 in the textbook).
      • Deformed beam under load will have the top fibers in compression and the bottom fibers in tension. Junction of the compression and tension sections is called the neutral axis.
      • Can find strain of any point if we know the distance the point or differential element is from the neutral axis.
      • Relationship between strain and distance from neutral axis is: epsilon_x = -(1/r)*y where r is the radius of the circle formed by the arc of the deformed beam. Thus the strain must be 0 at the neutral axis and increase with increasing distance from it, reaching a maximum at the surface of the member.
  • Homework problems assigned, due 04-17-08.
Lecture #18 (04-17-08):
  • Flexural Stress (associated reading: Section 7.3 in the textbook).
    • Variation of flexural stress with the distance from the neutral axis. Max flexural stress will occur at the surface of the material the furthermost from the neutral axis.
    • Resisting moment is a couple moment created by the resultants of the compressive stresses and the tensile stresses. Resisting moment is equal magnitude but opposite direction of the applied moment due to the external forces on the beam.
    • Value of stress at any distance from neutral axis can be found if the distance is known and the maximum value is known. Stress(x) = (y/c) * stress (max).
    • Definition of second moment of area and its relation to the resisting moment.
    • For a homogeneous material, 2 dimensional analysis (what we have been doing), the moment of inertia (which is based on mass) can describe the second moment of area. In statics we did this to describe the centroid. Your statics text will list the common formulas for moment of inertia about the center of gravity.
    • Definition of section modulus S = I/c.
    • Using formula for resisting moment, the section modulus S= M/stress. In wood, this is written as S= M/fb. Values of S for dimensioned lumber were written on board and calculations illustrated.
    • Example: Find the size of floor joists in a residential dwelling, using code values for live loading and ignoring dead load at this time. Conversion of pounds per square foot (psf) to pounds per lineal foot (plf). Find applied moment from load. Given a safe working bending stress, find the section modulus and thus the size of the member.

Lecture #19 (04-22-08):
  • Covered homework problems for first two sets that were returned.
  • Flexural Stress (associated reading: Section 7.3 in the textbook).
  • Example
    • Finished example problem finding the stress at the surfaces of the joist.
    • Did second example problem involving a non-symmetrical shape – a T beam made of wood was used.
    • Find the new neutral axis for the shape using what we learned about centroids.
    • Find the second moment of area (moment of inertia for our homogeneous 2 dimensional example) for our T beam. Use the parallel axis theorem to find the moment of inertia about the centroid of the new shape.
    • Calculate the stress at the top and bottom surfaces as well as at the junction of the T with the stem of the beam.
  • NOTE: There will be a quiz on next Tuesday (4/29) covering the section on torsion.
  • Homework problems assigned, due 04-24-08.
Lecture #20 (04-24-08):
  • Problems 6.38 and 6.39 solved in class.
  • Shear forces and bending moments (associated reading: Section 7.5 in the textbook).
  • Where are shear forces important?
    • Wood beams (horizontal shear), Thin webbed beams, Short beams, and Composite beams.
    • Calculating resisting Shear and Moments from the equilibrium equations.
    • Sign Conventions
    • Calculating shear and bending moment at any point on a beam.
    • Example:
      • Determining the shear and bending moment for a simple beam with uniformly distributed and concentrated loads at different points on the beam.
      • Review of what we learned in statics.
  • NOTE: There will be a quiz on next Tuesday (4/29) covering the section on torsion.
Lecture #21 (04-29-08):
  • Finished Example: Determining the shear and bending moment for a simple beam with uniformly distributed and concentrated loads at different points on the beam. Review of what we learned in statics.
  • Shear and bending and load relationships (associated reading: Section 7.6 in the textbook).
    • Examined the mathematical relationships between load and shear and then shear and bending moment.
    • Local maximum bending moment will occur when shear =0 (shear line crosses the 0 shear value) since slope of bending moment line = shear force at that point.
  • Quiz #3 administered.
  • Homework problems assigned, due 05-01-08.
Lecture #22 (05-06-08):
  • Shearing stresses in beams (associated reading: Section 7.7 in the textbook).
    • We used the moment diagram to find the maximum moment for a beam and used that to find the section modulus of the beam and thus the size. We can use the shear diagram we created to find the maximum shear.
    • Look at horizontal shear to develop the shear stress equations. Example: wood will exhibit horizontal shear due to its grain pattern. Since we have horizontal shear, we must have forces in the x direction to contend with. We know that we will have flexural stress that runs in this direction.
    • Went through the derivation of the formula relating horizontal shear stress to the resisting moment.
    • tau = dM/dx * (-1/It) * integral over area of (ty dy)
    • Integral is first moment of area between an evaluation point and the furthest surface of the member. We designate this as Q Since dM/dx = V, the formula for horizontal shear is tau = -VrQ/It.
    • Examined the shear stress distribution. Will be a maximum at the neutral axis and will be parabolic in shape. Limited to rectangular cross sections at this time.
    • Since we are generally interested in the maximum shear, the formula for this was derived for rectangular cross sections. tau_max = 1.5 V/A
    • But, since we know from earlier that tau_xy = tau_yx, then the horizontal and vertical shear stresses must be the same.
    • Did example with 2x4 under 500 lb center concentrated load to find max shear and compare to allowable and to find max moment and compare to allowable.
  • NOTE: Quiz on Tuesday May 13th
  • NOTE: It is proposed that instead of a final exam on the 29th of May, we have a second 75 minute exam on the last day of class which will be the 20th of May. In order to do this the weighting of the final grade must change. Originally it was 20% for the first exam and 40% for the final exam. We propose to make it 30% for each exam. Since this is not what was originally described to you as the way grades were to be determined, we must have unanimous agreement of the class for this change to take place. If and only if you do not want the change, you should e-mail me that you want the weighting to remain the same as originally in the syllabus by May 9th. jns@udel.edu
Lecture #23 (05-08-08):
  • Shearing stresses in beams (associated reading: Section 7.7 in the textbook).
    • Did another example problem with the same loading situation, but using a WF 6x9 steel beam. Illustrated the method of finding the shear stresses when the thickness of a member is not constant (web thickness vs. flange thickness).
    • Went over homework problems that students have had problems/questions about for the rest of the period. Preparation for the Quiz on Tuesday (05-13-08).
  • Homework problems assigned, due 05-13-08.
Lecture #24 (05-13-08):
  • Went over homework problems on bending stresses, finding moment of inertia for non-symmetrical beams and sizing beams based on section modulus.
  • Quiz #4
  • NOTE: Second 75-minute exam will be on Tuesday, May 20th.



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Revision Date: 05.14.08