ME 323 – Mechanics of Materials

Exam 1 ReviewFall 2019

Joshua Pribe

Exam Information

• Date: Wednesday, October 2

• Time: 8 – 10 PM

• Room: Lily 1105• See course website for seating arrangement

• Equation sheet is posted online; a hard copy will be provided during the exam

• In the exam, do NOT write on the back of the pages

• Format: 3 calculation problems; 1 multiple-part “conceptual” problem

2

Exam Coverage

• Lectures: 1 – 14

• Lecture Book: Chapters 1 – 9

• Homework: 1 – 5

• All material up to and including shear force and bending moment diagrams• Static equilibrium, stress and strain, factor of safety

• Generalized Hooke’s law

• Axial deformation and thermal effects, statically indeterminate problems

• Torsional deformation, statically indeterminate problems

• Shear force and bending moment diagrams

3

Ch. 1-4: Equilibrium, stress and strain, factor of safety

4

0F = ( ) 0O

M =Equilibrium:

Normal stress and strain:avg

F

A = avg

L

L =

Shear stress and strain:avg

V

A =

*

2

s

sL

= =−

Factor of safety: . .or yield stress

or

failur

allowa

eF S

actu ble sal tress=

P P

Single shear connection:avg

P

A = avg

2A

P =Double shear connection:

Review material: HW 1-2, Quiz 1, Week 2 Pre-Week Videos

Ch. 5: Generalized Hooke’s law

5

xz

yz

xy

xy

z

xz

y

G

G

G

=

=

=

( )2 1G

E

=

+( )

( )

( )

1 1

1 1

1 1

z

y y x z y x

z z x y z

x x y x y z

y

z

x

T TE E E E

T TE E E E

T TE E E E

= − + +

= − + +

= − − +

= − − +

= − + +

= − − +

Normal strains Shear strains Shear modulus

Review material: HW 2, Quiz 2, Week 2 Pre-Week Videos

Ch. 6-7: Axial deformation

6

Force-temperature-deformation equation:

P(1)

A B

x

FLTL

AEe + =

For the example below, 1 1, B AP u u eF = + = +

Recall our sign convention

For F:-Tension: F > 0-Compression: F < 0

For e:-Element elongates: e > 0-Element contracts: e < 0

Ch. 6-7: Axial deformation

Determinate problems: no redundant supports; solve for all forces using equilibrium equations

Indeterminate problems: at least one redundant support; need compatibility equation(s) to solve for all forces; force in each member depends on properties of all other members

Four steps:• FBDs and equilibrium

• Force-temperature-deformation equations for each element

• Compatibility equation(s) relating elongation of the elements• Often helpful to sketch the geometry of deformations (see Quiz 4)

• Solve for unknowns

Review material: HW 2-4, Quizzes 3-4, Weeks 2-4 Pre-Week Videos, Appendix A.3 of Lecture Book

7

Ch. 6-7: Axial deformation

8

Connected axial members:

Axial members supporting rigid bar:

Planar trusses:

11 1) )cos( sin(D Du ve = +

22 2) )cos( sin(D Du ve = +

33 3) )cos( sin(D Du ve = +

θ is always measured counterclockwise from the x axis at the pin!

Ch. 8: Torsional deformation

9

Shear strain and stress increase linearly with radial dimension ρ: ,d d

Gdx dx

= =

Find the stress from the torque:

max,o

p p

T rT

I I

= =

Torque-twist equation:p

TL

GI =

T(1)

A B

x

For the example below:1 1 1, orB A B AT T = + − = = +

Recall sign convention: T > 0 if torque is counter-clockwise (by the right-hand rule, the vector points out of the surface)

Ch. 8: Torsional deformation

Four steps for statically-indeterminate torsion problems:• FBDs and equilibrium

• Torque-twist equations for each element

• Compatibility equation(s) relating angle of twist of the elements

• Solve for unknowns

See Example 8.8 and HW 5 for practice drawing stress distributions

Review material: HW 3-5, Quiz 5, Week 4 Pre-Week Videos, Appendix A.3 of Lecture Book

10

Ch. 8: Torsional deformation

11

Connected shafts:

Gears: Angle of twist at the gears:

C C B Br r = −

x

Ch. 9: Equilibrium of beams

12

1

1

1

1

( ), (

) (

) (

)

) ( )

( ), ( ) (

x

x

x

x

dVp x V x V x p x dx

dx

dV x d

MV x M x M x

dxx+

= = +

= =

Geometric relations:

Point forces and moments (be careful with signs!):

Ch. 9: Equilibrium of beams

13

From last Friday’s quiz:

Review material: HW 5, Quiz 6 (only part a), Week 5 Pre-Week Videos

Flexural stress and transverse shear stress are NOT on Exam 1

General reminders

• Consistent sets of units

• Practice the conceptual questions and sample exams posted on the course website!

• Practice solving problems “backward” (e.g. given the angle of twist at a connector, find the applied torque) to check how well you understand the equations

14

Metric, option 1

Force N

Length m

Stress Pa

Metric, option 2

Force N

Length mm

Stress MPa(1 MPa = 1 N/mm2)

Remember: 1 MPa = 106 Pa, 1 GPa = 109 Pa

US units, option 1

Force lb

Length in

Stress psi

US units, option 1

Force kips

Length in

Stress ksi

Remember: 1 kip = 103 lb, 1 ksi = 103 psi