ME221 Lecture #16 1

ME 221 Statics

Lecture #16

Sections 6.6 – 6.7Final Exam Review

ME221 Lecture #16 2

Homework #6• Chapter 7 problems:

– 6, 19 & 26

• Chapter 6 problems– 3 & 6– Due Today

• MatLab Group Problems– 7.19, 7.26 & 6.15 – Due Today

ME221 Lecture #16 3

TrussesComposed of slender straight pieces connected together by frictionless pins where all the loads (no moments) are applied.

Each member will act as a two-force member (either in tension or compression). All the forces acting on a truss member are axial.

ME221 Lecture #16 4

Analysis of Trusses Using the Method of Joints

Ay

Ax

PAx

PAY

PBy

PBx

PCy

PCx

Cy

A

B

C

We need to solve for:

(1) - Internal forces

FAB, FAC, and FCA

(2) - Reactions

Ax, Ay and Cy

ME221 Lecture #16 5

Using Matrix Notation

100sin00

000cos10

000sin0sin

000cos0cos

01000sin

00101cos

Cy

Cx

By

Bx

Ay

Ax

P

P

P

P

P

P

y

y

x

BC

AC

AB

C

A

A

F

F

F

=

-1

Using manual calculations

Look for joints with 2 unknowns

ME221 Lecture #16 6

Example

1.2 m 1.2 m

6 kN

3 kN

E

BA

C D0.9

m

ME221 Lecture #16 7

6 kN

3 kN

E

BA

C

D441616

4 4

9

915

15

5

5

ME221 Lecture #16 8

1.8 m

12 kN 12 kN

4 @ 2.4 m=9.6 m

B D F H J

A C E G I

Method of SectioningIf the question is to find internal forces in selected members of the truss, then one can alternatively use the method of sectioning.

Example: Determine the force in members FG and FH

ME221 Lecture #16 9

1.8 m

12 kN 12 kN

B D F H J

A C E G IFGE

FGF

FHF

1.8 m

12 kN 12 kN

B D F H J

A C E G IFEG

FFG

FEH

000 Anyptyx MFF

ME221 Lecture #16 10

Trusses: 3-D (Space Trusses)• Consider all the members as two force members

• Generally, use the method of joints

• Determinate structures– Equation for determining 3j = m + r

– Joints are ball & socket

ME221 Lecture #16 11

3-D Truss Methodology

• Determine unit vectors based on structure geometry• Solve for reaction forces with entire structure• Draw FBD of joint• Sum forces at joint to determine member forces• Solve joint-by-joint or write out as a system of

equations– the latter method may also include reaction forces

ME221 Lecture #16 12

Example

ME221 Lecture #16 13

Final Exam

Wednesday, June 30, 10:20am

25% of course grade

ME221 Lecture #16 14

Final Exam Topics

Chapter 2: Vectors and Force Systems– Scalars & vectors– Laws of sine & cosine– Resolution of a vector into components– 3D coordinates & unit base vectors– Directional cosines– Scalar products of two vectors– 2D & 3D equilibrium problems

ME221 Lecture #16 15

Final Exam Topics

Chapter 3: Rigid Bodies– Calculating moments– Cross products– Moment of force about an axis– Moment of a couple– Equivalent force systems

ME221 Lecture #16 16

Final Exam Topics

Chapter 4: Distributed Forces– Center of mass– Centroids of areas & volumes– Centroids of composite bodies– Distributed loads on beams

ME221 Lecture #16 17

Final Exam Topics

Chapter 9: Moments of Inertia– Second Moment of Area; Ixx & Iyy

– Polar Moments; J0z

– Parallel Axis Theorem– Composite Areas

– Principal Second Moments; Ixy

ME221 Lecture #16 18

Final Exam Topics

Chapter 5: Equilibrium of Rigid Bodies– Equilibrium Equations– Supports– Free Body Diagrams– Two-Force Members– 3D Equilibrium

ME221 Lecture #16 19

Final Exam Topics

Chapter 7: Internal Forces in Structures– Internal Forces– Shear Forces– Bending Moments– Shear & Bending in Beams

ME221 Lecture #16 20

Final Exam Topics

Chapter 6: Analysis of Structures– Planar trusses– Method of joints– Method of sections

ME221 Lecture #16 21

Final Exam Format

4 to 5 Problems

At least one each:

Internal Forces – Chapter 7

Structures – Chapter 6

Others from previous material

ME221 Lecture #16 22

Other Topics

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