ENGR221 Lecture 13

8/4/2019 ENGR221 Lecture 13

1/33

Trusses and Machines

ENGR 221

February 26, 2003


2/33

Lecture Goals

7-2 Plane Trusses

7-3 Space Trusses

7-4 Frames and Machines


3/33

Example ProblemThe axis of the three-hinged arc ABC is a parabola with

vertex B. Knowing that P = 20 kN and Q =0 determine a) the

reactions at C, b) the components of the force exerted at on

segment AB.


4/33

Example Problem

What is a hinge?

At a hinge the moment is

zero and can translate theforces across the point.


5/33

Example Problem

Draw the free-body

diagram of the arch.

x Ax Cx

y Ay Cy

Ay Cy

A Cy

Cy

Ay

0

0 20 kN

20 kN

0 10 m 20 kN 3 m

6 kN

14 kN

F R R

F R R

R R

M R

R

R


6/33

Example Problem

Draw the free-body

diagram of the arch at B.

x Ax Cx

B Ay Ax

Ax

Ax

Cx

0

0 5 m 20 kN 2 m 2.5 m

2.5 m 14 kN 5 m 20 kN 2 m

12 kN

12 kN

F R R

M R R

R

R

R


7/33

ZeroForce members

When only two members form a non-collinear truss joint

and no external load or support reaction is applied to the

joint then the members must be zero-force members.

When three members form a truss joint for which two of the

members are collinear and the third is forms an angle with

the first two, then the non-collinear member is a zero-force

member provided no external force or support reaction isapplied to the joint, The collinear members carry equal

loads.


8/33

ZeroForce members

Is there a zero force

member in the truss?


9/33

ZeroForce members

Is there a zero force

member in the truss?


10/33

Example Problem

The awning structure

is supported by ball

and socket joints at A

and C and by a short

link at B. The 750 N

force is parallel to the

z-axis. Determine the

support reactions andthe force in each

member of the space

truss.

EGAD!

3-D Trusses!


11/33

3-D Trusses

The difference between 2-D and 3-D is that there

are three more equations we will be concerned

with and will require a bit more bookkeeping.

Instead of just

use the 6 sets of equations

x y z0 0 0F F M

x y z

x y z

0 0 0

0 0 0

F F F

M M M


12/33

Simple Truss

A truss is analysis by using m=3*j-6, where m is

number of members, j represents the number of

joints and 6 represents the external supportreactions.


13/33

Simple Truss

If m< 3j-6, then the truss is unstable and will

collapse under load.

If m> 3j-6, then the truss has more unknowns

than know equations and is an indeterminate

structure.

If m= 3j-6, ensures that a simple space truss is

rigid and solvable, it is neither sufficient nor

necessary to ensure that a non-simple plane truss

is rigid and solvable.


14/33

Example Problem

The awning structure

is supported by ball

and socket joints at A

and C and by a shortlink at B. The 750 N

force is parallel to the

z-axis. Determine the

support reactions andthe force in each

member of the space

truss.


15/33

Example Problem

Draw the free body diagram

A C B

x Ax Cx

y Ay Cy B

Ay Cy B

z Az Cz

Az Cz

0 750 N

0

0

0 750 N

750 N

F R R R j k

F R R

F R R R

R R RF R R

R R


16/33

Equilibrium ProblemExample

Take the moment about A.

A AW 1 AB B AC C0M r W r R r R

1

1

AW

AW 1

4.0 m 5.0 m 0.0 m

4.0 m 5.0 m 0.0 m

0 0 750 N

3750 N-m 3000 N-m 0

r i j k

i j k

r W

i j k

AB

AB B

B

B B

4.0 m 0.0 m 4.0 m

4.0 m 0.0 m 4.0 m

0 0

4.0 m 0 4.0 m

r i j k

i j k

r R

R

R i j R k


17/33


Take the moment about A.

A AW 1 AB B AC C0M r W r R r R

AC

AC C

Cx Cy Cz

Cz Cy

8.0 m 0.0 m 0.0 m

8.0 m 0.0 m 0.0 m

0 8 m 8 m

r i j k

i j k

r R

R R R

i R j R k


18/33


Using the summations of moments about A and look at the

components.

B B

Cz Cz

B Cy Cy

component

3750 N-m 4.0 m 0 937.5 N

component

3000 N-m 8.0 m 0 375 N

component

4.0 m 8.0 m 0 468.75 N

i

R R

j

R R

k

R R R


19/33


Using the summations of forces about

A and look at the components. The

results are

Unfortunately, RAx and RCx could notbe computed.

Ay

Az

component

468.75 N

component

375 N

j

R

k

R


20/33


21/33

Equilibrium ProblemExampleUsing the method of joints look at joint A

A AB AB AD AD

x Ax AB AD

y AD

AD

z AB

AB

Ax

0

0 0.7071 0.6247

0 468.75 N 0.7809

600.3 N

0 375 N 0.7071

530.33 N

750 N

F R T T

F R T T

F T

T

F T

T

R

So the bars are in compression

multiply by the unit vector to get the

final set of forces.


22/33

Equilibrium ProblemExampleUsing the method of joints look at joint C and the

values will be same due to symmetry.

C CB CB CD AD

x Cx CB CD

y CD

CD

z CB

CB

Cx

0

0 0.7071 0.6247

0 468.75 N 0.7809

600.3 N

0 375 N 0.7071

530.33 N

750 N

F R T T

F R T T

F T

T

F T

T

R

So the bars are in compression multiply by the unit

vector to get the final set of forces.


23/33

Equilibrium ProblemExampleUsing the method of joints look at joint D

z DB

DB

0 750 N 0.6247

1200.6 N

F T

T

Look at the y component of joint D

DB

2 2 2

DB

DB

0 m 5 m 4 m

0 m 5 m 4 m 6.403 m

0 m 5 m 4 m6.403 m 6.403 m 6.403 m

0.0 0.7809 0.6247

r i j k

r

i j k

i j k


24/33

3-D Trusses - Class Problem

The space truss is supported by ball and socket joints at D and

by a short link at A and C. Determine the support reactions

and the force in each member of the space truss.


25/33

Frames -Definition

Frames are structures that

always contain at least one

member acted on by forces

at three or more points.Frames are constructed and

supported so as to prevent

any motion. Frame like

structures that are not fully

constrained are called

machines or mechanisms.


26/33

Frames

In order to determine the

internal forces holding the

various parts of a frame

together we mustdismember the frame and

draw a free-body diagram

for each of the component

parts.


27/33

FramesExample

For the marine crane, which is

used in offshore drilling

operations, determine (a) the

force in link CD (b) the force inbrace AC and ( c) the force

exerted at on boom AB


28/33

Frames

Example

Draw the free-body diagram

x Ax

Ax

3 2

y Ay

Ay

0

0 N

0 90 x10 kg 9.81 m/s

882900 N or 882.9 kN

F R

R

F D R

D R

RAxRAyD

W


29/33

Frames

Example

Take the moment about A

A

Ay

0

15 m 882.9 kN 38 m

2236.68 kN

R 3119.58 kN

M

D

D

RAxRAyDW


30/33

Frames

Example

Cut along CD

A

CD

CD CD

0

15 m 882.9 kN 38 m

2236.68 kN 2236.68 kN T

M

T

T T

TCD


31/33

Frames

Example

Look at joint A

FACFAB

1 o

1 o

18 mtan 50.19

15 m

58 mtan 55.4135 m


32/33


33/33

Homework (Due 3/5/03)

Problems:

7-58, 7-60, 7-66, 7-68

Documents

ENGR221 Lecture 13