ENGI 1313 Mechanics I

Lecture 39: Analysis of Friction with Wedges Shawn Kenny, Ph.D., P.Eng.

Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland [email protected]

Lecture 39 Objective

 to illustrate by example equilibrium analysis of wedges with dry friction forces

2 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Wedge

 Simple Machine  Inclined plane  Small applied force to generate larger (orthogonal) forces for stability, lifting or moving objects

3 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Wedge (cont.)

 Analysis of Force P  FBD of wedge first then object  Analyze FBD with least # of unknowns

F 2

 

s N 2

4 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

5

Wedge (cont.)

 Block Lowered onto Wedge  If P > 0 (+) then wedge is stable  If P = 0 then wedge is self-locking  If P < 0 (-) then wedge is unstable • • Small  s Large 

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

6

Example 39-01

 The wedge is used to level the floor of a building. For the floor loading shown, determine the horizontal force P that must be applied to move the wedge forward. The coefficient of static friction between the wedge and the two surfaces of contact is μ s = 0.25. Neglect the size and weight of the wedge and the thickness of the beam.

© 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Example 39-01 (cont.)

 FBD

7 © 2007 S. Kenny, Ph.D., P.Eng.

2 kN 4 kN 4 kN 2 kN N 1 F 1 N 1 F 1 N 2 F 2 ENGI 1313 Statics I – Lecture 39 P A y A x

Example 39-01 (cont.)

 Beam Normal Reaction Force 

M O



0



4 8 N 1 kN



2

 

kN



4

 

kN

    

0 N 1



6 kN

2 kN N 1 F 1 4 kN 4 kN 2 kN A y A x 8 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

9

Example 39-01 (cont.)

 Wedge Force P  Impending motion

N 1



6 kN F

 

s N

  

F y



0 N 2 cos 15

 

F 2 sin 15

 

N 1



0 N 2 cos 15

  

s N 2 sin 15

 

N 1



0 N 2 cos 15

 

0 .

25 N 2 sin 15

 

6 kN



0 N 2



6 .

658 kN



6 .

66 kN

N 1 N 2



F 1



F 2 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39 P



= 15



Example 39-01 (cont.)

 Wedge Force P  Impending motion

N 1



6 kN



F x N 2



6 .

66 kN F

 

s N

N 1 F 1

  

0



F 1



F 2 cos 15

 

N 2 sin 15

 

P



0



s N 1

 

s N 2 cos 15

 

N 2 sin 15

 

P 0 .

25



6 kN

 

0 .

25



6 .

658 kN



cos 15

 

0

N 2



F 2

 

6 .

658 kN



sin 15



P



= 15

 

P



0 P



4 .

831 kN



4 .

83 kN

10 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Example 39-02

 The wedge blocks are used to hold the specimen in a tension testing machine. Determine the design angle θ of the wedges so that the specimen will not slip regardless of the applied load. The coefficients of static friction are μ A = 0.1 at A and μ B = 0.6 at B. Neglect the weight of the blocks.

11 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Example 39-02 (cont.)

 FBD  Test coupon  Wedge grip

N B F B P F B N B 12 © 2007 S. Kenny, Ph.D., P.Eng.

F A N A



F B ENGI 1313 Statics I – Lecture 39 N B

Example 39-02 (cont.)

 Analysis of Test Coupon   

F y



0

F B

2 F B



P



0

N B

F B



P 2

P F B N B 13 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Example 39-02 (cont.)

 Analysis of Wedge Grip

F B



P 2

  

F y



0 F A cos

 

N A sin

 

F B



0



A N A cos

 

N A sin

 

P 2



0 P



2

 

A N A cos

 

N A sin

 

N A F A 14 © 2007 S. Kenny, Ph.D., P.Eng.



F B N B ENGI 1313 Statics I – Lecture 39

Example 39-02 (cont.)

 Analysis of Wedge Grip

F B



P 2 P



2

 

A N A cos

 

N A sin

    

F x



F A sin

 

0



N A cos

 

N B



0

 

A N A sin

 

N A cos

 

F B



B



0



P



A N A



sin

 

N A 2



B



N A cos



cos

  

P 2



B



A N A sin

  

0

N A F A 15 © 2007 S. Kenny, Ph.D., P.Eng.



F B ENGI 1313 Statics I – Lecture 39 N B

Example 39-02 (cont.)

 Analysis of Wedge Grip

P



2

 

A N A cos

 

N A sin

 

N A

P



2



B



N A cos

  

A N A sin

 

F A



F B N B

2



B



N A cos

  

A N A sin

   

A N A cos

 

N A sin

 

N A

 

B cos

  

A



B sin

  

N A

 

A cos

 

sin

   

B tan

  

A

 

cos

   

1

 

B

 

A

 

A B

  

1

  

A



B



sin

 

1



0



.

6 0 .



1 0 .

1 0 .

6

    

25 .

3



16 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

Chapter 8 Problems

 No Problem Set  Problem 8-1 through 8-70 applicable • Omit 8-45 and 8-61  Answers on page 647-648

17 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39

References

 Hibbeler (2007)  http://wps.prenhall.com/esm_hibbeler_eng mech_1

18 © 2007 S. Kenny, Ph.D., P.Eng.

ENGI 1313 Statics I – Lecture 39