Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Chapter  5:  Deformation

Let  me  tell  you  the  secret  that  has  led  me  to  my  goal.  My  strength  lies  in  my  tenacity.

Louis  Pasteur

Testing  of  787  Dreamliner  wings.  (Courtesy  of  Boeing  Corp.)

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Moment-­‐‑Curvature  Relations In  terms  of  ordered  derivatives:

q

EI=

d4y

dx4

− V

EI=

d3y

dx3

M

EI=

d2y

dx2

θ =dy

dx

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Design  Procedure  5.1:  Deflection  by  Singularity  Functions

1.  Draw  a  free-­‐‑body  diagram  showing  the  forces  acting  on  the  system.

2.  Use  force  and  moment  equilibria  to  establish  reaction  forces  acting  on  the  system.

3.  Obtain  an  expression  for  the  load  intensity  function  for  all  the  loads  acting  on  the  system  while  making  use  of  Table  2.2.

4.  Integrate  the  negative  load  intensity  function  to  give  the  shear  force  and  then  integrate  the  negative  shear  force  to  give  the  moment.

5.  Make  use  of  Eq.  (5.9)  to  describe  the  deflection  at  any  location. 6.  Plot  the  following  as  a  function  of  x:

a.  Shear b.  Moment c.  Slope d.  Deflection

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Beams

Figure  5.1:    Cantilevered  beam  with  concentrated  force  applied  at  free  end.    

x

y

P

P

l

V M

x

a b

y

l

P

x

a x – a

y

P

V

M

(a)

(b)

Pb___lRA = RB = Pa___

l

Figure  5.2:    Free-­‐‑body  diagram  of  force  anywhere  between  simply  supported  ends.  (a)  Complete  beam;  (b)  portion  of  beam.

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Example  5.3

Figure  5.3:    Cantilevered  beam  with  unit  step  distribution  over  part  of  beam.  (a)  Loads  and  deflection  acting  on  cantilevered  beam;  (b)  free-­‐‑body  diagram  of  forces  and  moments  acting  on  entire  beam;  (c)  free-­‐‑body  diagram  of  forces  and  moments  acting  on  portion  of  beam.      

y

a

A B

C

bl

wo

RA = wob

(a)

(b)

(c)

a bwo

MA = wob a + b_2( )

a

x

V

M

x – awo

wob

wob a + b_2( )

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Example  5.4

Figure  5.4:    Pinned-­‐‑fixed  beam  with  concentrated  force  acting  anywhere  along  beam.  (a)  Sketch  of  assembly;  (b)  free-­‐‑body  diagram  of  entire  beam;  (c)  free-­‐‑body  diagram  of  part  of  beam.    

y

a

A C

bl

RC

(a)

(b)

MC

B

P

x

RA

a b

P

V

(c)

MA

RA

a

x

x – a

P

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Beam  Deflection

Table  5.1:  Deflection  for  common  cantilever  and  simply-­‐‑supported  beam  conditions.  See  also  Appendix  D.    

noitceeDgnidaoLfoepyT

y = P6EI

x a 3 x3 + 3 x2 a

When b = 0 , y = P6EI

3lx 2 x3

and ymax = y(l) =PL 33EI

xb

Py

l

ymaxa

y = wo24EI

4bx3 12bx2 a + b2

x a 4

When a = 0 and b = l, y = wo24EI

6l2x2 4lx 3 + x4

and ymax =wo l48EI

x

by

l

ymaxa wo

y = Mx 22EI

, ymax =Ml 22EIM

x

y

l ymax

y = P6EI

blx 3 x a 3 + 3 a 2x 2alx a 3x

lx

y

l

bP

a

Pal

Pbl

y = wo b24lEI

4 c + b2

x 3 lb

x a 4 x a b 4

+ x b3 + 6 bc2 + 4 b2 c + 4 c3 4l2 c + b2

x

y

l

a cb

wo

a + b–2wob–––l

c + b–2wob–––l

( )

( )

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Example  5.5

Figure  5.5:    Beam  fixed  at  one  end  and  free  at  other  with  moment  applied  to  free  end  and  concentrated  force  at  any  distance  from  free  end.  (a)  Complete  assembly;  (b)  free-­‐‑body  diagram  showing  effect  of  concentrated  force;  (c)  free-­‐‑body  diagram  showing  effect  of  moment.      

ya

AB

P

l

Cx

yl

(a)

Mo

ya

Px

x

yl, 1

yl, 2

(b)

(c)

yMo

l

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Stress  Elements

Figure  5.6:    Element  subjected  to  normal  stress.    

dx

z

dz

dy

Figure  5.7:      Element  subjected  to  shear  stress.

dx

dz

dydz

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Strain  Energy

Table  5.2:  Strain  energy  for  four  types  of  loading.      

Strain energy forspecial case whereall three factors are General expression

Loading type Factors involved constant with x for strain energy

Axial P, E, A U = P 2 l2EA U = P 2

2EAdx

Bending M,E, I U = M 2 l2EI

U = M 2

2EIdx

Torsion T, G, J U = T 2 l2GJ

U = T 22GJ

dx

Transverse shear V,G, A U = 3V 2 l5GA

U = 3V 25GA

dx(rectangular section)

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Example  5.8

Figure  5.8:    Cantilevered  beam  with  concentrated  force  acting  at  a  distance  b  from  free  end.  (a)  Coordinate  system  and  significant  points  shown;  (b)  fictitious  force,  Q,  shown  along  with  concentrated  force,  P.    

y

l

l

b

B CA

P

(a)

x

Q

bP

(b)

x

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Example  5.9

Figure  5.9:    System  arrangement.  (a)  Entire  assembly;  (b)  free-­‐‑body  diagram  of  forces  acting  at  point  A.    

(b)

PP2

P1

A Q

P

A

(a)

l, A1, E1

l, A2, E2

Fundamentals of Machine Elements, 3rd ed.�Schmid, Hamrock and Jacobson� © 2014 CRC Press�

Example  5.10

Figure  5.10:    Cantilevered  beam  with  90°  bend  acted  upon  by  horizontal  force,  P,  at  free  end.      

PA

BC

l

y

x

Q

h