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Failure of slender and stocky columns Dr Alessandro Palmeri <[email protected]>

Failure of slender and stocky columns (2nd year)

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Lecture slides on the mathematical derivation and application of the Euler's buckling load for slender column, as well as the Rankine's failure load.

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Page 1: Failure of slender and stocky columns (2nd year)

Failure  of  slender  and  stocky  columns  

Dr  Alessandro  Palmeri  <[email protected]>  

Page 2: Failure of slender and stocky columns (2nd year)

Teaching  schedule  Week Lecture 1 Staff Lecture 2 Staff Tutorial Staff 1 Beam Shear Stresses 1 A P Beam Shear Stresses 2 A P --- --- 2 Shear centres A P Basic Concepts J E-R Shear Centre A P 3 Principle of Virtual

forces J E-R Indeterminate Structures J E-R Virtual Forces J E-R

4 The Compatibility Method

J E-R Examples J E-R Virtual Forces J E-R

5 Examples J E-R Moment Distribution -Basics

J E-R Comp. Method J E-R

6 The Hardy Cross Method

J E-R Fixed End Moments J E-R Comp. Method J E-R

7 Examples J E-R Non Sway Frames J E-R Mom. Dist J E-R 8 Column Stability 1 A P Sway Frames J E-R Mom. Dist J E-R 9 Column Stability 2 A P Unsymmetric Bending 1 A P Colum Stability A P 10 Unsymmetric Bending 2 A P Complex Stress/Strain A P Unsymmetric

Bending A P

11 Complex Stress/Strain A P Complex Stress/Strain A P Complex Stress/Strain

A P

Christmas Holiday

12 Revision 13 14 Exams 15 2  

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Mo@va@ons  (1/5)  

•  Load-­‐carrying  structures  may  fail  in  a  variety  of  ways,  depending  upon:  –  Type  of  structure  (truss,  frame,  …)  –  Condi@ons  of  support  (pinned,  fixed,  …)  –  Loads  applied  (sta@c,  dynamic,  …)  –  Materials  used  (briQle,  duc@le,  …)  

•  Failures  are  prevented  by  designing  structures  so  that  maximum  stresses  (strength  criterion)  and  maximum  displacements  (s,ffness  criterion)  remain  within  admissible  limits  

3  

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Mo@va@ons  (2/5)  

4  

•  For  the  fans  of  The  Big  Bang  Theory:  –  Sheldon  and  Howard  have  got  this  seriously  wrong!  

•  You  can’t  use  the  Young’s  modulus  to  quan@fy  the  strength  of  material,  but  its  s,ffness!    

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Mo@va@ons    (3/5)  

•  S@ffness  and  strength  of  materials  –  In  the  stress-­‐strain  curve  for  a  duc@le  material  (e.g.  steel),  the  Young’s  modulus  E  defines  the  s@ffness,  while  the  yield  stress  σy  represents  the  strength  

5  

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Mo@va@ons    (4/5)  

•  S@ffness  criterion:    “Slender  Column”  

6  

•  Strength  criterion:  “Short  Column”  

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Mo@va@ons    (5/5)  

7  

Coventry  Cathedral  

ç  Slender  column  

Detail  of  the  support  è  

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Learning  Outcomes  

•  When  we  have  completed  this  unit  (2  lectures  +  1  tutorial),  you  should  be  able  to:  

– Derive  the  Euler’s  cri@cal  load  for  slender  pinned-­‐pinned  columns  in  compression  

– Predict  the  mode  of  failure  for  both  short  and  slender  columns  in  compression  

8  

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Further  reading  

•  R  C  Hibbeler,  “Mechanics  of  Materials”,  8th  Ed,  Pren@ce  Hall  –  Chapter  13  on  “Buckling  of  Column”  

•  T  H  G  Megson,  “Structural  and  Stress  Analysis”,  2nd  Ed,  Elsevier  –  Chapter  21  on  “Structural  Instability”  (eBook)  

9  

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Short  and  Slender  Struts  

10  

•  Increasing  the  length  of  a  strut  reduces  its  buckling  load  –  For  instance,  a  matchs,ck  is  reasonably  strong  in  compression  (lek),  but  a  longer  s,ck,  with  the  same  cross  sec@on  and  the  same  material,  would  be  weaker  and  buckles  in  compression  (right)  

–  The  slenderness  of  a  strut  plays  an  important  role  in  its  mode  of  failure  in  compression  

Page 11: Failure of slender and stocky columns (2nd year)

Buckling,  i.e.  Lateral  Instability  (1/2)  

11  

•  That  is,  if  a  column  is  rela@vely  slender,  it  may  deflect  laterally  when  subjected  to  a  compressive  force  P  (Fig  (a))  and  fail  by  bending  (Fig  (b)),  rather  than  failing  by  direct  compression  of  the  material  

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Buckling,  i.e.  Lateral  Instability  (2/2)  

12  

•  Pcrit  is  the  so-­‐called  cri,cal  buckling  load  –  If  the  axial  load  P  is  less  than  Pcrit,  bending  is  caused  by  lateral  loads  only  –  If  P  is  greater  than  Pcrit,  the  ruler  bends  even  without  lateral  loads  

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Euler’s  Cri@cal  Load  for  Pinned-­‐Pinned  Slender  Columns  

•  One  of  the  Learning  Outcomes  of  this  Unit  is  for  you  to  become  able  to  mathema@cally  derive  (and  remember  as  well)  the  expression  of  the  cri@cal  load  Pcrit  for  pinned-­‐pinned  slender  column  

•  Pcrit=PE  is  oken  called  Euler’s  buckling  load  –  Aker  the  Swiss  mathema@cian  Leonhard  Euler  (1707-­‐1783)  

!!Pcrit =

π 2EImin

L2

13  

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Mathema@cal  Deriva@on:  Bending  Equa@on  

•  What’s  the  equa@on  ruling  the  beam’s  downward  deflec@on,  uz(x),  for  a  given  bending  moment  diagram,  My(x)?  

•  We  used  this  second-­‐order  differen@al  equa@on  in  part  A  to  calculate  the  beam’s  deflec@on  under  transverse  loads…  

•  where,  as  usual:  –  E=  Young’s  modulus  –  Iyy=  Second  moment  of  area  about  the  horizontal  neutral  axis  

14  

EIyyd2uz (x)dx2

= −My (x)

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Mathema@cal  Deriva@on:  Sign  Conven@on  

•  Do  you  remember  from  last  year?  

15  

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Mathema@cal  Deriva@on:  P-­‐Delta  (1/2)  

•  What’s  the  bending  moment  My  in  this  circumstance?    

•  We  don’t  have  transverse  loads  this  @me  –  the  column  is  subjected  to  the  axial  load  P  only  

•  How  can  we  have  a  bending  moment?    

•  In  order  to  derive  the  expression  of  the  Euler’s  buckling  load,  we  need  to  assume  that  –  a  disturbance/imperfec@on  exists  in  the  column,  –  therefore  the  buckling  occurs  –  and  My  can  be  consistently  evaluated  by  using  the  equilibrium  

equa@ons  in  the  deformed  shape  16  

Page 17: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  P-­‐Delta  (1/2)  

17  

Deformed  shape  

Equilibrium  condi@on  P  

P   My=  P  uz  

EIyy  

z  

uz  

Page 18: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Buckling  Equa@on  (1/2)  

•  Knowing  the  bending  moment  My  in  the  deformed  shape:  

•  we  can  subs@tute  it  within  the  deflec@on  equa@on:  

 •  This  equa@on  can  be  rewriQen  as:  

•  Where  α  is  a  posi@ve  quan@ty,  given  by:  

    18  

My = Puz

EIyyd2uzdx2

= −Puz

d2uzdx2

+α 2uz =0

α = PEIyy

Page 19: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Buckling  Equa@on  (2/2)  

•  What  do  we  do  in  order  to  solve  an  ordinary  differen@al  equa@on?  

 

•  First,  we  find  the  general  solu,on,  which  contains  as  many  integra@on  constants  as  the  order  of  the  differen@al  equa@on  (two,  in  this  case)  

         

19  

uz= C

1cos(αx) + C

2sin(αx)

Page 20: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Boundary  Condi@ons  (1/2)  

•  Second,  we  apply  the  boundary  condi,ons  (BCs)  to  get  the  values  of  the  integra@ons  constants  for  the  par@cular  case  

–  For  two  unknown  constants,  C1  and  C2,  two  BCs  are  needed!    

•  For  a  pinned-­‐pinned  column,  the  BCs  read:  

•  uz=0  @  x=0  (i.e.  the  transverse  transla@on  is  prevented  at  the  lek-­‐hand  side  end)  

 

•  uz=0  @  x=L  (i.e.  the  transverse  transla@on  is  prevented  at  the  right-­‐hand  side  end  as  well)  

       

20  

EIyy

z

Page 21: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Boundary  Condi@ons  (2/2)  

21  

•  The  applica@on  of  the  first  BC  is  quite  straighrorward  

   uz(x) = C

1cos(αx) + C

2sin(αx)

uz= 0@ x = 0 1 20 1 0C C⇒ = × + ×

1 0C⇒ =

General  solu5on  

Boundary  condi5on  

Page 22: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Non-­‐Trivial  Solu@on  (1/3)  

22  

•  The  second  BC  does  require  more  effort  

•  Trivial  solu@on:  –  It  would  follow  y=0  for  any  value  of  the  abscissa  x  –  No  transverse  displacements  would  occur  (straight  column)  –  This  solu@on  is  therefore  unacceptable    

•  Non-­‐trivial  solu@on:      

uz(x) = C

1cos(αx) + C

2sin(αx)

uz= 0@ x = L 20 sin( )C Lα⇒ =

sin( ) 0Lα⇒ = L nα π⇒ =

nnLπα α⇒ = =

2 0C⇒ =

Page 23: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Non-­‐Trivial  Solu@on  (2/3)  

23  

•  Recalling  now  the  expression  of  the  parameter  a,  one  obtains:    

•  The  associated  modes  of  instability,  for  n=  1,  2,  3,  …,  are  sinusoidal  func@ons,  having  a  total  number  n  of  peaks  and  valleys  

       

PEIyy

= nπL

⇒ PEIyy

= n2 π 2

L2⇒ P

n= n2

π 2 EIyy

L2

n= 3

n= 2

n= 1

Page 24: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Non-­‐Trivial  Solu@on  (3/3)  

24  

•  Larger  values  of  the  buckling  load  are  associated  to  more  complicated  modes  of  instability  

•  Theore@cally,  these  modes  could  be  achieved  if  roller  supports  are  applied  at  the  points  of  contraflexure  

•  However,  in  prac@ce,  the  lower  value  P1  is  never  exceeded  

Page 25: Failure of slender and stocky columns (2nd year)

Mathema@cal  Deriva@on:  Euler’s  Buckling  Load  

•  The  actual  cri,cal  load,  i.e.  the  so-­‐called  Euler’s  buckling  load,  is  the  “engineering  solu@on”,  which  is  the  minimum  among  the  mathema@cal  solu@ons  P1,  P2,  P3,  …,  and  is  obtained  for  n=1  

•  Moreover,  in  order  to  be  truly  the  minimum,  you  must  use  the  minimum  value  of  the  second  moment  of  area,  which  might  not  be  Iyy  

•  The  laQer  expression  is  very  important  in  Structural  Engineering  –  You  are  requested  to  remember  it  –  You  must  be  able  to  derive  this  expression  as  well  

25  

Pcrit

= PE= P

1=π 2 EI

yy

L2

2min

E 2EIPL

π=

Page 26: Failure of slender and stocky columns (2nd year)

FEM-­‐Computed  Modes  of  Instability  

26  

•  Euler’s  buckling  load  (PE=  P1=  181  kN)  

•  Higher  buckling  load  in  the  orthogonal  direc@on  

•  (P4=  3,518  kN)  

XY X

Z

Y

ZXY X

Z

Y

Z

4min 330 cmI = = 4

max 6572 cmI

Horizontal  sway  

Ver0cal  deflec0on  

Page 27: Failure of slender and stocky columns (2nd year)

Effects  of  the  Boundary  Condi@ons  (1/2)  

27  

The  more  the  column’s  ends  are  restrained,  the  higher  is  the  buckling  load  

Similar  sinusoidal  shapes  are  

observed  for  different  BCs  

Page 28: Failure of slender and stocky columns (2nd year)

Effects  of  the  Boundary  Condi@ons  (2/2)  

28  

(a)  Pinned-­‐pinned  (b)  Can@levered  (c)  Fixed-­‐fixed  (d)  Propped  

L0  is  the  distance  between  two  consecu@ve  crosses  of  the  horizontal  axis  

Page 29: Failure of slender and stocky columns (2nd year)

Effec@ve  Length  •  It  is  useful  to  introduce  the  concept  of  equivalent  length,  Le=k  L  as  

the  length  of  a  pinned-­‐pinned  column  having  the  same  Euler’s  cri@cal  load  

•  We  therefore  must  know  the  value  of  the  coefficient  k  for  different  BCs  

29  

k=2  

k=1   k=0.7   k=0.5  

Can5levered  Pinned-­‐  pinned   Propped  

Fixed-­‐  fixed  

2min

E 2e

EIPL

π=

Page 30: Failure of slender and stocky columns (2nd year)

Stocky  Columns  (1/2)  •  If  we  divide  the  Euler’s  cri@cal  load  PE  by  the  cross  sec@onal  area  A,  we  get  

the  so-­‐called  Euler’s  cri@cal  stress  σcrit:  

   

•  This  is  the  maximum  normal  stress  which  is  allowable  to  prevent  buckling  instability,  and  is  inversely  propor@onal  to  the  square  of  the  equivalent          length  Le  

•  If  we  introduce  the  parameter  ρmin  as  the  minimum  radius  of  gyra@on  of  the  cross  sec@on,  and  then  the  slenderness  ra@o  λ=Le/ρmin,  the  above  equa@on  can  be  rewriQen  as:  

   

30  σcrit

=π 2 E I

minA( )

Le2

=π 2 E ρ

min2

Le2

= π 2 E

Le

ρmin( )2

= π 2

λ2Eρ

min=Imin

A

2min

crit 2e

EP EIA AL

πσ = =

Page 31: Failure of slender and stocky columns (2nd year)

Stocky  Columns  (2/2)  

31  

•  The  s@ffer  the  material,  i.e.  the  larger  the  Young’s  modulus  E,  the  higher  is  σcrit:  

•  The  larger  the  slenderness  ra@o  λ,  the  lower  is  σcrit,  i.e.  very  slender  columns  will  have  very  low  values  of  σcrit  

•  Conversely,  stocky  columns,  with  a  small  slenderness  ra@o  λ,  will  not  experience  the  buckling  failure,  as  the  yielding  of  the  material  is  likely  to  happen  first:  

2

crit 2 Eπσλ

=

crit y Material’s yield stressfσ > =

Page 32: Failure of slender and stocky columns (2nd year)

Strength  and  S@ffness  Criteria  (1/2)  

•  “Strength”  criterion  

•  Stocky  columns  tend  to  fail  because  the  elas@c  limit  of  the  material  is  reached  

•  The  safety  checks  is:      

•  “S,ffness”  criterion  

•  Slender  columns  tend  to  fail  because  the  elas@c  configura@on  is  unstable  

•  The  safety  check  is:      

32  

2min

E 2e

EIP PL

π< =y yP P f A< =

Both  must  be  sa0sfied

Page 33: Failure of slender and stocky columns (2nd year)

Strength  and  S@ffness  Criteria  (2/2)  

•  For  briQle  materials  such  as  concrete,  the  yielding  stress  fy  is  replaced  with  the  crushing  stress  fc  

•  The  safety  check  then  reads:  

   

33  

2min

E 2e

EIP PL

π< =P < Pc= f

cA

Both  must  be  sa0sfied

fc  

Page 34: Failure of slender and stocky columns (2nd year)

Strength  and  S@ffness  Criteria  

34  

•  The  Rankine’s  failure  load  PR  combines  these  two  different  criteria,  therefore  taking  into  account  both  material  and  geometrical  nonlineari@es  

•  PR=Py  for  λ=0  

•  PR  approaches  PE  as  λ  goes  to  infinity  

0 100 200 300 4000.0

0.5

1.0

1.5

2.0

l

PêPy Py  

PE  PR  

PR=PyPE

Py+ P

Eλ  

P/P y

 

Rankine  (1820-­‐1872)  was  a  Scoush  civil  engineer,  physicist  and  mathema@cian  

Page 35: Failure of slender and stocky columns (2nd year)

Ul@mate  Normal  Stress  

35  

•  …  experimentally  derived  (dots)  for  wide-­‐flange  steel  columns  

•  …  as  a  func@on  of  the  slenderness  λ=  k  L/ρmin  

λ  

Page 36: Failure of slender and stocky columns (2nd year)

Key  Learning  Points  1.  Columns  in  compression  may  fail  because  

–  Insufficient  bending  s@ffness:  è  Lateral  buckling  –  Insufficient  axial  capacity:  è  Yielding/Crushing  

2.  Euler’s  buckling  load  PE  depends  on:  –  Minimum  second  moment  of  area,  Imin  

–  Length  of  the  column,  L  –  Boundary  condi@ons  

3.  Interac@on  between  lateral  buckling  and  axial  capacity  can  be  taken  into  account  through  the  (approximate)  Rankine’s  formula  

36  

è  Effec@ve  length,  Le  

PR=PyPE

Py+ P

E

PE =π 2EImin

Le2