Structural Analysis Stability And Degrees Of Indeterminacy

This document is part of the notes written by Terje Haukaas and posted at www.inrisk.ubc.ca. The notes are revised without notice and they are provided “as is” without warranty of any kind. You are encouraged to submit comments, suggestions, and questions to [email protected]. It is unnecessary to print these notes because they will remain available online.

Stability and Degrees of Indeterminacy

A  structure  has  many  characteristics,  many  of  which  are  properties  of  the  structural   members.   This   document,   however,   describes   some   global   characteristics   that   are   important   in   linear   static   structural   analysis.   The   characteristics   are   stability   and   degree  of  static  and  kinematic  indeterminacy.  

Stability   Stability   implies   that   there   are   no   modes   of   deformation   with   zero   stiffness.   Such   modes   are   often   called   mechanisms   and   they   make   the   structure   unstable.   An   unstable   structure   will   collapse   even   without   load.   For   2D   structural   models   it   is   often  straightforward  to  see  that  a  structure  is  unstable.  Figure  1  shows  examples  of   unstable   structural   models.   The   two   hinges  at   the   top   of   the   frame   combined   with   the   pinned   supports   means   that   this   structure   will   collapse   sideways.   Even   one   hinge   would   be   sufficient   to   make   it   unstable.   The   truss   structure   in   Figure   1   is   also   unstable;  a  cross  brace  is  required  to  make  this  a  useful  structure.  

Terje Haukaas

University of British Columbia

Frame

Truss

www.inrisk.ubc.ca

 

Figure  1:  Unstable  frame  and  truss.  

Obviously,   great   care   must   be   exercised   to   avoid   the   presence   of   instabilities   in   structures.  Fortunately,  there  exist  indicators  that  expose  instabilities.  The  first  is  a   negative  degree  of  static  indeterminacy,  described  shortly.  However,  this  indicator   is   imperfect.   A   structure   can   be   statically   determinate   and   still   unstable.   Another   way  of  detecting  instabilities  is  to  model  the  structure  by  the  stiffness  method  and   attempt   to   solve   the   resulting   linear   system   of   equations.   If   the   stiffness   matrix   is   singular,  i.e.,  it  cannot  be  inverted,  then  instability  may  be  the  cause.    

Degree  of  Static  Indeterminacy  

A   structure’s   degree   of   static   indeterminacy   (DSI)   exposes   the   deficit   of   equilibrium   equations  compared  with  the  number  of  unknown  internal  forces  in  the  structure.   In   other   words,   for   a   statically   determinate   structure   (DSI=0)   it   is   possible   to   compute  the  section  forces  (M,  V,  N)  by  equilibrium  equations  alone.  More  advanced   methods  are  required  for  structures  that  are  statically  indeterminate.     Different   engineers   have   different   habits   when   it   comes   to   determining   the   DSI.   However,   every   rule   boils   down   to   counting   the   number   of   unknowns   and   comparing  it  with  the  number  of  available  equilibrium  equations.  In  this  document,   this  calculation  is  set  up  as  follows:  

DSI = ( f ! m + r ) " ( e ! j + h )  

 

(1)  

where  all  variables  are  non-­‐negative  integers  with  the  following  meaning:   • • • • • •

f  =  forces       m  =  members     r  =  restraints     e  =  equations     j  =  joints     h  =  hinges    

=  number  of  internal  force  in  each  member   =  number  of  members   =  number  of  restraints,  i.e.,  boundary  conditions   =  number  of  equilibrium  equations  per  joint   =  number  of  joints   =  number  of  hinges  or  other  section  force  releases    

The   number   of   internal   forces,   f,   in   each   member   depends   on   the   member   type.   A   truss   member   has   only   one   unknown   force:   the   axial   force.   Conversely,   a   frame   member  in  a  2D  structural  model  has  three  internal  forces:  axial  force,  shear  force,   and   bending   moment.   This   number   increases   from   three   to   six   for   3D   frame   members.  Table  1  summarizes  the  value  of  f  for  different  structures.     The   number   of   equilibrium   equations,   e,   per   joint   is   obtained   by   counting   the   orthogonal  directions  in  which  equilibrium  can  be  considered.  For  the  typical  case   of   2D   frame   structures   there   are   three   equilibrium  equations  per  joint:  horizontal,   Stability and Degrees of Indeterminacy

Page 2

Terje Haukaas

University of British Columbia

www.inrisk.ubc.ca

vertical,   and   angular   equilibrium.   For   a   joint   in   a   2D   structure   with   only   truss   members  entering,  i.e.,  member  without  bending  stiffness,  rotational  equilibrium  is   cancelled.  Table  1  summarizes  the  value  of  e  for  different  structures.   Table  1:  Forces  per  member  and  equations  per  joint.  

 

f  

e  

2D  truss  

1  

2  

2D  frame  

3  

3  

3D  truss  

1  

3  

3D  frame  

6  

6  

  The   number   of   restraints,   r,   is   obtained   by   counting   the   number   of   support   reactions.   Although   rather   trivial,   Figure   2   provides   an   overview   of   the   number   of   unknown   reaction   forces   for   different   kinds   of   2D   boundary.   The   arrows   in   the   figure  show  the  forces.  The  degrees  of  freedom  will  be  described  later.     Support type

Unknown forces

Degrees of freedom

!"

#"

$"

%"

Roller

%"

$"

!"#$%&'

$"

%"

Hinge

$"

&"

Fixed

Pinned

 

Figure  2:  Number  of  unknown  forces  and  degrees  of  freedom  for  some  2D  joint  types.   Stability and Degrees of Indeterminacy

Page 3

Terje Haukaas

University of British Columbia

www.inrisk.ubc.ca

The   number   of   hinges,   h,   is   obtained   by   counting   the   number   of   hinges   and   releases   in   the   structure.   Each   hinge   represents   one   release,   i.e.,   the   specification   of   one   internal  force.  The  typical  example  is  to  replace  a  moment  connection  with  a  hinge   so  that  the  bending  moment  becomes  known  and  equal  to  zero.  The  determination   of  h  can  sometimes  seem  difficult,  especially  when  section  forces  other  than  bending   moments  are  released.  Examples  will  help.     For   frames   the   number   of   members,   m,   and   joints,   j,   in   a   structural   model   is   subjective.   However,   the   subjectivity   does   not   affect   the   final   DSI.   Usually,   joints   are   identified   wherever   there   is   a   boundary   condition   or  a   bend   or   intersection   in   the   structure.   If   for   some   reason   the   analyst   places   a   joint   in   the   middle   of   a   frame   member  then  this  increases  j  and  m  in  a  way  that  leaves  DSI  unchanged.    

Internal  and  External  Degrees  of  Indeterminacy   Although   practically   irrelevant,   it   is   possible   to   divide   the   DSI   into   two   categories.   The   external   DSI   is   the   number   of   boundary   conditions,   i.e.,   number   of   unknown   reaction  forces  minus  the  number  of  global  equilibrium  equations.  For  2D  structures   there  are  three  global  equilibrium  equations.  For  3D  structures  there  are  six.  As  an   example,   a   2D   structure   with   one   fixed   support   and   one   pinned   support,   i.e.,   five   unknown   reaction   forces,   has   an   external   DSI   equal   to   two   because   there   are   only   three   global   equilibrium   equations.   If   the   total   DSI   is   greater   then   the   rest   are   internal  DSI.    

Degree  of  Kinematic  Indeterminacy  

While   DSI   provides   information   about   unknown   member   forces   the   degree   of   kinematic  indeterminacy  (DKI)  exposes  the  number  of  unknown  joint  displacements   and  rotations.  DSI  is  a  key  number  in  force-­‐based  structural  analysis  methods  and   DKI   is   the   key   figure   in   displacement-­‐based   methods.   In   fact,   the   DSI   is   the   size   of   the   flexibility   matrix   and   DKI   is   the   size   of   the   stiffness   matrix,   for   the   flexibility   methods  and  the  stiffness  method,  respectively.     The   DKI   is   easier   to   determine   than   the   DSI.   Even   a   computer   can   do   it   in   a   straightforward   manner.   This   is   why   the   stiffness   method   is   implemented   in   all   structural   analysis   software,   while   the   flexibility   method   is   not.   Essentially,   DKI   counts  the  number  of  degrees  of  freedom  (DOFs)  of  a  structure.  Each  joint,  usually   called   node   in   displacement-­‐based   methods,   has   a   pre-­‐defined   number   of   DOFs.   A   2D   structure   has   three   DOFs   per   node,   i.e.,   three   possible   directions   to   move:   horizontal,   vertical,   and   rotation.   Similarly,   a   3D   structural   model   has   six   degrees   of   freedom  per  node:  three  displacements  and  three  rotations.  For  truss  structures  the   rotational   DOFs   are   neglected   altogether   because   they   are   associated   with   zero   stiffness   from   the   truss   elements.   Some   structural   analysis   programs   deal   with   trusses   by   first   keeping   all   rotational   degrees   of   freedom   and   later   restraining   them   in  the  same  way  as  nodes  with  boundary  conditions  are  restrained.     When  doing  hand  calculations  there  are  two  exceptions  to  the  rule  that  every  node   has   equally   many   DOFs.   The   first   is   for   nodes   with   boundary   conditions.   For  

Stability and Degrees of Indeterminacy

Page 4

Terje Haukaas

University of British Columbia

www.inrisk.ubc.ca

example,   a   fixed   node   has   zero   DOFs.   Figure   2   provides   an   overview   for   2D   structures   of   the   number   of   DOFs   per   node   for   different   boundary   conditions.   The   second  exception  to  the  rule  appears  when  axial  deformations  are  neglected  in  the   analysis   of   frame   structures.   This   is   quite   common   in   hand   calculations   with   the   classical   stiffness   method   because   the   axial   stiffness   of   frame   members   is   usually   significantly   higher   than   the   bending   stiffness.   Neglecting   axial   deformations   requires   careful   consideration   of   the   DOFs   at   each   node,   which   is   difficult   for   a   computer.   Hence,   in   computer   analysis   it   is   easier   to   always   account   for   axial   deformations.   By   hand,   one   simply   removes   the   DOFs   that   will   experience   zero   displacement  when  the  members  do  not  deform  axially.    

Stability and Degrees of Indeterminacy

Page 5