a 1,000-word essay. In the essay, you must summarize what you consider to be the key concepts from the chapters covered, and you must include at least three photographs that you've taken to help explain the concepts.

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BENDING  

Chapter  8  

Perspec,ve   This  chapter  is  ul,mately  about  just  one  thing:       It  is  a  simple  equa,on  describing  the  development  of  normal   stresses  in  beams  from  internal  reac,on  moments.       But  there  are  obstacles.  Intricate,  mul,-­‐step  methods  are   required  to  find  the  terms  M  and  IZ,  and  since  normal  stresses   vary  over  a  beam  cross-­‐sec,on,  you  must  pay  aFen,on  to  where   on  the  cross-­‐sec,on  to  perform  the  calcula,on.       You  must  be  very  detail-­‐oriented  to  master  this  material.          

σ = − My IZ

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INTRODUCTION   8.1  

Coordinate  System   •  Note  the  coordinate  system  –  we  will  use  this  throughout  

•  The  x-­‐axis  begins  at  the  leO  and  runs  along  the  beam  length   •  The  z-­‐axis  is  transverse  and  at  the  cross-­‐sec,on  centroid   •  The  y-­‐axis  is  posi,ve  ver,cally  

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Side  View  /  Cross-­‐sec,on  View   •  You  will  switch  back  and  forth  between  these  perspec,ves  

Side  view  

Cross-­‐sec,on  view  

FLEXURAL  STRAINS    

8.2  

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Deriva,on   •  Flexural  strain  is  not  generally  calculated   •  The  rela,onship  between  overall  beam  deforma,on  and  

strain  is  used  to  derive  the  widely  used  stress  formula  

•  Note  that  the  deriva,on  assumes  constant  beam  curvature   which  is  almost  never  true  

•  Even  when  beam  curvature  varies,  the  equa,ons  derived   based  on  constant  curvature  are  generally  “close  enough”  for   most  engineering  purposes  

NORMAL  STRESSES   IN  BEAMS    

8.3  

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The  Equa,on   •  You  will  become  very  familiar  with  this  equa,on:  

•  The  terms:    

σ = − My IZ

σ = lognitudinal (perpendicular to cross-section) normal stress M = internal reaction bending moment, from the bending moment curve y = distance from the centroidal z-axis to the location of analysis IZ = moment of inertia (second moment of area) about z-axis

ANALYSIS  OF  BENDING   STRESSES  IN  BEAMS    

8.4  

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Divide  and  Conquer   •  That  is  the  advice  one  of  my  professors  imparted  for  this  

material:   –  Divide  problems  into  sub-­‐problems   –  Conquer  each  sub-­‐problem  as  part  of  the  overall  solu,on  

•  For  beam  bending:   –  The  flexure  formula  defines  the  sub-­‐problems   –  Focus  on  each  term   –  Work  towards  a  conceptual  understanding  of  what  each  term  means   –  Treat  each  term  as  a  small  problem  in  itself   –  Draw  on  past  experience   –  If  you  do  not  recall  material  typically  covered  in  Sta,cs  (finding  

centroids,  parallel  axis  theorem  etc.),  then  review  

Flexural  Formula  Term:  M   •  The  term  M  is  the  internal  reac,on  bending  moment   •  Since  this  varies  generally  with  loca,on  along  a  beam,  the  

first  task  is  determining  where  along  the  beam  to  analyze   •  Loca,ons  of  greatest  posi,ve  and  nega,ve  moment  are  the  

most  important,  for  example  -­‐6000  and  5625  here   •  Watch  for  problems  where  analysis  at  a  specific  loca,on  along  

the  beam  is  described  –  use  the  corresponding  M  value    

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Flexural  Formula  Term:  y   •  Now  you  shiO  focus  to  the  beam  cross-­‐sec,on   •  The  term  y  is  the  distance  from  the  neutral  axis  to  the  level  

where  you  want  to  calculate  stress   •  Note  that  y  has  a  sign  

•  Generally  the  top  and  boFom  levels  are  the  most  important   •  Watch  for  problems  where  analysis  at  another  level  is  

described  –  use  the  corresponding  y  value    

At  the  top  of  this  beam,  y  =  +7  in     At  the  boFom,  y  =  -­‐5  in     At  the  centroid,  y  =  0  

Flexural  Formula  Term:  IZ   •  This  is  another  beam  cross-­‐sec,on  term   •  It  is  the  moment  of  iner,a  (second  moment  of  the  area)  

about  the  z-­‐axis,  which  is  aFached  to  the  centroid  

•  I  will  only  assign  problems  where  IZ  can  be  found  using  the   composite  approach  and  parallel  axis  theorem  

IZ  is  calculated  about  this  axis  

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Flexural  Formula  Term:  -­‐   •  The  formula  contains  a  nega,ve  sign  for  an  important  reason  

–  The  proper  sense  of  the  resul,ng  stress  (tension,  compression)  is   accurately  predicted  if  all  the  sign  conven,ons  (sign  of  the  internal   reac,on  moment  M,  sign  of  the  posi,on  variable  y)  are  followed  

•  This  is  par,cularly  important  in  problems  with  beams  that  are   not  symmetric  top-­‐to-­‐boFom  

INTRODUCTORY  BEAM   DESIGN  FOR  STRENGTH    

8.5  

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A  Simple  Idea   •  As  a  first  approxima,on,  beam  strength  can  simply  be  

evaluated  by  comparing  the  maximum  normal  stress  from  the   flexure  formula  with  a  normal  stress  allowable  

•  The  sec,on  modulus  defini,on  simply  makes  this  process   simpler,  rela,ng  allowable  stress  directly  with  M  

•  Please  note  that  we  will  not  take  our  study  of  beam  flexure   any  further  than  this  sec,on.