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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SHEAR  STRESS  IN   BEAMS    

Chapter  9  

Perspec,ve   This  chapter  is  ul,mately  about  just  one  thing:       It  is  a  simple  equa,on  describing  the  development  of  shear   stresses  in  beams  from  internal  shear  forces.       The  difficul,es  are  that  intricate,  mul,-­‐step  methods  are   required  to  find  the  terms  V,  IZ,  and  Q,  and  since  shear  stresses   vary  over  a  beam  cross-­‐sec,on,  you  must  pay  aGen,on  to  where   on  the  cross-­‐sec,on  to  perform  the  calcula,on.       You  must  be  very  detail-­‐oriented  to  master  this  material.          

τ = VQ IZt

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INTRODUCTION   9.1  

Two  Stress  Components   •  There  is  a  stress  component  associated  with  each  of  the  

internal  reac,ons  in  beams  

–  Internal  moment  M  is  associated  with  normal  stress   –  Internal  shear  V  is  associated  with  shear  stress  

•  They  follow  different  paGerns  over  the  beam  cross-­‐sec,on  

–  Normal  stress  are  max  at  top  and  boGom,  zero  at  the  neutral  axis   –  Shear  stresses  are  max  at  the  neutral  axis,  zero  at  top  and  boGom  

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RESULTANT  FORCES   PRODUCED  BY  BENDING  

STRESSES    

9.2  

Which  Perspec,ve?   •  Both:  shear  stress  acts  on  both  the  beam  cross-­‐sec,on  and  on  

horizontal  planes  along  the  beam  length,  in  equal  magnitudes  

Cross-­‐sec,onal  faces   Horizontal  plane  

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Origin  of  the  Stress   •  If  bending  moment  is  different  at  two  cross-­‐sec,ons,  normal  

stress  is  different,  and  one  part  of  the  beam  is  trying  to  “push   away”  from  another  part  of  the  beam  

•  Shear  stresses  act  to  balance  this  varying  push-­‐off  force  

THE  SHEAR  STRESS   FORMULA    

9.3  

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A  Key  Formula   •  The  amount  of  shear  stress  is  calculated  using  the  following  

general  formula:  

•  Note  that  sec,ons  9.5,  9.6,  and  9.7  are  simply  applica,on  of   this  formula  to  par,cular  cross-­‐sec,onal  shapes  

•  Even  sec,on  9.8  is  a  special  interpreta,on  of  a  slight   modifica,on  of  this  formula  

•  Focus  on  learning  how  this  formula  works  

τ = VQ IZt

THE  FIRST  MOMENT  OF   AREA  Q    

9.4  

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Q   •  The  confusing  part  of  the  shear  formula  is  the  term  Q   •  The  first  step  in  finding  Q  is  establishing  the  point  (level  up  

and  down  on  the  beam  cross-­‐sec,on)  of  interest   •  Since  shear  stress  is  max  at  the  centroid,  this  is  a  logical  point   •  But  watch  for  problems  that  ask  for  analysis  of  other  points  

–  Three  points  of  interest  (a,  b,  c);  three  shaded  regions  (A’)  

Shaded  Region   •  With  the  “shaded  region”  established:     •  Where:  

–  The  term  “y  bar”  is  the  distance  from  the  centroid  of  the  shaded  region   to  the  overall  centroid  of  the  beam  cross-­‐sec,on  

–  The  term  A  is  simply  the  area  of  the  shaded  region   –  The  shaded  region  can  be  broken  up  with  the  result  summed  

Q = yA = yiAi i ∑

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SHEAR  STRESSES  IN   BEAMS  OF  RECTANGULAR  

CROSS  SECTION    

9.5  

Special  Case   •  Just  apply  VQ/IZt  to  a  beam  with  cross-­‐sec,on  b  x  h  (base  by  

height)  and  you  get:  

•  No,ce  the  parabolic  varia,on  over  the  cross-­‐sec,on  

•  The  maximum  shear  stress  is  at  the  neutral  axis,  where  y  =  0  

τ = 6V bh3

h2

4 − y2

⎛ ⎝⎜

⎞ ⎠⎟

τmax = 3 2 V A

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SHEAR  STRESSES  IN   BEAMS  OF  CIRCULAR  

CROSS  SECTION    

9.6  

Special  Case   •  Just  apply  VQ/IZt  to  a  beam  with  a  circular  cross-­‐sec,on  of  

diameter  d  and  at  the  center  you  get:  

    •  Just  apply  VQ/IZt  to  a  beam  with  a  hollow  circular  cross-­‐

sec,on  of  diameters  D  (outer)  and  d  (inner)  and  at  the  center   you  get:  

 

Q = 1 12 d3,t = d

Q = 1 12

D3 − d3( ),t = D − d

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SHEAR  STRESSES  IN  WEBS   OF  FLANGED  BEAMS    

9.7  

Special  Case   •  The  web  (ver,cal  piece)  of  an  I-­‐beam  carries  shear  stress   •  The  neutral  axis  is  where  the  loca,on  of  max  shear,  but  these  

beams  are  omen  built-­‐up  with  welds  to  aGach  the  flanges   •  The  green  shaded  area  is  then  of  interest,  even  though  the  

shear  stress  is  of  lower  magnitude   •  Note  that  the  relevant  value  of  t  here  is  tw,  not  bf  

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SHEAR  FLOW  IN  BUILT-­‐ UP  MEMBERS    

9.8  

Forget  the  t   •  If  we  just  drop  t  from  the  denominator  of  the  shear  stress  

equa,on,  we  get  a  quan,ty  called  shear  flow:  

•  The  units  and  interpreta,on  are  the  important  issues:   –  Units  are  shear  force  per  unit  of  beam  length   –  Used  to  evaluate  fasteners  and  welds  that  hold  built-­‐up  beams  

together  

•  We  will  go  no  further  with  this  chapter  …  

q = VQ IZ