Statics-MEM

MEM  330  Mechanics  of  Materials  II   HW  Assignments  

10. A  beam  with  a  thin-­‐walled  open  cross-­‐section  as  shown   is  subjected  to  an  inclined  load  that  produced  a  bending   moment  𝑀𝑀 = 5.0  N ∙ m  in   a   direction  𝜃𝜃 = −25°  (i.e.,   25°   counter-­‐clockwise  from  the  z-­‐axis).    The  thickness  of  the   wall  is  2  mm  throughout.   (a) Determine  the  centroid  of  the  cross-­‐section  in  terms  

of  𝑦𝑦  and  𝑧𝑧  as  shown.   (b) Compute  the  centroidal  moments  of  inertia,  𝐼𝐼 ,  𝐼𝐼 ,  and  

the  centroidal  product  of  inertia,  𝐼𝐼 .   (c) Determine  the  orientation  of  neutral  axis,  in  terms  of  

an  angle  𝛼𝛼  measured  clockwise  from  the  z-­‐axis.   (d) Determine   the   locations   and   the   magnitudes   of   the  

maximum   tensile   and   compressive   stresses   due   to   the  applied  moment  using  the  equation  

𝜎𝜎 = − 𝑀𝑀 𝐼𝐼 + 𝑀𝑀 𝐼𝐼 𝐼𝐼 𝐼𝐼 − 𝐼𝐼

𝑦𝑦 + 𝑀𝑀 𝐼𝐼 + 𝑀𝑀 𝐼𝐼 𝐼𝐼 𝐼𝐼 − 𝐼𝐼

𝑧𝑧  

(e) Determine   the   principal   directions,   𝜙𝜙  (measured   clockwise  from  the  z-­‐axis),  and  compute  the  principal  moments  of  inertia  𝐼𝐼  and  𝐼𝐼 .    Set  up  a   new   coordinate   system   Y-­‐Z   coinciding   with   the   principal   directions   such   that  𝐼𝐼 = 𝐼𝐼  and   𝐼𝐼 = 𝐼𝐼 .  

(f) Determine  the  orientation  of  neutral  axis,  in  terms  of  an  angle  Α  measured  clockwise  from   the  principal  Z-­‐axis.  Compare  the  value  of  Α  with  that  of  𝛼𝛼  computed  in  (c).  

(g) Compute   the   maximum   tensile   and   compressive   stresses   due   to   the   applied   bending   moment  using  the  equation  below  and  compare  with  those  computed  in  (d).  

𝜎𝜎 = − 𝑀𝑀 𝑌𝑌 𝐼𝐼

+ 𝑀𝑀 𝑍𝑍 𝐼𝐼

 

   

80#mm

120$mm

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z O25°

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z0

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