HW5 - 1 4.3 A cantilever beam of rectangular cross section...

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1 4.3. A cantilever beam of rectangular cross section is loaded by a force F di- rected along the diagonal AC of the section as shown in Figure P4.3. Show that the neutral axis in this case coincides with the other diagonal BD for all values of the dimensions a , b of the rectangle. F A B C D a b Figure P4.3 The force F can be resolved into the components Fa a 2 + b 2 ; Fa a 2 + b 2 shown in Figure P4.3.1(b). a b F FaL 2 2 a + b FbL 2 2 a + b Fb 2 2 a + b Fa 2 2 a + b (a) (b) (c) Figure P4.3.1 It follows that the bending moments M x , M y on a cross-section distance L from the free end are
2 M x = FbL a 2 + b 2 ; M y = FaL a 2 + b 2 , as shown in Figure P4.3.1(c). We also have I x = b 3 a 12 ; I y = a 3 b 12 , so E R x = M x I x = 12 FbL b 3 a a 2 + b 2 = 12 FL b 2 a a 2 + b 2 E R y = M y I y = 12 FaL a 3 b a 2 + b 2 = 12 FL a 2 b a 2 + b 2 , from equations (4.16, 4.17). Substitution into equation (4.7) then gives σ zz = Ey R x - Ex R y = 12 FLE ab a 2 + b 2 parenleftBig y b - x a parenrightBig . The neutral axis corresponds to the condition σ zz = 0 and hence y b = x a ,

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