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MECHANICS - EXAMPLE


Cantilever Beam Loaded
in Two Directions
  Example

 

Two distributed line loads act on a cantilever beam as shown in the diagram on the left. Both loads act through the center of the rectangular cross section in the directions shown. What is the maximum absolute bending stress in the wall?

   
    Solution


Moments at Wall due to
Distributed Loads

 

To determine the bending stresses, the maximum moment needs to known. Since this is a cantilevered beam, the maximum bending moment will occur at the fixed end, or the wall.

     Mz = Fy d  = -[(1.5)(0.5)] (1.5 + 0.75)
          = -1.6875 kN-m
     My = Fz d = [(1.5)(0.5)] (0.75)
          = 0.5625 kN-m

The moment of inertia around each axis is

      Iz = (0.1)3(0.06)/12 = 5×10-6 m4
      Iy = (0.06)3(0.1)/12 = 1.8×10-6 m4

     

Beam Cross-Section at Wall
 

The maximum moment at one of the four corners. From inspection, both the My and Mz moments will cause tension stress in the upper left corner, point A (don't forget that Mz is negative value). Similarly, both moments will cause compression stress at point D. These two points, A and D, will be the maximum tension and conpression stress, respectively. In this case, the magnitudes for both the maximum compression and maximum tension will be equal.

Maximum stress at point A, y = 0.05 m, z = 0.03 m, is

     
        = 26.25 + 16.875 MPa

     σA = 26.25 MPa

     
   
 
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