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Rudiy27
3 years ago
3

For a cantilever beam subjected to a triangular distributed load, what is the order of the internal moment expression of the bea

m as a function of the location

Physics
1 answer:
bearhunter [10]3 years ago
6 0

Answer:

The Order is 4th order

Explanation:

Let take the diagram on the first uploaded image as an example of triangular distributed load  

Let x denote the location

To obtain the vertical force acting at point a we integrate the function  

          \frac{dA_y}{dx} = w

    where A_y is the vertical force

    dx is the change in location

     w is a single component of force acting downwards

             A_y = \int\limits^L_0 {w} \, dx

           A_y = \int\limits^8_0 {\frac{x^2}{8} } \, dx

                 =  \frac{1}{8}  [\frac{8^3}{3} ]

                  = 21.33N

To obtain the moment of of inertia \

              M_A = \int\limits^L_0 {wx} \, dx

                     =\int\limits^8_0 {[\frac{x^2}{8} ]x} \, dx

                     = \frac{8^4}{32}

                   = 128 Nm

To obtain the shear force acting at a distance x from A

             V = A_y -\int\limits^x_0 {[\frac{x^2}{8} ]} \, dx

                  = 21.33 - \frac{x^3}{24}

   To obtain the moment about the triangular distributed load section that is at a distance x from A as shown on the diagram

       = >    dM = \int\limits^x_0 {V} \, dx

          =>        M_x -M_A = \int\limits^x_0 {[21.33- \frac{x^3}{24}]} \, dx

      =>           M_x = 128 + 21.33x -\frac{x^4}{96}

Looking at this equation as a polynomial we see that the order is 4 i.e x^4

             

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