Question

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

Answer 1

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 $x^4$