Chapter 9, Problem 9.2.1P

The equation of the deflection curve for a cantilever beam is

   $v(x)=\frac{m_0{x}^2}{2EI}$

1. Describe the loading acting on the beam.

Draw the moment diagram for the beam.

To determine

Load acting on the beam.

Answer to Problem 9.2.1P

No load acting on the beam. Only moment $M_0$ acting on the beam at $x=L$ .

Explanation of Solution

Given:

Equation of the deflection curve:

  $v\left(x\right)=\frac{M_0{x}^2}{2EI}$

The equation of the deflection curve for a cantilever beam,

  $v\left(x\right)=\frac{M_0{x}^2}{2EI}$

Here,

  $y$ , deflection at any point $x$ along the axis of beam

  $EI$ , flexural rigidity

  $M_0$ , Moment

Slope represents the first derivative of the deflection.

Therefore, differentiating the above w.r.t $x$ to get the slope.

  $y\text{'}=\frac{M_0\left(2x\right)}{2EI}$

  $y\text{'}=\frac{M_{0}x}{EI}$

Now, write the basic differential equation of deflection curve.

  $y\text{'}\text{'}=\frac{M}{EI}$

Here,

  $M$ , Bending Moment

  $EI$ , Flexural rigidity

And

  $y\text{'}\text{'}$ , Second derivative of deflection

Differentiate the obtained slope equation w.r.t $x$ ,

  $y\text{'}\text{'}=\frac{M_0}{EI}$

Therefore, the beam is subjected to positive bending moment $M_0$ .

Now, shear force to be calculated:

  $SF=\frac{dM}{dx}=0$

No shear force is acting on the beam.

Now, calculate load:

  $w=\frac{-d\left(SF\right)}{dx}$

  $w=0$

No load acting on the beam.

Therefore, only moment $M_0$ acting on the beam at $x=L$ .

To determine

Draw the moment diagram for the beam.

Answer to Problem 9.2.1P

Bending moment diagram:

  

Explanation of Solution

Given:

Equation of the deflection curve:

  $v\left(x\right)=\frac{M_0{x}^2}{2EI}$

Bending moment diagram subjected to positive bending moment $M_0$ .