What is the Point of Zero Moment?

Approach

There are many ways to find the point of zero moment:

• Analytical Approach. Create the moment equation of a segment. Then, equate the  moment to zero and solve for the position \(x\).
• Graphical Approach. Using geometry, plot the points and find the intersection point of the graph and the x-axis.

We will focus on the analytical approach to solve for the positions of zero moment.

Find the Segments

The first thing is to find which beam segment will have the point of zero moment.

We do this by analyzing the moment at both endpoints of each segment using equations or relating the diagrams. Afterward, we perform a simple test:

• If the moment changed from positive to negative or vice versa between its endpoints, then a point of zero moment occurred.
• If the moment remains the same for both endpoints (positive to positive or negative to negative), there is no point of zero moment.

In our example, there is  a change from positive to negative when analyzing the moment at segment \(CD\):

\(M_C=280.12kN•m\)

\(M_D=-135kN•m\)

From these, there is a point of zero moment between these two points \(C\) and \(D\).

Point of Zero Moment

The point of zero moment is where the location of inflection points is in its deflected shape. In our example, it occurs between points \(C\) and \(D\).

Like we did with the point of zero shear, we apply algebra: formulate the moment function for segment \(CD\), substitute \(M_{CD}=0\), and solve for \(x\):

\(M_{CD}=-18 x^2+222.75 x-407.25\)

\(0=-18 x^2+222.75 x-407.25\)

Since this is a quadratic function, we can apply the quadratic formula:

\(x=\frac{-(222.75) \pm \sqrt{(222.75)^2-4 (-18) (-407.25)}}{2 (-18)}\)

\(x=10.15\)

The point of zero moment is at a distance \(10.15m\) from the origin at \(A\).