Continuing from this example, let's learn how to find the point of zero moment:

*The solution presented is in SI. The author will update the post soon to reflect English units.*

The point of zero moment is the position along the beam in which the moment is zero. Let's explore different ways to calculate this position using functions or graphs.

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Learn moreContinuing from this example, let's learn how to find the point of zero moment:

*The solution presented is in SI. The author will update the post soon to reflect English units.*

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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.

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 \(D\).

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 \(A\).

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Created On

June 5, 2023

Updated On

February 23, 2024

Contributors

Edgar Christian Dirige

Founder

References

WeTheStudy original content

Revision

1.00

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