Blodgett’s Basics: When you only have a moment

Nov 1, 2008 12:00 PM, By OMER W. BLODGETT, ScD., P.E.

The moment of inertia of the cross-section of a structural member measures the resistance to rotation provided by the section.

Expressed as (I), the moment of inertia is required for many engineering calculations. To determine deflection, for example, we need to know the moment of inertia.

If we buy a rolled section from a steel mill, we can select the overall depth of the shape, and use various “look up” tables to find the foot-weight of the section that will provide the required moment of inertia (I), measured in inches to the fourth power (in⁴).

When we choose to design a weldment instead of buying a rolled section, we have the opportunity to optimize the design because there are an infinite number of combinations of material thicknesses and widths that can be used.

The section depth can be constant or variable, and different material strengths can be used. Once a preliminary configuration is established, we need to calculate the moment of inertia.

One approach to determining the moment of inertia for a preliminary design is by trial and error.

We put the dimensions of the section together on paper and do the calculations. If the calculated moment of inertia is not right, we must modify our preliminary design and re-calculate the moment of inertia.

This process can be repeated until the optimal configuration is established.

The problem with this approach is, if we come up with an idea and it's close, but not perfect, we have to throw away all our previous calculations and start over.

There is a better way, which I will describe.

With this approach, after the preliminary design is evaluated, we can easily modify our design, adding or subtracting components of the weldment design until we get the optimized result we want.

It provides a fast and efficient method that is ideal for weldments because it gives us flexibility.

This approach is based upon the parallel axis theorem. Using the parallel axis theorem for shifting the axis for a moment of inertia, the moment of inertia of the whole section about the reference line y-y is

where n is the distance from the base to the neutral axis of the cross-section.

Since
and

Then, substituting this back into equation (2), we obtain the following:

Note: The distance to the neutral axis (n) has dropped out of the equation.

Thus:

where:

I_n = the moment of inertia of a whole section about its neutral axis, n-n.

I_y = the sum of the moments of inertia of all elements about a common reference axis, y-y.

M = the sum of the moments of all elements about the same reference axis, y-y.

A = the total area, or sum of the areas of all elements of the section.

Using the relationship above, a table can be established and the relative contribution of each element of the preliminary weldment design can be added.

Most importantly, after the value of the moment of inertia is determined, additional sections can be added or subtracted to optimize the design.

Let's use an example to illustrate how this approach works.

Consider, for instance, a case in which the engineer has determined that a cross section will require a moment of inertia of 2700 in.⁴.

The configuration of the weldment shown in Figure 1 basically satisfies the geometric requirements of the application. But does it have the required moment of inertia of 2700 in.⁴?

Table 1 contains the seven columns needed to work out the problem.

The first column lists the individual components of the cross section, and the second column lists the dimensions of the components. The distance of the center of gravity for each component from the reference axis y-y then is listed in the third column. From this point forward, we can make the various calculations and add them.

After making the calculations and the populating the fields, we will total the computed values.

The value of I_g typically is small enough that it can be conservatively ignored. The greater the depth of any element relative to the maximum width of the section, the more significant its I_g value is likely to be.

In this case, I_g will be considered to provide a complete example. Therefore, the values for I_y and I_g are totaled as well.

At this point, all of the values necessary for incorporation into equation 3 above have been determined, as follows:

The distance from the base to the neutral axis (n) can be determined from the following:

For our example, the computed moment of inertia (2359 in.⁴) for the initial design is too low since the target value was 2700 in.⁴.

How will the moment of inertia change if we add element D, a 2 in. × 4 in. area, to the plate at the top, as illustrated in Figure 2?

To determine the new value for I, we can simply add the properties of element D into our previous table, as shown in Table 2.

The columns are re-totaled, and the following values obtained, carrying the column totals forward.

Now we have to enter only the properties of the added area, as shown in Table 2:

In this case, the moment of inertia exceeds requirement of 2700 in.⁴, so the 2 in. × 4 in. section has provided sufficient capacity. But, if it had not, we could simply continue to add capacity as needed.

The final weldment design would not have elements C and D: The actual weldment would utilize a top flange that is 4 in. by 8 in.

This simple and straightforward method allows the engineer to optimize weldment design, taking full advantage of the flexibility in design that joining sections by welding permits.