Can it be that by making an assembly stiffer, the designer can simultaneously make it weaker? In an example from my experience, the challenge the designer faced was pretty simple: a flat plate deflected too much when loaded— see Figures 1 and 2. A simple stiffness problem, he thought.
And the obvious solution? Add some stiffeners. A deeper section is a stiffer section. Without thinking too much about the practicality of what he was doing, the designer decided to double the depth of the section. The un-stiffened plate was in. thick—he decided he would double that and go to a section depth of 1- in.
To do this, he considered adding three stiffeners to each side of the plate, each … in. thick. With the goal in mind of doubling the section depth, the designer specified 3/8 in. tall stiffeners (see Figure 3). They would be welded in place, and that would double the depth of the section.
It was at about this stage of the design process that the designer telephoned me. Today, I don’t recall specifically what prompted him to call, but when I saw the design, I intuitively knew something was wrong. The design would be stiffer—with that I had to agree. But when I told the designer it would be weaker as well, he was perplexed. How could that be? Surely the stiffeners would make the assembly stiffer and stronger, he argued.
I decided to prove my point with the ultimate weapon: cold, hard, unforgiving and unemotional mathematics. First, the deflection (stiffness) of the panel could be predicted from the following:
The maximum stress (strength) of the panel could be predicted from this relationship:
where S= Section Modulus, also equal to C/I where C is the half depth of the section.
The stiffeners weren’t going to change P, L, or E, but they would change I and S.
Lines 1 and 2 in Figure 4 show the values for the original design, and the contemplated modified design. The increase in the moment of inertia (I) from 0.53 in.4 to 0.71 in.4 proves that the design with the stiffeners will be 1.35 times stiffer, or perhaps more importantly, it will deflect 26 percent less. That, of course, was the reason for adding the stiffeners, and did not come as any surprise.
The strength of the assembly—the maximum load it could withstand without yielding—was controlled by the section modulus (S) and as shown in the table, this value went down from an original value of 1.41 in.3 to 0.95 in.3. This resulted in a decrease in strength of 33 percent (e.g., 0.67 of the original value). Here it was: mathematic proof that the stiffeners would make the assembly stiffer, and weaker.
The reason for this behavior is related to the controlling cross-sectional properties: the moment of inertia versus the section modulus. Granted, the section modulus is dependent on the moment of inertia, but also on the half depth of the section, or “C”. For the section modulus to increase concurrently with the moment of inertia, “I” must increase faster than “C”.
The designer’s dilemma, of course, wasn’t solved yet: how could the panel be stiffened without the loss of strength? For this particular assembly (this will not apply to other configurations), the stiffener height could be increased to 1 in. and the increased stiffness would still result in decreased strength— see line 3 of Figure 4. While the stiffness is over three times that of the original design, the strength is still 7 percent less.
At a height of 1-1/8 in., the stiffness is four times as much, and the strength is essentially unchanged from the original design (theoretically, 4 percent greater). Further increases in stiffener height will simultaneously increase stiffness and strength, as can be seen in lines 5 and 6 of Figure 4. A 2-in. stiffener as described in line 6 would work, but would require more material than necessary, making it uneconomical.
In general, designers are usually concerned with either stiffness (rigidity) or strength—rarely are both equally important. Consider the following generalizations:
(Moment of Inertia Controls)
machine tool bases
(Section Modulus Controls)
If your design requires additional rigidity (a higher “I”), by all means add some stiffeners. But, be aware that the strength may be reduced. Lower strength may not pose a problem. However, if you need more strength in your design, your answer won’t come from adding stiffeners, but rather, by increasing the section modulus (“S”).
Can the designer make an assembly stiffer and simultaneously make it weaker? Sure, and now you know why.
|Omer W. Blodgett, Sc.D., P.E., senior design consultant with The Lincoln Electric Co., struck his first arc on his grandfather’s welder at the age of ten. He is the author of Design of Welded Structures and Desig of Weldments, and an internationally recognized expert in the field of weld design. In 1999, Blodgett was named one of the “Top 125 People of the Past 125 Years” by Engineering News Record. Blodgett may be reached at (216) 383-2225.|