# How can I determine the optimal cross section of a member to correct overdesign?

I am currently working on a bridge design issue. The main goal is to correct issues of overdesign. My professor said we can do this by selecting beams with different cross sections. My question is how can I determine what is the optimal cross sectional area for the force being applied to a specific beam. So far this is what I can understand. I must first calculate the amount of stress the member is experiencing using stress = force / area of the current beam used.

• From a practical point of view, over-optimization is not always a good thing for structures that have to last for a long period of time. You have to consider tail risks in the design based on rare extreme events. While the approaches suggested in the answers are OK for classroom exercises, I would strongly suggest exploring probabilistic approaches too. Nov 4 '18 at 21:13

For beam, it is often used to resist bending. Especially, when you design the bridge, the force is often in the trasverse direction, like pedestrian walking or car moving on the bridge.

The stress formula in the beam under bending is

$$\sigma = -\frac{My}{EI}$$

Where $$I$$ is the moment of inertia. The geometry of cross section will determine this term. For example, the rectangular cross section with $$h$$ height and $$b$$ width have $$I=\frac{bh^3}{12}$$

Of course, there are other stresses in the beam. For instance, tension force$$\sigma = \frac{F}{A}$$, flexural shear force, etc. You need to determine whether or not you need to consider them as they might be much smaller than the stress introduced by bending.

Now, to optimize your design. You need to know what is the maximum stress allowed in your bridge material and safety factor. Then, you can determine the minimum moment of inertia of the cross section. At this point, you need to choose whatever best cross section to meet the minimum moment of inertia requirement.

To make this easy to do by hand, start with a statically determinate structural design. Then (ignoring the self-weight of the bridge) the load in each part of the structure is independent of its stiffness.

When you know the loads, you can choose the size of each member to create a "fully stressed design" where the maximum stress in each member is equal to the safe limit for the stress in the material (including a safety factor, of course).

For a bridge, you probably can't ignore its self-weight completely, but you can iterate the calculations. Start with a guess of the self weight, calculate the loads, change the size of the members, recalculate the self weight based on the new size, and repeat till nothing changes.

If you are using computer software to optimize the design, that will work for statically indeterminate structures in a similar way, but by hand it is usually too much work to recalculate how the load paths through the structure change if you change the size of different parts of it by different amounts.