# Can the thickness of a guard post be derived purely from energy principles?

Consider an ordinary staircase with a banister rail supported by a wall at one end and a steel post of constant profile at the other end. The base of the post is embedded rigidly in the fabric of whatever structure the stairs are part of. The post must not bend beyond a set (small) amount, permanently deform, or fail to support the guard rail, in the event that someone falls downstairs or against it, and the maximum deflection of the post in the event of a fall is in any case to be small compared to its length (rough indication: 0.5–4.0% or say ~5–40 mm vs ~1000 mm).

The rail and post, and any fixings, are each thick enough and well enough secured, that they will not shear or become loosened by any plausible impact of a person (or persons, including anything they carry) falling down the stairs and against them at any point. Within these criteria, it's desired to make the post with a narrow profile, and therefore minimally thick.

From a pure engineering perspective (ignoring any building codes for now) is it sufficient to reason that any collision/impact will be elastic and therefore the minimum profile for the post can be derived by:

1. Estimating a maximum KE and PE for the falling persons,
2. Setting a maximum desired deflection of the post.
3. Reasoning on a pure energy basis that the post can be treated as a simple/ideal deflected cantilever beam, and its minimum profile is that which satisfies strain energy of the steel at maximum desired deflection >= maximum KE+PE of likely falling objects?

It seems to make sense and yet I am not too familiar with impulsive impact (more familiar with statics), and I'd like to check if there's a reason this would be an underestimate or must be treated with caution. (For example maybe a falling object would have CoG slightly higher than the actual rail height? Or other reasons might exist that this would lead to an insufficient result?)

• It all seems to make sense, except you have little detail on fixings, and I suspect they'll be the weakest point. Also, you don't mention the required longevity, environmental conditions, or consider fatigue from repeated stress on components. Finally, regarding CoG, AFAIK that's barely an engineering aspect; rather a requirement spec, depending on regs and/or client requirements – CL22 May 17 '16 at 8:15
• You also need to consider the physical dimensions of the restraint system. If you design something that is strong enough to survive the impact but acts like razor wire, that's probably not a good idea. – alephzero May 17 '16 at 11:58
• Both good points, thanks! Not sure how I will fix it but the fixing will be visible and checked to relevant code. Physical dimensions not an issue - the core will probably be 25mm steel cladded to 40-50mm with timber but any severe impact will almost certainly be to the side (which isn't size-limited and will be broad) not to the narrow edge. The issue is available space in one dimension not both, so I'm limited on one side only (thickness of balustrade rail as you'd look at it) not both dimensions – Stilez May 17 '16 at 13:19

The codes also define factors of safety, which a purely theoretical method such as the one you're suggesting can't generate. The Brazilian code at least also defines the allowable deflections at $\dfrac{L}{250}$, where $L$ is the railing's span.