Not Amenable To Easy Analysis

Not our project:

While discussing a project recently with our client, it became clear that part of our work was going to require arch analysis of a masonry lintel. (The project is a bit sensitive, so I’m going to be vague about the specifics of the location and owner, which fortunately does not require being vague about the engineering issues.)

In short, we have a big stone lintel, flat on the bottom, spanning across a window opening. We’re reasonably sure that the lintel is just stone, not stone hiding a steel beam, and there’s no other place for structure to be hiding from view. The other current issue of note is that there are cracks in the bottom of the lintel stone.

One of the biggest differences between new-building design and old-building analysis is that the first step in analysis is a lot more difficult. Before we calculate structural capacity, before we figure out loads, the first step is figuring out what structural mechanism is present. In a new building we decide this, but in an existing building we need to figure it out. All of the tools we have to analyze structures depend, to one degree or another, on assumptions about how loads are traveling through the structure, so we can’t use those tools until we have enough information to make the assumptions. Some examples should help clarify this:

• In order to analyze the capacity of a column in any material, we need to know (or make an assumption) about the end conditions. Whether a column is free to rotate between floors or is restrained against doing so has a large effect on its strength.
• Similarly, in order to analyze a beam, we need to know if it’s continuous at its ends with neighboring structure.
• In order to analyze a frame, we need to know whether the floors are stiff enough to constrain the columns to move together or not.

All three of those examples are about what is referred to in math as “boundary conditions.” The underlying equations for the analyses stay the same, but the boundaries can change the results.

Back to our lintel…cracks in the bottom of a masonry beam are very serious and probably indicate imminent collapse. Cracks at the bottom of a masonry arch are maybe not serious at all. So we have to know whether the lintel is a beam or an arch in order to analyze it meaningfully. Of course, it’s the same pieces of stone either way, so what I really mean is we have to know if the boundary conditions exist that allow arch analysis to make sense: Is there sufficient structure on either side of the proposed arch to take the lateral thrust that arches develop? Is there a continuous path for forces? Is there sufficient depth of masonry to allow for a curved thrust line? In this case, these conditions are present, so arch analysis is possible. It could still not work – the analysis could show that the forces are too great for the stone, for example – but at least it’s a starting point.

More on this topic: here.