Monthly Mechanics: Reinforced Concrete

This is Part 2 of a series about building materials. Read Part 1 here.

If you measure the #1 building material in the world by weight, steel and concrete are neck and neck. If you measure by volume, concrete wins hands down. The World Steel Association estimates that 1630 million metric tons of steel were produced worldwide in 2016 – a bit over 7000 million cubic feet. Meanwhile, the Royal Society of Chemistry estimates that 2000 million metric tons of concrete are placed each year – over 30,000 million cubic feet.

These are staggering numbers. If you put a year’s worth of concrete all together you could build a solid cube measuring over half a mile on each side. Obviously, concrete has the edge over steel by volume because it’s much less dense. Steel is over 90% iron, and its density is basically the same as pure iron. (Every structural engineer knows this density by heart: it’s 490 pounds per cubic foot.) But concrete is a complicated mixture of different sizes of rocks and sand, Portland cement, water, and various additives, and depending on how it’s mixed a builder can trap a significant amount of air inside or just a little bit. Thus the density of concrete varies.

Rocks might seem heavy, but in fact they don’t get much denser than 150 pounds per cubic foot, so this is the upper limit on the density of concrete. Lightweight mixtures get down to 90 pounds per cubic foot with more entrained air; they’re often specified when concrete doubles as the architectural finish and needs to look sleek. Specialty products, such as autoclaved aerated concrete, are actually so light that they float. But such lightness comes at a big financial cost and reduces the strength as well.

Speaking of strength, concrete as a building material has one serious shortcoming. It works wonderfully in compression, but has almost no capacity at all in tension. That’s why you will never see a concrete chandelier.

Engineers address this shortcoming by putting a little bit of steel inside the concrete. The result is a composite material called reinforced concrete, in which the concrete provides the compressive strength and the steel provides the tensile strength. If the concrete doesn’t need much tensile strength (a slab on grade for example), then a steel mesh or welded wire fabric may be sufficient. In other situations, reinforcing bars (aka rebar) are used. They come in sizes ranging from 3/8 inch diameter to over 1 inch diameter, and they always have bumps on their surface to prevent them from sliding through the hardened concrete.

ConcreteMath
Concrete beam cross section analysis. The striped region at the top is the effective concrete area. The dots at the bottom are the steel reinforcing.

In designing a reinforced concrete beam, engineers try to balance tension and compression. Recall that when a beam bends downward, the bottom is in tension and the top is in an equal amount of compression. Reinforcing steel goes in the tension zone. It takes a certain amount of concrete in the compression zone to balance the strength of the steel, and that amount is surprisingly small. (Most reinforced beams are tension controlled, which means the steel fails before the concrete does.) In other words, most of the concrete in a reinforced beam never feels any stress! Its only purpose is to hold the beam together.

UNreinforced concrete structures are rare today, but early builders took advantage of the material’s pure compressive strength to produce spectacular arches and shells. The Pantheon in Rome, finished in the year 128, remains the world’s largest unreinforced concrete dome. In the early 20th century, engineers like Félix Candela and Robert Maillart took unreinforced concrete to its artistic apex; many of their structures still survive.

Monthly Mechanics: Steel

Steel is an alloy of iron and carbon, and one of the strongest materials that can be produced economically. The history of steel as a structural material begins in the 18th century when cast iron was first used to build bridges. But iron is very ductile, which is not desirable. Carbon stiffens the molecular structure so the material barely deforms up to a very high stress.

Like most materials, steel gets a little longer when in tension and a little shorter when in compression. The change in length is proportional to the stress applied – if you plot the tension versus percent elongation in a steel member, the graph is a straight line. (See The Elastic Modulus for more information.) Moreover, when the stress is removed the steel member returns to its original shape, like an elastic band. Thus, steel is a linear elastic material.

All this is true up to a point. The linear elastic relationship breaks down above the yield stress. Once the stress applied exceeds the yield stress, the steel continues to deform but doesn’t return to its original shape, like melted plastic. If the stress is higher still, the steel eventually ruptures – complete failure. This is called the ultimate stress.

Steel Stress-Strain
Stress-strain graph for typical steel.

Structural engineers don’t like plastic behavior. It’s OK if a structure deforms a little under stress (skyscrapers bend in the wind; bridges sag with heavy traffic), but only if it returns to its original shape when the stress is removed. For this reason, structural engineers usually specify steel based on its yield stress, and they design to make sure this stress is never exceeded. In the US, the most common steel grade is A36, corresponding to a yield stress of at least 36,000 pounds per square inch. Developers and state highway departments, seeking designs for commercial buildings and major bridges, increasingly require steel with a yield stress of at least 50,000 pounds per square inch.

Steel has been mass produced since the early 19th century, and manufacturers have standardized a huge number of shapes, made by extruding molten steel through sets of rollers. They also produce steel plates of various depths, and custom curved beams. Here are some of the rolled shapes available:

Steel Shapes

Wide-flange beams, labeled with a W, such as W12x16 – the first number indicates the depth; the second number indicates the weight in pounds per foot.

Channels, labeled with a C, such as C10x30 – the first number indicates the depth; the second number indicates the weight in pounds per foot.

Angles, labeled with an L, such as L4x3x3/8 – the first and second number indicate the length of each leg; the third number indicates the thickness.

Hollow structural steel, labeled with an HSS, such as HSS8x4x1/4 – the first and second number indicate the depth and width; the third number indicates the thickness.

Monthly Mechanics: Where Do I Start?

I was describing my job to a new acquaintance, and he asked me, “When you start a new project, what’s the first thing you do?” Indeed, Monthly Mechanics has explored every part of the design process, but rarely discussed the order. So, here is a flowchart.

WhatsFirst

Step 1: Make a model. What are the components of your structure? Components of a building include the roof, the walls, the floors, and the foundation. Components of a bridge might include the deck, the stringers, the piers, and again the foundation. A rough sketch of the structure helps to identify what parts you need to design, and (importantly!) enables you to define the scope of work with your client.

There are archetypes for the most common components. A beam, such as a deck joist or roof rafter, is basically a line. A column is also a line, but it’s loaded at the top rather than on the side. A floor might be a rectangle. A column foundation is a point. It takes some practice to see the shapes – your best bet is to think about where the loads are coming from and how the loads push, pull, twist, or bend each component. You’ll notice what paths the loads take through your structure while building your model.

This is also the time to identify the failure modes you’ll need to check. Are the beams loaded along one side (like a joist supporting a floor, with loads only from above) or along two sides (like a bridge girder supporting a deck weight and a simultaneous wind load)? Are they continuous over a central support? (If so, you’ll need to check negative bending over the support in addition to positive bending at midspan.) Do they experience any tension or compression, or only flexure?

Step 2: Determine the loads and distribute them. Right away you’ll notice that different components experience the loads in different ways. For example, one- and two-family housing is designed with a live load of 40 psf. That weight is a uniform load on the plywood subfloor, and it’s distributed down to the floor joists according to their tributary area. Following the load path, a wall or column receives that same live load as a reaction from the joist ends – a concentrated load.

Once you’ve identified all the loads, apply load combinations. Check building codes to determine how your jurisdiction adds loads together, and calculate all possibilities. It’s prudent to determine the governing load combination for vertical loads (dead, live, snow) as well as for horizontal loads (wind, seismic). Some load combinations emphasize live load; others give extra weight to atmospherics like snow and wind; still others minimize downward forces to test for uplift.

Sometimes while you’re determining the loads, you’ll realize that your model is too simplified. In that case, return to Step 1.

Step 3: Figure out how to support the loads. If you’re designing a floor joist, you might start by looking at dimensional lumber. Choose a size and species – maybe a southern pine 2×10 – and calculate the stress for all the failure modes you’re checking – perhaps flexure, shear, and bearing at the support. (This is called the stress required.) Compare it with the beam’s strength – found in material-specific manuals such as the National Design Specification for Wood Construction. (The strength is also called the stress provided.) If the stress required exceeds the stress provided, then you’ll need to choose a stronger beam – maybe a 2×12, or an LVL, or a steel beam. If the stress provided exceeds the stress required by a long shot, then you can economize by choosing a smaller beam.

Design is an iterative process. Repeat Step 3 until you close in on the Goldilocks beam – the shallowest, lightest, or least expensive member whose strength still exceeds the stress required. Then proceed down the load path.

Anyone can build a bridge that stands, but it takes an engineer to build a bridge that barely stands.
-Colin Chapman (maybe)