The heart of RC design โ master how beams bend, when they fail, and how to calculate the exact amount of steel they need.
Chapter 4 is where design becomes real. Chapter 3 gave you the theory โ now you'll use it to calculate exactly how much steel a beam needs and how thick it should be. These tools are used every time a concrete floor, bridge, or parking garage is designed.
Flexural design answers one central question: given the loads a beam must carry, what cross-section dimensions and how much steel reinforcement are required so the beam is safe yet economical?
Think of a wooden shelf sagging under too many books. The shelf bends โ the top gets squeezed (compression) and the bottom stretches (tension). If the wood is weak in tension, it cracks at the bottom. Concrete is that same shelf โ strong in compression but brittle in tension. Steel bars embedded at the bottom take over the tension job. The entire art of Chapter 4 is figuring out how many steel "books" can act as tension carriers.
Every design check in this chapter comes down to one statement:
| Symbol | Meaning | Typical Value |
|---|---|---|
| Mu | Factored moment demand (from loads ร load factors) | Calculated from analysis |
| Mn | Nominal moment capacity of the section | Calculated from section geometry |
| ฯ | Strength reduction factor | 0.90 for flexure (tension-controlled) |
Before you can calculate beam capacity, you must understand how a beam fails. RC beams don't just snap โ they go through predictable stages as load increases. Knowing these stages is what separates engineers who understand the code from those who just apply formulas blindly.
Loading an RC beam is like inflating a balloon until it bursts. At first, everything stretches elastically and nothing cracks. As you add more load, the tension side cracks โ but the steel wires inside the balloon (the rebar) hold it together. Finally, either the steel wire yields (ductile failure, with warning) or the balloon skin ruptures suddenly (brittle concrete crushing). Engineers design for the first type, always.
At ultimate load, the concrete stress distribution above the neutral axis is a curved, parabolic shape. Whitney's brilliant insight: replace the complicated curve with an equivalent rectangle that gives the same total compression force and the same centroid location. This makes the math tractable without losing accuracy.
From equilibrium of the internal couple (C = T, and Mn = C ร moment arm):
Where d = effective depth (top fiber to steel centroid), b = beam width, As = steel area.
ฮตt โฅ 0.005 โ ฯ = 0.90ฮตt < 0.004 at nominal strengthThis section hands you the complete design toolkit: how to pick beam dimensions, how to find the required steel area, and how to verify ACI's ฯ limits. Get comfortable here โ this exact procedure repeats in almost every beam you'll ever design.
The reinforcement ratio ฯ (rho) is simply the fraction of the beam's cross-section that is steel. It is the single most important parameter controlling beam behavior.
Adjust material strengths to see how ACI limits shift. Enter your ฯ to check feasibility.
| Limit | Formula | Reason |
|---|---|---|
| ฯmin | max(3โf'c/fy , 200/fy) | Minimum steel to prevent sudden brittle failure right at cracking |
| ฯ for ฮตt=0.005 | (0.85ฮฒโf'c/fy)ยท[3/8] | Upper limit for full ฯ=0.90 (tension-controlled) |
| ฯmax (ฮตt=0.004) | (0.85ฮฒโf'c/fy)ยท[3/7] | Absolute ACI maximum โ beam still ductile at this limit |
Adjust the parameters โ the capacity and stress block update instantly.
There are two types of design problems: Analysis (section is given, find capacity) and Design (demand is given, find the section). Here is the step-by-step for each.
a = Asfy/(0.85f'cb)Mn = Asfy(d โ a/2)ฮตt = 0.003(dโc)/c where c = a/ฮฒโฯMnฯ โฅ ฯminRn = ฯfy(1 โ ฯfy/1.7f'c)Mu = ฯยทRnยทbยทdยฒ, solve for bยทdยฒAs = ฯยทbยทdThe formulas in ยง4.3 involve an implicit circular dependency โ you need As to find a, but you need a to find the required As. Design aids (tables and charts) break this loop by pre-computing useful coefficients so you can solve problems in one pass without iteration.
Think of design aids like a conversion chart. Instead of typing miles ร 1.609344 every time, you glance at a table and read off the answer. Design aids for beams are the same โ they encode the beam equations so you look up a coefficient, multiply, and you're done.
The design equation can be rewritten as:
This means: once you pick ฯ (and know f'c, fy), you can look up Rn in a table, then solve bยทdยฒ = Mu / (ฯยทRn) directly. No iteration needed.
Select material strengths to generate a design aid table โ just like Table A.7 in the textbook.
| ฯ (%) | Rn (psi) | Ku = ฯRn (psi) ฯ=0.90 | Strain ฮตt | Status |
|---|
Usage: Compute Mu (kยทft) โ convert to lbยทin โ solve bยทdยฒ = Mu/(ฯยทRn) in inยณ โ choose b, find d.
The curve shows how increasing ฯ gives diminishing returns in Rn โ the relationship is not linear because of the a/2 correction. Red dashed lines = ACI limits.
bยทdยฒ (deeper or wider beam). A larger ฯ allows a shallower beam but costs more steel.ฯMn โฅ Mu โ tables are for preliminary sizing, not final confirmation.The formulas in ยง4.3 give you As and d on paper โ but a real beam has to be physically built. Bars need space to fit. Concrete must flow around them. Cover protects steel from corrosion and fire. This section translates calculated numbers into a buildable beam.
You've calculated you need a 3-foot door opening โ but lumber comes in fixed sizes, hinges have specific positions, and you need room to swing the door. Practical beam design is the same: bars come in discrete sizes, beams are built to round dimensions, and code rules dictate minimum gaps. You don't get to specify 11.3 inches โ you round up to 12.
Cover is the distance from the outer face of the beam to the surface of the nearest bar. It protects steel from corrosion, fire, and bond failure near the surface.
| Exposure Condition | Min. Cover | Notes |
|---|---|---|
| Interior, not exposed โ bars #6โ#18 | 1ยฝ in | Typical interior beams & slabs |
| Interior, not exposed โ bars #5 and smaller | ยพ in | Slabs, walls with small bars |
| Exposed to weather โ bars #6โ#18 | 2 in | Exterior beams, retaining walls |
| Exposed to weather โ bars #5 and smaller | 1ยฝ in | |
| Cast against and in contact with earth | 3 in | Footings โ no formwork between bar and soil |
Design formulas use d โ but you specify h on drawings. The relationship depends on cover, stirrup size, and bar diameter:
Specify the beam's total depth and detailing โ get the effective depth d used in all Mn calculations.
ACI 318 requires clear spacing between parallel bars to satisfy all three rules simultaneously:
| Parameter | Guideline | Reason |
|---|---|---|
| Beam width b | b โ 0.4โ0.6 ร d | Efficient aspect ratio; common values 10, 12, 14, 16 in |
| Total depth h | h = d + 2.5 to 3.5 in | Cover + stirrup + half bar; round up to nearest inch |
| Min h for deflection | h โฅ L/16 (simple span) | ACI Table 9.3.1.1 โ avoids explicit deflection check |
| Bar layers | 1 or 2 preferred | More layers reduce d and complicate detailing |
Sometimes you cannot increase beam depth โ a parking garage has a fixed floor-to-floor height, or the architect has fixed the soffit. If a singly-reinforced beam can't carry Mu without exceeding ฯmax, adding compression steel A's near the top unlocks extra capacity within the same envelope by allowing more tension steel As at the bottom.
Imagine a pulley system where the main rope is already maxed out. You can't strengthen that rope โ but you can add a second rope on the other side of the pulley to share the load. Compression steel is that second rope: it opens up a new load path and lets the beam carry more moment in the same cross-section height.
The key question: is f's = fy valid? If the neutral axis is high (small c), the strain at A's depth may not reach yield strain.
f's = fy. Equilibrium gives:a = (AsโA's)fy / (0.85f'cยทb)f's = Esยทฮต's. Equilibrium becomes quadratic in c. More work โ but same principle.Asยทfy = 0.85f'cยทฮฒโcยทb + A'sยทfy โ solve for cฮต's = 0.003(cโd')/c โฅ fy/Es. If not, iterate with f's = Esยทฮต'sฮตt = 0.003(dโc)/c โฅ 0.004a = ฮฒโยทcMn = (AsโA's)ยทfyยท(dโa/2) + A'sยทfyยท(dโd')Add compression steel A's and watch how ฯMn grows beyond what a singly-reinforced section achieves at the same depth.
Almost every real building beam is cast monolithically with the floor slab above it. Under positive bending (sagging), the slab is on the compression side โ it adds a huge compression area. Ignoring this treats the beam as a wasteful little rectangle. The T-beam model correctly accounts for the slab's contribution, which can dramatically reduce the required beam depth or steel area.
Think of a steel I-beam. Its wide top flange handles compression efficiently โ the force is spread over a large area, keeping stress low. The narrow web carries shear. A T-beam is concrete's I-beam โ the monolithic slab is the flange, the rectangular stem below it is the web. You get the efficiency of the I-shape for free because the slab is already there.
a_trial = Asยทfy/(0.85ยทf'cยทbe)Asf = 0.85ยทf'cยท(beโbw)ยทhf / fyAsw = As โ Asfaw = Aswยทfy / (0.85ยทf'cยทbw)Mn = Asfยทfyยท(dโhf/2) + Aswยทfyยท(dโaw/2)ฮตt = 0.003(dโc)/c, where c = (hf + aw/2)ยท... โ or use c = a_total/ฮฒโEnter beam and slab geometry. The calculator finds be, auto-detects Case 1 vs 2, and computes ฯMn.
fr = 7.5โf\'c. This is the range of normal service loads for lightly loaded members.'
},
{
label:'Stage II โ Cracked Elastic',
color:'#f59e0b',
desc:'After cracking, before steel yields. Tension cracks form at the bottom. Cracked concrete carries ZERO tension โ only the steel does. The neutral axis shifts upward. Stress is still linear in the remaining concrete and in the steel. This is the range of most service-load conditions. We use the cracked transformed section here (Chapter 7 serviceability checks).'
},
{
label:'Stage III โ Ultimate (Nominal Strength)',
color:'#fb7185',
desc:'Steel has yielded; concrete is near crushing. The steel stress plateaus at fy. Concrete compressive stress is highly nonlinear. Whitney\'s insight: replace this curve with an equivalent rectangle of depth a = ฮฒโc and intensity 0.85f\'c. This gives the nominal moment capacity Mn. ACI design is based entirely on this stage.'
}
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// Section geometry
const bx=60, by=40, bw=100, bh=200;
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