Reinforced Concrete Beams and Slabs

Reinforced Concrete Beams and Slabs

Strength is a gate, not the finish: A beam with φM_n ≥ M_u can still be unacceptable because of shear, development, splice, minimum reinforcement, spacing, cover, deflection, or constructability.

Geometry and the Compression Block

For a singly reinforced rectangular beam under positive bending, measure effective depth d from the extreme compression face to the centroid of the tension steel. Overall depth h, distance to the bar surface, and clear cover are different dimensions. If bars occur in multiple layers, locate their combined centroid.

Under the ACI 318-14 rectangular stress-block model, a common singly reinforced calculation is

a = A_s f_y/(0.85f'_c b)

and

M_n = A_s f_y(d - a/2).

These equations do not decide the strength-reduction factor by themselves. Determine neutral-axis depth and tensile strain under the applicable ACI convention to establish whether the section is tension-controlled, in transition, or otherwise limited. Compression reinforcement in a doubly reinforced section requires strain compatibility and its actual stress; do not simply add A'_s f_y without checking whether it yields.

Worked Flexural Strength

A rectangular beam has b = 12 in, actual d = 21 in, A_s = 4.00 in², f'_c = 4.0 ksi, and f_y = 60 ksi. Factored demand is M_u = 300 kip-ft. The problem states that the strain check qualifies for φ = 0.90.

Compression-block depth is

a = 4.00(60)/[0.85(4.0)(12)] = 5.882 in.

Nominal moment strength is

M_n = 4.00(60)(21 - 5.882/2)

M_n = 4,334 kip-in = 361.2 kip-ft.

Design strength is

φM_n = 0.90(361.2) = 325.1 kip-ft.

The utilization is 300/325.1 = 0.923, so flexural strength passes under the stated assumptions. Comparing 300 kip-ft directly with 361.2 kip-ft would skip φ. Using h instead of d would incorrectly enlarge the lever arm. The section still must satisfy ACI reinforcement limits and detailing.

Shear After Flexure

Concrete and transverse reinforcement contribute to nominal one-way shear resistance under the applicable ACI expressions. Use the critical section, effective depth, concrete density, longitudinal reinforcement, axial effects, and stirrup geometry required by the provision.

Suppose a separate problem-given shear check has V_u = 140 kips, calculated V_c = 120 kips, and φ_v = 0.75. Required nominal stirrup contribution from equilibrium is at least

V_s ≥ V_u/φ_v - V_c

V_s ≥ 140/0.75 - 120 = 66.7 kips.

That result is only a required contribution. Select stirrup area, strength, legs, and spacing using ACI 318-14; then check minimum shear reinforcement, maximum spacing, maximum permitted strength, anchorage, and support regions. A calculated V_s = 0 would not automatically waive code minimums.

For slabs near columns, check two-way punching around the applicable critical perimeter. Openings, drop panels, column dimensions, edge/corner location, and unbalanced moment transfer can change demand and capacity. Beam one-way shear formulas are not a substitute.

Development, Splices, and Cutoffs

Flexural resistance exists only where reinforcement can develop the required stress. Development length depends on bar size, concrete strength, coating, top-bar condition, cover, spacing, confinement, and the applicable modification factors. Standard hooks, headed bars, or mechanical anchorage work only under their stated geometric and confinement conditions.

At a bar cutoff, extend reinforcement beyond the theoretical point as required and examine shear and crack implications. At supports, positive reinforcement may need continuity or anchorage for integrity and load reversal; negative bars require development into the support region. Lap-splice location, class, staggering, and confinement matter. Congested bars that cannot be placed with cover and spacing shown are not a valid detail.

Beam and Slab Behavior

A one-way slab primarily spans between two opposite support lines and can be analyzed as strips with actual continuity. A two-way slab distributes moment in two directions; use an ACI-permitted analysis method and distribute moment to the prescribed regions. Gravity loading typically produces bottom tension in positive midspan regions and top tension over continuous supports. Provide shrinkage-and-temperature reinforcement, edge/corner detailing, and support reinforcement as required rather than reinforcing only the largest positive strip moment.

Deflection checks may require immediate and long-term stiffness, cracking, sustained load, creep, shrinkage, and compression reinforcement. Strength-level M_u is not the service-load input for every deflection expression.

Complete ACI Workflow

1. Establish factored demands and service effects from the actual support/slab model.
2. Draw bars and calculate actual d; identify singly or doubly reinforced behavior.
3. Calculate M_n, tensile strain, applicable φ, and reinforcement limits.
4. Check one-way shear and, for slabs, punching shear.
5. Develop every required bar and design splices, cutoffs, hooks, and support anchorage.
6. Check minimum reinforcement, cover, spacing, crack control, and service deflection.
7. Verify the load path through bearings, joints, columns, and walls.

Do not import a later ACI edition's equation or detailing rule into a 2026 solution.