Reinforced Concrete Beam and Column: Method

Reinforced Concrete Beam and Column

What the tool does

The tool performs the strength design checks of a reinforced-concrete beam and column the way USACE EM 1110-2-2104 (Strength Design for Reinforced Concrete Hydraulic Structures) and ACI 318 lay them out: the nominal flexural strength of a rectangular or T beam by strain compatibility with the equivalent rectangular stress block, the net tensile strain that classifies the section and fixes the strength-reduction factor φ, the steel ratios and the minimum reinforcement, the one-way shear strength of the concrete and the stirrups with the spacing limits, the gross and cracked moments of inertia with the effective inertia for deflection, and, for a rectangular or circular column, the complete axial-load–moment interaction diagram with the factored load checked against it. Every requirement is printed as a plain comparison (φM_n ≥ M_u, φV_n ≥ V_u, the load point inside the design curve): the strength-reduction factor sits on the resistance side, as the code writes it, and the tool carries no load-level radio or margin boxes.

Conventions and units

Depths are measured from the compression face: d is the depth of the main tension layer, d′ that of the compression steel, and for a column each bar layer has a depth d_i from the face that is in compression under a positive moment. Compression is positive in the column diagram; M_n is taken about mid-depth (the plastic centroid of a symmetric section). A T beam has its flange on the compression face. The code expressions with √f′_c are written for psi; the library works in inches, psi, pounds and lb·in and converts, so an SI entry (MPa, mm, kN, kN·m) is handled through the site's unit dictionary and the results are printed in MPa, mm, kN and kN·m. The metric forms of the √f′_c expressions are quoted beside the U.S. ones (0.17λ√f′_c for 2λ√f′_c, 0.25√f′_c for 3√f′_c, 1.4 for 200). The dictionary has no kip·ft unit: enter a moment in lb·ft (150 kip·ft = 150,000 lb·ft) or kN·m, the results print kip·ft and kip·in.

Materials

The stress-block depth factor is β₁ = 0.85 for f′_c ≤ 4,000 psi, falling 0.05 for each 1,000 psi above that and not below 0.65 (ACI 318-19 Table 22.2.2.4.3). The modulus of elasticity of normalweight concrete is E_c = 57,000√f′_c psi (§19.2.2.1), the modular ratio n = E_s/E_c, and the modulus of rupture f_r = 7.5λ√f′_c (§19.2.3.1), λ being the lightweight-concrete factor. The reinforcement is elastic-perfectly plastic: f_s = E_sε_s capped at ±f_y, with ε_ty = f_y/E_s (0.00207 for Grade 60).

Strain compatibility and the equivalent stress block

At the nominal strength the extreme compression fibre reaches ε_cu = 0.003 and the strain varies linearly with depth, so a bar at depth d_i has ε_si = 0.003 (c − d_i)/c for a neutral-axis depth c. The concrete stress distribution is replaced by Whitney's uniform block of 0.85f′_c over the depth a = β₁c (ACI 318-19 §22.2.2.4, after Mattock, Kriz and Hognestad), acting at the centroid of the compressed area: a rectangle b·a, the flange plus the web of a T when a exceeds h_f, or a circular segment for a round column. A compression bar lying inside the block displaces concrete that the block already counts, so its net force is A′_s(f′_s − 0.85f′_c); EM 1110-2-2104 example C-3 makes that deduction while its 2016 example D.4 does not, and a checkbox selects either convention.

For a beam the tool finds the c at which the compressive and tensile forces balance, by bisection (the resultant force increases monotonically with c), and the moment of the forces is M_n. This one solution covers singly reinforced, doubly reinforced and T sections, a second tension layer below yield and compression steel that does not yield, without the case distinctions of hand calculation. The classical closed forms are printed beside it: a = A_sf_y/(0.85f′_cb) and M_n = A_sf_y(d − a/2) (EM 1110-2-2104 eq. D-5), the T-beam form with the overhanging-flange force C_f = 0.85f′_c(b_f − b_w)h_f, and the compression-steel yield check (ρ − ρ′) ≥ 0.85β₁(f′_c/f_y)(d′/d)·87,000/(87,000 − f_y) of example D.4.

Net tensile strain and φ

The net tensile strain ε_t = 0.003 (d_t − c)/c at the extreme tension layer classifies the section (ACI 318-19 §21.2.2): compression-controlled at or below ε_ty, tension-controlled at or above the tension-controlled limit (ε_ty + 0.003 in ACI 318-19, 0.005 in ACI 318-14), and in transition between, where φ varies linearly from φ_c (0.65 with ties, 0.75 with spirals) to φ_t = 0.90. The limits and the φ values are entered inputs with these defaults, so another edition or another code's values can be used. Beams must be tension-controlled (§9.3.3.1); the tool notes a section that is not.

The balanced steel ratio of a single rectangular layer follows from c_b = ε_cud/(ε_cu + ε_ty): ρ_b = 0.85β₁(f′_c/f_y)·ε_cuE_s/(ε_cuE_s + f_y), the familiar 87,000/(87,000 + f_y) form in psi; the ratio at the tension-controlled limit replaces ε_ty by that limit. The minimum flexural reinforcement is the larger of 3√f′_cb_wd/f_y and 200b_wd/f_y (§9.6.1.2). EM 1110-2-2104 additionally keeps hydraulic structures at ρ ≤ 0.25ρ_b without special study; the tool prints that value for information.

One-way shear

The concrete contribution is the simplified V_c = 2λ√f′_cb_wd (ACI 318-19 Table 22.5.5.1, the row for members with at least minimum stirrups; the detailed form with the size effect and ρ_w is not implemented), the stirrups add V_s = A_vf_ytd/s with f_yt at most 60,000 psi, V_s is capped at 8√f′_cb_wd (§22.5.1.2), and φV_n = φ(V_c + V_s) must reach V_u. Minimum stirrups A_v,min = max(0.75√f′_c, 50)b_ws/f_yt are required where V_u exceeds φλ√f′_cb_wd (§9.6.3), and the spacing may not exceed d/2 or 24 in, halved when V_s exceeds 4√f′_cb_wd (Table 9.7.6.2.2). The stirrup spacing needed for the entered V_u is printed with the check.

Cracked section and effective moment of inertia

The cracked transformed section ignores the concrete in tension, counts the tension bars at nA_s and the compression bars at (n − 1)A′_s, and places the neutral axis kd where the first moments balance; I_cr follows (b(kd)³/3 for a rectangle, flange and web for a T, a segment for a circle, plus the steel terms). The cracking moment is M_cr = f_rI_g/y_t. At a service moment M_a the effective moment of inertia is given in Branson's form, I_e = (M_cr/M_a)³I_g + [1 − (M_cr/M_a)³]I_cr ≤ I_g (ACI 318-14 eq. 24.2.3.5a), and in Bischoff's form adopted by ACI 318-19, I_e = I_cr/[1 − (⅔M_cr/M_a)²(1 − I_cr/I_g)] for M_a > ⅔M_cr. The deflection itself is E_cI_e in the elastic formula of the span (the site's Beam Deflection tool takes EI directly).

Column interaction diagram

For a column the same strain compatibility is evaluated at a sweep of neutral-axis depths from c = ∞ (uniform ε_cu, the squash load P_o = 0.85f′_c(A_g − A_st) + f_yA_st, ACI 318-19 eq. 22.4.2.2) down to a small c (large tension), each giving P_n = ΣF and M_n = ΣF·(h/2 − y); the pure-tension point P_nt = −f_yA_st closes the curve. The bars are entered as layers (depth, number, bar area) or as n bars on a circle, whose bars at equal depths are merged into layers; the circular section's block is a circular segment with its exact area and centroid. The nominal curve is capped at P_n,max = 0.80P_o (ties) or 0.85P_o (spirals) for accidental eccentricity (Table 22.4.2.1), and the design curve is φP_n, φM_n with φ from ε_t at the extreme tension layer at every point and the cap φP_n,max. The characteristic points are printed: P_o, the balanced point (ε_t = ε_ty), the tension-controlled limit, pure bending and pure tension, in the form of EM 1110-2-2104 Appendix C.4 / 1992 Appendix E.

The factored load (P_u, M_u) is checked radially: the ray from the origin through the load point is extended to the design curve and the utilization is the ratio of the two distances, U = |(M_u, P_u)|/|(φM_n, φP_n)|, so U ≤ 1 means the point lies inside the curve at constant eccentricity. The design moment strength at the entered P_u and the design axial strength at the entered M_u are printed as well. The section drawing shades the block of the design-curve point on that ray and colours the bars that are in tension there.

Limits

The tool checks a section: slenderness and moment magnification (enter the magnified moment), biaxial bending, torsion, punching shear, development and splice lengths, bar spacing, cover and crack control are not included, nor are EM 1110-2-2104's hydraulic load factors. The simplified V_c is used throughout; the concrete contribution is not reduced for axial tension, nor increased for compression. A T beam is analysed with the flange in compression. The strain limits and φ are taken as entered; the tool does not enforce the beam requirement ε_t ≥ ε_ty + 0.003 beyond a note.

Sources

The method and every worked value the tool is checked against come from the two editions of USACE EM 1110-2-2104 (1992 with Change 1 of 2003, and 2016), public U.S. government documents: the singly reinforced beam, the slab reinforced in both faces with non-yielding compression steel, the interaction-diagram points, the doubly reinforced wall and the coastal floodwall stem of their appendices. ACI 318-19 is cited for the φ factors, strain limits, β₁, E_c, f_r, A_s,min, V_c, V_s and I_e expressions, each a single stated provision; no code table is reproduced. The bar areas of the size select are the ASTM A615 nominal dimensions. An independent numpy fibre integration of every section (2,000 fibres) confirms the library's strain-compatibility results to 0.1 %.