Tools › Civil and Geotechnical › Concrete Design › Reinforced Concrete Beam and Column › Example
12 × 24 in beam with 3 No. 8 bars and a 16 × 16 in tied column with 8 No. 8 bars: flexure, shear, cracked section and the interaction diagram by strain compatibility
| Tool | Reinforced Concrete Beam and Column |
| Source | U.S. Army Corps of Engineers (2016). Strength Design for Reinforced Concrete Hydraulic Structures. Engineer Manual EM 1110-2-2104, Department of the Army, Washington, DC. Appendix C (strain-compatibility examples C.2, C.3 and the interaction points of C.4) and Appendix D (D.4, D.6); the 1992 edition's Appendices C, D, E and F; American Concrete Institute (2019). Building Code Requirements for Structural Concrete (ACI 318-19). ACI, Farmington Hills, MI (φ, β_1, strain limits, A_s,min, V_c, V_s and I_e provisions). |
| Level | Undergraduate |
Problem
A rectangular beam 12 in wide and 24 in deep (d = 21.5 in) of 4,000 psi concrete carries 3 No. 8 Grade 60 bars and No. 3 stirrups at 10 in; the factored loads are M_u = 150 kip·ft and V_u = 40 kip, the service moment 100 kip·ft. A 16 × 16 in tied column of the same materials has 8 No. 8 bars as 3-2-3 at depths 2.5, 8 and 13.5 in and carries P_u = 400 kip with M_u = 120 kip·ft. Find the nominal and design moment strengths, the net tensile strain and φ, the steel ratios and minimum reinforcement, the shear strength and stirrup limits, the cracked and effective moments of inertia, the column interaction diagram and the utilization of the factored load.
Inputs to Enter
- Materials f′_c 4,000 psi, f_y 60 ksi, E_s 29,000 ksi, λ 1.0; φ_t 0.90, φ_c 0.65 (ties), φ_v 0.75, tension-controlled limit 0.005, ε_ty = f_y/E_s
- Beam: b 12 in, h 24 in, d 21.5 in, 3 No. 8 (A_s 2.37 in²), A_v 0.22 in² at s 10 in, f_yt 60 ksi, M_u 150,000 lb·ft, V_u 40 kip, M_a 100,000 lb·ft
- Column: 16 × 16 in, tied, layers 3 bars at 2.5 in, 2 at 8 in, 3 at 13.5 in (0.79 in² each), P_u 400 kip, M_u 120,000 lb·ft
Answers
- β_1 = 0.85, E_c = 3,605 ksi, n = 8.044, f_r = 474 psi; a = A_s f_y/(0.85 f′_c b) = 3.485 in, c = 4.100 in, ε_t = 0.01273 (tension-controlled, φ = 0.90)
- M_n = A_s f_y (d − a/2) = 2,809.5 kip·in = 234.1 kip·ft, φM_n = 210.7 kip·ft ≥ 150 kip·ft (M_u/φM_n = 0.712); ρ = 0.00919, ρ_b = 0.02851, A_s,min = 0.860 in²
- V_c = 2√f′_c b_w d = 32.63 kip, V_s = A_v f_yt d/s = 28.38 kip, φV_n = 45.76 kip ≥ 40 kip; s_max = d/2 = 10.75 in, A_v,min = 0.100 in² at 10 in
- I_g = 13,824 in⁴, kd = 6.828 in, I_cr = 5,377 in⁴, M_cr = 45.5 kip·ft; at M_a = 100 kip·ft I_e = 6,175 in⁴ (Branson) / 5,698 in⁴ (Bischoff)
- Column: P_o = 1,228.1 kip, φP_n,max = 0.65 × 0.80 P_o = 638.6 kip, balanced c_b = 7.99 in with P_b = 360.7 kip and M_b = 268.2 kip·ft, pure bending M_o = 186.8 kip·ft
- Ray through (120 kip·ft, 400 kip) meets the design curve at (135.8 kip·ft, 452.8 kip): utilization U = 0.883 ≤ 1; φM_n at P_u = 147.1 kip·ft, φP_n at M_u = 513.6 kip
Notes
The tool's default inputs are this case; press Submit. The default is illustrative (no printed example has these sections) and is checked in test_rc_beam_column.php (224 checks) against an independent numpy fibre integration of the sections at 2,000 fibres (References/RC-Beam-Column/checks/rc_check.py) and the closed forms; the same test reproduces the USACE EM 1110-2-2104 appendix examples to their printed digits: the 1992 and 2016 singly reinforced beams (φM_n 1,616.8 and 1,649.9 kip·in), the slab reinforced in both faces with non-yielding compression steel (26,208 and 26,515.2 kip·in), the interaction points of Appendix E / C.4 (M_n 2,404.7 / 2,463.5 kip·in, P_b 218.62 / 331.52 kip, M_b 3,390.65 / 4,120.55 kip·in), the 18 × 18 in column of Appendix F (146.1 and 145.9 ft·kip at 142.9 kip), the doubly reinforced wall D.4 and the floodwall stem D.6 (V_c 55.4 kip, φM_n 242 kip·ft).