Tools › Structures › Beams › Continuous Beam and Moment Distribution › Example
Three equal 20-ft spans of a W12x26 under 1.0 kip/ft dead and 1.5 kip/ft patterned live load: stiffness solution, Hardy Cross table, three-moment equations and the live-load envelope
| Tool | Continuous Beam and Moment Distribution |
| Source | National Aeronautics and Space Administration (1975). Astronautic Structures Manual, Volume I. NASA TM X-73305, Marshall Space Flight Center, Huntsville, AL. Section B4.2.4 (three-moment equation, Table B 4.2.4-1) and Section B5.1.1-B5.1.2 (fixed-end moments Table B 5.1.1-1, the moment-distribution procedure and Sample Problem 1); Merriman, M. (1876). On the Theory and Calculation of Continuous Bridges. D. Van Nostrand, New York (the equal-span coefficients, pp. 77-80). |
| Level | Undergraduate |
Problem
A W12x26 (I = 204 in^4, E = 29,000 ksi) is continuous over three equal 20-ft spans on pinned supports and carries 1.0 kip/ft dead load on every span and 1.5 kip/ft live load that may occupy any set of spans. Find the reactions, the support moments and the span maxima with the live load everywhere, reproduce them by moment distribution and by the three-moment equations, find the envelope of moments, reactions and deflections over the eight live-load patterns, redistribute the negative moments by 10 percent, and compare the design moment with 139.5 kip-ft and the deflection with L/360.
Inputs to Enter
- Spans 20 / 20 / 20 ft, I 204 in^4 shared, E 29,000 ksi; supports 1-4 pin / roller, no springs, no settlement
- Dead load uniform 1,000 lb/ft on every span (span 0), live load uniform 1,500 lb/ft on every span; case shown dead + live on all spans; pattern family every combination; factors 1.0 / 1.0
- Influence line of the reaction at support 2; redistribution 10 percent; allowable moment 139.5 kip-ft (0.9 x 50 ksi x 37.2 in^3, illustrative); deflection limit L/360; distribution tolerance 0.001, 50 cycles
Answers
- Live load everywhere (w = 2.5 kip/ft): R_1 = R_4 = 0.4 w L = 20.00 kip, R_2 = R_3 = 1.1 w L = 55.00 kip; M_2 = M_3 = -0.1 w L^2 = -100.0 kip-ft; end-span maximum 0.08 w L^2 = 80.00 kip-ft at 0.4 L = 8.00 ft, middle span 0.025 w L^2 = 25.00 kip-ft; largest shear 30.00 kip; largest deflection 0.804 in at 8.92 ft from the end support (Gere's coefficient 0.0069 w L^4/EI = 0.806 in)
- Moment distribution: K = I/L = 10.2 in^4/ft, DF 0.5 / 0.5 at supports 2 and 3, FEM = w L^2/12 = 83.33 kip-ft, converged in 10 cycles to M_2 = -100.0 kip-ft (0.03 percent of the largest FEM from the stiffness value); three-moment equations 4.868e-4 M_1 + 1.947e-3 M_2 + 4.868e-4 M_3 = -0.2434 rad at support 2 (and its mirror) give M_2 = M_3 = -100.0 kip-ft
- Envelope over the 8 patterns: end-span maximum 92.45 kip-ft with live on spans 1 and 3 (M_2 = -70.0, R_1 = 21.5 kip, 92.45 = 21.5^2 / (2 x 2.5)); M_2 = -110.0 kip-ft and R_2 = 58.00 kip with live on spans 1 and 2 (-0.1 w_D L^2 - 0.1167 w_L L^2, 1.1 w_D L + 1.2 w_L L); middle-span maximum 55.00 kip-ft with live on span 2 only (0.025 w_D L^2 + 0.075 w_L L^2); largest deflection 1.016 in
- Redistribution 10 percent: M_2 = M_3 = -90.0 kip-ft, end-span maximum 84.05 kip-ft at 8.20 ft, middle span 35.00 kip-ft; design moment 110.0 kip-ft, ratio 110.0 / 139.5 = 0.789; deflection ratio 1.016 / 0.667 = 1.52 (L/360 exceeded under the service envelope)
Notes
The tool's default inputs are this beam; press Submit. The beam is illustrative (stated on the page); every coefficient is a closed form derived from the three-moment equation and checked by test_continuous_beam.php (312 checks) against an independent numpy stiffness code, the NASA TM X-73305 Sample Problem 1 moment-distribution table and Merriman's 1876 printed examples.