Continuous Beam and Moment Distribution: Method

Continuous Beam and Moment Distribution: Method, Sign Conventions, Moment Distribution and the Three-Moment Equation

What the tool does

The tool analyses a straight beam continuous over up to eight spans. Each span has its own length and, if wanted, its own moment of inertia (a stepped beam); the supports are pins or rollers, fixed ends, free ends (which make the end span a cantilever), vertical springs, and any of them may carry a rotational spring and a prescribed settlement. Loads are entered span by span as uniform or partial uniform loads, linearly varying (ramp or triangular) loads, point loads and couples, in a dead-load table and a live-load table, each with its own factor. For the case chosen the tool prints the reactions, the support moments, the shear and bending-moment diagrams with every maximum located, the slopes and deflections; then it works the same case by Hardy Cross moment distribution as the handbook table and by Clapeyron's three-moment equations, both of which must reproduce the support moments. The live load is then patterned over the spans (every combination, or the alternate and adjacent-span patterns of the building codes) and the extremes are collected as an envelope; an axle group can be stepped along the beam; the influence line of a reaction, a bending moment or a shear is drawn; the negative support moments can be redistributed by an entered percentage; and the governing moment and deflection are compared with an allowable moment and a span/n limit as plain ratios.

Sign conventions

x runs from the left end of the beam. Spans and supports are numbered from the left (support 1 is the left end, span k lies between supports k and k + 1). Distributed loads and point loads act downward when positive; a couple is positive clockwise, which raises the sagging moment just to its right. A settlement is positive downward. The internal bending moment is sagging positive (tension on the bottom); the shear is V = dM/dx; the deflection is reported downward positive and the slope θ = dy/dx is positive when the beam rises to the right. Reactions are upward positive. A support moment is the internal bending moment at the support, so a continuous beam under downward load has negative (hogging) support moments; at a rotational spring the moment jumps by the spring moment and both sides are listed. In the moment-distribution table the end moments follow Cross's convention, clockwise on the end of the member positive: at the left end of a span the internal moment equals the end moment, at the right end it is its negative.

The stiffness solution

Each span is a Hermite beam element of the Euler–Bernoulli theory (plane sections, linear elastic material, small deflections, no shear deformation) with two unknowns at each support, the deflection and the rotation. The element stiffness matrix is the exact one of a prismatic member, so the discretisation introduces no error: the solution is exact within the beam theory. The loads enter as equivalent nodal loads, the integrals of the load intensity against the shape functions; for the polynomial loads used here a four-point Gauss rule integrates them exactly, and the moments among them are the fixed-end moments of the handbook tables (uniform load w L²/12, point load P a b²/L² and P a² b/L², triangular load w L²/30 and w L²/20, and so on: NASA TM X-73305 Table B 5.1.1-1 lists them). A pin prescribes the deflection (zero, or the settlement), a fixed end also the rotation, a vertical spring adds its stiffness k to the diagonal of its deflection, a rotational spring adds k_r to the diagonal of its rotation, and a free end prescribes nothing. The equations of the free unknowns are solved by Gaussian elimination with pivoting; a beam that cannot be held (a single pin, no vertical support) is reported as a mechanism.

From the end forces of each element the shear and the moment along the span are written as sums of singularity terms c⟨x − a⟩ⁿ, one per load edge, and the slope and deflection follow by integrating those terms in closed form from the known rotation and deflection at the left support. Within each interval between load edges the shear is a polynomial of degree at most two, so its zeros, where the moment is extreme, are found exactly; the deflection extremes sit where the slope vanishes and are found the same way. The diagrams are drawn from 100 stations per span plus both sides of every load edge.

Moment distribution

Hardy Cross's method (1930) solves the same equations by successive approximation without writing them. Every joint is first locked against rotation; each loaded span then carries its fixed-end moments. The stiffness factor of a member end is K = I/L (the rotational stiffness 4 E I/L divided by 4 E, which is common to every member when E is shared; the tool uses E I/L internally so that spans with different E I are handled, and prints I/L), the distribution factor of each member end at a joint is DF = K/ΣK over the members meeting there, and the carry-over factor to the far end of a prismatic member is one half. A rotational spring at a joint takes a share DF = (k_r/4E)/ΣK of the balancing moment and carries nothing over; a column fixed at its far end is such a spring with k_r = 4 E I_c/L_c, which is how the tool reproduces the NASA sample problem of a beam with a column. A cantilever end span has no stiffness (its end is free): its root moment is the static moment of its loads, entered as a fixed value with DF = 0, and the joint distributes the unbalance among the other members. A settlement enters as the chord-rotation fixed-end moment −6 E I ψ/L at both ends of the span, ψ = (Δ_right − Δ_left)/L clockwise positive.

The table is written in the simultaneous form of NASA TM X-73305 Section B5.1.2: in every cycle all free joints are balanced at once (balancing moment = −DF × the sum of the moments at the joint that are out of balance, which after the first cycle is the sum of the carry-overs just received), then every balancing moment sends half of itself to the far end of its member. A fixed end is never balanced; it accumulates carry-overs. A pinned end joint has a single member and DF = 1, so it is balanced to zero each cycle (the modified stiffness 3/4 K of the textbooks shortens the work by hand; the tool keeps the plain method so that the printed table matches the handbook layout). The cycles stop when the largest balancing moment of a cycle is below the entered fraction of the largest fixed-end moment, and the final moment at each member end is the column sum. Because every balance restores joint equilibrium and every carry-over is the exact response of the far end, the sums converge to the stiffness solution; the page prints the largest difference, which is the truncation of the cycles. Translational springs and unsupported interior nodes involve joint translation (sidesway) and are left to the stiffness solution.

The three-moment equation

Clapeyron's equation (1857) expresses the continuity of the slope at a support between two spans in terms of the three support moments. With the moments sagging positive, the flexural rigidity E I of each span kept (so that stepped beams are covered), A x̄ the first moment of the simple-span bending-moment diagram of the loaded span about its far support, and Δ the downward settlements, the equation at support j between spans 1 (to its left) and 2 (to its right) is

M_j−1 L₁/E I₁ + 2 M_j (L₁/E I₁ + L₂/E I₂) + M_j+1 L₂/E I₂ = −(6 A₁ x̄₁/L₁)/E I₁ − (6 A₂ x̄₂/L₂)/E I₂ + 6 (Δ_j − Δ_j−1)/L₁ + 6 (Δ_j − Δ_j+1)/L₂.

The right-hand side is the slope discontinuity that the support moments must close, in radians; the handbook form multiplies through by E and writes L/I and 6 A x̄/(I L) (NASA TM X-73305 Section B4.2.4 tabulates the 6 A x̄/L terms as K₁ and K₂ for the usual loads: w L³/4 for a uniform load, P a (L² − a²)/L for a point load, and so on). The tool integrates the simple-span moment diagram exactly instead of using a table, so any combination of loads is covered. A pinned end has M = 0, a fixed end is written with an imaginary zero-length span (2 M₀ L₁/E I₁ + M₁ L₁/E I₁ = −(6 A x̄/L)/E I₁ + 6 (Δ₁ − Δ₀)/L₁, with A x̄ now about the near support), and a cantilever's root moment is known from statics and moved to the right-hand side. The equations are printed with their numerical coefficients and solved; springs are outside this form.

Pattern live loading and the envelope

The live load that maximises one effect does not occupy every span: the maximum sagging moment in a span comes with live load on that span and on every second span away from it, the maximum hogging moment at a support and the maximum reaction there with live load on the two spans adjoining it. The tool keeps the dead load on every span and places the live load on either every subset of spans (2ⁿ patterns, the true envelope), the code patterns (all spans; alternate spans in both parities; each pair of adjacent spans, as ACI 318 6.4.2 prescribes and the AISC commentary describes, both cited only), or the entered set. Each pattern is solved exactly, the extremes of the moment, the shear and the deflection are collected at 40 stations per span plus the exact extremes of every span, and the governing pattern is named. The classical coefficients of three equal spans follow: with live load on the two outer spans the end-span sagging moment is largest (support moments −0.05 w_L L², end reaction 0.45 w_L L), with live load on two adjacent spans the common support moment is −0.1167 w_L L² and the reaction 1.2 w_L L, with live load on the middle span alone its sagging moment is 0.075 w_L L².

Moving load

An axle group (loads and spacings) is placed with its first axle at 401 positions from one group length before the left end to the right end, so that every axle enters and leaves; axles off the beam are dropped, the beam is solved with the dead load on, and the moment, shear and reaction extremes are collected with the position of the group that produces them. The absolute maximum moment under a group of concentrated loads on a simple span occurs when the resultant and the nearest axle are equidistant from mid-span; the step of one four-hundredth of the travel locates it to that accuracy. An impact or dynamic allowance is entered as the live factor.

Influence lines

The influence line of a quantity is its value for a unit downward load as a function of the load position. The tool places the unit load at 60 positions per span and solves each, which for a reaction equals the Müller-Breslau construction (the deflected shape of the beam when the support is removed and displaced by one unit in the direction of the reaction), the check used in the test script. For a bending moment at a section the ordinate is the moment there, for a shear it is the shear there with the jump of one unit at the section. A concentrated load contributes its value times the ordinate, a uniform load its intensity times the area under the curve over the loaded length; the areas by span are printed, and a span whose area is negative should be left unloaded for the maximum (loaded for the minimum).

Moment redistribution

Because a continuous beam is statically indeterminate, a ductile section at a support can yield and shed moment to the spans without collapse; the codes allow the elastic negative moments to be reduced by a percentage, the positive moments being increased so that statics is preserved. The tool reduces every negative support moment of the case shown by the entered percentage and recomputes the span moments as the simple-span moment diagram plus the straight line between the adjusted support moments. ACI 318-19 6.6.5 permits up to 1000 ε_t percent, at most 20 percent, when the net tensile strain ε_t is at least 0.0075; AISC 360-16 B3.3 permits 10 percent for compact beams with the positive moment raised by one tenth of the average negative moments. Both are cited only; the percentage is the entered value.

Reading the results

The Model section restates the spans, supports and factored loads and draws them (Figure 1). The Stiffness Solution gives the reactions and support moments, the per-span extremes with their positions, the headline values and the shear, moment and deflection diagrams (Figures 2 to 4). The Moment Distribution table has one column per member end (labelled by its joint and far joint) and per rotational spring, the rows K, DF, FEM, then the balance and carry-over of each cycle and the final sum; the support moments from the table are compared with the stiffness solution. The Three-Moment Equation section prints each equation with its numerical coefficients and the solution. The Envelope tables name the governing live-load pattern for every span and support, with the envelope diagrams (Figures 5 and 6). The Moving Load, Influence Line and Redistribution sections follow when selected. The Comparison writes the ratio of the design moment (the largest of the case shown, the envelope and the moving load) to the allowable moment and, span by span, the ratio of the maximum deflection to L/n; no factor of safety is applied, the loads and the allowable are compared at the level they are entered.

Limits

Shear deformation is neglected (a few percent for spans shorter than about ten depths); the beam is straight, the material elastic (apart from the entered redistribution), and there is no axial load (a beam-column amplifies the moments; see the Beam-Column tools). A change of section inside a span is entered as two spans with an unsupported interior node (stiffness solution only). Lateral-torsional buckling, the section stresses and the strength of the member are not checked here; the Beam Stress and Lateral-Torsional Buckling tools take the moments and shears found. The moving load is quasi-static.