Beam Deflection: Method

Beam Deflection: Method, Sign Conventions and the Catalogue of Cases

What the tool does

The tool analyses a straight prismatic beam on one span under transverse loads and couples: it reports the support reactions, the shear and bending-moment equation of every segment with the substituted numbers, the extreme moments and their positions, the slope and deflection along the beam (their extremes, the end slopes, the values at a chosen point), the stiffness of the beam as a spring, four diagrams, and a margin of safety on the maximum deflection against an allowable. Five support arrangements and eighteen catalogue load cases cover the classical beam tables; two further options accept any number of point loads, uniform loads, triangular loads and couples on a simply supported or a cantilever beam and add their effects by superposition.

Theory

The beam is the Euler–Bernoulli beam of elementary strength of materials: plane sections remain plane and normal to the deflected axis, the material is linear elastic, deflections are small, and shear deformation is neglected. With x along the beam from the left end and y the upward deflection, the load intensity q(x) (upward positive), the shear V, the bending moment M (sagging positive), the slope θ = dy/dx and the deflection are linked by

dV/dx = q,   dM/dx = V,   E I d²y/dx² = M,

so V = ∫q dx, M = ∫V dx, E I θ = ∫M dx + C₁ and E I y = ∫E I θ dx + C₁x + C₂. Each load is written as a singularity (Macaulay) function c ⟨x − a⟩ⁿ, which is zero for x < a and c (x − a)ⁿ beyond: a couple is a term of order −2, a point force of order −1, a uniform load of order 0 and a linearly varying load of order 1 (a block or a ramp that ends inside the span is closed by the opposite term at its end). Integration of such a term is a shift of its order, so V, M, θ and y are exact piecewise polynomials, which is why the tool can print the equation of every segment with its numerical coefficients.

The support reactions and the two constants are the unknowns of a small linear system: the shear and the moment must vanish beyond the right end (force and moment equilibrium of the whole beam), the deflection is zero at every support and the slope is zero at a fixed end. A statically determinate beam (simply supported, cantilever, overhanging) has as many conditions as reactions plus constants; the fixed-fixed beam and the propped cantilever have one or two more reactions, and the extra kinematic conditions supply them, so no separate compatibility or moment-distribution step is needed and the answer is exact for every case. The maximum deflection and the moment extremes are located by scanning the exact curve at 2,000 stations and refining the extreme by golden-section search; for the catalogue cases the textbook closed form of the same quantity is printed beside it.

Sign conventions

x is measured from the left end; for a cantilever or a propped cantilever the fixed end is the left end; for the overhanging beam the supports are at 0 and L and the overhang a extends beyond the right support, so the beam is L + a long. Point loads P and distributed loads w act downward (a negative value acts upward). Reactions are reported upward positive; a fixing moment is reported as the internal bending moment at the fixed end, negative when the beam hogs there (tension on top), which is the usual case under downward load. The bending moment is sagging-positive (tension on the bottom). The deflection δ is reported downward positive, so that the usual answer is a positive number; the slope θ = dy/dx is positive when the beam rises toward the right, so a simply supported beam under downward load has θ_A < 0 and θ_B > 0, and a cantilever has a negative slope at its free end.

Couples: in the two end-moment cases M₀ is positive when it bends the beam concave up (sagging) at the end where it acts, which is clockwise at the left support of the simple beam and counter-clockwise at the free end of the cantilever; the formulas of Gere's Table G-2 case 7 (couple at the left support) and the cantilever-with-end-couple case of Table G-1 then apply with their printed signs, the simply supported beam deflecting downward and the cantilever's free end rising. In the superposition tables a couple is positive clockwise: it raises the bending moment by its value just to its right.

The catalogue cases

Simply supported: uniform load (δ_max = 5wL⁴/384EI at mid-span, M_max = wL²/8, θ = wL³/24EI); point load at a (M_max = Pa(L − a)/L under the load, δ under the load Pa²(L − a)²/3EIL, δ_max at √(a(a + 2b)/3) from the support nearer the longer segment a, Pab(a + 2b)√(3a(a + 2b))/27EIL); partial uniform load from a to b (reactions and the position of zero shear from statics, no simple closed form for δ_max); triangular load rising from zero at the left support to w at the right (R_A = W/3, R_B = 2W/3 with W = wL/2, M_max = 2WL/9√3 at L/√3, δ_max = 0.01304 WL³/EI at 0.5193L); end moment at the left support (θ_A = M₀L/3EI, θ_B = M₀L/6EI, δ_max = M₀L²/9√3EI at L(1 − 1/√3), δ at mid-span M₀L²/16EI); two point loads P at a and P₂ at b (symmetric equal loads: M = Pa between the loads, δ_max = Pa(3L² − 4a²)/24EI).

Cantilever (fixed at the left end): point load at the free end (δ = PL³/3EI, θ = PL²/2EI, M_A = −PL); point load at a (δ_end = Pa²(3L − a)/6EI, δ under the load Pa³/3EI, θ_end = Pa²/2EI); uniform load (δ = wL⁴/8EI, θ = wL³/6EI, M_A = −wL²/2); partial uniform load from a to b, obtained from Gere's case of a load from the fixed end to a distance c, δ_end = wc³(4L − c)/24EI, applied to b and to a and subtracted; moment at the free end (δ = M₀L²/2EI, θ = M₀L/EI, constant moment M₀); triangular load rising from zero at the free end to w at the fixed end (δ = WL³/15EI = wL⁴/30EI, θ = WL²/12EI, M_A = −WL/3, W = wL/2).

Fixed at both ends: central point load (R = P/2, M_ends = −PL/8, M_centre = PL/8, δ = PL³/192EI); point load at a with b = L − a (R_A = Pb²(3a + b)/L³, R_B = Pa²(a + 3b)/L³, M_A = −Pab²/L², M_B = −Pa²b/L², M under the load 2Pa²b²/L³, δ under the load Pa³b³/3EIL³, δ_max for a ≥ b at 2aL/(3a + b) equal to 2Pa³b²/3EI(3a + b)²); uniform load (M_ends = −wL²/12, M_centre = wL²/24, δ = wL⁴/384EI).

Propped cantilever (fixed at the left end, simply supported at the right): point load at a from the fixed end, with the AISC formulas written in terms of the distance L − a from the simple support (R_fixed = P(L − a)(3L² − (L − a)²)/2L³, R_simple = Pa²((L − a) + 2L)/2L³, M_A = −Pa(L − a)(2L − a)/2L², δ_max by the two branches of the table according to whether the load is nearer the simple support or the fixed end; a central load gives δ_max = 0.009317PL³/EI at 0.4472L from the simple support and M_A = −3PL/16); uniform load (R_fixed = 5wL/8, R_simple = 3wL/8, M_A = −wL²/8, maximum sagging moment 9wL²/128 at 5L/8 from the fixed end, δ_max = 0.005416wL⁴/EI at L(15 − √33)/16 = 0.5785L from the fixed end; AISC prints the coefficient as 1/185, 0.2 % low, and the tool evaluates the table's own deflection curve instead).

Simply supported with an overhang a beyond the right support: point load at the end of the overhang (R_A = −Pa/L downward, R_B = P(L + a)/L, M over the support −Pa, the span rises with its extreme at L/√3 equal to PaL²/9√3EI, the end deflects Pa²(L + a)/3EI downward); uniform load over the span and the overhang (R_A = w(L² − a²)/2L, R_B = w(L + a)²/2L, maximum sagging moment w(L + a)²(L − a)²/8L² at x = (L/2)(1 − a²/L²), moment over the support −wa²/2, end deflection wa(4a²L − L³ + 3a³)/24EI, which is negative, the end rising, when the overhang is short).

Superposition tables: any number of rows, each a point load P at a, a uniform load w from a to b, a triangular load rising from zero at a to w at b, or a clockwise couple M at a, on a simply supported beam of span L or a cantilever of length L. The response is the sum of the responses to the rows, exactly as for a single load.

Reading the results

The Loads and Supports table restates the model in the display units. The Reactions are signed as described above. The Shear and Moment Equations table gives V(x), M(x) and E I y(x) for every segment between load and support positions, with x in the display length unit; the moment extremes that follow distinguish the maximum sagging moment, the maximum hogging moment and the moment of largest magnitude, which is the design moment. The Slopes and Deflections section gives the maximum deflection with its position, the end slopes in radians, the values at the entered point x, and the stiffness k = (total transverse load)/δ_max, a convenient spring constant for a beam that supports a mass or another member; it is not defined for a loading without a transverse resultant. The Textbook Closed Forms section prints the published formulas of the case with the substituted numbers; their results equal the exact values above to round-off and serve as the hand check. The Diagrams are drawn from the exact equations; the deflection is plotted downward positive so the curve hangs like the beam. The Margin of Safety compares the magnitude of the maximum deflection with the allowable: L/360, L/240 or L/180 of the span between the supports (the International Building Code's Table 1604.3 limits for floors and plastered ceilings, roofs and floors in general, and roofs without ceilings respectively), or a value entered directly; deflection is a serviceability check, so it is written at limit (service) load with no factor of safety.

Limits

Shear deformation is neglected: for a span shorter than about ten times the depth, or for sandwich and built-up members with a soft web, the Timoshenko correction V/(κGA) adds a few percent or more to the deflection. The beam is prismatic (constant E I), on a single span (continuous beams over three or more supports are not offered), without axial load (a beam-column amplifies the deflection; see the Beam-Column tool), and the material remains elastic: the tool does not check stresses, which the Beam Stress tool reports from the moments and shears found here. Loads are static; the stiffness k is the static stiffness, usable for a lumped-mass natural frequency only with the appropriate effective mass.