Beam Stress: Method

Beam Stress Recovery and Load Conventions

Problem Statement

A beam analysis, a hand load calculation or a finite element model delivers the six section resultants at a station of a straight member: the axial force, two shears, the torque and two bending moments. The question for the stress analyst is what the fibers of the section see. For a section drawn on the screen or loaded from My Sections, this tool evaluates the normal stress at every vertex, names the fibers in maximum tension and maximum compression, finds the neutral axis, converts the stresses to strains, adds the transverse and torsional shear stresses, and writes margins of safety against the entered allowables. Any number of load cases run at once; the equations are written out for one of them and all are tabulated.

Normal Stress

Plane sections remain plane, so the normal stress is linear over the section, σ = P / A + k_x x + k_y y with x and y measured from the centroid. The two slopes follow from the definitions of the bending moments as resultants of the stress. With the member axis z toward the viewer and the right-hand rule about the drawing axes, M_x = ∫σ y dA and M_y = −∫σ x dA, and solving the two equations with the product of inertia kept gives

σ = P / A + (M_x I_yy + M_y I_xy) y / (I_xx I_yy − I_xy²) − (M_y I_xx + M_x I_xy) x / (I_xx I_yy − I_xy²)

which reduces to M_x y / I_xx − M_y x / I_yy when the drawing axes are principal axes. Because the stress is linear, its largest and smallest values over a polygon are at vertices, so the tool evaluates every vertex of the outer boundary and of the holes and reports the two extremes with their coordinates. The neutral axis is the line σ = 0; it passes through the centroid only when the axial force is zero and is inclined to the x axis whenever M_y or I_xy is not zero. The section properties are exact for the drawn polygon (Green's theorem) and come from the same library as the Section Properties tool.

Stress Symbols

The symbols follow the chosen convention. With the aerospace convention the applied stresses are written in Bruhn's and Peery's notation, lower-case f with a subscript (f_t and f_c for the critical tension and compression fibers, f_sx, f_sy and f_sv for the transverse shear components and their resultant, f_st for the torsional shear, f_s for the shear sum used in the margins, f_vm for the von Mises bound), and the allowables are capital F (F_tu, F_ty, F_cy, F_su) as in MIL-HDBK-5. With the vector, Nastran and civil conventions the applied stresses are σ and τ with the same subscripts in capitals (σ_T, σ_C, τ_Vx, τ_Vy, τ_V, τ_T, τ, σ_vm), and the allowables are still F. The largest normal-stress magnitude of the two critical fibers is f_peak or σ_peak.

Restrained Bending

The unsymmetrical-bending formula gives the stress for whatever moments the section carries; it says nothing about supports. A free beam under M_x alone bends in a plane inclined to the vertical whenever I_xy is not zero, deflecting sideways as well as down, and that sideways curvature is where its large stresses come from. A stringer riveted to a wing skin, or a cap held by a web, cannot deflect sideways. Its curvature in x is then zero, the fiber strain is proportional to y alone, σ = P / A + k y, and substituting that into the definitions of the moments gives M_x = k I_xx and M_y = −k I_xy: the section carries M_y = −M_x I_xy / I_xx about y, supplied by the skin through the rivet line, and the stress is the symmetric formula σ = P / A + M_x y / I_xx exactly. The same ratio holds between the shears, since V = dM / dz, and the shear flow follows from d(σ)/dz of the reduced formula, so V_y Q_x / (I_xx t) is exact there too. Restraining y-translation gives the mirror result with M_x = −M_y I_xy / I_yy and σ = P / A − M_y x / I_yy.

The Restraint choice on the form selects the free beam or one restrained translation. Under a restraint the results page lists the moment and shear the section carries, the reaction the restraint must supply (the applied value is still what the Given section shows), and the reduced formula in the convention's symbols. On Peery's Z-section (Chapter 7, Example 1, I_xy = −240 in⁴) a moment M_x of 100,000 lb-in gives 2,045 psi at the web-flange corner as a free beam, with the tip of the tension flange in compression, and 865 psi uniform across the flange when sideways deflection is restrained: a factor of 2.4, which is why the restraint matters for Z-stringers. The restraint must be real, however: a twist restraint alone does not suppress lateral curvature, and a stringer that is one small area of a closed box is better analyzed as part of the whole box section, about the box's own axes, where its own I_xy plays no part. A beam restrained in both directions does not bend (σ = P / A) and is not offered.

Load Conventions

The equation above is written in the vector convention. The texts and the finite element codes do not all use it, and the differences are the usual source of a sign error, so the tool lets the loads be entered in any of four published conventions and converts them to the vector set internally. Each convention is a renaming and a sign map; the results page states the convention, draws its figure and writes the bending equation in its own symbols.

Vector, right-hand rule (Allen and Haisler, Megson): z along the member, x and y in the section, every load the right-hand-rule resultant on the +z face. A positive M_x puts the +y fiber in tension and a positive M_y puts the +x fiber in compression. This is the form in which loads come off a free-body diagram or a nodal force balance.

Aerospace fiber rule (Bruhn, Peery, Niu): z along the member and the same section axes, but a positive bending moment produces compression on the positive side of the axis it is about, in both planes. This carries the beam-diagram habit, sagging positive with compression on top, into two planes; the texts write σ = −(M_x I_yy − M_y I_xy) y / D − (M_y I_xx − M_x I_xy) x / D. Relative to the vector set, M_x has the opposite sign and M_y the same.

Nastran CBAR and CBEAM element forces. The element x axis runs along the member from end A to end B and the section lies in the element y–z plane; Plane 1 is x_e–y_e and Plane 2 is x_e–z_e. The element forces are section resultants (a uniform moment reads the same at both ends), and the stress recovery is σ = F_x / A − M₁ y_e / I₁ − M₂ z_e / I₂, so a positive moment compresses the positive side of its plane in both planes. The tool takes the drawing's x axis as y_e and its y axis as z_e, so I₁ = I_yy, I₂ = I_xx and I₁₂ = I_xy of the drawing, M₁ maps to the vector M_y and M₂ to minus the vector M_x. The convention was checked against the figures of the NX Nastran Element Library Reference and the recovery routine of the public-domain NASTRAN-95 source, both filed in the References folder; with I₁₂ the tool reproduces the stresses Nastran prints at its recovery points, which is a useful check before trusting the critical-fiber values. Lines from the FORCES IN BAR ELEMENTS and FORCES IN BEAM ELEMENTS blocks of an .f06 can be pasted into the box above the Nastran table: each CBAR line gives end A and end B, each CBEAM station one row, and subcase numbers are kept in the labels. The numbers take the units of the table's first row.

Civil beam diagram (Eurocode 3 and AISC names): the strong axis is horizontal in the drawing and the strong-axis moment (M_y in Eurocode 3, M_x in AISC) is sagging positive, tension at the bottom. The weak-axis moment (M_z in Eurocode 3, M_y in AISC) is positive when it compresses the +x side, the vertical shear follows the "left up, right down" rule of the shear diagram, and the axial force is tension positive. Draw the section with the web vertical.

Whichever convention is chosen, the results name the critical fibers by their coordinates and say which is in tension and which in compression, so a sign slipped on input shows at once as the wrong fiber in tension.

Strain

The strain at a critical fiber is σ / E for an elastic section. When a Ramberg-Osgood exponent is entered with the yield strengths, the plastic part 0.002 (σ / F_y)ⁿ is added, with F_ty on the tension side and F_cy on the compression side. That is the uniaxial strain of the fiber at the stress the linear theory gives; it does not redistribute the stress when the fiber yields, which the Cozzone tool does for plastic bending.

Plastic Bending

The strain section reads a Ramberg-Osgood strain at the elastic stress, which is only consistent below the proportional limit. The Plastic bending box computes the consistent answer: plane sections stay plane, so the strain is linear, ε = ε₀ + κ_x y − κ_y x; the stress at each point follows the Ramberg-Osgood curve at that strain, ε = f / E + 0.002 (f / F_y)ⁿ inverted by Newton, with F_ty in tension and F_cy in compression; and the three unknowns ε₀, κ_x, κ_y are solved by Newton's method so that the stress integrated over a raster of the section (about 20,000 cells) equals P, M_x and M_y. A restrained translation holds the curvature in that direction at zero and the moment about the other axis becomes an output, the restraint's share. The page then gives the centroid strain, the curvatures, the critical fibers with their strain and stress, the zero-strain axis, a vertex table with the elastic stress beside, and a second stress figure. Below the proportional limit the result equals the elastic one; above it the outer-fiber stress is lower than M c / I and the strain higher than the elastic estimate, and for an unsymmetric section the neutral axis moves as the section yields. The margins stay on the elastic stresses. Not offered: a plastic-bending allowable moment (Cozzone's F_b), residual stresses after unloading.

Shear and Torsion

The transverse shear stress is taken at the centroidal axes by V Q / (I t), with Q the first moment of the half section and t the width at the axis, both stored by the section library. That is where the shear stress is largest for compact and I-shaped sections and it is an average across the width; for an unsymmetrical section (I_xy ≠ 0) it is approximate, and for a thin-walled open section loaded away from the shear center the twist that results is not included. The torsional shear stress comes from the Saint-Venant solution: with Prandtl's stress function φ, which solves ∇²φ = −2 in the section and is zero on the boundary, the shear stress is (T / J) |∇φ| and J = 2 ∫φ dA. The finite-difference solution that gives J also gives the largest gradient c_T, which lies on the boundary, so τ_max = T c_T / J; for a round bar c_T is the radius and for a thin rectangle its thickness, the familiar T r / J and T t / J. At a re-entrant corner without a fillet the Saint-Venant stress is singular and the grid value there grows with refinement; the tool leaves such corners out of c_T and says so, since the corner value is a fillet question, not a section-constant question.

Combined Stress and Margins

The two shear stresses are added as magnitudes and combined with the larger normal stress magnitude in the von Mises form, as if the three maxima acted at one point. They do not (the normal stress peaks at a vertex, the transverse shear at the centroidal axis, the torsional shear on the boundary away from corners), so the combined value is an upper bound; where it governs, the stresses at the actual point should be combined by hand. Margins are written against F_tu and F_ty at the tension fiber, F_cy at the compression fiber, F_su for the shear sum and F_ty for the combined stress, each only when its allowable is entered. An ultimate allowable is compared with the ultimate load and a yield allowable with the limit load: the form asks whether the loads entered are ultimate or limit and for the factor of safety (1.5 by default), and the margins move the stresses to the other level, multiplying by the factor for the ultimate margins when limit loads were entered, or dividing by it for the yield margins when ultimate loads were entered. An ultimate load entered is never multiplied by the factor of safety, and the factor stays active with ultimate loads entered because the yield margins divide by it to recover the limit load. A separate yield factor FS_yld multiplies the limit-load stresses in the yield margins (1.15 by default, the factor several certification agencies require on limit load for no detrimental permanent set; 1.0 writes them at limit load). Each margin box names the level of its allowable (ULT for F_tu and F_su, LIM for the yield checks), whatever level the loads were entered at. Column buckling of the member and crippling of the compression flange are separate problems with their own tools.

Several Load Cases

The load table takes any number of rows, or the pasted Nastran lines supply them. Every row is run through the same section and the summary table lists its critical fibers, strains, shear stresses, combined stress and lowest margin, with the governing case marked (lowest margin, or largest normal stress when no allowable is entered). The equations are written out for the row named in "Worked Example Row", so any case can be checked by hand without re-entering it.