Box Beam Shear Flow

Box Beam Shear Flow

Box Beam Shear Flow: Bredt–Batho Theory, Multi-Cell Compatibility and the Shear Centre

What the tool computes

A thin-walled closed section (a wing box, a torque box, a fuselage ring with floors) carries shear loads and torque by shear flow q = τ t, the shear force per unit length of wall, which is the natural variable when the wall is so thin that the stress is uniform through its thickness. The tool takes the section as numbered nodes joined by walls, finds the closed cells itself, and returns the shear flow in every wall, the rate of twist, the torsion constant J, the shear centre and the wall shear stresses for any combination of shear loads V_x, V_y and torque T. The analysis is the classical one of Bredt (1896) and Batho (1916), set out for aircraft sections by Kuhn (NACA TN 691, 1939) and, in the notation the tool follows, by Megson (Aircraft Structures for Engineering Students, Sections 17.3, 18.1, 20.3 and 23.3–23.5).

Conventions

x is to the right and y up in the plane of the section; the beam axis z points at the reader, so a counter-clockwise torque is positive. The flow in a wall is positive from the wall's first node to its second; the flow around a cell and the rate of twist dθ/dz are positive counter-clockwise. The cells are numbered I, II, … in the order the search finds them and are traversed counter-clockwise, so a wall whose from→to direction runs against the cell is marked with a prime in the cell table. The shear loads act at the load point P or, when that box is checked, through the shear centre S.

Walls, arcs and cells

A wall is a straight line between its two nodes unless an arc length is entered, in which case it is drawn as a circular arc of that length bulging away from the section (a D-nose, a rounded end). Handbook problems usually state a curved wall by its peripheral length and the area it encloses with its chord; when both are entered the drawing follows the length and the stated area is used in the cell area and in the twist equations, so a non-circular nose is represented exactly as far as Bredt–Batho theory sees it. Each wall may have its own shear modulus G; it then enters every twist integral through the modulus-weighted thickness t* = t G/G_ref (Megson eq. 23.9), where G_ref is the reference modulus entered on the form. The cells are the bounded faces of the planar graph formed by the walls; a wall that bounds no cell (a web hanging below an open bay, a stiffener) carries only the open-section flow.

Torsion of one cell

For a single cell the shear flow is constant around the wall and the torque it carries is T = 2A q, where A is the area enclosed by the wall mid-line (Bredt–Batho; Megson eq. 18.1). The rate of twist follows from the shear strain energy or from the warping being single-valued around the cell: dθ/dz = (q/(2A G)) ∮ ds/t (eq. 18.4), so the torsion constant is J = 4A²/∮(ds/t) and the stress in each wall is τ = q/t, largest in the thinnest wall.

Torsion of several cells

With N cells each cell carries its own constant flow q_R, and a wall shared by two cells carries the difference of their flows. The torque is T = Σ 2A_R q_R (Megson eq. 23.4), one equation for N unknowns; the rest come from compatibility: the cross-section keeps its shape, so every cell twists at the same rate,

dθ/dz = (1/(2A_R G_ref)) [ q_R δ_R − Σ_S q_S δ_RS ],

where δ_R = ∮_R ds/t* around cell R and δ_RS = ∫ ds/t* along the walls it shares with cell S (Megson eq. 23.6, written here for any arrangement of cells, not only consecutive ones, by summing over the walls of each cell as the book recommends for its Problem P.23.4). The tool prints the N + 1 equations as a matrix with the cell flows and G_ref dθ/dz as unknowns and solves it; the same system with a unit torque gives J = T/(G_ref dθ/dz). Booms do not enter the pure-torsion problem.

Shear loads: the open-section flow

A shear load produces direct stresses that vary along the beam, and their gradient is what feeds the shear flow. In the idealized section the direct stress is carried by concentrated booms B_r (stringers, spar caps) and the walls carry only shear, so the flow is constant between booms and changes at each boom by its ∂P_r/∂z; with the walls carrying direct stress (t_D = t) the flow varies along every wall, parabolically along a straight one. In both cases the general expression is Megson eq. 20.6,

q_b = −[(V_x I_xx − V_y I_xy)/(I_xx I_yy − I_xy²)] [∫ t_D x ds + Σ B_r x_r] − [(V_y I_yy − V_x I_xy)/(I_xx I_yy − I_xy²)] [∫ t_D y ds + Σ B_r y_r],

with x, y measured from the centroid of the direct-stress area and the section properties computed from that area (booms, or booms plus walls in the thin-wall approximation). A closed section is statically indeterminate once for every cell, so the tool cuts one wall per cell at its mid-length, where q_b = 0, and accumulates the flow outward from the cuts along the remaining tree of walls; the cuts are chosen on the top skins when the geometry allows, as Megson recommends for well-conditioned equations. The flow arriving at the last node must vanish (the section properties are centroidal), which the program checks.

Closing flows, shear centre and the torque of the loads

The cut section is made whole again by adding a constant flow q_s,0,R around each cell. When the loads act through the shear centre the section does not twist, so for every cell Σ_S C_RS q_s,0,S = −∮_R q_b ds/t* with the same coefficients C_RR = δ_R, C_RS = −δ_RS as in torsion (Megson eq. 23.10 with dθ/dz = 0). The shear centre is the point through which the resultant of that zero-twist flow passes: solving for a unit V_y alone, the moment of the flows about the origin is x_S; for a unit V_x alone it is −y_S (Section 23.5). A section whose direct-stress area lies on one line (two booms on a web) has only one shear-centre coordinate and can carry only the load normal to that line.

Loads applied elsewhere are moved to S; what is left is the torque T_eff = T + (x_P − x_S) V_y − (y_P − y_S) V_x, carried by the multi-cell torsion solution above. The total flow in every wall is the sum of the zero-twist shear flow and the torque flow, and the rate of twist is T_eff/(G_ref J). This is the same answer as Megson's direct method (eqs 23.10 and 23.11 with the twist as an unknown and the moment equation about the load point); the tool solves both ways and the test script confirms they agree to machine precision. The flows must balance the applied loads — Σq L along each direction equals V_x, V_y and the moment of the flows about the origin equals T + x_P V_y − y_P V_x — and the last table prints that check.

Stresses and margin

The shear stress in a wall is τ = q/t, evaluated where |q| is largest along the wall (an end or the vertex of the parabola when the wall carries direct stress). With a shear allowable entered, the margin of safety is F_su/τ_max − 1 at ultimate level; limit loads are multiplied by the factor of safety first. The shear stress in a boom-stiffened panel is often checked against the buckling or diagonal-tension allowable of the panel rather than the material shear strength; that check is in the Diagonal Tension Web tool.

Reading the figures

Figure 1 draws the section to scale with the walls (line width proportional to thickness), the booms as blue discs, the cells shaded and labelled, the shear centre S, the centroid C of the direct-stress area and the load point P; at the middle of every wall an arrow shows the direction of the total flow and the number beside it its magnitude. Figure 2, present when shear loads are entered, shows the open-section flows q_b with the cuts marked, which is the intermediate result a hand analysis would compare.

Limits

The theory assumes thin walls (stress uniform through the thickness, mid-line geometry), an undistorted cross-section (closely spaced ribs or frames) and free warping; a built-in root restrains warping and raises the flange stresses and the web shear near it (Kuhn's TN 691 gives the correction in his formulas 22 and 23). Taper, cut-outs, buckled webs carrying diagonal tension, and the stress concentration where a boom meets a wall are not covered. Curved walls are drawn as circular arcs; a handbook nose of stated length and area is represented exactly in the twist equations and areas, but its drawn shape is the arc. Walls that cross each other, duplicate straight walls between the same nodes and sections without a closed cell are refused (an open section belongs to the Open Section Shear Flow tool). Results are elastic; the shear modulus enters the stresses only through the ratios G/G_ref and the twist through G_ref.

Verification

The test script reproduces Bruhn's single- and two-cell hexagonal torque boxes (A6.8 and A6.10), Megson's Examples 18.1, 20.4, 23.2 and 23.3 and Problems P.23.3, P.23.4 and P.23.5, Kuhn's NACA TN 691 Examples 6 to 11 (a D-section with a semicircular nose, a two-cell torsion tube and the same tube with two and four flanges, including its shear centres), the Bredt formulas for rectangular and circular tubes, the finite-difference Saint-Venant torsion constant of the Section Properties library for a thick-walled rectangle, and numpy solutions of every compatibility system. Megson's printed closing flows for Example 23.3 (5.5, 10.2, 16.5 N/mm) do not satisfy the book's own moment equation; the tool and the numpy solution of the printed equations agree with each other (4.85, 9.26, 16.0 N/mm), and the final wall flows differ from the book's by at most a few per cent.