Weld Group Stress

Weld Group Stress

Welds as Lines: Method, Conventions and Allowables

A fillet-weld group loaded off its centroid carries a direct shear and, through the eccentricity, a torsion of the group in its plane and bending out of its plane. The classical hand method (Blodgett, Design of Welded Structures, 1966; Shigley Chapter 9) treats every weld as a line of unit width along its root, so the group's properties are those of a line figure: the area is the throat width times the total length, and the second moments are the line's second moments times the throat. Each stress component is then a simple formula, and the components are added as vectors at the weld ends, where the resultant is largest. The method is elastic and linear: the plates are rigid, the weld is a spring per unit length, and every component of stress on the throat, whether a true shear or the "bending" normal component, is counted as shear on the throat and compared with a shear allowable. This is how Shigley, Blodgett and the AISC elastic (vector) method work; the AISC instantaneous-centre method (Manual Part 8, Tables 8-4 to 8-11), which fits the measured load-deformation curve of a fillet weld (Lesik and Kennedy 1990) and rotates the group about an iterated centre, gives larger strengths, so the elastic result here is conservative for ductile steel groups.

Geometry and conventions

x to the right, y up, z toward the viewer (out of the weld plane, on the attachment's side); moments by the right-hand rule, so M_z is counter-clockwise positive, positive M_x puts the +y welds in tension and positive M_y the −x welds. Positive F_z pulls the attachment away from the base. The seven standard configurations are Shigley's Tables 9-1 and 9-2 placed as the figures draw them: the vertical reference weld on x = 0 and the bottom weld on y = 0, so case 1 is the line from (0, 0) to (0, d), case 4 the channel with its vertical weld on x = 0 and horizontals from x = 0 to b at y = 0 and y = d, case 7 a circle centred on the origin. The entered-segments configuration takes any set of straight welds (x₁, y₁, x₂, y₂), each with its own leg; curved welds are entered as a few straight pieces. The load point (x_P, y_P) is given in the same frame, with z_P its height above the plane: the tool finds the centroid itself and moves the loads to it, so the eccentricity never has to be measured by hand.

Properties of the line figure

With throat t = 0.707 h (Shigley's rounding of 1/√2) and weld lengths l_i, the throat area is A = Σ 0.707 h_i l_i, the centroid G the area-weighted centre of the lines, and the second moments of the lines about G are I_x = Σ t_i(l_i Δy_i²/12 + l_i e_yi²), I_y likewise with Δx and e_x, I_xy = Σ t_i(l_i Δx_i Δy_i/12 + l_i e_xi e_yi) and J = I_x + I_y, where Δx, Δy are the projections of a weld and e its centre's offset from G. For a uniform leg these are 0.707 h times Shigley's unit values J_u and I_u, which the page prints beside the actual values; the tool reproduces every closed form of Tables 9-1 and 9-2 (for example J_u = (8b³ + 6bd² + d³)/12 − b⁴/(2b + d) for the channel and d²(3b + d)/6 for the rectangle in bending). The formulas take the welds as lines, which is Blodgett's approximation: a weld's own second moment about its length axis (of order t³ l) is dropped, and the throat is laid flat in the plane of the group. For a circle the tool uses a 360-sided polygon, within 0.01 % of 2π r t, 2π r³ t and π r³ t.

Stresses

The loads at the load point become, at the centroid, the same forces and the moments M_G = M + r × F with r from G to the load point: M_zc = M_z + F_y(x_P − x_G) − F_x(y_P − y_G), M_xc = M_x + F_z(y_P − y_G) − F_y z_P, M_yc = M_y − F_z(x_P − x_G) + F_x z_P. A cantilever bracket loaded by F_y at an arm z_P from the wall (Shigley Example 9-5) is therefore entered as the force at z_P, and the tool produces the bending moment. At a point of the weld lines:

Primary shear τ′ = V/A, uniform over the group and parallel to the in-plane force (Shigley Eq. 9-5).

Secondary shear τ″ = M_zc r/J, perpendicular to the radius r from G, largest at the point farthest from G (Eq. 9-6). The two in-plane terms add as vectors; the resultant is largest where τ″ points along τ′, which is why the corners nearest the load side of a channel govern in Example 9-1 while the free ends, where τ″ nearly opposes τ′, carry less.

Bending from the out-of-plane loads: F_z/A plus M c/I, with c the distance from the centroidal axis (Shigley Section 9-4, Eq. 9-8). For a group whose product of inertia is not zero (an L, an unequal channel) the tool uses the general two-axis formula σ = a x′ + b y′ with a = −(M_yc I_x + M_xc I_xy)/(I_x I_y − I_xy²) and b = (M_xc I_y + M_yc I_xy)/(I_x I_y − I_xy²), which reproduces the applied moments exactly; it reduces to M c/I when I_xy = 0. This component is a normal stress on the throat in the geometry of the joint, but the method follows Shigley in treating it as shear on the throat, as the AISC code does for fillet welds.

Resultant τ = (τ_x² + τ_y² + τ_z²)^½ at every weld end; each term is linear along a straight weld, so the maximum of the resultant lies at an end, and the tool lists every end, marks the critical one and draws the vectors there.

Allowables and margins

AISC code allowable (Shigley Tables 9-4 and 9-6). Shear on the throat of a fillet weld is limited to 0.30 F_EXX, F_EXX the electrode classification strength (60 ksi for E60XX, 70 for E70XX, and so on), a factor of 1.44 against the distortion-energy shear yield of the weld metal. The allowable force per unit length f = 0.707 h (0.30 F_EXX) is Table 9-6 (5.57 kip/in for a 3/8-in E70 weld), the required leg h_req = h τ/τ_all because every stress scales with 1/h, and n_w = τ_all/τ is the factor by which the loads could grow. The margin printed is τ_all/τ − 1.

AISC 360 LRFD and ASD. The nominal strength of the weld metal is F_nw = 0.60 F_EXX on the effective throat (J2.4), with φ = 0.75 (design strength 0.45 F_EXX against factored loads) or Ω = 2.00 (allowable 0.30 F_EXX against service loads, the same number as the code allowable). The optional directional increase (1.0 + 0.50 sin^1.5 θ) raises the strength of a weld loaded across its axis up to 1.5 times; AISC writes it for a linear weld group of uniform leg loaded through its centroid, and the tool applies it with θ taken at the critical point between the resultant stress and the weld axis there, which for an eccentric group is an approximation on the conservative elastic method. The instantaneous-centre method is not offered.

Yield check. Shigley's static-load method compares the resultant throat stress with the distortion-energy shear yield S_sy = 0.577 S_y of the weld metal (Table 9-3: 50 ksi for E60, 57 for E70, 67 for E80, 77 for E90, 87 for E100, 107 for E120), with a design factor n_d chosen by the analyst (3.0 in Example 9-5). The page prints n_y = S_sy/τ, the leg size that gives exactly n_d, and a margin S_sy/(n_d τ) − 1. The code allowable and the AISC strengths carry their own factors, so n_d does not enter them.

Base metal. The same force per unit length that stresses the throat also shears the base metal along the leg h, so τ_bm = 0.707 τ (Shigley Example 9-2 writes F/(2 h l) for a double fillet). It is checked against shear yielding 0.60 S_y with φ = 1.00 or Ω = 1.50 (0.40 S_y, Shigley Table 9-4) and, when the ultimate strength is entered, shear rupture 0.60 F_u with φ = 0.75 or Ω = 2.00. Tension or bending in the attachment itself (Shigley checks the shank at 0.60 S_y) depends on the member's section and is not part of the weld-group analysis.

Fatigue

Shigley's Section 9-7 treats the weld like any other steel part with a shear endurance limit S_se = k_a k_b k_c S_e′, where S_e′ = 0.5 S_ut of the weaker of the weld metal and the attachment (100 kpsi above 200 kpsi), k_a = 39.9 S_ut^−0.995 (S_ut in kpsi) is the as-forged surface factor used for the weld surface, k_b = 1 for a uniform shear on the throat and k_c = 0.59 for shear. The fatigue stress-concentration factor K_fs of Table 9-5 (1.2 reinforced butt weld, 1.5 toe of a transverse fillet, 2.7 end of a parallel fillet, 2.0 T-butt with sharp corners) multiplies the stress. The tool takes the entered loads as the maximum of a cycle with load ratio R, so τ_a = K_fs τ (1 − R)/2 and τ_m = K_fs τ (1 + R)/2, and gives the Goodman factor 1/n_f = τ_a/S_se + τ_m/S_su with S_su = 0.67 S_ut; a completely reversed load (R = −1) gives n_f = S_se/τ_a, Shigley's Example 9-6 (n_f = 4.72 for a 1000-lbf reversed load on two 3/8-in by 2-in E6010 fillets on 1018 steel). This is a design-office estimate; welded steel structures under many cycles are rated by the detail categories of AISC 360 Appendix 3 or AWS D1.1, which this tool does not include.

Reading the results

The figure draws the welds, the centroid G, the load point P with its in-plane force, and at the critical point the primary shear (green), the secondary shear (orange) and their resultant (purple) to one scale. Table 1 lists the weld lines and Table 2 the stress components at every weld end with the critical point in bold; the equations below it repeat Shigley's chain for that point, A = 0.707 h Σ l, J = 0.707 h J_u, τ′ = V/A, τ″ = M r/J, τ_b = M c/I and the resultant. The default case is Shigley's Example 9-1 (10th ed.): the 25-kN half of a 50-kN load carried by one plate into a 200-mm channel through 6-mm fillets in a channel pattern with b = 56 mm and d = 190 mm, the load 100 mm outboard of the channel edge. The page reproduces the book's A = 1,280 mm², x̄ = 10.4 mm, J = 7.07 × 10⁶ mm⁴, M = 2,760 N·m, τ′ = 19.5 MPa and τ″ = 41.0 MPa at the free ends and 37.3 MPa at the corners; adding the vectors, the resultant is 43.9 MPa at the two corners on the channel edge (where τ″ has a component along τ′) and 37.1 MPa at the free ends.

Limits

Fillet welds only, treated as lines of throat 0.707 h with the welds' own thickness neglected; groove welds and partial-penetration welds need their effective throat and the base-metal rules of AISC J2. The elastic method assumes rigid connected parts and a linear weld; it is conservative against the instantaneous-centre method for eccentric in-plane loads and does not model prying, plate bending or the stiffening of the compression side under out-of-plane bending (Shigley, like Blodgett, takes the neutral axis through the centroid). Minimum and maximum weld sizes (AISC Table J2.4, J2.2b), the minimum length of 4 h, end returns and the 100 h effective-length limit for longitudinal welds are the detailer's checks and are not applied here.