Anchor and Base Plate Design: Method

Anchor and Base Plate Design: Method, Conventions and the Equations Used

What the tool does

The tool checks a group of anchors in concrete and the column base plate that loads them, by load and resistance factor design. The anchor relations are those of ACI 318-19 Chapter 17, the concrete capacity design (CCD) method of Fuchs, Eligehausen and Breen (1995), written out as mathematics with the section numbers; the bearing relation is ACI 318-19 Section 22.8; the plate relations are the statics of a bearing block and of plate cantilevers, the method of AISC Design Guide 1. The code and the design guide are cited, none of their tables or figures is reproduced, and every resistance factor φ, the coefficients k_c and k_cp and the cracking factors ψ are inputs whose defaults are the code values. Each failure mode is reported as nominal strength, design strength, factored demand, their ratio and a margin of safety M.S. = design / demand − 1.

Conventions

The origin is the centre of the base plate, x runs along the plate length N (the column web, depth d) and y along the plate width B (the flanges). The four concrete edges are named L (toward −x), R (+x), B (−y) and T (+y); an edge distance is measured from the outermost line of anchors on that side, and a blank distance means no edge there. P_u is the factored axial force, compression positive (negative is uplift). M_ux is the moment that compresses the +x edge of the plate, so that the anchors at −x go into tension; M_uy compresses the +y edge. V_ux and V_uy are the shear components acting on the concrete in +x and +y, taken through the centroid of the anchor group. The loads are factored by the strength combinations before they are entered; the tool applies no load factor. Relations with empirical coefficients are evaluated in pounds, pounds per square inch and inches; with SI units selected the inputs are converted through the unit dictionary and the coefficients are shown converted to newtons, megapascals and millimetres (k_c = 24 becomes 10.04, 17 becomes 7.11, 16 becomes 3.91, the shear coefficients 7 and 9 become 0.58 and 3.77, and 160 becomes 13.3; the SI edition of the code prints the rounded values 10, 7, 3.9, 0.6, 3.7 and 13, so its results differ from these by up to 4 percent).

Anchor forces

Plane sections (default). The plate is taken as rigid. The stress under it is a plane, σ = a + b x + c y, compression positive. The concrete carries σ where it is positive, over the part of the plate rectangle on the compressed side of the neutral axis; anchor i carries the tension T_i = −n A_se σ(x_i, y_i) where σ is negative, with the modular ratio n = E_s / E_c and E_c = 57,000 √f′_c (psi); an anchor in the compressed zone carries nothing. The three constants follow from the three equilibrium equations ΣF = P_u, ΣM = M_ux, ΣM = M_uy, which the tool solves by Newton's method with the contact area and its moments integrated exactly over the clipped polygon. This is the cracked transformed-section analysis of a reinforced concrete section applied to the plate, valid for any combination of axial force and two moments, including uplift. It is an assumption: a flexible plate moves the compression resultant toward the column and raises the anchor tension, and prying is not included.

Rectangular bearing block. For a moment about one axis the base-plate design method assumes that the concrete reaches its design bearing strength f_p,max uniformly over a length Y at the compressed edge. With q_max = f_p,max B and the tension anchor line at f from the plate centre, e = M_u / P_u and e_crit = N/2 − P_u / (2 q_max):

small moment, e ≤ e_crit: Y = N − 2e, q = P_u / Y, no anchor tension;

large moment, e > e_crit: T + P_u = q_max Y and q_max Y (N/2 − Y/2 + f) = P_u f + M_u, which give Y = (f + N/2) − √[(f + N/2)² − 2 (P_u f + M_u) / q_max] and T = q_max Y − P_u.

A negative quantity under the root means the plate is too small for the moment. The tension T is shared equally by the anchors of the line farthest from the compressed edge. The block always serves the plate thickness check when the moment is uniaxial; it serves the anchor checks only when it is selected as the tension model. The plastic block sits at the very edge of the plate, so its lever arm is longer and its anchor tension lower than that of the elastic wedge of the plane-section analysis.

Steel strength (17.6.1, 17.7.1)

A_se = (π/4)(d_a − 0.9743/n_t)² for a threaded anchor with n_t threads per inch, or the entered area. N_sa = A_se f_uta with f_uta not above 1.9 f_ya or 125 ksi. V_sa = 0.6 A_se f_uta for a cast-in headed or hooked bolt and for a post-installed anchor, A_se f_uta for a welded headed stud, times 0.80 where the anchors pass through a built-up grout pad. φ = 0.75 in tension and 0.65 in shear for ductile steel. The most loaded anchor is checked in tension; the shear is divided equally among the anchors.

Concrete breakout in tension (17.6.2)

N_cbg = (A_Nc / A_Nco) ψ_ec,N ψ_ed,N ψ_c,N ψ_cp,N N_b, for the group of anchors that are in tension.

A_Nco = 9 h_ef² is the projected area of one anchor far from edges, a square of side 3 h_ef. A_Nc is the area of the union of those squares, one centred on each tension anchor, cut off at the concrete edges; the tool computes the union exactly for any layout, which reproduces the rectangle (c_a1 + s₁ + 1.5 h_ef)(c_a2 + s₂ + 1.5 h_ef) of a regular group and never exceeds n A_Nco.

N_b = k_c λ_a √f′_c h_ef^1.5 with k_c = 24 for cast-in and 17 for post-installed anchors; for cast-in headed anchors with 11 in ≤ h_ef ≤ 25 in, N_b = 16 λ_a √f′_c h_ef^5/3 (the checkbox turns this form off). At 11 in the two forms differ by 0.6 percent; at 25 in the second is 14 percent higher.

ψ_ec,N = 1 / (1 + e′_N / (1.5 h_ef)), with e′_N the distance from the resultant tension to the centroid of the tension anchors, taken in x and in y and multiplied. ψ_ed,N = 0.7 + 0.3 c_a,min / (1.5 h_ef), not above 1. ψ_c,N = 1.0 in cracked concrete, 1.25 (cast-in) or 1.4 (post-installed) where analysis shows no cracking at service load. ψ_cp,N = c_a,min / c_ac, not below 1.5 h_ef / c_ac, for a post-installed anchor in uncracked concrete without supplementary reinforcement, otherwise 1.

Where three or four edges are closer than 1.5 h_ef, h_ef is replaced in all of these relations by the greater of c_a,max / 1.5 and s / 3, with c_a,max the largest of those edge distances and s the largest spacing in the group (17.6.2.1.2). f′_c is limited to 10,000 psi for cast-in and 8,000 psi for post-installed anchors. φ = 0.70 for cast-in anchors without supplementary reinforcement; the seismic factor 0.75 multiplies the concrete tension strengths when it is entered.

Pullout and side-face blowout (17.6.3, 17.6.4)

N_pn = ψ_c,P N_p with N_p = 8 A_brg f′_c for a headed anchor, 0.9 f′_c e_h d_a for a hooked bolt with 3 d_a ≤ e_h ≤ 4.5 d_a, and the qualified value for a post-installed anchor; ψ_c,P = 1.4 uncracked, 1.0 cracked. A blank head area is taken as that of a heavy-hex head or nut, width across flats 1.5 d_a + 1/8 in, less the shank.

A headed anchor in tension with h_ef > 2.5 c_a1 can blow out the side face: N_sb = 160 c_a1 √A_brg λ_a √f′_c, multiplied by (1 + c_a2/c_a1)/4 at a corner with c_a2 < 3 c_a1 (the ratio kept between 1 and 3). For anchors along the edge spaced closer than 6 c_a1, N_sbg = (1 + s / (6 c_a1)) N_sb with s the distance between the outer anchors, against the sum of their tensions.

Concrete breakout in shear and pryout (17.7.2, 17.7.3)

V_cbg = (A_Vc / A_Vco) ψ_ec,V ψ_ed,V ψ_c,V ψ_h,V V_b, with A_Vco = 4.5 c_a1², A_Vc the width of the 1.5 c_a1 projections of the anchors of the line along the edge (cut by the side edges) times the lesser of 1.5 c_a1 and the member thickness h_a, and

V_b = the lesser of 7 (l_e/d_a)^0.2 √d_a λ_a √f′_c c_a1^1.5 and 9 λ_a √f′_c c_a1^1.5, l_e = h_ef ≤ 8 d_a.

ψ_ed,V = 0.7 + 0.3 c_a2 / (1.5 c_a1) ≤ 1; ψ_h,V = √(1.5 c_a1 / h_a) ≥ 1; ψ_ec,V = 1 / (1 + e′_V / (1.5 c_a1)); ψ_c,V = 1.0 cracked without edge reinforcement, 1.2 or 1.4 with it, 1.4 uncracked. In a narrow member of limited thickness (both c_a2 and h_a below 1.5 c_a1) c_a1 is limited to the greatest of c_a2/1.5, h_a/1.5 and s/3. For shear parallel to an edge the strength is twice the value for shear toward that edge with ψ_ed,V = 1.

Each shear component is checked against the edge it pushes toward and the two edges it runs along. Where the anchors stand in more than one line, two cases are checked: the front line under its share of the shear (the shear divided equally among all anchors), and the rear line under the whole shear on its own, deeper, breakout surface. Where both components act the code gives no combination rule, and the tool adds the two governing ratios. Pryout: V_cpg = k_cp N_cpg, k_cp = 1.0 for h_ef below 2.5 in and 2.0 otherwise, N_cpg the tension breakout strength of all the anchors without eccentricity.

Tension and shear together (17.8)

With r_N and r_V the largest demand-to-design ratios in tension and in shear: where r_V ≤ 0.2 the full tension strength is allowed, where r_N ≤ 0.2 the full shear strength, and otherwise r_N + r_V ≤ 1.2. The table reports that rule as a ratio to its limit; the 5/3 power form r_N^5/3 + r_V^5/3 ≤ 1 is printed beside it and drawn on the interaction chart.

Base plate

Bearing: f_p,max = φ 0.85 f′_c √(A₂/A₁) with φ = 0.65, A₁ = B N and √(A₂/A₁) not above 2 (so the nominal stress does not exceed 1.7 f′_c). Under axial load alone f_p = P_u / (B N); with a moment the block gives f_p, and in the large-moment case the block is at f_p,max by assumption, so that row of the table is not a check. With moments about both axes the peak pressure of the plane-section analysis is compared with f_p,max, which is conservative.

Thickness: the plate cantilevers beyond the column lines by m = (N − 0.95 d)/2 and n = (B − 0.8 b_f)/2; under axial load the yield-line length between the flanges λn′ = λ √(d b_f)/4, with λ = 2√X / (1 + √(1 − X)) ≤ 1 and X = [4 d b_f / (d + b_f)²] P_u / φP_p, is a third candidate, and l is the largest. The plate moment per unit width is f_p l²/2, or f_p Y (l − Y/2) where the bearing length Y is shorter than l. On the tension side the most loaded anchor bends the plate about the flange centre line at the lever x, spread over a width 2x: M_pl = T x / (2x). The required thickness follows from φ F_y t²/4 = M_pl with φ = 0.90: t = √(4 M_pl / (φ F_y)), which for the uniform cantilever is the familiar t = l √(2 f_p / (0.9 F_y)).

Shear transfer: by the anchors alone; by friction first, φ μ C with C the compression under the plate, the anchors taking the rest; or by a shear lug, whose bearing strength is φ k_brg f′_c A_ef ψ_brg,sl (k_brg = 1.7 and φ = 0.65 by ACI 318-19 17.11.2), with A_ef = b h_ef,sl, h_ef,sl the lesser of the depth below the grout h − G and twice the lug thickness, and ψ_brg,sl = 1 + 4 P_u / (B N f′_c) ≤ 2 under axial compression or 1 + P_u / (n N_sa) under uplift (P_u negative, n anchors in tension). The lug plate bends as a cantilever under V_u acting at G + h_ef,sl/2. Concrete breakout in front of the lug (17.11.3) is checked for every edge the shear pushes toward or runs along, with the lug at the centre of the plate: c_a1 from the bearing face to the edge, A_Vc = (lug width + 1.5 c_a1 each side) × (h_ef,sl + 1.5 c_a1) − A_ef cut by the side edges and the member thickness, V_b = 9 λ_a √f′_c c_a1^1.5, V_cb,sl = (A_Vc / A_Vco) ψ_ed,V ψ_c,V ψ_h,V V_b, doubled with ψ_ed,V = 1 for an edge parallel to the shear, φ = 0.65. The moment of the lug couple is not added to the anchor tensions; add it to M_u.

Verification

FEMA P-751 (2012), Section 14.3, works two single anchors by these relations (as ACI 318-08 Appendix D): a 1/4-in cast-in anchor embedded 2 in, φN_sa = 1,392 lb, 0.75 φN_b = 2,254 lb and φV_sa = 724 lb; and a 5/8-in post-installed anchor embedded 9 in, 13,515 lb, 14,151 lb and 7,030 lb with the interaction sum 0.99. The tool's library reproduces each printed value. No public document was found that works a group of anchors near edges to printed digits, so the projected areas, every modification factor and both forms of N_b and V_b are checked on forty seeded groups and forty seeded shear lines against an independent numpy computation (areas by inclusion and exclusion), the plane-section analysis against scipy on twenty-four seeded plates with equilibrium satisfied to 10⁻⁹ of the load, and the bearing block against the roots of its quadratic and both equilibrium equations.

Not covered

Anchor reinforcement and its development; bond strength of adhesive anchors; the ductility, stretch-length and overstrength requirements for earthquake loading beyond the entered 0.75 factor; minimum spacings and edge distances against splitting; torsion on the anchor group; prying and plate flexibility; the welds, the column and the footing. A post-installed anchor is governed by its product evaluation report, whose values are entered here.