Masonry Wall and Pier: Method

Masonry Wall and Pier: Method, Equations, Tables and Conventions

Scope and sources

The tool applies the working-stress (allowable-stress) method of the joint Army, Navy and Air Force manual Masonry Structural Design for Buildings, TM 5-809-3 / NAVFAC DM-2.9 / AFM 88-3 Chap. 3 (October 1992), a public-domain U.S. government work. Its chapter 5 gives the material assumptions, the transformed-section equations and the allowable stresses; chapter 6 the walls under axial load and out-of-plane bending; chapter 7 the shear walls and piers; chapter 8 the lintels; chapter 9 the columns and pilasters; and its Tables 5-12 to 5-14 the anchor-bolt allowables, which the manual says were developed from the ACI 530 / ASCE 5 criteria of its day. The manual covers reinforced masonry; the unreinforced-wall element uses the allowable-stress forms of TMS 402 (ACI 530) as mathematics, with that standard cited and its tabulated allowable tensions entered rather than reproduced. Where the current TMS 402 differs from the manual the difference is recorded on this page as a difference, not a correction: the manual's values stand as the tool's defaults.

Every check prints the demand-to-capacity ratio U and the margin M.S. = 1/U − 1. When the load combination includes wind or seismic forces (the checkbox) every allowable stress and resisting moment is increased by one third, as footnote 1 of Table 5-7 allows and as the manual's examples do, so that the unity equations keep their 1.0 limit.

Materials (chapter 5)

Assumed strengths (5-5): f′_m = 1,350 lb/in² for hollow units and 1,500 lb/in² for solid units in type M or S mortar, 1,000 lb/in² for type N mortar; a prism-tested value up to 3,500 lb/in². Allowables (Table 5-7): axial F_a = 0.20 f′_m R for walls (eq 5-24), flexural F_b = f′_m/3, shear √f′_m ≤ 50 lb/in² without shear reinforcement and 3√f′_m ≤ 120 lb/in² with reinforcement for the entire shear, bearing 0.25 f′_m on the full area and 0.30 f′_m on one third of the area or less; steel F_s = 24,000 lb/in² for Grade 60 (Table 5-8); E_m = 1,000 f′_m, E_v = 0.4 E_m; f_r = 2.5 √f′_m for deflection (5-3b). The stress-reduction factor is R = 1 − (12h/40t_n)³ with h in feet and the nominal thickness t_n in inches (eq 5-25), not used at the top and bottom of a wall where buckling is not a concern.

Differences in TMS 402-16 (cited only): E_m = 900 f′_m for concrete masonry and 700 f′_m for clay masonry (offered as a select); F_b = 0.45 f′_m for reinforced masonry; F_s = 32,000 lb/in² for Grade 60; the wall axial allowable (0.25 f′_m A_n + 0.65 A_st F_s)[1 − (h/140r)²] with the (70r/h)² form beyond h/r = 99; and the in-plane shear allowables written on M/(Vd) with an axial term. These are differences of edition, entered where the tool lets you.

Transformed cracked section (5-4)

Plane sections, no tension in the masonry, n = E_s/E_m. For a rectangular section with steel ratio p = A_s/(bd) the neutral-axis ratio satisfies k²/2 + npk − np = 0, so k = √(np² + 2np) − np and j = 1 − k/3 (eqs 5-9, 5-10). Stresses: f_m = 2M/(k j b d²), f_s = M/(A_s j d) (eqs 5-12, 5-13); resisting moments M_rm = F_b k j b d²/2 and M_rs = F_s A_s j d; the balanced-stress ratio p_e = n/[2(F_s/F_b)(n + F_s/F_b)] (eq 5-11; 0.0027 at f′_m 1,350 lb/in², Table 5-9). When the compression block is deeper than the face shell t_s of a hollow unit, the T-section forms apply with the web's compression neglected: k_T = [np + ½(t_s/d)²]/[np + t_s/d], j_T = [6 − 6(t_s/d) + 2(t_s/d)² + (t_s/d)³/(2pn)]/[6 − 3(t_s/d)], M_rmT = F_b[1 − t_s/(2k_Td)] b t_s j_T d, M_rsT = F_s A_s j_T d (eqs 5-15 to 5-20). The manual notes the difference from the rectangular section is usually about 1 % at f′_m = 1,350 lb/in²; the tool prints both and the section rule chooses.

The manual's assumed unit dimensions (Table 5-1), equivalent thicknesses (Table 5-2), areas effective in axial compression and in-plane shear (Table 5-3) and wall weights (Table 5-5) are shipped: the 6, 8, 10 and 12 in units have design thicknesses 5.625, 7.625, 9.625 and 11.625 in, face shells 1, 1¼, 1⅜ and 1½ in, one-bar depths 2.81, 3.81, 4.81 and 5.81 in and a shear width b_w of 7½ in per cell; the spacing tables are interpolated between the listed spacings.

Reinforced wall out of plane (chapter 6)

The wall is a simple span of height h between its supports. With P per unit length at eccentricity e at the top and a lateral load w, the reaction at the bottom is R_a = wh/2 ± Pe/h (eq 6-1) and the moment at x above the bottom M(x) = ±Pe x/h + w x(h − x)/2 (eq 6-4), additive or opposing as the wind reverses; its stationary point is x = h/2 ± Pe/(wh) (eq 6-8) and the tool takes the largest |M| over the height. The design moment is M × s on one bar spacing (eq 6-5), the resisting moments those of the section of width s (limited to 6t_n and 48 in). Axial stress f_a = [P + w₂(h − x)]/A_e (eq 6-10) on the effective area. Unity equations: masonry f_a/F_a + M/M_rm ≤ 1 (eq 6-13) and reinforcement M/M_rs − f_a/F_a ≤ 1 with the minimum axial stress (eq 6-15; the manual notes f_a = 0 is conservative). Out-of-plane shear at the bottom f_v = R_as/(b_wd) (eq 6-17). The P-Δ effect is neglected when h/t_n ≤ 24 (6-5); above it the manual asks for the mid-height deflection with the P-Δ moment (eqs 6-18, 6-19), which the tool does not compute: it says so.

Unreinforced wall (TMS 402 forms)

On the net section (solid from t, or entered): f_a = P/A_n, f_b = M/S_n with the same moment diagram as above; F_a = ¼ f′_m[1 − (h/140r)²] for h/r ≤ 99 and ¼ f′_m(70r/h)² beyond; F_b = ⅓ f′_m; f_a/F_a + f_b/F_b ≤ 1; net tension f_b − f_a against the entered allowable flexural tension F_t; the virtual eccentricity M/P against the kern t/6; and the axial force against P_e/4 with P_e = π²E_mI_n/h² (1 − 0.577e/r)³. The default F_t of 25 lb/in² is the TMS 402 Table 8.2.4.2 value for hollow ungrouted units in type M or S portland cement–lime mortar, quoted as a single fact; other cases are entered.

Shear wall or pier (chapter 7)

f_v = V/(td) with t the actual or the equivalent thickness (eq 7-5). With M = Vh/2 for a pier fixed top and bottom or Vh for a cantilever (Figure 7-1), the allowable without shear reinforcement is F_vm = ⅓(4 − M/Vd)√f′_m ≤ 80 − 45 M/Vd when M/Vd is below 1 and √f′_m ≤ 35 lb/in² otherwise (eqs 7-1 to 7-2a); with horizontal reinforcement carrying the whole shear, F_vs = ½(4 − M/Vd)√f′_m ≤ 120 − 45 M/Vd or 1.5√f′_m ≤ 75 lb/in² (eqs 7-3 to 7-4a), A_v = Vs/(F_sd) (eq 7-6), spacing not above d/2 nor 48 in and vertical bars of at least A_v/3. The pier bends as a vertical cantilever beam of width t and depth d − cover with the end bars as the tension flange; axial stress P/(td) at the bottom without R; the entered overturning stress f_o = M_oC/I_n of the whole panel; unity f_b/F_b + f_a/F_a + f_o/F_a ≤ 1 as in the manual's example 2. Deflection and rigidity: Δ = Vh³/(12E_mI) + 1.2Vh/(E_vA) fixed, Vh³/(3E_mI) + 1.2Vh/(E_vA) cantilever, k = 1/Δ (eqs 7-8 to 7-10); the Figure 7-5 charts are these equations with V = 10⁶ lb, E_m = 1.35 × 10⁶ lb/in² and t = 12 in (Δ = 0.0617(h/d)³ + 0.1852(h/d) and 0.2469(h/d)³ + 0.1852(h/d), eqs 7-13, 7-14).

Lintel or beam (chapter 8)

Loads: the uniform dead load (the lintel's weight and the wall it carries uniformly) and live load; with arching (8-2a), only the 45° triangle of wall above the opening, weight w′L²/4 with its apex at L/2, so that M = w′L³/24 and V = w′L²/8; a concentrated load at height h′ spreads at 60° over 2h′ tan 30° = 1.155h′ as the uniform load w_p = P′/(1.155h′) on the length a = h′ tan 30° + L/2 − x′ of the span it reaches (8-2b). The tool integrates the shear and moment diagrams along the span and reproduces the manual's closed forms. Flexure as in chapter 5 with A_sb = p_ebd telling which material governs; shear f_v = V/(bd) at the support against √f′_m ≤ 50 lb/in², or with stirrups A_v = Vs/(F_sd) (spacing ≤ d/2 and 24 in) against 3√f′_m ≤ 120 lb/in²; the depth that needs no stirrups V/(F_vb). Deflection with I_e = (M_cr/M_max)³I_g + [1 − (M_cr/M_max)³]I_cr ≤ I_g (eq 8-1), M_cr = f_rI_g/Y_t (eq 8-2), I_cr = b(kd)³/3 + nA_s(d − kd)² (eq 8-3), the midspan deflection by the virtual-work integral of the moment diagram, and Δ = Δ_short + 3Δ_long (eq 8-4) against L/600 not above 0.3 in (8-3). Bearing: length at least 8 in (eq 8-5), stress 2R/A_brg for an end free to rotate (eq 8-6) or R/A_brg when restrained, against 0.25 f′_m; lateral support of the compression face within 32b (8-6). Continuous lintels (example 3 of the manual, ACI moment coefficients) are not modelled: enter the end shear and check the stirrups with the beam element.

Bearing and anchor bolts

Bearing: 0.25 f′_m on the full area, 0.30 f′_m when the loaded area is one third of the supporting area or less and the edges are at least a quarter of the loaded side apart, interpolated between (Table 5-7, footnote 7). Anchor bolts: the manual's Tables 5-12 to 5-14 are reproduced by B_a = 0.5 A_p√f′_m on the projected cone A_p = πl² with l the lesser of the embedment and the edge distance, less half of any area shared with a neighbouring bolt, not above 0.2 A_bf_y; B_v = 350(f′_mA_b)^1/4 not above 0.12 A_bf_y, reduced linearly from an edge distance of 12 bolt diameters to zero at 1½ in; b_a/B_a + b_v/B_v ≤ 1; embedment at least 4 d_b. The four coefficients are inputs so that the TMS 402-16 values (1.25 on the cone, 0.6 and 0.36 on the steel, 350 on crushing with its added pry-out and edge terms) can be compared.

Column or pilaster (chapter 9)

f_a = P/[bt + (n − 1)A_s] (eq 9-1); F_a = [0.18 f′_m + 0.65 p_g F_sc] R with p_g = A_s/A_g, F_sc = 0.4 f_y and R = 1 − (h′/40t_n)³ on the effective height Kh (eq 9-2; R = 1 at the top); f_b = 6M/(bt²) for an uncracked section (f_a ≥ f_b, eq 9-3) or 2M/(bd²jk) for a cracked one (eq 9-4); f_s = M/(A_stjd) (eq 9-6); f_a/F_a + f_b/F_b ≤ 1 (eq 9-8). Checked at the top (P, Pe) without the increase and at mid-height (P plus the upper half's weight, Pe/2 and the lateral moment w s h²/8) with it, as in example 9-6; the manual's example assumes a cracked section at every stage, which the tool follows in the unity equations while printing the uncracked value beside it. Reinforcement 0.005 A_g to 0.04 A_g, least nominal dimension 12 in (8 in at half the allowable axial stress), ties per 9-2c (not checked).

Conventions and limits

Units: lengths, forces, line loads, areas and stresses go through the site dictionary; the lateral pressure, the wall weight per unit area, the per-length net-section properties and the unit weight are plain numbers labelled lb/ft², in²/ft, lb/ft³ in USCS and Pa, mm²/m, kN/m³ in SI, not converted by a unit-system switch. The wall moment is printed per unit length of wall (lb·ft/ft or N·m/m) and the design moment per bar spacing. Stresses display in lb/in² (psi) and MPa. Not covered: the P-Δ moment of slender walls (h/t_n above 24), continuous lintels, torsion, seismic detailing (TM 5-809-10), the rigidity analysis of walls with openings (the pier element gives each pier's own rigidity), strength design.

Checks

The library reproduces the manual's design examples 6-1 (12 in wall: R_a 330.3 lb/ft, x 13.2 ft, M 2,182 ft-lb/ft, k 0.308, j 0.897, M_rmT 4,147, M_rsT 4,611 ft-lb, f_a 38.3, F_a 212 lb/in², unity 0.93), 6-3 (stiffener: x 7.96 ft, M 8,872 ft-lb, F_a 253), 7-1 and 7-2 (rigidities 0.3461 / 0.0790 / 0.211 / 0.3726 in; piers B, C, D: f_v 3.1 / 7.9 / 20.9, F_vm 35 / 40.4 / 44.7, f_b 53.9 / 67.9 / 73.4, f_s 2,511 / 3,737 / 4,718, unity 0.25 / 0.23 / 0.30), 8-1 (M 257 ft-lb, M_rm 1,028, M_cr 564, Δ 0.0057 in, f_v 6.36, f_brg 7.3), 8-2 (w_p 2,165 plf, V 5,361 lb, d_reqd 19.01 in, M 13,814 ft-lb, M_rs 14,945, I_cr 2,384, I_e 2,745 in⁴, Δ 0.242 in, f_brg 176), 8-3 (M_si 14,944, f_v 41.02, Max F_v 110, l_c 20.33 ft) and 9-6 (f_a 109 / 114, F_a 627 / 610, f_b 158 / 457 / 79, f_s 7,907, unity 0.90), every cell of Tables 5-12 to 5-14 and the chapter-5 tables, with an independent numpy script for the closed forms; the test is test-scripts/test_masonry_wall_pier.php. Slips in the manual found on the way: Table 7-1 lists C_B 7.53 ft and I_n 213.77 ft⁴ but the text computes f_oB with 7.94 ft and 237.42 ft⁴; example 9-6 adds 0.17 + 0.35 as 0.48 and 0.19 + 0.18 as 0.37; pier D's k is rounded to 0.27 for 0.26 (f_b 73.4 for 75.6); the summary of example 8-2 says "2-#14 bars" for 2-#4; pier B's M/Vd prints 1.38 for 1.375.