Masonry column and pilaster

A reinforced concrete masonry (CMU) column or pilaster, fully grouted, designed by strength design to TMS 402-22 Chapter 9 with the column requirements of Chapter 5, the seismic tie rules of Chapter 7 and ASCE 7-22 2.3.1 load combinations. The calculation checks the axial strength with its slenderness reduction, axial load with biaxial bending by strain compatibility (the neutral axis rotated to the direction of the applied moment) with the strength reduction factor set by the strain, shear in each direction, the reinforcement limits and the lateral ties.

The calculation is by strength design only, so the LRFD / ASD switch is not shown. Allowable stress design (TMS 402-22 Chapter 8) is not offered; a calculation saved with ASD is designed by strength design.

Edition. The design basis is TMS 402-22 / TMS 602-22. The text of the 2022 code is not on hand. Section numbers below are 402-22 numbers taken from TMS's own 402-16 to 402-22 cross-reference, and the changes of the 2022 edition come from the TMS webinar handouts on it. The wording of the provisions was read in TMS 402-13 / 602-13 (code and commentary), so an equation number is written as a 402-13 number unless it is marked otherwise. What remains unverified is listed under the limitations.

Inputs of the masonry column #

• Section:
  • Column or pilaster. A column is an isolated member (TMS 402 2.2), and the column limits of 5.4.1 apply. A pilaster is built into a wall; it is designed as the rectangle alone, with the wall flanges neglected (conservative, as FEMA P-1051 Section 13.1.6 does), and the column limits are not applied.
  • The section from nominal unit sizes (8 to 32 in., the actual size is the nominal less 3/8 in.) or entered as actual dimensions. b is the width along x and t the depth along y; Mx bends about x (the depth t resists it).
  • The clear height between supports (self-weight and the drawing) and the effective height for bending about each axis. The effective height is your input: TMS 402 Commentary Figure CC-5.4-1 (number inferred) takes the clear height for a member braced at both ends and twice the height for a cantilever, and allows the distance between points of inflection only where data show reliable restraint (1.3).
• Materials: f'm entered or by the unit strength method (TMS 602 Table 2); at least 1,500 psi, and a value above 4,000 psi is used as 4,000 psi (4.3, Table 4.3.1). The mortar sets the modulus of rupture of the moment magnifier (Table 9.1.9.1, fully grouted), the column of the unit strength table and the SDC D to F mortar rule. Grade 60 or Grade 40 bars.
• Reinforcement:
  • The vertical bar size and the bars per face along b and along t (corner bars included): 2 and 2 give 4 bars, 3 and 3 give 8 bars. The face to bar centre distance (3.81 in. is the centre of the cell of an 8 in. unit).
  • The least dimension and the area of the gross grout space (the cell holding a bar), and the bars per cell, for the limits of 6.1.3. Take them from the units you specify.
  • The exposure (masonry cover, 6.1.5.1).
  • The lateral ties: 1/4 in. wire, #3 or #4, their spacing, and whether they are in grout or in the mortar joints. Wire ties are taken at 60 ksi (A1064 wire, limited by 9.1.9.2.1); bar ties at the bar grade.
  • Whether the ties count as shear reinforcement (Vns with the two tie legs): your choice.
  • A pilaster only: whether the compression bars are laterally tied. Untied bars are neglected in compression (9.3.2(e)) and the tie rules are not applied.
• Loads at the critical section: one row per load type (D, L, Lr, S, R, W, E): the axial compression P, the largest first-order moments Mx and My along the height, and the shears Vx and Vy. The symmetric section resists either sign; W and E act in both directions. SDS adds Ev = 0.2SDS D to the seismic combinations. The optional self-weight adds the whole column weight to D at the critical section (conservative).
• Second-order moments (TMS 402 9.3.3 bases member forces on an analysis and has no column moment magnifier of its own):
  • Moment magnifier: the procedure of the wall provisions, 9.3.4.4.3 (402-13 Eqs. 9-31 to 9-33), applied about each axis: ψ = 1/(1 − Pu/Pe), Pe = π² Em Ieff / h², Ieff = 0.75 In below the cracking moment (fr of Table 9.1.9.1) and Icr above it (Eq. 9-28 with the bars of the tension face, the axial load not credited). ψ multiplies the largest first-order moment, with no Cm reduction (conservative). Pu ≥ Pe is unstable.
  • Entered: the moments come from your second-order analysis (a note says so).
  • TMS strength design has no minimum eccentricity: the 0.80 factor of the axial strength covers it (402-13 Eq. 9-19 and its commentary).
• Seismic detailing: the seismic design category, whether the column is part of the seismic-force-resisting system and whether it supports discontinuous stiff elements.

Auto-design (editor toolbar) varies the nominal section (width and depth), vertical bars (size and bars per face) and ties (size and spacing). Objectives: lowest cost index (units, grout, reinforcement); least reinforcement. Every other input is kept; a section entered by its actual dimensions is kept as entered; see Auto-design.

Checks performed #

Maximum reinforcement. TMS 402-22 deletes the maximum flexural reinforcement check for columns and pilasters, and the check is gone from this calculation along with its R ≤ 1.5 input (a saved calculation that has it still opens). The strength reduction factor of Table 9.1.4 takes its place: heavily reinforced sections are penalised by φ.

The key results list the section, f'm, the net area, the bars (Ast and ρ), φPn and the φ of the governing point. The drawings (on screen, in the report and on the PDF / DXF drawing sheet) are the column section with the bars, the tie, the moment axes, the bar positions and the overall dimensions; one φPn–φMn interaction diagram (not to scale), with φ by Table 9.1.4 and the 9.1.4.4.2 limit, about x (solid) and, for an unsymmetric section, about y (dashed), with the Mux and Muy of every combination and the governing one at its resultant moment; and the column elevation over the clear height with the bars, the ties at their spacing (the first at half a spacing), the axial loads and the effective heights. Crossties, unit joints and webs, splices, dowels, anchor bolts and the bearing at the base are not drawn.

Limitations #

Not covered:

• Allowable stress design (TMS 402-22 Chapter 8) and lightly loaded columns (5.4.2).
• Bearing of the column on its support, anchor bolts, splices and development of the bars, grout strength (f'g ≥ f'm), the grout space between bar and unit (6.1.4.5).
• Partially grouted columns (TMS 402 requires columns fully grouted), clay masonry, bundled bars, bars not around the perimeter.
• The flanges of a pilaster (neglected, conservative) and the wall it supports.
• The moment gradient along the height: the moments are the largest along the height, magnified without a Cm reduction.
• The overstrength factor Ω0: enter Emh = Ω0QE as E where ASCE 7-22 12.4.3 requires it.

Simplifications:

• The section is the full rectangle (fully grouted), the bars do not displace masonry in the stress block, as the TMS examples do.
• The bars are at one face to bar centre distance on all four faces, equally spaced along each face.
• The moment magnifier is the wall procedure of 9.3.4.4.3 applied to each axis, with the axial load not credited in Icr and Mcr (both permitted, not required).

How the 2022 edition was checked. The 2022 code text is not on hand. The strength reduction factor, the 9.1.4.4.2 limit, the deleted maximum reinforcement, Eq. 9-28 and the bar size rules come from TMS's own handouts on the 2022 changes. The section numbers come from TMS's cross-reference. The wording of these provisions was read in the 402-13 edition (page numbers of that edition) and is taken as unchanged because neither source lists a change: column limits (402-22 5.4.1; 402-13 5.3.1, pp. C-61 to C-62), bar spacing and cover (6.1.4.2, 6.1.5.1; 402-13 6.1.3.2, 6.1.4.1, pp. C-67 to C-68), seismic ties (7.4.3.2.1, 7.4.3.2.5, 7.4.4.2.1, pp. C-92 to C-94), design assumptions (9.3.2, p. C-134), axial and shear strength (9.3.3.1; 402-13 9.3.4.1, pp. C-140 to C-141) and the moment magnifier (9.3.4.4.3 to 9.3.4.4.5; 402-13 9.3.5.4.3 to 9.3.5.4.5, pp. C-145 to C-146).

Still to verify in TMS 402-22 / 602-22 (marked (to confirm against the code text) in the code and the check references):

• the wording of every provision: the section numbers come from the cross-reference, not from the code, and the 2022 equation numbers other than Eq. 9-28 are not confirmed;
• whether 402-22 added a moment magnifier or a second-order provision for columns;
• that No. 9 is no longer a strength design limit: the handouts say the bar size rules are now the same for both design methods and list the three rules above, and the No. 11 ceiling is carried over from 402-13 6.1.2.1 through the cross-reference;
• the number of Commentary Figure CC-5.4-1 (inferred from 402-13 CC-5.3-1) and the net shear area of Table 4.4.5 (for a fully grouted column it is taken as the net area);
• the 135° or 180° hooks of the ties in SDC D and above (402-16 and later);
• the definition of the transverse reinforcement ratio of 7.4.3.2.5 (taken as two tie legs over the tie spacing times the larger side);
• the 4,000 psi upper limit of f'm.

Verification examples #

Verification status: 43 values compared in 11 examples (5 published, 6 hand calculations from the cited equations): 40 within tolerance, 3 recorded deviations. Code basis: written from an older edition, with the differences noted. Not yet compared line by line with the target edition.

Published examples (golden tests in backend/tests/test_verify_masonry_column.py):

• The Masonry Society, Strength Design of Masonry night school, session 5 Design of Columns and Pilasters & System Behavior (J. Hochwalt, March 2021, TMS 402-16), column example: 20 ft, 16 in. × 16 in. CMU column, f'm = 2,000 psi, (4) #6 centered in the cells (d = 11.81 in.). Free from TMS.
• FEMA P-1051 (NEHRP Recommended Seismic Provisions: Design Examples, 2015 edition, TMS 402-13), Section 13.1.6: the 16 in. square pilaster with (4) #6 bars and #3 ties (d = 12 in.) spanning 28 ft under out-of-plane wind, flanges neglected. Free from FEMA.
• The Masonry Society, webinar handout Design of Masonry Shear Walls (R. Bennett, 2018, TMS 402-16), Example SD-1: the shear of a 2 ft pier, entered as a 24 in. × 7.625 in. section bending about y. Free from TMS.

Documented deviations (docs/verification/deviations.json):

• TMS402-22-PHI-COL: the TMS handout is written to 402-16 with a flat φ = 0.90 for axial load and flexure. Its nominal values match; its φ-values at pure axial load and at the balanced point differ because 402-22 Table 9.1.4 gives 0.65 there.
• FEMA-PIL-2L: the FEMA example takes only the two tension bars (φMn = 520,059 lb-in. = 43.3 kip-ft). ILX solves the strain compatibility with all four bars; the second layer is in tension, and φMn = 44.3 kip-ft (2 % higher). The TMS column example makes the same comparison (47.3 kip-ft with two bars, 48.8 kip-ft with both), and ILX matches its both-layer value.

Hand calculations in closed form check the remaining paths:

The behaviour tests also compare the stress block of a rotated neutral axis with a fine numerical integration of the section (within 0.3 %). Not yet compared with a published example: a biaxial column (no free published TMS example was found), the seismic tie rules, and a TMS 402-22 example (the TMS / NCMA Masonry Designers' Guide and the TMS Strength Design of Masonry book are not free; the free handouts are written to 402-16 and use the old φ).