Masonry wall out-of-plane

A reinforced concrete masonry (CMU) wall that spans vertically between lateral supports and carries axial load and out-of-plane wind or seismic pressure, designed by strength design to TMS 402-22 Section 9.3.4 (wall design for out-of-plane loads). The design section is the mid-height of a one-foot strip of wall. The moment includes the P-delta effect by either procedure of 9.3.4.4, and the service deflection is limited to 0.007h (9.3.4.5). Load combinations follow ASCE 7-22: 2.3.1 and 2.3.6 for strength, 2.4.1 and 2.4.5 for the deflection.

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 Eq. 9-28, the one confirmed for this chapter. What remains unverified is listed under the limitations.

Inputs of the out-of-plane wall #

• Wall:
  • The nominal thickness: 6, 8, 10 or 12 in. units (specified 5.625, 7.625, 9.625, 11.625 in.; face shells 1.0 in. for 6 in. units and 1.25 in. for the others, the ASTM C90 minimums).
  • The wall height h between the lateral supports. The wall is simply supported at the top support and at the base, the case 9.3.4.4.2 is written for.
  • An optional parapet above the top support. Its weight adds to the axial load at mid-height. Its lateral load is ignored for the mid-height moment unless you tick Parapet lateral load reduces Mu: then its cantilever moment at the top support, w hp²/2, reduces the mid-height moment by w hp²/4. Ignoring it is common practice (FEMA P-1051 13.1.5.2.5) and conservative for the wall; the parapet itself is not designed here.
  • The grout: fully grouted, or grouted only at the reinforced cells.
• Vertical reinforcement:
  • The bar size and spacing s. One bar per cell, so s is at least 8 in.; multiples of 8 in. fit the units.
  • The bar position: one bar centred (d = tsp/2), one bar at a depth d from the bearing face, or one bar at each face at a given distance from the face. The bearing face is the face the eccentric axial load leans toward (positive e). Pressure that bends the wall the same way as the eccentric load puts the far face in tension, so the bar depth is d; pressure the other way uses tsp − d. Bars in compression are neglected (not laterally tied, 9.3.2).
  • For a partially grouted wall, the grouted width per grouted cell: the cell with its webs and grout, taken full depth between the face shells. FEMA P-1051 uses 8 in.; the CMHA CMU-TEC-002-23 tables for 8 in. units match 8.2 in.
• Materials: f'm entered or by the unit strength method (TMS 602 Table 2); Grade 60 or 40 bars; the mortar type and cement for the modulus of rupture (Table 9.1.9.1). For a partially grouted wall the table says to interpolate between hollow and grouted masonry. You choose the basis: by net area, as FEMA P-1051 13.1.5.2.5 does, or by the fraction of cells grouted, its simpler alternative.
• Section properties: computed, or entered per foot from the NCMA / CMHA tables (An, In, Sn). Computed: the two face shells over the full length plus the grouted width per grouted cell, full depth. FEMA P-1051 enters the TEK 14-1B values.
• Loads (one row per load type D, L, Lr, S, R, W, E):
  • P is an axial line load at the top (plf), with its eccentricity e from the wall centre line (positive toward the bearing face). The eccentric moment at mid-height is Σ P e / 2.
  • w is the out-of-plane pressure over the height (psf). W and E act in both directions. For seismic, enter E = 0.4 SDS Ie times the wall weight (ASCE 7-22 12.11.1), at least 0.1 times the wall weight.
  • The wall weight (psf) is dead load. The weight above the mid-height section counts: Pw = w (h/2 + parapet).
  • SDS adds Ev = 0.2SDS D to the seismic combinations (ASCE 7-22 12.4.2.2). It is 0 by default; a note says so while E is present.
• Design options:
  • The P-delta procedure: the iterative mid-height deflection of 9.3.4.4.2 or the moment magnifier of 9.3.4.4.3. The same procedure computes the service deflection (9.3.4.5.1 or 9.3.4.5.2).
  • The axial load used in Mcr and Icr: the combination's own axial load (default), the unfactored dead load (FEMA P-1051), or none. The code permits the axial load in both (9.3.4.4.4 commentary, 9.3.4.4.5).
  • The seismic design category, optional: it adds the prescriptive vertical bar spacing of a nonparticipating wall (7.4.3.1 / 7.4.4.1) as an advisory.

All values are per foot of wall, and the steps give the per-bar values where useful.

Auto-design (editor toolbar) varies the nominal thickness, the grout (at the bars or full) and the vertical bars (size #4 to #8 and spacing 8 to 120 in.). Objectives: lowest cost index (units, grout, reinforcement); least reinforcement. The bar position, f'm, the height, the loads (including the wall weight you entered) and the procedure are kept; update the wall weight for a new thickness or grout. See Auto-design.

Checks performed #

Maximum reinforcement. TMS 402-22 deletes the maximum flexural reinforcement check for walls loaded out of plane, and the check is gone from this calculation. The strength reduction factor of Table 9.1.4 takes its place: heavily reinforced walls are penalised by φ.

The key results list the procedure, Mu with P-delta, φ at the design point, φMn, the factored deflection and the service deflection against 0.007h. The drawings (on screen, in the report and on the PDF / DXF drawing sheet) are:

• the wall section: the wall from the base to the top of the parapet, the bars where the calculation puts them, the lateral supports, the eccentric axial load at the top with its eccentricity, the pressure band (values by load type, both ways) and the deflected shape of the governing combination (exaggerated) with δu;
• the moment diagram along the height for the governing combination (P-delta included; the deflected shape taken as a half sine for the diagram, which equals the code's value at mid-height), at its own vertical scale;
• a plan section of a length of wall: face shells, the grouted cells (or full grout), the bars at their spacing and the thickness.

The roof and floor framing, the anchorage of the wall, the foundation, splices and dowels, horizontal reinforcement and the parapet design are not part of the calculation and are not drawn.

Limitations #

Not covered:

• Allowable stress design (Chapter 8).
• Support conditions other than simple supports top and bottom: a fixed base, a cantilever wall, a wall spanning two ways or horizontally between pilasters. 9.3.4.4.2 sends those to mechanics; use the engineer's own moments elsewhere.
• Concentrated loads and their distribution (5.1.1), openings, jambs and pilasters.
• In-plane loads (see the masonry shear wall calculation), unreinforced walls, clay masonry.
• The parapet as a cantilever, the support reactions and the wall anchorage (ASCE 7-22 12.11.2).
• Horizontal reinforcement, splices and development.

Simplifications:

• The mid-height section is the design section, as 9.3.4.4.2 states. With a large eccentricity and a small lateral load the largest moment may be elsewhere (402-13 commentary); the moment diagram shows it.
• Lateral loads uniform over the height; axial loads at the top support (floor loads at mid-height are entered as top loads, which is conservative for the eccentric moment).
• The out-of-plane shear uses γg = 1.0 and dv = tsp.
• Running bond.

Still to verify in TMS 402-22 (the code text is not on hand; marked (to confirm against the code text) in the steps where it bears on a step):

• 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;
• 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;
• whether the 402-22 commentary still writes Mn with Pu (402-13 Commentary 9.3.5.2) now that the design point is at Pu/φ;
• the axial stress limit (402-13 Eq. 9-26) and whether it covers the moment magnifier as well;
• the moment magnifier's Ieff rule (0.75 In / Icr);
• the modulus of rupture table, the net shear area out of plane (Table 4.4.5, taken as An here), the γg of out-of-plane shear and dv;
• the effective compressive width of 5.1.2 and the face-shell thicknesses (ASTM C90-22).

Verification examples #

Verification status: 71 values compared in 29 examples (19 published, 10 hand calculations from the cited equations): 69 within tolerance, 2 recorded deviations. Code basis: written from an older edition, with the differences noted. Not yet compared line by line with the target edition.

Golden tests in backend/tests/test_verify_masonry_slender_wall.py:

• FEMA P-1051 (NEHRP Recommended Seismic Provisions: Design Examples, 2015 edition, Chapter 13, TMS 402-13), Section 13.1.5.2: the 28 ft side walls of the one-story warehouse. 8 in. CMU grouted at 24 in., #7 at 24 in. centred, f'm = 2,000 psi, 2 ft parapet, roof 200 plf at e = 7.32 in., wall 65 psf, seismic out-of-plane 37 psf, SDS = 1.43 (0.614D and 1.486D). FEMA enters In = 355 in.⁴/ft, Sn = 93.2 in.³/ft and An = 51.3 in.²/ft from NCMA TEK 14-1B and uses 1.0D in Mcr and Icr; the test does the same. Free from FEMA.
• CMHA CMU-TEC-002-23 (the successor of NCMA TEK 14-1B), Table 2A: net section properties per foot of 8 in. walls, face-shell bedded. Free from CMHA.
• The Masonry Society, webinar handout Major Structural Design Changes in TMS 402/602-22 (R. Bennett, October 2022), slide 19: the largest axial load of a tension-controlled wall, f'm = 2 ksi, Grade 60. The one source here written to 402-22. Free from TMS.

FEMA iterates until the deflection changes by less than 5 % (1.2 % and 1.7 % at its last step); ILX iterates to 0.1 %, which raises Mu by 0.2 % (0.614D) and 0.9 % (1.486D). FEMA's Icr arithmetic gives 42.1 in.⁴/ft; Eq. 9-34 with its own numbers gives 41.7. The wall's bars are centred, so 402-22 Eq. 9-28 gives the same Icr as FEMA's equation and Mcr, Mu, Pe and the deflections do not change; only the φ of the 1.486D combination does. The ductility values of the example are no longer compared, because 402-22 deletes that check. Both FEMA and ILX find the 1.486D case governing. By the moment magnifier FEMA is about 5 % over its design strength and ILX about 8 % over.

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

• TMS402-22-PHI-OOP: FEMA uses the flat φ = 0.90 of 402-13. At Pn = Pu/φ the 1.486D case has εt = 0.00478, in the transition region of Table 9.1.4 (0.90 starts at 0.00507), so φ = 0.876 and φMn = 57.3 in.-kip/ft against 58.8. With the iterative Mu of 58.2 in.-kip/ft the wall that passes in FEMA is now 1.5 % over; with the moment magnifier it is over in both.
• ASCE7-EV-ASD: FEMA's service deflection uses D + 0.7E. ASCE 7-22 2.4.5 adds 0.7Ev = 0.14 SDS D, so with SDS = 1.43 the axial load is 1.2D and the deflection 4.09 in. against 3.85 in. The deflection example is also run with SDS = 0, which reproduces FEMA (3.87 in.).

Hand calculations in closed form (written out in the test file):

Not yet compared with a published example: a complete TMS 402-22 out-of-plane example (the TMS / NCMA Masonry Designers' Guide is not on hand; the free handouts with full wall examples are written to 402-16 and use the old φ), a partially grouted wall whose stress block passes the face shell, and the iterative service deflection.