Masonry lintel and beam

A reinforced concrete masonry (CMU) lintel or beam over an opening, designed by strength design (Sections 5.3 and 9.3, ASCE 7-22 2.3.1) or allowable stress design (Chapter 8, ASCE 7-22 2.4.1) to TMS 402-22. The lintel is a fully grouted simple span between its bearings. The calculation builds the load the lintel carries (its own weight, the wall above, floor and roof loads, point loads), with or without arching action, then checks flexure, shear, the reinforcement limits, the deflection, the bearings and the anchorage of the bottom bars.

Allowable stress design is the LRFD / ASD switch. It takes the cracked transformed section with Fs of 32 ksi (Grade 60) or 20 ksi (Grade 40) and 0.45 f'm in flexure, the shear of Chapter 8, and bearing at 0.25 f'm. The strength design rules for maximum reinforcement and for 1.3 Mcr become notes. It rests on TMS 402-13 Chapter 8, not on 402-22, checked against CMHA TEK 17-01D's tables (2011 MSJC, the same equations); section numbers are marked to verify. A calculation saved with ASD before this option was designed by strength design and is now designed by allowable stress.

Edition. The design basis is TMS 402-22 / TMS 602-22, but the code text is not on hand. Section numbers come from TMS's own 2016 to 2022 cross-reference list. The provisions that changed come from TMS's 2022 webinar handouts and the free Masonry Designers' Guide 2022 sample (Example 12.4-2A). The rest was checked against TMS 402-13 / 602-13 on 2026-11-02 and carried over. An equation number appears as a 402-22 number only where TMS confirms it; otherwise it reads "402-13 Eq.". What remains unconfirmed is listed under the limitations.

Arching action is your judgment. Whether the masonry above arches over the opening depends on conditions the calculation cannot see (control joints, openings, the masonry beside the opening). It is a checkbox, off by default, with the usual conditions in its hint and in a note. The calculation checks the two it can: running bond and enough masonry above the lintel.

Inputs of the masonry lintel #

• Lintel:
  • The nominal width: 6, 8, 10 or 12 in. units (specified widths 5.625, 7.625, 9.625 and 11.625 in.).
  • The number of courses, 1 to 8 (8 in. each). The lintel is designed fully grouted. The actual depth is 8 in. per course less a 3/8 in. joint (15.625 in. for two courses), or you enter it (no more than 8 in. per course).
  • The bond: running bond, or stack bond (not running bond). It sets the modulus of rupture and arching needs running bond.
• Materials:
  • f'm, entered or by the unit strength method (TMS 602-22 Table 2). At least 1,500 psi; above 4,000 psi the value used for nominal strengths is 4,000 psi (Table 4.3.1).
  • The mortar type (M or S, N) and the mortar cement (portland cement / lime or mortar cement; masonry cement or air-entrained). Together with the bond they give the modulus of rupture of Table 9.1.9.1 (tension parallel to the bed joints, fully grouted): 267, 200, 160 or 100 psi in running bond, 335 psi for the continuous grout section in stack bond.
  • Grade 60 or 40 bars; coarse or fine grout (1/2 or 1/4 in. of grout between the bars and the units, 6.1.4.5).
• Reinforcement:
  • The bottom bar size (#4 to #9), 1 or 2 bars per course, in 1 or 2 courses (the second course 8 in. above the first). Two bars in a course sit against the grout clearance at each side of the grout space.
  • The distance from the soffit to the centre of the lowest bars (bars on chairs in the lintel unit, usually 2 to 4 in.); d follows from it.
  • Optional top bars (for uplift), the same distance below the top.
  • Optional stirrups: single bars with 180° hooks (9.3.3.2.3(a)), #3 to #5, at a spacing of 2 in. or more.
  • The embedment of the bottom bars past each face of the opening (the bearing plus the extension into the grouted course over the jamb), for the development check.
  • Whether the soffit is exposed to earth or weather (cover, 6.1.5.1).
• Span and bearings:
  • The clear span (the opening) and the bearing length at each end. The span is the clear span plus half the required bearing length at each end (5.3.1.1, as the Masonry Designers' Guide 2022 states it); the lintel is analysed as a simple span between the bearing centres. The bearing length you enter is taken as the required one, so enter what the lintel needs, not all the masonry beside the opening.
  • The f'm of the supporting masonry (0 = the lintel's) for the bearing check.
  • The spacing of the lateral supports of the top (compression face), 0 when it is braced continuously (5.3.1.3).
• Loads (D, L, Lr, S, R, W, E):
  • The lintel self-weight: the weight of the fully grouted masonry per ft² of elevation (NCMA TEK 14-13B, TEK 17-1D Table 1) times the nominal depth.
  • The masonry above the lintel: its height (top of the lintel to the top of the wall) and weight (psf).
  • Arching action develops (checkbox). On: the wall load is the 45° triangle over the effective span, apex at half the span (NCMA TEK 17-1D); loads applied at or above the apex and point loads outside the triangle are not on the lintel (the steps and a note list them). Off: the full height of masonry above is a uniform load and every superimposed load is on the lintel.
  • Uniform loads applied within the triangle (when arching): applied in full, as without arching (TEK 17-1D Figure 4, philosophy 1, the default), or the part within the triangle, w (ℓ − 2y)/ℓ, dispersed at 45° over the span (philosophy 2). TMS 402 does not address it.
  • Uniform loads (plf) with the height above the lintel where they are applied (blank = the top of the wall).
  • Point loads (lb) with the position from the left face of the opening, the height above the lintel (blank = the top of the wall) and the bearing length along the wall. With dispersion on (default), a point load spreads at 2 vertical : 1 horizontal (5.1.1.1) over its bearing plus its height and is applied as a uniform load P / (bearing
    • height), the part beyond a support going to it directly (TEK 17-1D Figure 3). Off: it acts over its bearing, or as a concentrated load.
• Code options (the engineer's choices):
  • The shear design section: d/2 from the face of the support, the face (your choice), or the centre of the bearing. 5.3.1.5 allows d/2 only where (a) the support reaction compresses the end regions of the beam and (b) no concentrated load lies between the face and d/2. Otherwise it requires the maximum shear, and the calculation uses the shear at the reaction. Condition (a) is a checkbox, on by default (true for a lintel bearing on the masonry below; turn it off for a beam hung from or framed into its support). Condition (b) is read from the point loads, and a dispersed point load counts by its position. A note states which applied.
  • Mu/(Vu dv) for shear: taken as 1.0 (the TMS 402 commentary permits it for beams; conservative) or computed at the design section (positive, at most 1.0).
  • The deflection limit (span / n, 600 prefilled for beams supporting unreinforced masonry, 4.6; 0 = not checked) and whether the 8d exemption of 5.3.1.6.1 applies.

The depths in shear are not an option: TMS 402-22 settles them. The net shear area uses d (Table 4.4.5). dv stays the actual depth of the member, in Mu/(Vu dv), Vns and the stirrup rules, as the Masonry Designers' Guide 2022 example does (a saved value of the earlier shear depth option is ignored).

Auto-design (editor toolbar) varies the number of courses, the bottom bar size, the bars per course, the courses with bars and the stirrups (size and spacing). Objectives: lowest cost index (units, grout, reinforcement); shallowest; least reinforcement. The width, span, bearings, loads, the arching assumption and the code options are kept; see Auto-design.

Checks performed #

The key results list the lintel, the effective span, the load model, Mu / φMn, Vu / φVn and the factored reactions. The load steps show the effective span and every part of the load model with its value (and the loads not on the lintel). The drawings (on screen, in the report and on the PDF / DXF sheet) are the lintel loading (the loads on the lintel by type over the effective span, as the calculation applies them), the opening elevation (jambs, bearings, the lintel courses with the bars and their embedment, the stirrups, the wall above and the arching triangle when assumed), the lintel section (bars, stirrup, d and h) and the shear and moment diagrams of the governing combination. Unit shapes, splices, jamb reinforcement and the masonry below the bearings are not drawn.

Limitations #

Not covered:

• Deep beams (5.3.2): NOT FEASIBLE with a hint. The commentary allows a shallower lintel that carries the masonry above as load; use fewer courses.
• Continuous or fixed-end beams, a lintel that is part of a continuous bond beam (designed here as a simple span), cantilevers.
• Partially grouted beams, clay masonry, AAC. TMS 402-22 allows partially grouted beams, but a beam needs full grout where shear reinforcement is required; this calculation designs full grout only.
• Allowable stress design (Chapter 8).
• Axial load on the beam (9.3.3.2.1: a note), torsion, the development of the top bars, the grout strength (f'g ≥ f'm is assumed), the masonry supporting the jambs.
• The end limits of the point load dispersion of 5.1.1.1 (half the wall height, a movement joint, an opening, the spacing of the loads): a note asks you to check them.

Simplifications:

• The bearing area is the bearing length times the full width (A2 = A1).
• Two bars in a course are placed against the grout clearance; the stirrup is a single leg at the centre.
• The grout space of a course is the clear width (width less two face shells) times 7.625 in. less a bottom shell taken as the face-shell thickness (lintel unit geometry varies by manufacturer).
• The deflection is the immediate deflection; creep is not added (the commentary notes it).

Unverified against the code text. TMS 402-22 itself is not on hand. The section numbers come from TMS's cross-reference list, the 5.3.1.1 span rule and the depth used in Vns from the Masonry Designers' Guide example, the rest of the changed provisions from TMS's 2022 handouts. The cross-reference leaves these open, marked (to confirm against the code text) in the check references: the bearing section 4.4.4 (its row is blank, the number is inferred), the subsections of 5.3.1.1 (marked "Deleted?"), the section of the 8 in. nominal depth limit, φ = 0.60 for bearing, and the subsection of 6.1.3 for each size rule. The deep beam definition is carried over from 402-13.

Verification examples #

Verification status: 116 values compared in 29 examples (23 published, 6 hand calculations from the cited equations): 106 within tolerance, 10 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_lintel.py):

• The Masonry Society, Masonry Designers' Guide 2022 (TMS 402/602-22), free sample, Example 12.4-2A: a 16 ft opening, 8 in. CMU lintel 48 in. deep, and its rework with heavier loads. It is the only published example on hand written to the 2022 code.
• The Masonry Society, webinar handout Strength Design of Masonry (October 2017, 402-16 numbering), beam example: a 10 ft opening, 24 in. deep.
• The Masonry Society, webinar handout Design of Beams (R. Bennett, February 2021, TMS 402-16), which works the TMS Strength Design of Masonry guide's Examples 6.3.1.8 and 6.3.1.9: a 16 ft opening, 8 in. CMU beam 48 in. deep, f'm = 2,000 psi, Type S PCL mortar, 8 in. bearings, beam weight 81 psf. Free from TMS.
• NCMA (CMHA) TEK 17-01D, ASD of Concrete Masonry Lintels, design example: 12 in. CMU, 5 ft 4 in. opening, 4 in. bearings, 10 ft of masonry above, 1,000 plf at the top of the wall; Case 1 with arching, Case 2 a 24 in. lintel without. Only the loads, moments and shears are compared (the TEK's capacities are allowable stress design). Free from CMHA.

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

• MDG-12.4-2A-PRINT: printing errors in Example 12.4-2A. For 2 No. 4 it prints ρ = 0.00092 (that is 1 No. 5; 2 No. 4 give 0.00119). In the rework it prints 1,640,000 lb-in. as Mn (by hand Mn = 1,826,000 lb-in.; 1,643,000 is φMn). The Mn of 2 No. 4 is printed as 1,038,000 lb-in. (by hand 1,032,000, within the test tolerance), and the page writes 0.007 b for 0.0007 b.
• TMS-RHOMAX-16: Example 6.3.1.8 uses the 402-16 limit ρmax = 0.00952; 402-22 requires a tension-controlled beam, 0.00711 at f'm = 2 ksi.
• TMS-2017-VNS-D: the 2017 handout takes d = 20 in. in Vns and asks for Av = 0.101 in.²; the 2021 and 2022 TMS examples use the actual depth, which gives 0.084 in.² here.
• TEK17-1D-L57: the TEK computes its moments with the span rounded to 5.7 ft (Mlintel 357 lb-ft, Case 2 Mmax 61,601 lb-in.); ILX uses 5.67 ft (353 lb-ft, 60,866 lb-in.).
• TEK17-1D-TRI: the TEK text takes the 45° triangle (height half the effective span) but its example multiplies the wall weight by 3.5 ft (half the span plus 8 in.): 221 plf at the apex, Mwall 598, Mmax 955 lb-ft, Vmax 566 lb. ILX follows the text: 179 plf, 478, 831 lb-ft, 502 lb.

The TMS examples of 2021 and 2022 take Anv = b d and dv = the overall depth, the rule TMS now states, so the spacing limit of 24 in. that an earlier version of this calculation recorded as a difference now matches.

The hand calculations in closed form (ML-H1 to ML-H6) check the remaining paths:

Not on disk and not free (research/DOCUMENTS_NEEDED.md): TMS 402-22 / 602-22, the TMS Strength Design of Masonry guide (its lintel and beam examples beyond the webinar slides) and the full Masonry Designers' Guide (TMS 402/602-22).