PBDPy

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MIDAS ASCE 41-17 Column — Methodology

• Per-column flexure check — Eq. \( 7\text{-}36 \)
• Per-column shear check — Eq. \( 7\text{-}37 \)
• Reference: ASCE/SEI 41-17 §10.4.2.3 and §10.4.2.4

Action classification (§10.4.2.4.1)

• Per §10.4.2.4.1, the axial-load ratio \( P/(A_g f'_{cE}) \) determines whether column flexure is treated as deformation- or force-controlled. The same ratio is also the axial-axis row selector for Table 10-10a (linear procedure m-factors).
• \( P/(A_g f'_{cE}) \le 0.5 \) → deformation-controlled; m-factor from Table 10-10a applies (Eq. \( 7\text{-}36 \))
• \( P/(A_g f'_{cE}) > 0.5 \) → force-controlled; set \( m = 1 \) and compare \( \kappa\,M_{CL,P} \) vs \( M_{UF} \) (Eq. \( 7\text{-}37 \))
• Table 10-10a footnote (a) governs linear interpolation between the listed corner values; it is not the source of the action-classification threshold.

Splice check (§10.3.5, Eq. 10-1a + 10-1b)

• Two-step check unique to columns
• Step (a) — use available lap length \( l_b \)
• Step (b) — reduce by \( (2/3)\,d \) to model bond degradation in the plastic-hinge zone
• If either step gives \( f_s < f_{yE} \) → column is controlled by inadequate splicing; solver uses the lower half of Table 10-10a

Shear strength Eq. (10-3)

• \( k_{nl} \) — ductility reduction per Eq. (10-3): \( 1.0 \) at \( \mu \le 2 \), \( 0.7 \) at \( \mu \ge 6 \), linear between. \( \mu \) is taken from the flexural demand (\( \mu \approx M_{UD}/M_{CE} \)); the performance level only sets a fallback \( \mu \) when the flexural demand is unavailable
• \( \alpha \) — \( 1.0 \) for \( s/d \le 0.75 \), linear to \( 0 \) at \( s/d = 1.0 \)
• \( M/(Vd) \) clamped to \( [2,\,4] \) per §10.4.2.3.1
• \( N \) (compression) clamped at \( 0 \) for net tension
• Mode-classification \( V_{yE}/V_{Col0E} \) — expected materials with \( k_{nl} = 1 \)
• Shear acceptance — lower-bound materials with ductility-demand \( k_{nl} \) (Eq. 10-3)

Table 10-10a lookup (rectangular w/ seismic hoops)

• Three classification axes — bilinear interpolation:
  • axial ratio \( P/(A_g f'_{cE}) \)
  • transverse-rebar ratio \( \rho_t \)
  • shear-demand ratio \( V_{yE}/V_{Col0E} \)
• Three shear-bin slices — \( V_{yE}/V_{Col0E} < 0.6 \), \( 0.6\text{–}1.0 \), \( \ge 1.0 \)
• Two halves per slice — NOT splice-controlled vs splice-controlled
• Footnote (b) — splice-controlled \( m \) never exceeds the non-splice value
• Component type dropdown — Primary (lateral-resisting) vs Secondary (gravity-only); IO shared across both, LS / CP differ
• Table 10-10b (circular columns w/ spirals) — to-be-developed in a follow-up

Table 10-10a — m-factor corner values

• Source: ASCE/SEI 41-17 Table 10-10a (rectangular columns w/ seismic hoops per ACI 318)
• Bilinear interpolation between listed corner values
• Footnote (b) — splice-controlled \( m \) never exceeds the corresponding non-splice value
• \( V_{yE}/V_{Col0E} \) floored at \( 0.2 \) per §10.4.2.3.1
• \( \rho_t \le 0.0175 \) cap; \( \le 0.0075 \) when ties are not adequately anchored in the core

Capacity-design shear demand

• \( i,\,j \) — ends of the MIDAS column element (double-curvature, both ends hinging)
• Constant section + reinforcement over the clear height → \( M_{CE,i} = M_{CE,j} = M_{CE,P} \) (capacity depends only on section, rebar, axial load)
• Grid takes a single \( M_{CE} \) per column — supply explicitly for governing cases, or leave blank to auto-estimate from a simplified P-M envelope

Engineering note:

• "top / bot" \( \equiv \) "\( i / j \)" — for a vertical column, \( i \) is typically the bottom node and \( j \) the top node
• For different reinforcement at the two ends or a meaningful axial-load gradient, ask and I'll add per-end \( M_{CE,i} \) / \( M_{CE,j} \) input columns to the grid

Status taxonomy

• OK — \( \mathrm{DCR} \le 1.0 \)
• N/C — \( \mathrm{DCR} > 1.0 \)
• — — insufficient input

Caveats

• Eq. \( (10\text{-}3) \) is empirical and dimensionally non-strict — calibrated for SI inputs (MPa, mm, N) returning kN-scale results
• The simplified P-M envelope is a screening tool only — verify governing columns with a rigorous P-M diagram (e.g., spColumn)

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MIDAS ASCE 41-17 LDP — Methodology

• Per-location flexural evaluation of RC beams
• Linear (LDP / LSP) procedures
• Reference: ASCE/SEI 41-17, ACI 318-19

Acceptance Equation (§7.5.2.2)

Deformation-controlled flexure — Eq. \( 7\text{-}36 \) requires:

• \( Q_G = \max\bigl(1.1(D + S_D + 0.25L),\; 0.9D\bigr) \) per Eq. \( 7\text{-}1 \), \( 7\text{-}2 \)
• \( Q_E \) — §7.2.5.1 Procedure 1 (100/30) orthogonal combination of the MIDAS response-spectrum envelopes: \( Q_E = \max(R_M + 0.30\,R_O,\; R_O + 0.30\,R_M) \)
• \( R_M,\,R_O \) — RS-X and RS-Y major-axis moment envelopes
• \( Q_{CE} = M_n \) — ACI 318-19 §22.3 with \( \phi = 1.0 \) using expected material strengths per §10.3.2.2 (\( f'_{cE} = 1.5\,f'_c \), \( f_{yE} = 1.25\,f_y \))

m-Factor Row Selectors (Table 10-13)

• m-factor read from Table 10-13 using three row-selectors, reproduced in the per-location result table for auditability:
  • \( (\rho - \rho')/\rho_{bal} \) — thresholds \( \le 0.0 \) / \( \ge 0.5 \)
  • Transverse-reinforcement conformance — C / NC per §10.4.2.2
  • \( V_d/(b_w d \sqrt{f'_{cE}}) \) — thresholds \( \le 0.25 \) / \( \ge 0.5 \) in MPa-equivalent units (\( \le 3 \) / \( \ge 6 \) in psi form)

Capacity-Design Shear \( V_d \) (§10.4.2.4.1 item 1)

• The shear that drives the row selector is the capacity-design shear — not the elastic envelope from analysis:

• \( V_p = (M_{CE,-} + M_{CE,+})/L_n \) — plastic-mechanism shear at full hinging of both beam ends
• \( V_G = 1.1\,(V_{DL} + V_{SDL} + 0.25\,V_{LL}) \) — coexisting gravity shear from the §7.2.2 Eq. \( 7\text{-}1 \) combination
• Per-case shears pulled from MIDAS when available; otherwise the solver falls back to the user-supplied \( V \) column, treated as already-aggregated gravity shear

Table 10-13 Conditions i–iv (footnote b)

• Footnote (b): "Where more than one of conditions i, ii, iii, and iv occurs for a given component, use the minimum appropriate numerical value from the table."
• Solver evaluates all four conditions and applies the minimum m-factor from those that trigger
• Cond. i (flexure) is the default — used only when none of ii / iii / iv triggers

Condition i — Beams controlled by flexure

• Default when \( V_p < V_n \) and no splice / embedment deficiency
• \( m \) read from the flexure rows of Table 10-13 using the three row-selectors above

Condition ii — Beams controlled by shear (§10.4.2.4.2)

• Compare \( V_p \) against the expected shear strength \( V_n = V_c + V_s \):

• Per §10.3.4, \( s > d/2 \) → \( V_s \) reduced by effectiveness factor \( 2(1 - s/d) \)
• \( s > d \) → \( V_s \) drops to zero
• Cond. ii triggers when \( V_p \ge V_n \)
• \( m \) from the Cond. ii row, spacing bin \( s \le d/2 \) vs \( s > d/2 \)

Condition iii — Inadequate development / splicing in span (§10.3.5 Eq. 10-1a)

• Class B tension lap splice required length: \( 1.3\,l_d \) (\( l_d \) per ACI 318-19 §25.4.2.4)
• Maximum developable stress at the splice:

• Cond. iii triggers when \( f_s < f_{yE} \) — the available splice cannot deliver yield
• Provide \( l_b \) on the form; \( l_b = 0 \) disables the check
• \( m \) from the Cond. iii row of Table 10-13, same tight / loose spacing bins as Cond. ii

Condition iv — Inadequate joint embedment (§10.3.5 Eq. 10-2)

• Deformed straight bar terminating with short embedment into a beam-column joint:

• Cond. iv triggers when \( f_s < f_{yE} \)
• Two compounding effects:
  • \( m \) taken from the Cond. iv row of Table 10-13 (\( m = 2 / 2 / 3 \) for IO / LS / CP, no spacing bin)
  • Positive-moment capacity \( M_{CE,+} \) recomputed using \( f_s \) in place of \( f_{yE} \) per §10.3.5 item 4
• Cells with degraded \( M_{CE,+} \) are flagged with an asterisk in the result table
• Provide \( l_e \) on the form; \( l_e = 0 \) disables the check
• Strictly, Eq. \( 10\text{-}2 \) requires clear cover \( \ge 3\,d_b \)

Footnote (b) — Multi-Condition Acceptance

• When multiple conditions trigger, the governing \( m \) is the minimum across triggered conditions, applied to its respective \( M_{CE} \)
• For Cond. iv, the worked-example Interpretation (c) is used: the minimum \( m \) is multiplied by the degraded \( M_{CE,+} \) — the most-conservative reading of the otherwise-silent footnote

Status taxonomy

• PASS — \( \mathrm{DCR} \le 1.0 \)
• SPLICE-GOVERNED — \( \mathrm{DCR} \le 1.0 \) with §10.3.5 splice reduction applied (lap-spliced sections at plastic-hinge zones)
• FAIL (LS) — \( \mathrm{DCR} > 1.0 \); beam exceeds the active hazard. \( \mathrm{DCR} > 2.0 \) is flagged with a "section replacement / FRP" note — m-factor benefit alone cannot bridge the gap
• DATA MISSING — non-numeric or insufficient input (\( b \), \( d \), \( f'_{cE} \), \( f_{yE} \), or rebar)

Caveats

• All calculations subject to verification by the responsible licensed engineer
• Default Cond. ii m-factors are conservative ASCE 41-17 values — verify against the spec version in force for the active project

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Linear Procedure (LSP / LDP) — Methodology

• One calculator per element — beam, column, slab, joint
• Reference: ASCE/SEI 41-17 §7.5.2.2 and Chapter 10; page numbers are the printed pages
• Table values checked against the printed page through the Structural Mind vault (qaqc/linear_vault_check.py)

Acceptance (§7.5.2.2, p. 79)

• Deformation-controlled action — expected strength with the m-factor:

• Force-controlled action — lower-bound strength, no m-factor:

• Both are printed as strict inequalities — a DCR of exactly \( 1.000 \) does not pass
• \( \kappa \) — knowledge factor, §6.2.4; entered by the user, never defaulted
• Secondary m-factors apply only to a component left out of the linear model's lateral stiffness (§7.2.3.3, p. 64; §7.5.1.1, p. 77)
• IO is printed once per row and shared by both component types
• Values between listed rows — linear interpolation (footnote a of each table); inputs outside the listed range are clamped to it

Beam — Table 10-13 (p. 158)

• Condition i — bilinear in \( (\rho-\rho')/\rho_{bal} \) and \( V/(b_w d \sqrt{f'_{cE}}) \) (MPa units)
• \( V \) — limit-state shear per §10.4.2.4.1, not the elastic analysis shear
• Conforming ties: hoops at \( \le d/3 \) in the hinge region and, at moderate or high ductility demand, \( V_s \ge 0.75\,V \) (footnote c)
• Footnote b — where several conditions apply, the minimum \( m \) governs

Column — Table 10-10a (p. 155)

• Rectangular columns; circular columns (Table 10-10b) are not implemented
• Bilinear in \( N_{UD}/(A_g f'_{cE}) \) and \( \rho_t \) inside the \( V_{yE}/V_{ColOE} \) band; \( V_{ColOE} \) from Eq. (10-3), p. 154
• Splice-controlled where \( f_s \) from Eq. (10-1a) or (10-1b), p. 149, falls below \( f_{yE} \)
• \( P/(A_g f'_{cE}) > 0.5 \) — PBDPy takes flexure as force-controlled with \( m = 1 \). This is a PBDPy rule, not an ASCE 41-17 provision; it overrides the printed \( \ge 0.7 \) rows

Slab — Table 10-16 (p. 163)

• Slab-column connections are keyed on \( V_g/V_o \) and continuity reinforcement only
• \( V_g \) — gravity shear on the slab critical section; \( V_o \) — direct punching shear strength; both as defined by ACI 318 (footnote c)
• Continuity = Yes where the continuous bottom bars through the column cage each way are \( \ge 0.5\,V_g/(\phi f_y) \); for a post-tensioned slab, at least one tendon each way through the cage (footnote d)
• — = footnote e: the action is force controlled and takes Eq. (7-37)
• Condition iii is force controlled as a primary component and deformation controlled as a secondary one
• The \( > 0.6 \) rows are exclusive — \( V_g/V_o = 0.6 \) exactly reads the \( 0.6 \) row

Joint — Table 10-14 (p. 159) and Table 10-12 (p. 158)

• Joint shear strength, §10.4.2.3.2 (pp. 155–156):

• \( A_j \) = joint depth × joint width; the depth is the column dimension in the direction of framing
• Joint width — the smallest of: the column width; the beam width plus the joint depth; twice the smaller distance from the beam axis to the column side
• \( \lambda = 0.75 \) lightweight, \( 1.0 \) normal-weight; \( \gamma \) from Table 10-12
• Conforming joint hoops: spaced at \( \le h_c/2 \) within the joint
• \( P \) — design axial force on the column above the joint; \( A_g \) — gross area of the joint (footnote b)
• Every primary cell is 1 — a primary joint takes no ductility credit, so Eq. (7-36) reduces to \( \kappa\,V_J > V \)

Caveats

• All calculations subject to verification by the responsible licensed engineer
• Footnote a does not say how to interpolate on two axes at once; bilinear interpolation is the reading used here
• Open audit findings on the beam and column lookups are listed in CLAUSES.md