Analytical, Numerical, and Four-Axis Assessment of a Triaxial (0°/60°/120°) Hexagonal Reinforcement System for Two-Way Reinforced Concrete Slabs

1. Introduction

Conventional two-way reinforced-concrete slabs are predominantly detailed with two orthogonal reinforcement families. Previous experimental and analytical studies have shown that reinforcement arrangement can influence slab stiffness, strength, deformation, and the development of tensile membrane action [16]. Three-way reinforcement has also been investigated previously; Aoyagi, Yamada, and Takahashi examined three-way reinforced-concrete containment models and reported changes in shear rigidity and large-deformation energy absorption [26]. Accordingly, the present study does not claim that three-way reinforcement is itself unprecedented.

The research problem addressed here is narrower and more quantitative: whether a planar 0°/60°/120° reinforcement network can be described by a consistent set of transformation rules and governing limits that permit controlled reduction of total reinforcement while maintaining the required strength, serviceability, and detailing performance. The central contribution is therefore the integration of geometry, reinforcement quantity, structural verification, and load-transfer descriptors into a single decision framework.

A key methodological requirement is to distinguish four categories of statements: (a) identities derived from the adopted mathematical idealization; (b) observations calibrated from the existing FE matrix; (c) proposed research parameters requiring further validation; and (d) requirements that remain mandatory under the governing concrete design code. This distinction is maintained throughout the revised manuscript.

2. Research Gap, Hypothesis, and Objectives

The principal hypothesis is that the 0°/60°/120° network provides a more directionally distributed reinforcement field than a conventional orthogonal grid and may therefore permit a lower total reinforcement quantity for selected slab geometries. The hypothesis does not assume that steel quantity alone controls capacity; instead, the admissible reduction is governed by the most demanding structural or detailing requirement.

The objectives are to: (1) derive the density and spacing transformation for a triaxial mesh; (2) establish idealized directional stiffness and moment relationships; (3) formulate a governing reinforcement envelope; (4) compare matched S1–S10 ABAQUS configurations; (5) identify the role of spacing, effective depth, and load-transfer geometry; and (6) demonstrate the procedure on a 1.83 × 1.83 m two-way slab matched directly to four of the original ABAQUS specimens.

3. Research Methodology and Evidence Structure

Method	Primary task	Output	Evidence status
M1—Analytical / mathematical	Topology, density, symmetry, spacing and geometric descriptors	Closed-form equations and dimensionless ratios	Derived within adopted model
M2—Code/design reference	Minimum steel, spacing, strength, serviceability, shear, punching and detailing	Reference design envelope	Code-based where applicable
M3—Nonlinear FE	ABAQUS/Standard comparison of orthogonal and triaxial configurations	P_u, deflection, cracking and stress redistribution	Numerical evidence
M4—Comparative calibration	Matched S1–S10 ratios and parametric trends	R_P, ΔP and governing trends	Model-specific calibration
ACI CODE-318-25 is used as the current reference for conventional reinforced-concrete design and detailing. The triaxial-specific equations in this paper are research equations; they are not presented as provisions contained in ACI CODE-318-25 [15]. Shrinkage and temperature reinforcement, shear, punching shear, crack control, anchorage, cover, durability, and constructability remain independent checks.

4. Four-Axis Theoretical Framework

4.1. Axis I—Topology, Density, and Spacing

Let s_o be the spacing of each reinforcement family in the conventional 0°/90° system and s_h the spacing of each family in the 0°/60°/120° system. For equal bar area A_b, the reinforcement density per unit slab area is proportional to the number of families divided by their spacing.

ρ_o = 2A_b/s_o (1)

ρ_h = 3A_b/s_h (2)

R_h = ρ_h/ρ_o = 3s_o/(2s_h) (3)

s_h = [3/(2R_h)] s_o (4)

R_h	Interpretation	Equivalent spacing
1.0	Equal total steel	1.5 s_o
0.6	Intermediate research level	2.5 s_o
0.5	Nominal 50% research target	3.0 s_o

The transformation in Equation (4) is a quantity-equivalence relation, not a maximum-spacing provision. If the resulting spacing exceeds an applicable detailing or crack-control limit, the spacing must be reduced and the actual reinforcement quantity recalculated.

s_f = min(s_h, s_3w) (5)

s_3w ≤ min(2h, 450 mm, 1.85d) (6)

The 2h and 450 mm terms are retained as reference two-way slab spacing limits. The 1.85d term is a proposed geometric research bound and is not identified as an ACI requirement.

Figure 1 shows the bar-orientation geometry of the two systems schematically, before the density and spacing transformation is derived analytically.

Figure 1. Schematic bar-orientation comparison between the conventional 0°/90° orthogonal grid and the proposed 0°/60°/120° triaxial mesh (illustrative geometry, not to scale).

Figure 1. Schematic bar-orientation comparison between the conventional 0°/90° orthogonal grid and the proposed 0°/60°/120° triaxial mesh (illustrative geometry, not to scale).

Figure 2. Equivalent triaxial spacing as a function of the adopted total reinforcement ratio (s_o = 200 mm).

Figure 2. Equivalent triaxial spacing as a function of the adopted total reinforcement ratio (s_o = 200 mm).

4.2. Axis II—Reinforcement Quantity and Idealized Directional Response

For equal total steel, the reference orthogonal reinforcement ratio is distributed among three equally spaced directions. Within the adopted idealized membrane/mesh formulation, the directional stiffness and moment expressions are rotation-invariant.

ρ_3 = (2/3)ρ_o (7)

E_hex(φ) = E_s ρ_3 Σ cos⁴(φ − kπ/3) = 0.75E_sρ_o (8)

m_n(φ) = m_o[cos²φ + cos²(φ−60°) + cos²(φ−120°)] = 1.5m_o (9)

Equations (8) and (9) are idealized mathematical results at equal total steel. They should not be interpreted as direct proof that a real cracked reinforced-concrete slab develops 1.5 times the conventional capacity.

2T cos30° = H → T = H/√3 = 0.577H (10)

The coefficient 1/√3 is a geometric projection coefficient associated with the adopted triangular force-transfer idealization. It is kept separate from the 1.5 moment relation to avoid double counting. The practical value k=0.60 is retained only as a research hypothesis. With a secondary conventional component represented by αA_m, the arithmetic relation is:

R_h = k/(1+α) (11)

For k=0.60 and α=0.20, Equation (11) gives R_h=0.50. This arithmetic demonstrates internal consistency of the proposed target but does not establish α=0.20 as a universal fraction for two-way slabs.

4.3. Axis III—Structural Resistance and Serviceability

Reinforcement quantity is treated as an input variable rather than a direct surrogate for structural capacity. The principal numerical comparison is defined as:

R_P = P_u,triax/P_u,ref (12)

ΔP = (R_P − 1)×100% (13)

Each candidate configuration must satisfy flexural resistance, one-way shear, punching shear where applicable, deformation limits, crack-width requirements, reinforcement strain limits, concrete compression/damage criteria, anchorage and development, cover, minimum reinforcement, and constructability. The triaxial network does not automatically eliminate shrinkage-and-temperature reinforcement.

4.4. Axis IV—Effective Depth and Load-Transfer Geometry

Because the triaxial system may occupy multiple layers or crossing levels, effective depth must be defined consistently. For a three-layer idealization:

d_1 = h − c_c − d_b/2 (14)

d_2 = h − c_c − 1.5d_b (15)

d_3 = h − c_c − 2.5d_b (16)

d_avg ≈ h − c_c − 1.5d_b (17)

The bar-diameter envelope 0.06≤d_b/d≤0.07 is retained as a research parameter rather than a code limit. A geometric load-diffusion descriptor is defined by:

θ_d = arctan(h_d/r_d) = arctan(2h_d/D_d) (18)

This descriptor is not interchangeable with an STM strut angle. The S9 value of approximately 23.6° is treated as a model-specific observation and not as evidence for replacing the conventional STM methodology with a universal triaxial angle.

Figure 3. Spacing versus the geometric load-diffusion descriptor for d = 125 mm. The 25° line is shown only as a reference to the discrete STM idealization discussed in the text.

Figure 3. Spacing versus the geometric load-diffusion descriptor for d = 125 mm. The 25° line is shown only as a reference to the discrete STM idealization discussed in the text.

5. Governing Reinforcement Envelope

The framework is completed by expressing the admissible reinforcement ratio as the maximum requirement generated by the independent design determinants:

R_adopted ≥ max(R_M, R_Δ, R_cr, R_V, R_p, R_detail) (19)

Determinant	Symbol	Primary check	Status in present study
Flexure	R_M	M_Ed ≤ M_Rd	Mandatory
Deflection	R_Δ	Serviceability deformation	Mandatory
Cracking	R_cr	Crack width/control	Mandatory
One-way shear	R_V	V_Ed ≤ V_Rd	Mandatory
Punching shear	R_p	V_u ≤ V_c / design resistance	Mandatory where applicable
Detailing/minimum	R_detail	Minimum steel, spacing, cover, anchorage	Mandatory

This formulation is the central design logic of the study: the 50% target is not a fixed allowable reduction. It is a candidate quantity that remains admissible only when all governing requirements are satisfied.

6. ABAQUS Matrix and Comparative Calibration

The S1–S10 models are treated as an internal numerical matrix rather than as ten independent physical experiments. The matrix includes slab thicknesses of 67, 76, 180, and 200 mm and compares orthogonal controls with triaxial configurations. The reported comparisons are reproduced below.

Specimen	h (mm)	Layout	s (mm)	P_u (kN/m2)	Matched interpretation
S1	76	Orthogonal control	120	36	Reference
S2	76	Triaxial half-ratio	360	36	R_P=1.000 vs S1
S3	76	Triaxial same-ratio	180	37	R_P≈1.028 vs S1
S4	67	Orthogonal	120	28	Reference
S5	67	Triaxial half-ratio	360	28	R_P=1.000 vs S4
S6	180	Orthogonal	120	190	Reference
S7	180	Triaxial same-ratio	180	190	R_P=1.000 vs S6
S8	200	Orthogonal double	60	235	Reference
S9	200	Triaxial half-ratio	360	225	R_P=0.957 vs S8
S10	200	Triaxial same-ratio	180	231	R_P=0.983 vs S8

Figure 4. Ultimate resistance reported for the S1–S10 ABAQUS matrix.

Figure 4. Ultimate resistance reported for the S1–S10 ABAQUS matrix.

Figure 5. Normalized resistance of the matched FE comparisons.

Figure 5. Normalized resistance of the matched FE comparisons.

Comparison	P_u,triax (kN/m2)	P_u,ref (kN/m2)	R_P	ΔP
S2/S1	36	36	1.000	0.0%
S3/S1	37	36	1.028	+2.8%
S5/S4	28	28	1.000	0.0%
S7/S6	190	190	1.000	0.0%
S9/S8	225	235	0.957	−4.3%
S10/S8	231	235	0.983	−1.7%

The observed matched-pair range is approximately 0.957≤R_P≤1.028. These values are useful for calibration of the present numerical framework, but they are not proposed as universal acceptance limits because the matrix has not yet been independently validated by physical tests and has not demonstrated mesh convergence.

7. Scientific Status of the Proposed Determinants

Level	Content	Interpretation
I—Derived	s_h=[3/(2R_h)]s_o; equal-steel s_h=1.5s_o; E_hex=0.75E_0; m_n=1.5m_0; 1/√3 projection coefficient	Mathematical consequences of the adopted idealization
II—Numerically calibrated	S1–S10 resistance ratios and S9 response	Observations from the present FE matrix
III—Research hypotheses	R_h=0.50; k=0.60; 1.85d spacing bound; bar-diameter envelope	Require parametric and experimental validation
IV—Mandatory checks	Flexure, shear, punching, minimum steel, cracking, deflection, anchorage, cover, durability and constructability	Remain required irrespective of topology

The distinction “Derived ≠ Calibrated ≠ Proposed ≠ Code-mandated” is retained as a governing editorial principle for the paper.

8. Parametric Interpretation and Sensitivity

The following curve is an analytical sensitivity plot generated from the equations of the framework. It is intended to reveal governing trends and is not additional experimental evidence.

Figure 6. Illustrative spacing envelope versus slab thickness using d≈0.833h; the 1.85d term is a research hypothesis.

Figure 6. Illustrative spacing envelope versus slab thickness using d≈0.833h; the 1.85d term is a research hypothesis.

The spacing curve demonstrates a nonlinear practical consequence of reinforcement reduction: as R_h decreases, the quantity-equivalent spacing increases rapidly. Consequently, a reduction strategy that is feasible from a steel-volume perspective can become infeasible from a spacing or crack-control perspective. This is why the governing envelope in Equation (19) is necessary.

9. Final Interpretation of the 1.83 × 1.83 m Demonstration

The 1.83 × 1.83 m demonstration is treated here as a simply supported square reinforced-concrete slab supported continuously along all four edges. This boundary condition is consistent with the ABAQUS models in the reference study, in which U_z = 0 is imposed along each line of edge nodes while slab rotation remains free. The reference models all have a plan dimension of 1.83 × 1.83 m and use four slab thicknesses of 67, 76, 180, and 200 mm. [16,17]

The full slab span L = 1.83 m is used for the four-sided simply supported panels. For the present screening design, the design bending demand in each principal direction is evaluated from the square two-way plate response, and the reinforcement is then selected using a preliminary code-based flexural check with d = h − 25 mm, together with a minimum-reinforcement check. The resulting reinforcement layouts are preliminary code-designed counterparts to the original ABAQUS configurations and are not treated as replacements for the subsequent nonlinear finite-element verification.

The four proposed code-designed cases were deliberately matched to four direct counterparts among the ten original ABAQUS slabs. Case 1 corresponds directly to S4 (h = 67 mm, w_u = 28 kN/m²), Case 2 to S1 (h = 76 mm, w_u = 36 kN/m²), Case 3 to S6 (h = 180 mm, w_u = 190 kN/m²), and Case 4 to S10 (h = 200 mm, w_u = 231 kN/m²). These values are the reported ultimate FE pressures in the original study. [16,17]

The matched code-based layouts are summarized below. The reinforcement areas are expressed per metre width in each principal reinforcement direction for the two-way code-designed slabs.

Case	Direct ABAQUS counterpart	h (mm)	Target FE pressure w_u (kN/m2)	Original ABAQUS reinforcement	Code-designed reinforcement	A_s,req (mm2/m)	A_s,prov (mm2/m)
Case 1	S4	67	28	Ø4.76 @ 120 mm, two orthogonal families	Ø8 @ 120 mm, two-way	339	418.9
Case 2	S1	76	36	Ø4.76 @ 120 mm, two orthogonal families	Ø8 @ 130 mm, two-way	349	386.7
Case 3	S6	180	190	Ø4.76 @ 120 mm, two orthogonal families	Ø10 @ 130 mm, two-way	566	604.2
Case 4	S10	200	231	Ø4.76 @ 180 mm, three families at 0°/60°/120°	Ø10 @ 120 mm, two-way	606	654.5

The original ABAQUS study specifies Ø4.76 mm reinforcement. S1 and S4 use orthogonal reinforcement at 120 mm spacing, while S6 also uses the orthogonal configuration. S10 belongs to the triaxial/hexagonal group and uses the 180 mm arrangement with three reinforcement directions. [16,17]

For the original ABAQUS reinforcement, the area of one Ø4.76 mm bar is approximately 17.80 mm². Thus, for 120 mm spacing, the reinforcement intensity of one orthogonal family is approximately 148.3 mm²/m, and the two-family total is approximately 296.6 mm²/m. For the S10 three-family arrangement at 180 mm spacing, the corresponding intensity is approximately 98.9 mm²/m per family, or approximately 296.6 mm²/m when the three families are considered together.

A direct quantity comparison is therefore obtained as follows:

Case	ABAQUS model	ABAQUS total reinforcement area (mm2/m)	Code model	Code total reinforcement area (mm2/m)	Difference relative to ABAQUS
Case 1	S4	296.6	Ø8 @ 120, two-way	837.8	+182.5%
Case 2	S1	296.6	Ø8 @ 130, two-way	773.3	+160.7%
Case 3	S6	296.6	Ø10 @ 130, two-way	1208.3	+307.4%
Case 4	S10	296.6	Ø10 @ 120, two-way	1309.0	+341.4%*
* The Case 4 comparison should be interpreted with caution because S10 uses three reinforcement families, whereas the proposed code-designed Case 4 uses two orthogonal families. Therefore, the total steel-area comparison describes total embedded steel quantity, but it does not represent identical reinforcement topology.

The comparison indicates that the reinforcement quantities embedded in the original ABAQUS models are substantially smaller than those obtained from the present preliminary code-based flexural screening. This difference should not, by itself, be interpreted as evidence that either approach is incorrect. The original models were constructed primarily as nonlinear finite-element configurations for studying the influence of reinforcement topology and nominal steel quantity. The model set deliberately varies reinforcement topology and nominal ratio among S1–S10 rather than applying a single conventional code-design procedure to all ten slabs. The reported matrix includes orthogonal, triaxial/hexagonal half-ratio, triaxial/hexagonal same-ratio, and double-ratio configurations. [16,17]

Consequently, the appropriate interpretation of the present demonstration is not that the code-designed reinforcement should simply replace the original ABAQUS reinforcement. Instead, the two configurations should be treated as two different reinforcement design bases: (1) the original research configurations used in the ABAQUS simulations, and (2) newly generated code-based reinforcement layouts having the same slab dimensions, thicknesses, and target loading levels. The code-designed configurations should subsequently be re-modelled in ABAQUS using the same nonlinear material formulation, element types, reinforcement representation, embedded interaction, mesh-convergence procedure, and simply supported boundary conditions used for the reference models. The original study employs C3D8R concrete elements and T3D2 reinforcement elements with embedded interaction/perfect bond. [16,17]

The resulting comparison therefore establishes a more controlled framework for evaluating the proposed reinforcement concept. For each thickness level, the principal variables are held constant at a plan dimension of 1.83 × 1.83 m and a specified target pressure, while the reinforcement quantity and topology are varied. The principal response measures should then be the ultimate load, central deflection, crack-pattern development, principal concrete strains/stresses, and reinforcement stress/strain history. The ABAQUS results for the original configurations provide the reference response, while the code-designed models provide an independent reinforcement baseline.

Accordingly, the 1.83 × 1.83 m demonstration does not support direct transfer of a single bar size or spacing from one slab geometry to another. Rather, it demonstrates that the reinforcement arrangement must be re-derived for each thickness and loading condition, after which the proposed layouts must be verified through mesh-converged nonlinear ABAQUS analysis. In particular, any conclusion regarding reinforcement reduction in the triaxial/hexagonal configuration should be based on equal-dimensional, equal-thickness, and equal-loading comparisons with the corresponding orthogonal and code-designed models, followed by numerical and, where possible, experimental validation.

10. Discussion

The reorganized results indicate that the principal scientific question is not whether three reinforcement directions can be drawn within a slab, but whether their topology can be transformed into a reproducible design procedure. The exact density relation is the most robust part of the framework because it follows directly from counting reinforcement families per unit spacing. By contrast, the 50% reduction target is conditional and must be tested against spacing, minimum reinforcement, flexure, shear, punching, serviceability, and detailing.

The analytical stiffness and moment expressions provide a mechanistic basis for studying directional symmetry, but they are idealized. Real reinforced-concrete slabs include cracking, bond, concrete tension stiffening, nonlinear compression, anchorage, membrane action, boundary restraint, and redistribution after yielding. The FE matrix provides a first numerical calibration, but the present evidence does not isolate the proposed confinement or membrane mechanisms directly from ABAQUS stress and energy fields.

The S9 result is particularly useful for framing the load-transfer discussion. Its reported geometric/strut angle of approximately 23.6° is close to, but below, a 25° discrete STM reference angle, while the modeled resistance remains relatively high. The scientifically defensible interpretation is not that 23.6° should replace 25°, but that a single discrete STM angle should not be treated as a universal acceptance criterion for a continuum triaxial slab. Direct stress-trajectory extraction and dedicated STM comparisons are required.

11. Limitations and Required Validation

• Mesh convergence has not yet been demonstrated for the ten FE models.

• Only one underlying CDP material input was used; specimen-specific concrete strengths were not independently varied as a controlled parameter.

• Independent experimental validation of the triaxial specimens is absent.

• The proposed confinement and strain-energy mechanisms were not directly extracted from ABAQUS stress and energy fields.

• The FE models assume perfect bond and do not reproduce shrinkage, drying, or curing microcracking.

• The effective-depth basis should be unified before final publication, particularly for S9 d/s and angle calculations.

• The proposed 1.85d spacing bound, k=0.60, and 50% reduction target require parametric studies and physical testing before generalized design use.

Future validation should vary slab thickness, reinforcement ratio, bar diameter, spacing, aspect ratio, boundary restraint, and load position. Measurements should include crack widths, reinforcement strains in all three directions, principal concrete stresses, membrane/hoop forces, load-deflection response, and full-field deformation.

12. Conclusions

1. The revised paper establishes a four-axis framework that connects reinforcement topology, quantity, structural verification, and load-transfer geometry.

2. The exact quantity-equivalence relation is s_h=[3/(2R_h)]s_o; consequently, equal total steel corresponds to 1.5s_o, R_h=0.60 corresponds to 2.5s_o, and the nominal R_h=0.50 target corresponds to 3s_o before spacing caps.

3. The idealized equal-steel relations E_hex=0.75E_0 and m_n=1.5m_0 are mathematical results of the adopted mesh model and should not be interpreted as direct experimental strength multipliers.

4. The 50% total-steel reduction and k=0.60 are research hypotheses rather than code provisions or universal design limits.

5. The governing reinforcement requirement should be obtained from the maximum of flexural, deformation, cracking, shear, punching, and detailing/minimum-reinforcement requirements.

6. The S1–S10 FE matrix provides model-specific calibration, including R_P≈1.000 for S1/S2 and S4/S5, R_P≈1.000 for S6/S7, and R_P≈0.957 for S9/S8.

7. The 1.83 × 1.83 m demonstration, matched to four of the original ABAQUS specimens under four-edge simply supported conditions, shows that preliminary code-based reinforcement quantities substantially exceed the reinforcement embedded in the reference finite-element models; any claimed reduction must therefore be verified through matched nonlinear re-analysis rather than assumed from steel quantity alone.

8. The principal research contribution is therefore the integrated determinant framework and its explicit separation of derived equations, numerical observations, research hypotheses, and code-mandated checks.