Effects of Member Size, Shear Span-to-Depth Ratio, and Longitudinal Reinforcement Type on the Shear Behavior of Lightweight Concrete Beams: A Numerical Study

Abstract

The shear response of lightweight concrete beams depends on the member size, shear span-to-depth ratio, and longitudinal reinforcement type. Based on checks against two previously reported LC60 beam tests and three-dimensional nonlinear finite element analysis, this study evaluated these parameters within the investigated model series. Increasing the effective depth from 85 to 340 mm substantially increased the absolute shear capacity, while the nominal ultimate shear strengths remained approximately 0.895, 0.951, and 0.937 MPa and showed no pronounced classical monotonic decrease. Increasing the shear span-to-depth ratio from 1.5 to 5.0 shifted the computed response from arch action toward beam action and reduced the nominal ultimate shear strength by up to 75.9%. Replacing steel reinforcement with lower-modulus GFRP reinforcement reduced the stiffness and ultimate shear strength, although the difference between the two reinforcement types decreased with the member size. These results provide a numerical basis for understanding the effects of the member size, shear span-to-depth ratio, and longitudinal reinforcement type on the shear response of LC60 lightweight concrete beams.

1. Introduction

Lightweight concrete offers low density, high specific strength, and reduced structural dead load, making it attractive for long-span bridges, prefabricated structures, and marine transportation infrastructure. However, because of the porous nature of lightweight aggregates and the characteristics of the aggregate–matrix interface, its elastic modulus, tensile properties, and fracture toughness are generally lower than those of normal-weight concrete of comparable strength. Lightweight concrete members are therefore more susceptible to diagonal cracking, localized damage, and brittle failure under shear. In a previous study [1], small-scale beams without stirrups were tested to systematically examine the effects of the lightweight aggregate type, glass fiber content, and steel or GFRP longitudinal reinforcement on shear behavior. The results showed that concrete incorporating fly ash-based artificial core–shell lightweight aggregates exhibited favorable cracking resistance and shear performance, whereas the low elastic modulus of GFRP bars increased the longitudinal deformation and crack width and reduced both the cracking load and shear capacity [1]. That study, however, considered only a single member size and a limited range of shear spans; the applicability of its findings to structural-scale members therefore remains to be established.

Size-dependent shear behavior has been reported for conventional and FRP-reinforced concrete beams [2,3,4,5,6,7,8,9,10]. Whether a classical decrease in nominal shear strength occurs in the present LC60 system is treated as a research question. The shear span-to-depth ratio also governs the transition between arch action and beam action, and numerical studies have shown that increasing the ratio shifts the reinforced concrete beam response away from shear-dominated behavior [11,12,13].

In addition to the member size and shear span-to-depth ratio, the longitudinal reinforcement type and axial stiffness influence the shear behavior. Steel-, GFRP-, CFRP-, and BFRP-reinforced beams have shown depth-dependent responses, while the lower elastic modulus of FRP reinforcement can produce larger longitudinal strains and wider cracks [8,9,10,14,15,16]. Studies of lightweight and recycled aggregate concrete further show that the aggregate system and reinforcement detailing can alter the shear and flexural response [17,18,19,20]. These studies provide context, but they do not directly validate the present LC60 finite element model.

For beams made with lightweight aggregate concrete, there is therefore a lack of systematic studies that assess the member size, shear span-to-depth ratio, and longitudinal reinforcement stiffness within a single material system, under geometrically similar conditions, and without the contribution of stirrups. In particular, it remains unclear, in quantitative terms, whether low longitudinal reinforcement stiffness further increases crack opening and aggravates the reduction in shear strength in large members and at large shear span-to-depth ratios.

Building on the previous experimental study [1], the present work combines experimental validation with three-dimensional nonlinear finite element analysis to develop numerical models of lightweight aggregate concrete beams reinforced with either steel or GFRP bars, and to systematically investigate the effects of the member size, shear span-to-depth ratio, and longitudinal reinforcement stiffness on the shear behavior. By examining the nominal shear strength, crack evolution, longitudinal reinforcement strain, concrete damage, and the development of diagonal struts, the study elucidates the mechanism linking reduced longitudinal reinforcement stiffness to increased longitudinal deformation, wider diagonal cracks, diminished shear transfer across the cracked zone, and reduced shear strength. The coupled relationships among the size effect, shear span-to-depth ratio, and longitudinal reinforcement stiffness are then identified, and an approach for evaluating the shear behavior of structural-scale core–shell lightweight aggregate concrete beams is established to support the design of lightweight concrete members and the refinement of existing shear models.

2. Materials and Experimental Program

2.1. Materials

The LC60 core–shell lightweight aggregate concrete used in the previous study [1] was adopted in the present work. The binder consisted of P.O 52.5 ordinary Portland cement, Class I fly ash, and silica fume. Shale lightweight fine aggregate with a particle size of 0–3 mm was used as the fine aggregate, and fly ash-based artificial core–shell lightweight aggregate with a particle size of 5–10 mm was used as the coarse aggregate. A polycarboxylate high-range water-reducing admixture with a water reduction ratio of 27% was used, and tap water served as the mixing water. The basic properties of the shale lightweight aggregate and core–shell lightweight aggregate used in the previous study are listed in Table 1.

Table 1. Main physical and mechanical properties of the lightweight aggregates [1].

Property	Shale Lightweight Fine Aggregate	Core–Shell Lightweight Aggregate
Particle size/mm		0~3	5~10
Apparent density/(kg·m−3)		1482	1596
Bulk density/(kg·m−3)		631	910
Cylinder compressive strength/MPa		—	12.6
1 h water absorption/%		12.9	19.0
Single-particle splitting strength/MPa		—	10.58
Particle porosity/%		—	46.5

The mixture proportions of the LC60 core–shell lightweight aggregate concrete are given in Table 2. Because the core–shell lightweight aggregate has a relatively high water absorption, it was prewetted to a saturated surface-dry condition before mixing to minimize further water uptake by the aggregate and the associated effect on concrete workability. After casting, the concrete was cured at room temperature for 28 d. The mixture proportions and material properties are presented in Table 2 and Table 3, respectively.

Table 2. Mixture proportions of LC60 core–shell lightweight aggregate concrete (kg/m³) [1].

Concrete	Cement	Fly Ash	Silica Fume	Shale Lightweight Fine Aggregate	Core–Shell Coarse Aggregate	Water	Water Reducer
LC60	450	120	30	310	665	142	18.5

Table 3. Basic properties of LC60 core–shell lightweight aggregate concrete at 28 d.

Compressive Strength/MPa	Tensile Strength/MPa	Elastic Modulus/GPa	Slump/mm	Apparent Density/(kg·m−3)
71.50	3.31	20.5	137.8	1808.1

2.2. Beam Geometry

The two baseline beams reported in Ref. [1] were singly reinforced rectangular beams tested under three-point bending, with one specimen for each reinforcement configuration. Each beam was 1100 mm long with an 80 mm × 110 mm cross-section and an effective depth of d = 85 mm. The clear shear span was a = 425 mm, measured from the inner face of the 50 mm wide support contact region to the loading line and calculated as (900 − 50)/2. One 10 mm diameter steel or GFRP bar was placed in the bottom tension zone, the concrete cover was 20 mm, and no stirrups were provided. The geometry and the parameters required to define and check the finite element models are shown in Figure 1 and Table 4; the complete experimental program is reported in Ref. [1].

Figure 1. Geometry and cross-sectional parameters of the beam specimens.

Table 4. Design parameters of the test beams [1].

Specimen	Concrete	Bar Type	Diameter/mm	Tensile Strength/MPa	Elastic Modulus/GPa	Effective Depth d/mm	Shear Span-to-Depth Ratio a/d
LC60-S	LC60	Steel	10	550	200	85	5
LC60-G	LC60	GFRP	10	1200	40	85	5

2.3. Testing Procedure

The beams were subjected to monotonic three-point loading over a 900 mm support spacing. LVDTs were installed at midspan and at both supports, as shown in Figure 2. The influence of support movement was removed from the reported midspan displacement using the support LVDT measurements.

Figure 2. Test setup and instrumentation layout.

Loading was initially applied under load control. At approximately 80% of the theoretically estimated peak load, the loading mode was switched to displacement control and maintained until failure. Load and corrected midspan displacement were recorded continuously. The baseline cracking and peak load results used for deterministic model checking are reported in Ref. [1]; no estimate of specimen-to-specimen variability is claimed.

2.4. Finite Element Modeling

To extend the ranges of the member size and shear span-to-depth ratio beyond those covered by the previous experiments, three-dimensional nonlinear finite element models of LC60 lightweight concrete beams without stirrups were developed in ABAQUS 2026. The LC60-S and LC60-G baseline tests reported in Ref. [1] were used for model checking. Subsequent configurations were numerical parametric predictions for the investigated LC60 system.

2.4.1. Concrete

The LC60 concrete was modeled as a homogeneous continuum using C3D8R solid elements and the Concrete Damage Plasticity (CDP) model. Individual coarse and fine aggregates, mortar, and aggregate–mortar interfacial transition zones were not explicitly represented. The reported 5–10 mm aggregate size characterizes the experimental mixture and is not a geometric parameter of the finite element model. Table 5 lists the adopted concrete properties.

Table 5. Finite element parameters of LC60 core–shell lightweight aggregate concrete.

Ec/GPa	ν	ρ/(kg·m−3)	${\mathit{f}}_{\mathit{c}}^{\prime }$/MPa	ft/MPa	Element Type
20.5	0.20	1808.1	71.50	3.31	C3D8R

The CDP plasticity parameters used in the simulations were a dilation angle of 38°, a flow potential eccentricity of 0.1, f_b0/f_c0 = 1.16, K_c = 0.6667, and a viscosity parameter of 0.0001. This parameter set was adopted for the present simulations; its uniqueness was not established through inverse identification.

The compressive behavior was described using the modified Collins stress–strain model reported in Ref. [21], as expressed in Equations (1)–(5):

$${\mathit{\sigma }}_{\mathit{c}}={\mathit{f}}_{\mathit{c}}^{\prime }{\displaystyle \frac{\mathit{n}({\mathit{\epsilon }}_{\mathit{c}}/{\mathit{\epsilon }}_{\mathit{c}0})}{\mathit{n}\mathit{-}1\mathit{+}{({\mathit{\epsilon }}_{\mathit{c}}/{\mathit{\epsilon }}_{\mathit{c}0})}^{\mathit{nk}}}}$$

(1)

$${\mathit{\epsilon }}_{\mathit{c}0}=2{\mathit{f}}_{\mathit{c}}^{\prime }/{\mathit{E}}_{\mathit{c}}$$

(2)

$$\mathit{n}=0.8\mathit{+}{\mathit{f}}_{\mathit{c}}^{\prime }/17$$

(3)

$$\mathit{k}=\left\{\begin{array}{cc}0.67\mathit{+}{\mathit{f}}_{\mathit{c}}^{\prime }/62,\hfill & {\mathit{\epsilon }}_{\mathit{c}}/{\mathit{\epsilon }}_{\mathit{c}0}\mathit{>}1\hfill \\ 1.0,\hfill & {\mathit{\epsilon }}_{\mathit{c}}/{\mathit{\epsilon }}_{\mathit{c}0}\mathit{\le }1\hfill \end{array}\right.$$

(4)

$${\mathit{\epsilon }}_{\mathit{c}}^{\mathit{in}}={\mathit{\epsilon }}_{\mathit{c}}\mathit{-}{\mathit{\sigma }}_{\mathit{c}}/{\mathit{E}}_{\mathit{c}}$$

(5)

where σ_c and ε_c are the uniaxial compressive stress and total compressive strain, respectively; ${\mathit{f}}_{\mathit{c}}^{\prime }$ is the concrete compressive strength; ε_c0 is the strain at peak compressive stress; E_c is the elastic modulus; n and k are curve shape parameters; and ${\mathit{\epsilon }}_{\mathit{c}}^{\mathit{in}}$ is the compressive inelastic strain. The resulting stress–strain relationship was converted into the compressive inelastic strain required by the ABAQUS CDP model.

The tensile response follows the strain-based softening law adopted from Ref. [21], as expressed in Equations (6)–(8). Because this formulation is not regularized by element characteristic length, mesh-objective post-cracking behavior is not claimed. A fracture energy-based crack-opening formulation would provide a more rigorous regularization of localized cracking.

σ_t = E_cε_t,   0 ≤ ε_t ≤ ε_t0

(6)

$${\mathit{\sigma }}_{\mathit{t}}={\mathit{f}}_{\mathit{t}}{({\mathit{\epsilon }}_{\mathit{t}0}/{\mathit{\epsilon }}_{\mathit{t}})}^{0.4},{\mathit{\epsilon }}_{\mathit{t}}\mathit{>}{\mathit{\epsilon }}_{\mathit{t}0}$$

(7)

ε_t0 = f_t/E_c

(8)

where σ_t and ε_t are the uniaxial tensile stress and total tensile strain, respectively; f_t is the concrete tensile strength; and ε_t0 is the strain at the peak tensile stress. The tensile stress–strain relationship was further converted into the cracking strain and tensile damage variables required by the CDP model.

2.4.2. Reinforcement

The longitudinal steel and GFRP bars were modeled using T3D2 truss elements and coupled to the concrete with an Embedded Region constraint under a perfect bond assumption. Steel was represented by a bilinear elastoplastic law with E_s = 200 GPa. GFRP was represented as linear elastic with E_f = 40 GPa. Table 4 retains the experimentally reported GFRP tensile strength of 1200 MPa [1]. A reference strength of 900 MPa and the corresponding strain of 2.25% were used only for post-processing assessment; they were not implemented as an ABAQUS rupture criterion, damage initiation rule, stiffness degradation rule, or element deletion criterion. The model does not represent reinforcement concrete slip, material-specific surface and anchorage behavior, or their effects on local crack opening and stiffness.

2.4.3. Concrete Beam Model

The geometry and parameters of the finite element beam models are summarized in Table 6. The supports were modeled as three-dimensional rectangular blocks. For B1, each support measured 80 mm× 50 mm× 20 mm. The transverse support width was scaled with a beam width to 160 mm for B2 and 320 mm for B4, while the 50 mm longitudinal contact length was unchanged. Loading was applied through a cylindrical steel rod with an unchanged diameter across B1, B2, and B4. The loading rod/beam and support/beam interfaces used surface-to-surface contact with finite sliding and isotropic Coulomb friction (μ = 0.3). Reinforcement concrete interaction used Embedded Region under the perfect bond assumption.

Table 6. Finite element modeling parameters for the beams.

Model ID	Variable Type	h/mm	b/mm	d/mm	c/mm	a/mm *	a/d	As/mm2
B-S-Grid-8	Steel	110	80	85	20	425	5.0	78.2 (1φ10)
B-S-Grid-10	Steel	110	80	85	20	425	5.0	78.2 (1φ10)
B-S-Grid-12	Steel	110	80	85	20	425	5.0	78.2 (1φ10)
B-G-Grid-8	GFRP	110	80	85	20	425	5.0	78.2 (1φ10)
B-G-Grid-10	GFRP	110	80	85	20	425	5.0	78.2 (1φ10)
B-G-Grid-12	GFRP	110	80	85	20	425	5.0	78.2 (1φ10)
B1-S-A5	Steel	110	80	85	20	425	5.0	78.2 (1φ10)
B2-S-A5	Steel	220	160	170	40	850	5.0	312.8 (4φ10)
B4-S-A5	Steel	440	320	340	80	1700	5.0	1251.2 (4φ20)
B2-S-A1.5	Steel	220	160	170	40	255	1.5	312.8 (4φ10)
B2-S-A3	Steel	220	160	170	40	510	3	312.8 (4φ10)
B2-S-A4	Steel	220	160	170	40	680	4	312.8 (4φ10)
B1-G-A5	GFRP	110	80	85	20	425	5.0	78.2 (1φ10)
B2-G-A5	GFRP	220	160	170	40	850	5.0	312.8 (4φ10)
B4-G-A5	GFRP	440	320	340	80	1700	5.0	1251.2 (4φ20)

Note: In B-S-Grid-10, S denotes steel reinforcement and Grid-10 denotes a mesh size of 10 mm. In B2-G-A5, B2 indicates a twofold scaling of the beam dimensions, G denotes GFRP reinforcement, and A5 denotes a shear span-to-depth ratio of 5. *: a denotes the clear shear span from the inner face of the 50 mm wide support contact region to the loading line; for the baseline beam, a = (900 − 50)/2 = 425 mm.

The boundary conditions reproduced a simply supported beam. At the left support, U1, U2, and U3 were constrained while UR1 was released. At the right support, U1 and U2 were constrained and U3 and UR1 were released. Loading was displacement-controlled at the loading rod reference point. A Static, General step with geometric nonlinearity enabled was used, with total step time 1 and initial increment 0.001.

3. Results and Discussion

3.1. Validation of the Finite Element Model

The 8, 10, and 12 mm baseline comparisons were used to select a practical global mesh for load–displacement and peak load response, as shown in Figure 3. As shown in Table 7, a 10 mm global mesh was then held fixed for consistency. The results indicate that the selected mesh size provides good agreement.

Figure 3. Comparison of experimental and numerical load–displacement curves for different mesh sizes: (a) steel-reinforced beam; (b) GFRP-reinforced beam.

Table 7. Comparison of peak loads for models with different mesh sizes.

Model ID	Load/kN (Error)
LC60-Steel	11.8
B-S-Grid-8	13.56 (14.92%)
B-S-Grid-10	12.16 (3.05%)
B-S-Grid-12	12.00 (1.70%)
LC60-GFRP	10.18
B-G-Grid-8	8.48 (16.70%)
B-G-Grid-10	10.38 (1.96%)
B-G-Grid-12	9.88 (2.95%)

Figure 4 compares the experimental failure patterns of the LC60-S and LC60-G beams with the SDEG contour lines of the corresponding 10 mm reference model. SDEG is used solely as a scalar indicator of stiffness degradation to approximately assess the similarity to the failure characteristics of the experimental beams. Overall, the damage characteristics of the two show a high degree of similarity.

Figure 4. Comparison of experimental failure patterns and finite element SDEG contours: (a) experimental failure of LC60-Steel and LC60-GFRP; (b) SDEG contour of LC60-S; (c) SDEG contour of LC60-GFRP.

3.2. Member Size Effect

The B1, B2, and B4 models form a controlled member size series rather than a strictly geometrically similar size effect series. The longitudinal reinforcement ratio was maintained at approximately ρ_l ≈ 1.15%. Table 6 uses 1ϕ10, 4ϕ10, 4ϕ20 for both reinforcement types, with total areas of 78.2, 312.8, and 1251.2 mm². Departures from strict similitude include the discrete bar arrangement, unchanged 50 mm longitudinal support contact length, unchanged loading rod diameter, and fixed 10 mm mesh.

3.2.1. Cracking and Shear Strengths

For symmetric three-point loading, the shear force in each shear span was calculated as V = P/2. The cracking and ultimate shear forces and their nominal strengths were evaluated using Equations (9) and (10). The numerical first cracking loads were P_cr = 3.982 kN for LC60-S and 3.879 kN for LC60-G, compared with experimental values of 2.90 and 2.50 kN [1]. The finite element values are higher by 37.3% and 55.2%, respectively; therefore, the models reproduce peak load response more accurately than the exact onset of first cracking.

v_cr = V_cr/(bd)

(9)

v_u = V_u/(bd)

(10)

where b is the beam width and d is the effective depth, both in mm; V_cr and V_u are in N; and v_cr and v_u are in MPa. The calculated results for beams of different sizes are given in Table 8.

Table 8. Cracking and ultimate shear strengths of beams with different sizes.

Model	Vcr/N	vcr/MPa	Vu/N	vu/MPa
B1-S-A5.0	1991	0.29	6083	0.895
B2-S-A5.0	9392	0.35	25,863.5	0.951
B4-S-A5.0	35,455	0.33	101,972.5	0.937

Absolute cracking and ultimate shear forces increased with the member size. Increasing the effective depth from 85 to 340 mm increased the ultimate shear force from 6083 to 101,972.5 N, while the nominal ultimate shear strengths were approximately 0.895, 0.951, and 0.937 MPa. The nominal values do not show a pronounced classical monotonic decrease over the investigated range; the results are therefore described as member size-dependent responses under the adopted boundary conditions.

3.2.2. Load–Displacement Curves and Normalized Response

The load–displacement curves and normalized curves for beams of different sizes are shown in Figure 5a and Figure 5b, respectively. As the member size increased, the peak load rose from 12.166 to 203.945 kN and the displacement at the peak load increased from 5.4707 to 18.3148 mm, indicating substantial increases in both absolute shear capacity and ultimate deformation capacity. After normalization, the curves for the different sizes were nearly coincident before cracking and throughout the pre-peak nonlinear stage, suggesting that changing the member size did not appreciably alter the initial stiffness degradation pattern or the overall crack development process.

Figure 5. (a) Load–displacement curves for beams of different sizes; (b) normalized load–displacement curves.

3.2.3. Structural Stiffness and Global Energy Dissipation

The initial and post-cracking stiffnesses of beams with different sizes are compared in Figure 6. As the beam dimensions increased, the initial stiffness K_i increased from 5.849 to 27.869 kN/mm and the post-cracking stiffness K_cr increased from 1.582 to 7.554 kN/mm. Relative to B1-S-A5.0, K_i increased by 121.1% and 376.5% for B2-S-A5.0 and B4-S-A5.0, respectively, while K_cr increased by 122.8% and 377.5%. These results show that increasing the member size substantially enhances the overall beam stiffness, primarily because the larger cross-section increases the sectional moment of inertia and therefore improves resistance to deformation during loading.

Figure 6. Comparison of stiffness for beams of different sizes.

The area W under the structural load–displacement curve was integrated from loading onset to a common descending branch endpoint at approximately 0.90 P_u. The global structural energy dissipation index G_s was normalized by the reference sectional area A_ref = bd, as defined in Equations (11)–(13). This index is a whole-member structural response measure and not an intrinsic Mode-I fracture energy parameter.

$$\mathit{W}={\displaystyle \mathit{\int }\mathit{P}}\mathit{d}\mathit{\delta }$$

(11)

A_ref = bd

(12)

G_s = W/A_ref = W/(bd)

(13)

The global structural energy dissipation indices of beams with different sizes are shown in Figure 7. Their absolute magnitudes depend on the common post-peak integration endpoint, and no endpoint sensitivity analysis was performed. The localization and post-peak energy trends are consistent with quasi-brittle damage processes, but the CDP formulation combines plasticity and stiffness degradation and the nominal strength does not show a classical monotonic size effect reduction. The simulations therefore do not establish an exclusive fracture mechanics description.

Figure 7. Global structural energy dissipation index for beams of different sizes.

3.2.4. Final Failure Characteristics

The SDEG contours at the ultimate load are shown in Figure 8. They indicate numerical damage localization rather than measured physical crack width. A dominant diagonal damage localization band developed between the loading region and the support in each model, while the extent of the high-damage region increased from B1 to B4.

Figure 8. Final SDEG failure patterns for beams of different sizes: (a) B1-S-A5; (b) B2-S-A5; (c) B4-S-A5.

3.3. Effect of Shear Span-to-Depth Ratio

3.3.1. Cracking and Shear Strengths

The calculated V_cr, v_cr, V_u, and v_u values for models with different shear span-to-depth ratios are presented in Figure 9. As a/d increased from 1.5 to 5.0, both the cracking shear force and nominal cracking shear strength decreased. Taking a/d = 1.5 as the reference, v_cr decreased by 41.0%, 58.6%, and 66.2% for the other three models. The ultimate shear force and nominal ultimate shear strength followed a similar trend, with v_u decreasing by 57.5%, 72.3%, and 75.9%, respectively. These results demonstrate that increasing a/d markedly reduces both the cracking and ultimate shear performance of the beams, with a more pronounced effect on the ultimate shear strength.

Figure 9. Comparison of beams with different shear span-to-depth ratios: (a) shear force; (b) shear strength.

3.3.2. Load–Displacement Curves and Normalized Response

The load–displacement curves and normalized curves for beams with different shear span-to-depth ratios are shown in Figure 10a and Figure 10b, respectively. As a/d increased from 1.5 to 5.0, the displacement at the peak load increased from 2.59 to 10.26 mm, whereas the ultimate load decreased from approximately 214.56 to approximately 51.72 kN. After normalization, the curves for the different a/d values showed a high degree of similarity, with comparable trends through cracking and subsequent nonlinear development. This indicates that the shear span-to-depth ratio primarily governs the load level and deformation scale, while having a relatively limited effect on the normalized evolution of deformation.

Figure 10. (a) Load–displacement curves for beams with different shear span-to-depth ratios; (b) normalized load–displacement curves.

3.3.3. Structural Stiffness and Global Energy Dissipation

The structural stiffnesses and global energy dissipation indices of beams with different shear span-to-depth ratios are compared in Figure 11. As a/d increased from 1.5 to 5.0, both the initial and post-cracking stiffness decreased. The global energy index was evaluated to the same approximate 0.90 P_u endpoint for all models and is not an intrinsic material fracture energy.

Figure 11. Comparison of beams with different shear span-to-depth ratios: (a) stiffness; (b) global structural energy dissipation index.

3.3.4. Final Failure Characteristics

The SDEG contours at final failure are shown in Figure 12. The a/d = 1.5 model is interpreted as an arch action/deep beam regime, whereas the larger a/d models progressively approach beam action [22]. The region near a/d ≈ 3 is discussed as a numerical transition region rather than a universal mechanistic boundary.

Figure 12. Final SDEG failure patterns for beams with different shear span-to-depth ratios: (a) B2-S-A1.5; (b) B2-S-A3; (c) B2-S-A4; (d) B2-S-A5.

3.4. Effect of Longitudinal Reinforcement Type

3.4.1. Cracking and Shear Strengths

The cracking shear force V_cr and ultimate shear force V_u were extracted from the finite element results for the steel- and GFRP-reinforced beams, and the nominal cracking strength v_cr and nominal ultimate shear strength v_u were used to compare the size-dependent behavior of beams with different longitudinal reinforcement. As shown in Figure 13, both V_cr and V_u increased substantially with the member size for the two reinforcement types, whereas the changes in v_cr and v_u were much smaller than the corresponding increases in shear force. Steel- and GFRP-reinforced beams therefore exhibited broadly similar size-dependent trends.

Figure 13. Comparison between steel- and GFRP-reinforced beams: (a) shear force; (b) nominal shear strength.

At a given member size, V_cr of the GFRP-reinforced beams was 2.59%, 11.0%, and 1.14% lower than that of the corresponding steel-reinforced beams, while v_cr was unchanged for B1 and lower by 11.43% and 3.03% for B2 and B4, respectively. Except for the intermediate-size beam, the differences between the two reinforcement types at cracking were small and did not vary monotonically with size. This suggests that reducing the longitudinal reinforcement stiffness has a limited effect on the initial formation of diagonal cracks, and that cracking shear resistance remains governed primarily by the tensile properties of the concrete and the cross-sectional geometry. By contrast, the effect of the reinforcement type was more evident at the ultimate stage. Relative to the corresponding steel-reinforced beams, V_u of the GFRP-reinforced beams was lower by 14.62%, 11.41%, and 6.92%, while v_u was lower by 14.61%, 11.58%, and 7.45%, respectively. As the member size increased from B1 to B4, the relative difference in ultimate shear performance between the two reinforcement types progressively decreased. Lower longitudinal reinforcement stiffness therefore weakens shear transfer after diagonal cracking, but its relative influence on ultimate shear performance diminishes with increasing member size.

3.4.2. Load–Displacement Curves and Normalized Response

The load–displacement and normalized curves for steel- and GFRP-reinforced beams are shown in Figure 14. Under the intentional perfect bond assumption, the computed differences are associated primarily with the imposed reinforcement elastic and axial stiffness. The model suppresses slip and cannot quantify material-specific bond or anchorage effects on local cracking and stiffness.

Figure 14. Load–displacement behavior of beams at different sizes: (a) load–displacement curves; (b) normalized load–displacement curves.

After normalization, the curves for both reinforcement types showed broadly similar model series trends. The GFRP-reinforced models had lower peak load and larger displacement at peak load. These deterministic differences are interpreted within the adopted perfect bond model and are not evidence of an independently isolated bond–slip mechanism.

3.4.3. Structural Stiffness and Global Energy Dissipation

The numerical stiffnesses and global structural energy dissipation indices are shown in Figure 15. At each member size, the GFRP-reinforced beams had lower initial and post-cracking stiffness than the steel-reinforced beams. The global energy index was evaluated to the common post-peak endpoint and should not be interpreted as intrinsic material fracture energy.

Figure 15. Effect of longitudinal reinforcement type: (a) stiffness comparison; (b) global structural energy-dissipation comparison.

3.4.4. Final Failure Characteristics

The SDEG contours at final failure for the steel- and GFRP-reinforced beams are shown in Figure 16. Dominant diagonal damage localization in the shear span supports classification of the computed failure as shear-dominated. The maximum GFRP strain remained well below the reference tensile strain of 2.25%, so GFRP rupture did not govern. For the steel-reinforced beams, the available evidence does not exclude local yielding; complete absence of flexural action is therefore not claimed. Because perfect bond suppresses slip and SDEG does not measure physical crack opening, the model does not directly prove a bond–slip or physical crack transfer mechanism.

Figure 16. Final SDEG failure patterns: (a) B1-S-A5; (b) B2-S-A5; (c) B4-S-A5; (d) B1-G-A5; (e) B2-G-A5; (f) B4-G-A5.

4. Mechanisms and Design Context Comparison

4.1. Member Size Effect

A classical shear size effect is commonly characterized by decreasing nominal strength with increasing effective depth. Here, the member size is evaluated through the numerical B1-B4 series while recognizing departures from strict geometric similitude and the fixed modeling assumptions.

Increasing the effective depth from 85 to 340 mm increased the absolute shear capacity, while the nominal ultimate shear strengths remained approximately 0.895, 0.951, and 0.937 MPa. The longitudinal reinforcement ratio remained approximately ρ_l ≈ 1.15%, but the local longitudinal bearing dimensions, loading rod diameter, and absolute mesh size were not scaled.

The B1–B4 results are therefore interpreted as member size-dependent numerical responses under the adopted boundary conditions. No classical monotonic nominal strength size effect reduction is claimed, and local bearing effects cannot be completely separated from the computed response.

4.2. Effect of Shear Span-to-Depth Ratio

For a fixed member size and reinforcement definition, Equation (14) is retained as an empirical descriptor of the computed trend over 1.5 ≤ a/d ≤ 5. It is not presented as a universal mechanistic or design relationship.

$${\mathit{v}}_{\mathit{u}}(\mathit{\lambda })={\mathit{v}}_{\mathit{u},{\mathit{\lambda }}_0}{(\mathit{\lambda }/{\mathit{\lambda }}_0)}^{\mathit{-}\mathit{n}}$$

(14)

The a/d = 1.5 model lies in an arch action/deep beam regime, whereas the larger a/d models progressively approach beam action. The region near a/d ≈ 3 is interpreted as a numerical transition region. Equation (15) and its coefficient of determination in Equation (16) describe only the sampled numerical interval.

v_u = 1.6716(λ/3)^−1.22

(15)

The coefficient of determination for the fitted relationship is given in Equation (16):

R² = 0.998

(16)

The finite element results and the empirical curve are shown in Figure 17. As a/d increased from 1.5 to 5.0, the ultimate shear force and nominal ultimate shear strength decreased. The relation summarizes the computed points, including the deep beam case, and should not be extrapolated beyond the investigated range.

Figure 17. Comparison between fitted and finite element results for the shear span-to-depth ratio effect.

4.3. Effect of Longitudinal Reinforcement Type and Axial Stiffness

To compare the two reinforcement definitions, the dimensionless longitudinal reinforcement stiffness η_l and the longitudinal reinforcement ratio ρ_l were defined by Equations (17) and (18), respectively:

η_l = ρ_lE_l/E_c

(17)

ρ_l = A_s/(bd)

(18)

where ρ_l is the longitudinal reinforcement ratio, E_l is the longitudinal reinforcement modulus, E_c is the concrete modulus, A_s is the reinforcement area, and b and d are the beam width and effective depth. The B1, B2, and B4 models use A_s = 78.2, 312.8, and 1251.2 mm², respectively, with ρ_l ≈ 1.15%. With E_c = 20.5 GPa, η_l,steel ≈ 0.1122 and η_l,GFRP ≈ 0.0224 at each member size.

For a fixed member size and shear span-to-depth ratio, Equation (19) provides a two-point description between the steel and GFRP configurations:

$${\mathit{v}}_{\mathit{u}}({\mathit{\eta }}_{\mathit{l}})={\mathit{v}}_{\mathit{u},{\mathit{\eta }}_0}{({\mathit{\eta }}_{\mathit{l}}/{\mathit{\eta }}_0)}^{\mathit{p}}$$

(19)

The reference stiffness is the value for the steel-reinforced beam of the same size. Because η_l,steel/η_l,GFRP = 5, the p = 0.098, 0.075, and 0.045 remain unchanged. They are descriptive two-point stiffness sensitivity indices rather than regression constants, material constants, or a universal scaling law.

The model series trend is shown in Figure 18. The longitudinal reinforcement ratio was maintained at approximately 1.15% across B1, B2, and B4, so the decreasing steel–GFRP difference is not attributed to a change in the reinforcement ratio. It may reflect the member size together with constitutive assumptions, fixed mesh size, perfect bond, discrete reinforcement arrangement, and local loading/support geometry. The relationship is restricted to the investigated LC60 system, 1.5 ≤ a/d ≤ 5, effective depths of d = 85–340 mm, and the modeled reinforcement configurations [20].

Figure 18. Fitted relationship for the effect of longitudinal reinforcement stiffness.

4.4. Equation-Level Code Comparison

For the steel-reinforced LC60 beams without transverse shear reinforcement, nominal concrete shear resistance was evaluated using ACI CODE-318-25 [23]. The governing equation was

$${\mathit{V}}_{\mathit{c}}=[0.66{\mathit{\lambda }}_{\mathit{s}}\mathit{\lambda }{({\mathit{\rho }}_{\mathit{w}})}^{1/3}\sqrt{{\mathit{f}}_{\mathit{c}}^{\prime }}\mathit{+}{\mathit{N}}_{\mathit{u}}/(6{\mathit{A}}_{\mathit{g}})]{\mathit{b}}_{\mathit{w}}\mathit{d}$$

(20)

The reinforcement ratio and size effect factor were evaluated as

ρ_w = A_s/(b_wd)

(21)

$${\mathit{\lambda }}_{\mathit{s}}=\mathit{min}[1.0,\sqrt{2/(1\mathit{+}0.004\mathit{d})}]$$

(22)

The LC60 calculation inputs were ${\mathit{f}}_{\mathit{c}}^{\prime }=71.5\mathrm{MPa},\mathit{\lambda }=0.75,{\mathit{N}}_{\mathit{u}}=0$. No axial load was applied, so the axial load contribution did not affect the calculated resistance. The concrete strength term was limited according to ACI CODE-318-25:

$$\sqrt{{\mathit{f}}_{\mathit{c}}^{\prime }}\mathit{\le }8.3\mathit{ }{\mathrm{MPa}}^{1/2}$$

(23)

The prescribed lower-bound and maximum limits were checked and did not govern. Nominal equation-level values were used without a strength reduction factor. For symmetric three-point loading, P_ACI = 2V_c. These unnumbered code equations preserve the manuscript’s existing Equations (1)–(19) sequence.

Table 9 provides the baseline equation-level comparison with ACI CODE-318-25, EN 1992-1-1:2023 [24], and fib Model Code 2020 [25]. Table 10 extends the ACI comparison to the steel-reinforced LC60 series for which the sectional provision is applicable.

Table 9. Comparison of experimental and numerical peak loads with equation-level code predictions.

Beam	Reinf.	Code	Pexp (kN)	PFE (kN)	Pcode (kN)	Pcode/Pexp	Pcode/PFE	Error vs. Exp. (%)	Error vs. FE (%)	Remarks
LC60-S	Steel	ACI CODE-318-25	11.800	12.166	12.612	1.069	1.037	+6.9	+3.7	LWC factor λ = 0.75
LC60-S	Steel	EN 1992-1-1:2023	11.800	12.166	18.989	1.609	1.561	+60.9	+56.1	ddg = 16 mm
LC60-S	Steel	fib Model Code 2020	11.800	12.166	17.999	1.525	1.479	+52.5	+47.9	/
LC60-G	GFRP	ACI CODE-440.11-22	10.180	10.380	N/A	N/A	N/A	N/A	N/A	Not applicable
LC60-G	GFRP	EN 1992-1-1:2023 Annex R	10.180	10.380	N/A	N/A	N/A	N/A	N/A	Not applicable
LC60-G	GFRP	fib Model Code 2020	10.180	10.380	N/A	N/A	N/A	N/A	N/A	Not applicable

Note: For symmetric three-point loading, P_code = 2V_code. Error = (prediction − reference)/reference × 100%. Unit material factors were used for equation-level comparison; the values are not reliability-calibrated design strengths. GFRP-LC60 provisions outside the stated scope are reported as N/A.

Table 10. Extended comparison of FE peak loads with ACI CODE-318-25 predictions for steel-reinforced LC60 beams.

Model	d (mm)	a/d	PFE (kN)	PACI (kN)	PACI/PFE	Error vs. FE (%)	ACI Applicability/Remarks
B1-S-A5	85	5.0	12.166	12.612	1.037	+3.7	Sectional one-way shear
B2-S-A5	170	5.0	51.727	50.448	0.975	−2.5	Sectional one-way shear
B4-S-A5	340	5.0	203.945	185.766	0.911	−8.9	Sectional one-way shear
B2-S-A4	170	4.0	59.44	50.448	0.849	−15.1	Sectional shear; approximate FE value
B2-S-A3	170	3.0	91.14	N/A	N/A	N/A	Sectional shear; increasing arch action
B2-S-A1.5	170	1.5	214.56	N/A	N/A	N/A	Deep beam; ACI Chapter 23 STM required

Note: The ACI equation-level values use ACI code for non-prestressed members without minimum shear reinforcement. The LC60 concrete strength is 71.5 MPa, λ = 0.75, and the longitudinal reinforcement ratio is approximately 1.15%.

For the a/d = 5 steel-reinforced series, the ACI predictions remain within approximately 9% of the finite element peak loads over d = 85–340 mm. As a/d decreases, P_ACI/P_FE decreases and the sectional prediction becomes increasingly conservative relative to the computed peak load. The a/d = 1.5 model lies in the ACI deep beam domain; the conventional sectional equation is not applied and the strut-and-tie model therein is required. The comparison is a design context reference, not a validated design rule or reliability-calibrated safety margin. Previous experiments on FRP-reinforced concrete beams without transverse reinforcement reported depth-dependent reductions in nominal shear strength for steel-, GFRP-, CFRP-, and BFRP-reinforced series [9,10]. Tests on BFRP-reinforced short beams also showed that the effective depth, reinforcement ratio, and shear span-to-depth ratio affected the measured response [15]. These studies provide qualitative context.

5. Limitation

The study is limited by one specimen per experimental configuration and direct model checking against only two baseline beams; the larger-size and other-a/d models therefore remain numerical extrapolations. The Embedded Region constraint imposes perfect bond and does not represent reinforcement concrete slip, material-specific interface behavior, or anchorage effects. The LC60 concrete was modeled as a homogeneous continuum without explicit aggregate particles or interfacial transition zones. A fixed 10 mm mesh and strain-based tensile softening were used, so damage localization and the structural energy index may remain mesh-dependent. The longitudinal reinforcement ratio was maintained at approximately ρ_l ≈ 1.15% across B1, B2, and B4 using 1ϕ10, 4ϕ10, 4ϕ20, although the discrete arrangement is not a strict geometric mapping. Support transverse width was scaled with beam width, whereas the 50 mm longitudinal contact length and loading rod diameter were fixed. Equation (19) is a two-point descriptive relation, and the structural energy index was integrated to approximately 0.90 P_u. No classical monotonic decrease in nominal shear strength was observed over d = 85–340 mm. These limitations restrict the conclusions to the investigated LC60 system, reinforcement configurations, boundary conditions, and parameter ranges.

6. Conclusions

The main findings of this paper are as follows:

(1) A three-dimensional nonlinear finite element model was checked against the load–displacement response, peak load, and damage pattern of two previously reported LC60 baseline beams. The remaining configurations are numerical parametric predictions within the adopted model assumptions.

(2) Increasing the effective depth from 85 to 340 mm substantially increased the absolute shear capacity, while the nominal ultimate shear strengths remained approximately 0.895, 0.951, and 0.937 MPa. The results show a member size-dependent response but no pronounced classical monotonic reduction in the nominal shear strength over the investigated range.

(3) Increasing the shear span-to-depth ratio from 1.5 to 5.0 reduced the computed shear response and shifted the load transfer mechanism from an arch action/deep beam regime toward progressively stronger beam action. The power law relation is an empirical descriptor of the sampled numerical interval.

(4) At an approximately constant 1.15% longitudinal reinforcement ratio, GFRP-reinforced models exhibited lower stiffness and ultimate shear strength than the steel-reinforced models. The decreasing difference with the member size is a numerical model series trend associated with the reinforcement type and axial stiffness and is not presented as a universal scaling law.

(5) ACI CODE-318-25 equation-level predictions for the a/d = 5 steel-reinforced series remained within approximately 9% of the finite element peak loads over d = 85–40 mm. The sectional prediction became increasingly conservative as a/d decreased, and the a/d = 1.5 model requires deep beam strut-and-tie treatment. All findings are restricted to the investigated LC60 system, model assumptions, boundary conditions, and parameter ranges.