Generalized Shear Correction Factor for Non-Homogeneous Beam Cross-Sections with an Embedded Steel Core

1  Introduction

The growing demand for environmentally responsible construction solutions continues to reshape modern building technologies, stimulating the development and implementation of lightweight, low-carbon, and resource-efficient material systems [1–3]. Among these, expanded perlite has gained a particularly stable position as a readily available mineral aggregate enabling lightweight cementitious composites, thermal insulation components, and fire-related protection layers [4–6]. Recent studies underline that perlite-based mixtures can be engineered to provide a favorable balance between sustainability and performance, although their achievable mechanical response depends strongly on the binder system, the processing route, and the adopted composite architecture [7–9]. In parallel, perlite has been incorporated into polymer-based and hybrid porous composites to enhance mechanical functionality or thermostability in multi-phase systems, demonstrating the broader potential of porous architectures beyond conventional concrete technology [10–12].

In addition to mechanical and thermal performance, recent studies have increasingly reported environmental impact indicators of perlite-based materials, including embodied energy, carbon footprint, and recyclability potential [13–18]. These analyses highlight the relevance of lightweight porous aggregates in low-carbon construction strategies, particularly when combined with durability-oriented design approaches. The structural applicability of perlite-based composites must therefore be assessed not only in terms of mechanical response but also in relation to sustainability-driven performance metrics.

While the use of perlite is frequently motivated by weight reduction and thermal efficiency, the structural adoption of porous eco-materials requires reliable prediction of deformation mechanisms, especially under combined bending and shear [19–22]. The key difficulty is that porous matrices are inherently heterogeneous: their microstructure contains nonuniform pore sizes, irregular void geometries, density gradients, and local necking-like effects, which collectively influence mechanical properties and stress redistribution [23–25]. Such variability directly affects the effective stiffness fields that govern global response and, in particular, the transverse shear mechanism that may dominate serviceability limits in short or moderately slender members [26,27]. For these reasons, porosity characterization and microstructure-informed modeling are increasingly recognized as indispensable components of modern durability-oriented structural assessment [23,28–30].

The microstructure of expanded perlite is characterized by irregular, often closed-cell porous architectures formed during rapid thermal expansion. The resulting morphology includes thin-walled shells, intergranular voids, and density gradients, which strongly influence stiffness, strength, and local stress redistribution mechanisms. Recent microstructure-resolved investigations emphasize that pore topology and phase connectivity play a decisive role in the effective elastic response of perlite-based composites.

Transverse shear deformation is typically represented using reduced-order beam theories, most notably the Timoshenko formulation, which introduces an effective shear stiffness through a shear correction factor [31–33]. Although such representations are well established in classical solid mechanics and plate/beam theory, their standard correction factors assume homogeneous continua and do not account for stiffness fluctuations inside porous matrices or multi-phase cross-sections [34,35]. This is critical because even moderate errors in shear stiffness may propagate into inaccurate deflection predictions, biased stiffness identification, and unreliable assessment of structural condition under operational loads [36,37]. From the viewpoint of Structural Durability & Health Monitoring, this is not a cosmetic modeling detail: monitoring frameworks often interpret changes in global stiffness or displacement response as indicators of degradation, damage evolution, boundary-condition shifts, or long-term environmental impacts [38–40]. Therefore, shear-consistent reduced-order models are necessary to avoid misclassification of response changes in durability-related decision processes [41–44].

The modeling challenge becomes even more pronounced for hybrid beams, in which a porous matrix is combined with a stiff embedded reinforcement. One promising configuration is a beam composed of perlite-based matrix material strengthened by an embedded rolled steel I-section. Such members may offer an attractive compromise between sustainability (lightweight porous matrix) and robustness (steel load-carrying core), with potential relevance for prefabricated components and multi-functional elements [1,39–41]. At the same time, the steel insert introduces a strong stiffness contrast and a sharply non-homogeneous cross-section, leading to shear stress redistribution, energy localization near interfaces, and potential sensitivity to environmental actions such as moisture and temperature variations [45–50]. In this context, thermo-elastic effects and nonuniform stiffness fields may additionally influence effective shear correction factors, especially when material parameters depend on exposure conditions or functional gradients [35,51]. As a result, adopting classical homogeneous correction factors may become questionable for such hybrid cross-sections, particularly in serviceability-driven use cases.

Perlite-based composites have also been explored in structural load-bearing elements such as lightweight masonry blocks, prefabricated panels, and reinforced concrete members subjected to thermal and mechanical actions. Although primarily adopted for weight reduction and insulation performance, these systems demonstrate that perlite-containing materials can participate in structural load transfer when properly reinforced. This motivates the present investigation of hybrid perlite–steel beam configurations.

Existing research has addressed parts of this problem, but the available approaches remain fragmented across different communities and scales. In civil engineering practice, effective material descriptions for perlite-based lightweight concretes are often derived from empirical or semi-empirical relationships and homogenized stiffness estimates [19–22,52]. In the broader mechanics community, mathematical homogenization frameworks and representative volume concepts have been developed to connect microstructural features to effective stiffness properties [53,54]. Complementarily, microstructure-resolved numerical models have been used to capture local stress concentrations and quantify effective responses for porous elastic systems with random inclusion distributions or reconstructed microgeometries [27,55]. However, a direct translation of these high-resolution approaches to the shear correction factor required by Timoshenko-type beam models is not always straightforward, especially when the cross-section contains multiple phases with sharp stiffness discontinuities.

In recent years, data-driven strategies have also emerged, including machine-learning and deep-learning methods for parameter estimation, optimization, and predictive assessment in construction-related materials and geotechnical applications [39,56]. For durability and monitoring applications, digital twin concepts provide an additional layer of integration, enabling model updating and performance forecasting under changing environmental conditions [40,57]. Yet, even within digital twin workflows, reduced-order models remain the computational backbone for repeated evaluations, and their reliability hinges on consistent identification of effective stiffness parameters, including transverse shear stiffness. This highlights the need for a robust, transparent, and computationally efficient framework that links heterogeneous cross-sectional stiffness fields to an equivalent shear correction factor suitable for monitoring-oriented beam models.

To address these limitations, the present work proposes an energy-consistent analytical–numerical methodology for evaluating the shear correction factor ks in non-homogeneous beam cross-sections consisting of a porous perlite-based matrix and an embedded steel I-core. The approach extends the classical strain-energy equivalence idea by explicitly incorporating a spatially varying shear modulus field and evaluating the heterogeneous shear-energy integral using a pixel/voxel discretization of the cross-section [58–60]. This makes it possible to quantify how stiffness contrast, porous heterogeneity, and phase distribution govern shear strain energy localization and the resulting effective transverse shear stiffness. The framework is compatible with multi-phase and microstructure-informed material descriptions and can be used as a practical component of stiffness identification and model updating procedures relevant to Structural Durability & Health Monitoring.

Finally, it is worth noting that perlite-based composites remain a particularly compelling case study due to their broad availability and established use in both cementitious and polymeric systems, as well as their role in fire-related performance contexts [60–65]. However, the same methodology is directly transferable to other porous eco-materials and hybrid structural members, including configurations with different porous aggregates or alternative reinforcement topologies [34,66–70]. By providing an explicit connection between heterogeneous shear strain energy and the macroscopic shear correction factor, the present work supports more reliable serviceability prediction and monitoring interpretation for lightweight hybrid beams.

2  Materials and Methods

A prismatic beam of length L, subjected to a transverse shear force V, is considered. The cross-section A consists of a non-homogeneous matrix material (perlite-based composite or cementitious matrix) with an embedded steel core in the form of a rolled I-section. The cross-sectional geometry is constant along the beam axis, while the material properties vary spatially within the section due to the stiffness contrast between the matrix and steel, and potentially due to matrix heterogeneity.

Within the framework of Timoshenko beam theory, the transverse shear strain is expressed as

γxz(y,z)=θ(x)−∂w(x)∂x,(1)

where w(x) denotes the transverse displacement of the beam axis and θ(x) is the rotation of the cross-section. The effective transverse shear stiffness is written as

Kxz=ksGrefA,(2)

where ks is the shear correction factor and Gref is a reference shear modulus introduced to maintain a Timoshenko-type representation.

In this work, the reference shear modulus is defined as the area-averaged modulus of the analyzed cross-section,

Gref=1A∫AG(y,z)dA,(2a)

where G(y,z) is the local shear modulus field. This definition is used consistently throughout the workflow and in the reported numerical values. Consequently, Gref is configuration-dependent: it increases in hybrid sections due to the contribution of the high-modulus steel region and changes in heterogeneous cases due to the spatial variability of the matrix modulus.

Although the present study adopts the area-averaged shear modulus as the reference, alternative normalization choices would rescale the numerical value of ks but would not alter the physical shear stiffness (GA)eff=ksGrefA. Therefore, the physically meaningful quantity remains the effective shear stiffness, while ks depends on the adopted normalization convention. For the actual shear stress distribution τxz(y,z), the transverse shear strain energy stored in the cross-section is given by

Ureal=12∫Aτxz2(y,z)G(y,z)dA,(3)

where G(y,z) denotes the local shear modulus, varying spatially due to heterogeneity.

The shear stress distribution resulting from cross-sectional equilibrium is expressed as

τxz(y,z)=Vϕ(y,z),(4)

where ϕ(y,z) is the normalized shear-stress shape function satisfying

∫Aϕ(y,z)dA=1.(5)

Substitution into Eq. (3) yields

Ureal=V22∫Aϕ2(y,z)G(y,z)dA.(6)

In the Timoshenko formulation, the transverse shear energy is written as

UT=V22ksGrefA.(7)

The shear correction factor ks is identified by enforcing energy equivalence

Ureal=UT.(8)

Using Eqs. (6)–(8), the shear correction factor for a non-homogeneous cross-section becomes

ks=1AGref∫Aϕ2(y,z)G(y,z)dA.(9)

For a homogeneous material G(y,z)=Gref, Eq. (9) reduces to the classical expression.

The function ϕ(y,z) depends only on geometry and equilibrium constraints. In the classical form used for prismatic sections, it may be written as

ϕ(z)=Q(z)Ib(z),(10)

where Q(z) is the first moment of area above the level z, I is the second moment of area about the neutral axis, and b(z) is the local width at height z. In the present study, Eq. (10) is interpreted as a geometry-driven representation and is evaluated numerically for complex hybrid sections (matrix + embedded I-core), where b(z) and Q(z) reflect the combined area distribution of all phases.

The spatial distribution of shear modulus is defined as

G(y,z)={Gm(y,z),(y,z)∈Amatrix,Gs,(y,z)∈Asteel,(11)

where Gs is the steel shear modulus and Gm(y,z) corresponds to the matrix material. For porous matrices such as perlite composites, the modulus may depend on local density:

Gm(y,z)=G0f(ρ(y,z)ρ0),(12)

with f(⋅) calibrated experimentally.

To evaluate Eq. (9), the cross-section is discretized into surface elements (pixels/voxels) of area ΔAi. The integral is approximated by

∫Aϕ2GdA≈∑i=1Nϕi2GiΔAi.(13)

Accordingly, the discrete estimate becomes

ks=1AGref∑i=1Nϕi2/GiΔAi.(14)

The discretization naturally accommodates sharp stiffness jumps (steel vs. matrix) and can include void regions by introducing a material indicator field χi∈{0,1}, where χi=0 corresponds to pores/voids (excluded from A and from the summation).

The reference modulus is introduced to keep Eq. (2) consistent with a Timoshenko-type representation. In this work, Gref is taken as the area-averaged shear modulus over the analyzed cross-section, so that the computed ks directly reflects the deviation caused by heterogeneity and the steel core. Alternative definitions (e.g., area-averaged ⟨G⟩ or maximum phase modulus) can also be adopted, and their influence may be assessed within a sensitivity analysis.

For comparison, the classical shear correction factor for a homogeneous rectangular cross-section is adopted

ksref=56.(15)

The relative deviation due to heterogeneity is expressed as

Δks=ks−ksrefksref.(16)

3  Results

This section presents the numerical identification of the shear correction factor ks for a hybrid beam cross-section composed of a porous matrix and an embedded rolled steel I-section. The investigated configuration is a rectangular beam of dimensions b×h=0.20m×0.30 m with a centrally embedded IPE 200 core (Fig. 1). The adopted dimensions correspond to representative proportions commonly used in medium-span prefabricated beam elements and follow standard sectional geometry definitions for IPE 200 profiles according to European structural steel specifications. The chosen configuration is intended as a realistic structural scenario rather than a laboratory-scale specimen, enabling interpretation within practical design contexts.

Figure 1: Cross-section of the hybrid beam composed of a porous matrix (perlite-based composite) and an embedded rolled steel I-section (IPE 200): geometry, coordinate system, and phase domains Amatrix and Asteel.

The evaluation follows the energy-equivalence formulation (Fig. 2), combining the geometry-driven shear-stress function ϕ(y,z) with the heterogeneous shear modulus field G(y,z) and voxel-based numerical integration (Table 1).

Figure 2: Computational workflow of the proposed energy-consistent identification of the shear correction factor ks for heterogeneous cross-sections: modulus field definition, evaluation of ϕ(y,z), voxel-based energy integral, and calculation of ks and Δks.

The main outputs include the identified ks, its deviation from the classical benchmark ksref=5/6, the localization of heterogeneous shear energy ϕ2/G, and monitoring-oriented implications for deflection-based stiffness interpretation.

The analyzed hybrid cross-section is shown in Fig. 1 and summarized in Table 2. The matrix occupies the full 0.20m×0.30 m rectangular envelope, while the IPE 200 steel section is placed centrally, introducing a sharp stiffness contrast between steel and the porous matrix.

The adopted material parameters and reference modulus Gref are reported in Table 3. For numerical evaluation of the heterogeneous energy integral (Eq. (14)), the cross-section is discretized into pixels of area ΔAi (Table 1), ensuring stable ks values across grid refinements. Notably, because Gref is computed over the whole cross-section, its value reflects the steel–matrix composition and thus directly affects the numerical value of ks reported for each case, while the physical energy localization is governed by the field G(y,z) in Eq. (14).

The spatial distribution of the shear modulus G(y,z) is illustrated in Fig. 3. Two matrix descriptions are considered: (i) a uniform matrix modulus Gm, and (ii) a density-dependent heterogeneous modulus Gm(y,z) reflecting spatial variability typical of porous eco-materials. In both cases, the steel phase retains a constant modulus Gs, resulting in a highly non-homogeneous stiffness field.

Figure 3: Spatial distribution of the shear modulus G(y,z) in the hybrid cross-section: (a) uniform matrix modulus Gm; (b) heterogeneous matrix modulus Gm(y,z) derived from a density-dependent relation.

The normalized shear-stress shape function ϕ(y,z), evaluated for the matrix-only and hybrid sections, is shown in Fig. 4. Since the shape function is reconstructed from purely geometric quantities (i.e., Q(z), b(z), and I), its spatial pattern is identical for both sections when the cross-sectional outline is unchanged. Consequently, the embedded I-section does not alter the equilibrium-based shear-transfer shape ϕ(y,z); instead, the effect of stiffness heterogeneity enters exclusively through the energetic weighting ϕ2/G (Eq. (14)), which changes the local contribution to shear strain energy and therefore the extracted effective shear stiffness.

Figure 4: Geometry-driven normalized shear-stress function ϕ(y,z) for both matrix-only cross-section and hybrid cross-section with embedded IPE 200 steel core.

This distinction is essential: while the shear-stress shape function ϕ(y,z) is entirely geometry-driven, the effective shear stiffness is obtained from its energetic projection, in which ϕ(y,z) is weighted by the local shear compliance 1/G(y,z). As a result, differences in material stiffness affect the contribution of individual regions to the total shear strain energy, without altering the equilibrium-based stress shape, and thus govern the identified value of ks.

The heterogeneous shear-energy density indicator ϕ2(y,z)/G(y,z) is presented in Fig. 5. The maps reveal pronounced localization of shear strain energy within limited regions of the matrix phase, particularly in zones combining high shear demand with relatively low matrix stiffness. The embedded steel core reduces energy contributions inside the steel phase due to the high modulus Gs, but simultaneously enhances localization in surrounding matrix regions, especially near phase boundaries where ϕ remains significant.

Figure 5: Heterogeneous shear-energy density indicator ϕ2(y,z)/G(y,z) in: (a) homogenous matrix-only cross-section; (b) heterogenous matrix-only cross-section; (c) hybrid homogenous cross-section with embedded IPE 200 steel core; (d) hybrid heterogenous cross-section with embedded IPE 200 steel core.

To quantify the role of each phase, the energy partition measures are summarized in Table 4. The results demonstrate that the global transverse shear response of the hybrid member may remain matrix-dominated, even though a stiff steel core is present. This observation supports the need for heterogeneous shear correction rather than relying on classical homogeneous factors, especially for monitoring-oriented stiffness interpretation.

To quantify the relative contribution of each material phase to the transverse shear response, the total shear-energy indicator is partitioned into phase-specific components. Based on the energetic formulation adopted in this study, the relevant quantity is the geometry-based shear-stress shape function ϕ(y,z), weighted by the local shear compliance 1/G(y,z).

The partial shear-energy indicators associated with the steel and matrix phases are defined as

Ssteel=∫Asteelϕ2(y,z)Gs(y,z)dA,Smatrix=∫Amatrixϕ2(y,z)Gm(y,z)dA.(17)

These quantities represent the relative energetic weight of each phase under transverse shear, while the common multiplicative constants (such as the applied shear force or the factor 1/2) cancel out when normalized measures are introduced. The total shear-energy indicator follows as the sum of the phase-specific contributions,

Stot=Ssteel+Smatrix.(18)

The relative participation of each phase in the overall shear response is then expressed in terms of normalized energy fractions,

ηsteel[% ]=100SsteelStot,ηmatrix[% ]=100SmatrixStot.(19)

By construction, these measures depend solely on the spatial overlap between the geometry-driven shape function ϕ(y,z) and the local shear compliance distribution. Consequently, differences between homogeneous and heterogeneous configurations arise exclusively from the material-dependent weighting 1/G(y,z), while the equilibrium-based stress shape remains unchanged.

The identified ks values obtained from Eq. (14) are summarized in Table 5 for the considered matrix descriptions. For the uniform matrix modulus case, the hybrid cross-section yields a ks value that deviates from the classical benchmark 5/6, indicating that the embedded steel core and stiffness contrast affect effective shear stiffness even when the matrix itself is assumed homogeneous. When matrix heterogeneity is introduced via Gm(y,z), the deviation becomes more pronounced, reflecting additional energy localization in weaker matrix regions. The parametric trends are illustrated in Fig. 6, where ks is shown as a function of stiffness contrast and matrix stiffness level.

Figure 6: Parametric trends in the identified shear correction factor: (a) ks as a function of stiffness contrast Gs/Gm; (b) deviation Δks vs. matrix stiffness level and heterogeneity intensity.

In this sense, the classical value ks=5/6 should be regarded as a geometric reference rather than a robust approximation for hybrid porous members. The results confirm that the effective shear stiffness is controlled by heterogeneous energy redistribution rather than by geometry alone.

Fig. 7 shows the mid-span deflection error arising from adopting the classical value ks=5/6 instead of the identified heterogeneous ks. The error is reported for the considered matrix scenarios and illustrates that even moderate deviations in ks may lead to systematic bias in displacement predictions.

Figure 7: Serviceability and monitoring implication of using the classical shear correction factor: mid-span deflection error resulting from assuming ks=5/6 instead of the identified heterogeneous value.

To relate the identified shear correction factors to serviceability and monitoring-oriented interpretation, a simply supported beam of span L subjected to a concentrated transverse load P applied at mid-span is considered. The mid-span deflection is decomposed into a bending part (Euler–Bernoulli) and a shear part (Timoshenko), such that

w(ks)=wb+ws(ks).(20)

The bending contribution at mid-span is governed by the Euler–Bernoulli expression

wb=PL348EIref,(21)

where E is the effective Young’s modulus associated with bending and I is the second moment of area about the bending axis. In the present study, the bending stiffness EIref is evaluated using the classical transformed-section approach, accounting for the stiffness contrast between matrix and steel phases. Thus, bending heterogeneity is incorporated at the sectional level, ensuring methodological consistency with the shear stiffness identification. While the shear correction factor requires explicit energetic weighting due to compliance localization, the bending stiffness is governed by the standard sectional modulus integration and does not involve an additional correction factor.

The shear contribution is expressed in terms of the effective shear stiffness (GA)eff=ksGrefA and reads

ws(ks)=PL4ksGrefA,(22)

where A is the reference shear area of the cross-section and Gref is the reference shear modulus used in the identification procedure.

To quantify the bias introduced by adopting the classical value k0=5/6 instead of the identified value ks, we define the relative mid-span deflection error as

εw[% ]=100w(k0)−w(ks)w(ks).(23)

With this convention, εw<0 indicates that using k0 underestimates the deflection (i.e., the model appears artificially stiff in serviceability terms).

Finally, to express the same effect in terms of monitoring-oriented stiffness interpretation, we introduce an apparent stiffness bias based on the inverse proportionality between stiffness and deflection, i.e., K∝1/w. The bias is therefore defined as

εK[% ]=100(w(ks)w(k0)−1).(24)

A positive εK corresponds to an overestimation of stiffness when the classical value k0 is used in place of the identified heterogeneous value.

This has direct implications for monitoring workflows where measured deflections are interpreted through reduced-order models: if shear deformation is underestimated, the system may appear artificially compliant, which could be incorrectly attributed to bending stiffness degradation or damage accumulation. The monitoring-oriented metrics reported in Table 6 quantify these potential biases and highlight the importance of shear-consistent stiffness identification for hybrid perlite–steel members.

Although the detailed numerical results are presented for a representative configuration (IPE 200 embedded in a 0.20 m × 0.30 m section), the governing Eq. (14) allows interpretation in terms of non-dimensional parameters. In particular, the shear correction factor may be expressed as a function of:

•   the steel-to-matrix area ratio α=Asteel/A,

•   the geometric aspect ratio β=b/h,

•   and the relative position parameter e/h describing eccentricity of the steel core.

Increasing α raises the area-averaged modulus Gref while simultaneously reducing the compliance contribution within the steel domain. Since the energetic integral is compliance-weighted, the matrix region remains dominant unless α approaches unity. Therefore, the reduction in ks is expected to scale nonlinearly with α, particularly in heterogeneous matrices.

Eccentric placement modifies the geometric shear-stress distribution ϕ(y,z), leading to asymmetric energy localization and further reduction of effective shear stiffness in the weaker region.

The aspect ratio b/h influences the shear-stress distribution through the classical Q/Ib dependence, and thus affects the overlap between high-shear zones and low-modulus regions.

4  Discussion

From a structural health monitoring perspective, the proposed framework may be extended toward an inverse identification strategy. Measured mid-span deflections under controlled loading could be decomposed into bending and shear components. Assuming bending stiffness is independently known or identified from modal analysis, the residual shear contribution may be used to estimate an updated effective ks.

Changes in the identified ks over time may indicate modifications in the spatial distribution of the shear modulus field G(y,z), such as moisture-induced softening, cracking, or corrosion-induced stiffness redistribution. By coupling the present forward energetic formulation with inverse optimization procedures, it becomes possible to localize damage or degradation zones within heterogeneous composite members.

The proposed formulation clearly separates two fundamentally different contributions to the transverse shear response. The normalized shear-stress shape function ϕ(y,z), introduced in Eqs. (4) and (5), is determined exclusively by the cross-sectional geometry through the classical equilibrium relations involving I, Q(z), and b(z). Consequently, for a fixed outer contour of the cross-section, ϕ(y,z) remains identical regardless of material heterogeneity or the presence of an embedded steel profile. This behavior is consistently observed in Fig. 4, where the shape function exhibits the same spatial distribution for matrix-only and hybrid configurations sharing the same geometric outline.

In contrast, the effective shear stiffness and the associated shear correction factor ks are governed by the energetic weighting introduced in Eq. (14), in which the geometry-driven function ϕ(y,z) is scaled by the local shear compliance 1/G(y,z). This distinction is essential for the interpretation of the results: material heterogeneity does not alter the equilibrium-based shear-stress shape but redistributes the contribution of individual regions to the total shear strain energy. The maps shown in Fig. 5 directly reflect this mechanism, revealing strong variations in the indicator ϕ2/G even when ϕ(y,z) itself remains unchanged.

The mesh convergence study summarized in Table 1 demonstrates that the identified values of ks are numerically stable with respect to the pixel discretization used to evaluate the energy integral in Eq. (14). As the pixel resolution is refined, the changes in ks diminish rapidly, indicating that the dominant energetic features of the shear response are sufficiently resolved. This observation is particularly relevant for heterogeneous cases, where localized regions of high ϕ2/G may develop. The convergence results confirm that the proposed pixel-based integration scheme captures these localized effects without introducing mesh-dependent artifacts, thereby ensuring the reliability of the reported trends.

The role of the spatial distribution of the shear modulus G(y,z) is highlighted by the comparison between uniform and heterogeneous matrix descriptions shown in Fig. 3. When a constant matrix modulus is assumed, the energetic weighting remains relatively uniform across the matrix domain. Introducing a spatially varying modulus, derived from the density-dependent relation in Eq. (12), leads to pronounced energetic localization, as evidenced by the distributions in Fig. 5b,d.

The heterogeneous matrix case considered in this study represents a spatially varying modulus field derived from density–modulus relations. In practical materials, heterogeneity may exhibit structured patterns, such as weaker cores, surface degradation layers, or casting-induced density gradients. Different spatial distributions would alter the overlap between high-shear zones and compliant regions, potentially leading to distinct reductions in ks. For example, a weak core located near the neutral axis may have a smaller influence on shear energy compared to a weak cover layer coinciding with high ϕ regions.

These results illustrate that the effective shear response is controlled not merely by average material properties but by the spatial overlap between regions of high shear demand (large ϕ) and reduced local stiffness. The steel phase, characterized by a very high shear modulus (Table 3), contributes only marginally to the total shear energy, despite its structural presence. This effect is a direct consequence of the 1/G weighting and explains why the introduction of a stiff steel core does not necessarily lead to a steel-dominated shear response.

The phase-wise energy partition summarized in Table 4 provides a quantitative interpretation of the observations made in Fig. 5. The low values of ηsteel obtained for both hybrid cases indicate that the matrix phase governs the transverse shear response, even when a steel I-section is embedded within the cross-section. This finding underscores the energetic nature of the shear correction factor: regions with low shear compliance dominate the equilibrium stress field, whereas regions with higher compliance dominate the energy balance.

From a modeling perspective, this result emphasizes that the effective shear stiffness cannot be inferred from geometric considerations or stiffness contrasts alone. Instead, it emerges from the combined effect of geometry-driven stress distribution and material-dependent energetic weighting, as formalized in Eqs. (17)–(19). The identified values of ks summarized in Table 5 reveal substantial deviations from the classical reference value ks=5/6, particularly for heterogeneous configurations.

The very low value obtained for the heterogeneous hybrid case H2 (ks=0.187) is consistent with the adopted normalization strategy and the observed energy localization. Since Gref is defined as the area-averaged modulus of the entire cross-section, the presence of the high-modulus steel region increases Gref compared to the matrix modulus alone (Table 5). At the same time, the energetic integral in Eq. (14) is compliance-weighted through 1/G(y,z), which shifts the dominant shear energy contribution to the weaker matrix regions, as visualized by the ϕ2/G maps in Fig. 5d and quantified by the matrix-dominated energy partition in Table 4. The combination of (i) elevated Gref due to steel and (ii) strong compliance-driven localization in the heterogeneous matrix naturally leads to a markedly reduced extracted ks, i.e., a low effective shear stiffness when expressed in the Timoshenko form ksGrefA.

Fig. 6a shows that ks decreases systematically with increasing stiffness contrast between steel and matrix phases, while Fig. 6b highlights the dependence of Δks on the reference shear modulus and the degree of heterogeneity. These trends confirm that the classical shear correction factor should be regarded as a geometric benchmark applicable primarily to homogeneous sections. For hybrid or porous members with spatially varying material properties, the effective shear correction factor is inherently problem-specific and must be identified through an energetically consistent procedure.

The practical consequences of adopting an inappropriate shear correction factor are demonstrated in Fig. 7 and Table 6, where the mid-span deflection error and the apparent stiffness bias are evaluated for a simply supported beam subjected to a central point load. The results show that even moderate deviations in ks can lead to non-negligible errors in deflection predictions. In monitoring-oriented applications, such discrepancies may be misinterpreted as changes in structural stiffness or damage progression.

The presented results therefore highlight the importance of using a shear correction factor that is consistent with the actual material heterogeneity of the member. In the context of eco-materials and hybrid structures, where material properties may evolve over time due to environmental exposure or degradation, the proposed methodology offers a physically grounded framework for updating shear-related parameters in reduced-order models. The systematic nature of the observed trends across matrix-only and hybrid configurations further supports that the reduction of ks originates from physically consistent energy redistribution rather than from numerical artifacts or normalization choices.

From a durability perspective, hybrid perlite–steel members may require protective measures against environmental exposure, particularly moisture ingress and steel corrosion. While the present study focuses on elastic shear response, the proposed framework may support durability-oriented assessments by enabling updated stiffness identification under changing material parameters. Protective coatings, adequate concrete cover, and moisture control strategies remain essential design measures to ensure long-term performance of such composite elements.

In real field conditions, material properties may evolve due to moisture variations, temperature effects, or microcracking phenomena. The proposed model assumes linear elastic behavior and static loading conditions, and therefore does not capture time-dependent degradation mechanisms. Nevertheless, its computational efficiency and transparent energetic basis make it suitable for integration into digital twin or monitoring workflows, where shear stiffness parameters may be periodically updated based on measured structural response.

The present framework is derived from strain-energy equivalence within the Timoshenko beam theory. It does not aim to replace full 3D finite element modeling, but to provide a consistent reduced-order parameter identification procedure. For moderate slenderness ratios, Timoshenko theory has been shown in numerous studies to accurately capture transverse shear effects in heterogeneous beams when appropriate shear correction factors are employed. The current work focuses on identifying this parameter in an energetically consistent manner.

For very short beams or cases involving strong 3D stress concentrations, higher-order or full 3D models may be required. The proposed formulation is therefore best interpreted as a reduced-order stiffness identification tool rather than a substitute for detailed finite element analysis.

5  Conclusions

This study presented an energetically consistent framework for identifying the transverse shear correction factor ks in heterogeneous and hybrid beam cross-sections. By explicitly separating the geometry-driven shear-stress shape function from the material-dependent energy weighting, the proposed method provides a clear physical interpretation of how spatial variations in shear modulus influence the effective shear stiffness.

The study provides quantitative evidence that area-averaged normalization combined with compliance-weighted energy integration may lead to substantial deviations from classical shear correction factors in heterogeneous hybrid systems. The findings demonstrate that even limited volumetric fractions of stiff inclusions may alter reference stiffness measures while leaving the shear response matrix-dominated. This highlights the necessity of energetically consistent parameter identification in sustainability-driven structural design.

The results demonstrate that, for a fixed cross-sectional outline, the shear-stress shape function remains purely geometric, while deviations in ks arise solely from heterogeneous energy redistribution. In hybrid sections with an embedded steel core, the shear response may remain dominated by the matrix phase, despite the high stiffness of the steel component. This behavior cannot be captured by classical homogeneous shear correction factors.

Parametric studies and deflection-based assessments further showed that adopting the classical value ks=5/6 may introduce systematic errors in serviceability predictions and monitoring-based stiffness identification. The proposed pixel-based energy integration approach therefore constitutes a practical and robust tool for evaluating shear correction factors in modern hybrid and eco-material structures, where material heterogeneity plays a decisive role in the transverse shear response.

The presented results should be interpreted as a generalized framework demonstration rather than a geometry-specific numerical case. The energetic formulation is directly applicable to arbitrary hybrid cross-sections and heterogeneous modulus fields, provided that the spatial distribution G(y,z) is known or identified. Future studies may extend the approach toward systematic parametric campaigns and experimental validation.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, Tomasz Garbowski and Anna Szymczak-Graczyk; methodology, Tomasz Garbowski and Anna Szymczak-Graczyk; software, Tomasz Garbowski; validation, Zijadin Guri and Ilir Canaj; formal analysis, Anna Szymczak-Graczyk and Ilir Canaj; investigation, Tomasz Garbowski and Zijadin Guri; resources, Zijadin Guri and Ilir Canaj; data curation, Zijadin Guri and Ilir Canaj; writing—original draft preparation, Tomasz Garbowski; writing—review and editing, Anna Szymczak-Graczyk; visualization, Anna Szymczak-Graczyk and Tomasz Garbowski; supervision, Anna Szymczak-Graczyk and Tomasz Garbowski; project administration, Anna Szymczak-Graczyk and Tomasz Garbowski. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available from the corresponding author upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

References

1. Rashad AM. A synopsis about perlite as building material—A best practice guide for Civil Engineer. Constr Build Mater. 2016;121:338–53. doi:10.1016/j.conbuildmat.2016.06.001. [Google Scholar] [CrossRef]

2. Abed AT, Éva L. A comprehensive study of perlite in building materials: balancing sustainability and performance. J Therm Anal Calorim. 2025;150(14):10627–43. doi:10.1007/s10973-025-14410-6. [Google Scholar] [CrossRef]

3. Dzięcioł J, Szlachetka O. Waste or raw material? Perlite concrete as part of a sustainable materials management process in the construction sector. Sustainability. 2024;16(16):6818. doi:10.3390/su16166818. [Google Scholar] [CrossRef]

4. Koukouzas NK, Dunham AC, Scott PW. Suitability of Greek perlite for industrial applications. Appl Earth Sci. 2000;109(2):105–11. doi:10.1179/aes.2000.109.2.105. [Google Scholar] [CrossRef]

5. Jia G, Li Z, Liu P, Jing Q. Preparation and characterization of aerogel/expanded perlite composite as building thermal insulation material. J Non Cryst Solids. 2018;482:192–202. doi:10.1016/j.jnoncrysol.2017.12.047. [Google Scholar] [CrossRef]

6. Ariyaratne IE, Ariyanayagam A, Mahendran M. Bushfire-resistant lightweight masonry blocks with expanded perlite aggregate. Fire. 2022;5(5):132. doi:10.3390/fire5050132. [Google Scholar] [CrossRef]

7. Cojocaru A, Isopescu DN, Maxineasa SG. Perlite concrete: a review. IOP Conf Ser Mater Sci Eng. 2023;1283(1):012003. doi:10.1088/1757-899x/1283/1/012003. [Google Scholar] [CrossRef]

8. Topçu İB, Işıkdağ B. Effect of expanded perlite aggregate on the properties of lightweight concrete. J Mater Process Technol. 2008;204(1–3):34–8. doi:10.1016/j.jmatprotec.2007.10.052. [Google Scholar] [CrossRef]

9. Singh M, Garg M. Perlite-based building materials—A review of current applications. Constr Build Mater. 1991;5(2):75–81. doi:10.1016/0950-0618(91)90004-5. [Google Scholar] [CrossRef]

10. Qiao Y, Bian X, Ke H, Zhao Z, Zhang X, Zhang X, et al. Phenolic-based porous composite with embedded short carbon fiber/hollow spheres for mechanical properties and thermostability. Compos Commun. 2025;55:102305. doi:10.1016/j.coco.2025.102305. [Google Scholar] [CrossRef]

11. Ciminli AT, Bulut HA. A new approach for lightweight polymer concrete production: determination of the influence of resin and perlite types on the mechanical performance of polymer concrete. Constr Build Mater. 2025;490:142608. doi:10.1016/j.conbuildmat.2025.142608. [Google Scholar] [CrossRef]

12. Mattausch H, Laske S, Cirar K, Flachberger H, Holzer C. Fundamental investigations of reinforcement of expanded perlite in polypropylene. In: New developments in polymer composites research. New York, NY, USA: Nova Science Publishers; 2013. p. 213–26. [Google Scholar]

13. Szlachetka O, Dzięcioł J. Expanded perlite in civil engineering: a review of its potential for low-carbon and circular construction. Sustainability. 2026;18(3):1479. doi:10.3390/su18031479. [Google Scholar] [CrossRef]

14. Jing Z, Zong W, Zhang J, Zhang Y. Tracking interprovincial embodied carbon flows of cement in China: an energy life cycle perspective. Energy Ecol Environ. 2025;10(6):723–44. doi:10.1007/s40974-025-00377-5. [Google Scholar] [CrossRef]

15. Alwan Z, Jones P. The importance of embodied energy in carbon footprint assessment. Struct Surv. 2014;32(1):49–60. doi:10.1108/ss-01-2013-0012. [Google Scholar] [CrossRef]

16. Cardoso M, Mata TM, Monteiro H, Varum H, Martins AA. Influence of concrete composition on the carbon footprint and embodied energy of a frame structure. In: Caetano NS, Felgueiras MC, editors. The 9th International Conference on Energy and Environment Research; 2022 Sep 12–16; Porto, Portugal. p. 59–67. doi:10.1007/978-3-031-43559-1_6. [Google Scholar] [CrossRef]

17. Jiang L, Wang Q, Hu S, Zhao X, Wang W, Zhang Y, et al. Expanded perlite immobilization enhances long-term survival and mineralization activity of microbial spores in cementitious environments. Appl Environ Microbiol. 2026;92(2):e02456–25. doi:10.1128/aem.02456-25. [Google Scholar] [PubMed] [CrossRef]

18. Yeşilyurt A, Uysal N. Evaluation of properties of polypropylene matrix composites reinforced with perlite, expanded perlite, silanized expanded perlite, and talc fillers. J Appl Polym Sci. 2025;142(44):e57695. doi:10.1002/app.57695. [Google Scholar] [CrossRef]

19. Erdem TK, Meral Ç, Tokyay M, Erdoğan TY. Use of perlite as a pozzolanic addition in producing blended cements. Cem Concr Compos. 2007;29(1):13–21. doi:10.1016/j.cemconcomp.2006.07.018. [Google Scholar] [CrossRef]

20. Sengul O, Azizi S, Karaosmanoglu F, Ali Tasdemir M. Effect of expanded perlite on the mechanical properties and thermal conductivity of lightweight concrete. Energy Build. 2011;43(2–3):671–6. doi:10.1016/j.enbuild.2010.11.008. [Google Scholar] [CrossRef]

21. Demirboğa R, Örüng İ, Gül R. Effects of expanded perlite aggregate and mineral admixtures on the compressive strength of low-density concretes. Cem Concr Res. 2001;31(11):1627–32. doi:10.1016/S0008-8846(01)00615-9. [Google Scholar] [CrossRef]

22. Jedidi M, Benjeddou O, Soussi C. Effect of expanded perlite aggregate dosage on properties of lightweight concrete. Jordan J Civ Eng. 2015;9(3):378–91. doi:10.14525/jjce.9.3.3071. [Google Scholar] [CrossRef]

23. da Silva MTQS, Perretto F, do Rocio Cardoso M, Mazer W. Porosity: some characterization techniques. Mater Today Proc. 2023;1:296. doi:10.1016/j.matpr.2023.03.716. [Google Scholar] [CrossRef]

24. Taherishargh M, Belova IV, Murch GE, Fiedler T. The effect of particle shape on mechanical properties of perlite/metal syntactic foam. J Alloys Compd. 2017;693:55–60. doi:10.1016/j.jallcom.2016.09.168. [Google Scholar] [CrossRef]

25. Burriesci N, Arcoraci C, Antonucci P, Polizzotti G. Physico-chemical characterization of perlite of various origins. Mater Lett. 1985;3(3):103–10. doi:10.1016/0167-577X(85)90008-4. [Google Scholar] [CrossRef]

26. Alexa-Stratulat SM, Taranu G, Toma AM, Olteanu I, Pastia C, Bunea G, et al. Effect of expanded perlite aggregates and temperature on the strength and dynamic elastic properties of cement mortar. Constr Build Mater. 2024;438:137229. doi:10.1016/j.conbuildmat.2024.137229. [Google Scholar] [CrossRef]

27. Chen K, Qin H, Ren Z. Establishment of the microstructure of porous materials and its relationship with effective mechanical properties. Sci Rep. 2023;13(1):18064. doi:10.1038/s41598-023-43439-6. [Google Scholar] [PubMed] [CrossRef]

28. Khanna P, Mukulam AM, Teja KV, Meena T. Study on durability properties of perlite incorporated concrete. Int J Civ Eng Technol. 2018;9(10):1545–53. [Google Scholar]

29. Szlachetka O, Dzięcioł J, Dohojda M. Mechanical properties of perlite concrete in context to its use in buildings’ external walls. Materials. 2024;17(23):5790. doi:10.3390/ma17235790. [Google Scholar] [PubMed] [CrossRef]

30. Buljak V, Oesch T, Bruno G. Simulating fiber-reinforced concrete mechanical performance using CT-based fiber orientation data. Materials. 2019;12(5):717. doi:10.3390/ma12050717. [Google Scholar] [PubMed] [CrossRef]

31. Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. Warsaw, Poland: Arkady; 1962. [Google Scholar]

32. Donnell LH. Beams, plates and shells. New York, NY, USA: McGraw-Hill; 1976. [Google Scholar]

33. Carrera E, Giunta G, Petrolo M. Beam structures: classical and advanced theories. Chichester, UK: John Wiley & Sons; 2011. doi:10.1002/9781119978565. [Google Scholar] [CrossRef]

34. Graczyk J, Gajewski T, Garbowski T. Shear correction factor for porous eco-materials: mechanical characterization of a heterogeneous medium. Buildings. 2026;16(4):688. doi:10.3390/buildings16040688. [Google Scholar] [CrossRef]

35. Lim TK, Kim JH. Thermo-elastic effects on shear correction factors for functionally graded beam. Compos Part B Eng. 2017;123:262–70. doi:10.1016/j.compositesb.2017.05.031. [Google Scholar] [CrossRef]

36. Garbowski T, Pawlak TG, Szymczak-Graczyk A. Efficient load-bearing capacity assessment of a degraded concrete manhole using sectional homogenization. Materials. 2024;17(23):5883. doi:10.3390/ma17235883. [Google Scholar] [PubMed] [CrossRef]

37. Staszak N, Szymczak-Graczyk A, Garbowski T. Elastic analysis of three-layer concrete slab based on numerical homogenization with an analytical shear correction factor. Appl Sci. 2022;12(19):9918. doi:10.3390/app12199918. [Google Scholar] [CrossRef]

38. Ksit B, Szymczak-Graczyk A, Pilch R. Numerical simulation of the impact of water vapour and moisture blockers in energy diagnostics of ventilated partitions. Materials. 2022;15(22):8257. doi:10.3390/ma15228257. [Google Scholar] [PubMed] [CrossRef]

39. Li X, Wang F, Yi S, Yi H, Cheng G, Chen Z. Investigation on the long-term damage evolution of a strip coal pillar. Sci Rep. 2025;15:35688. doi:10.1038/s41598-025-19546-x. [Google Scholar] [PubMed] [CrossRef]

40. Ksit B, Szymczak-Graczyk A, Orlik-Kożdoń B, Garbowski T. Numerical and surrogate modeling of drying processes in building envelopes under variable climatic conditions. Energy. 2025;339:139014. doi:10.1016/j.energy.2025.139014. [Google Scholar] [CrossRef]

41. Cornaggia A, Garbowski T, Cocchetti G, Ferrari R, Rizzi E. Optimised structural modelling for inverse analysis parameter identification relying on dynamic measurements. In: 9th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering; 2023 Jun 12–14; Athens, Greece. p. 4234–48. doi:10.7712/120123.10715.20669. [Google Scholar] [CrossRef]

42. Garbowski T, Cocchetti G, Cornaggia A, Ferrari R, Rizzi E. Inverse analysis investigation by Gaussian processes optimisation of a historical concrete bridge relying on dynamic modal measurements. In: Proceedings of the 9th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2023); 2023 Jun 12–14; Athens, Greece. p. 4249–64. doi:10.7712/120123.10716.21212. [Google Scholar] [CrossRef]

43. Cornaggia A, Ferrari R, Zola M, Rizzi E, Gentile C. Signal processing methodology of response data from a historical arch bridge toward reliable modal identification. Infrastructures. 2022;7(5):74. doi:10.3390/infrastructures7050074. [Google Scholar] [CrossRef]

44. Ferrari R, Zola M, Cornaggia A, Rizzi E. Advanced signal processing methodology of vibration response data toward Structural Health Monitoring purposes. J Phys Conf Ser. 2024;2647(18):182040. doi:10.1088/1742-6596/2647/18/182040. [Google Scholar] [CrossRef]

45. Thanaraj DP, Anand N, Arulraj P, Al-Jabri K. Investigation on structural and thermal performance of reinforced concrete beams exposed to standard fire. J Build Eng. 2020;32:101764. doi:10.1016/j.jobe.2020.101764. [Google Scholar] [CrossRef]

46. Fletcher I, Welch S, Torero J, Carvel R, Usmani A. Behaviour of concrete structures in fire. Therm Sci. 2007;11(2):37–52. doi:10.2298/tsci0702037f. [Google Scholar] [CrossRef]

47. Li Z, Zhou X, Shen B. Fiber-cement extrudates with perlite subjected to high temperatures. J Mater Civ Eng. 2004;16(3):221–9. doi:10.1061/(asce)0899-1561(2004)16:3(221). [Google Scholar] [CrossRef]

48. Szymczak-Graczyk A. Rectangular plates of a trapezoidal cross-section subjected to thermal load. IOP Conf Ser Mater Sci Eng. 2019;603(3):032095. doi:10.1088/1757-899X/603/3/032095. [Google Scholar] [CrossRef]

49. Sayadi A, Neitzert TR, Charles Clifton G. Influence of poly-lactic acid on the properties of perlite concrete. Constr Build Mater. 2018;189:660–75. doi:10.1016/j.conbuildmat.2018.09.029. [Google Scholar] [CrossRef]

50. Al-Jadiri RSF, Abed Al-Wahab Ali M, Frayyeh QJ. Study some mechanical and thermal properties of reinforced perlite concrete. Key Eng Mater. 2022;924:233–42. doi:10.4028/p-9os233. [Google Scholar] [CrossRef]

51. Buljak V, Bruno G. Numerical modeling of thermally induced microcracking in porous ceramics: an approach using cohesive elements. J Eur Ceram Soc. 2018;38(11):4099–108. doi:10.1016/j.jeurceramsoc.2018.04.041. [Google Scholar] [CrossRef]

52. Garbowski T, Cornaggia A, Gajewski T, Grabski JK, Mrówczyński D. Inverse-based multi-step numerical homogenization for mechanical characterization of converted corrugated board. Compos Struct. 2025;373:119701. doi:10.1016/j.compstruct.2025.119701. [Google Scholar] [CrossRef]

53. Babu KP, Mohite PM, Upadhyay CS. Development of an RVE and its stiffness predictions based on mathematical homogenization theory for short fibre composites. Int J Solids Struct. 2018;130-131:80–104. doi:10.1016/j.ijsolstr.2017.10.011. [Google Scholar] [CrossRef]

54. Gitman IM, Askes H, Sluys LJ. Representative volume: existence and size determination. Eng Fract Mech. 2007;74(16):2518–34. doi:10.1016/j.engfracmech.2006.12.021. [Google Scholar] [CrossRef]

55. Anoukou K, Brenner R, Hong F, Pellerin M, Danas K. Random distribution of polydisperse ellipsoidal inclusions and homogenization estimates for porous elastic materials. Comput Struct. 2018;210:87–101. doi:10.1016/j.compstruc.2018.08.006. [Google Scholar] [CrossRef]

56. Dzięcioł J, Sas W. Perspective on the application of machine learning algorithms for flow parameter estimation in recycled concrete aggregate. Materials. 2023;16(4):1500. doi:10.3390/ma16041500. [Google Scholar] [PubMed] [CrossRef]

57. Szymczak-Graczyk A, Korentz J, Garbowski T. Digital twin model for predicting hygrothermal performance of building materials from moisture permeability tests. Materials. 2025;18(18):4360. doi:10.3390/ma18184360. [Google Scholar] [PubMed] [CrossRef]

58. Isaksson P, Krusper A, Gradin PA. Shear correction factors for corrugated core structures. Compos Struct. 2007;80(1):123–30. doi:10.1016/j.compstruct.2006.04.066. [Google Scholar] [CrossRef]

59. Hadavinia H, Gordnian K, Karwatzki J, Aboutorabi A. Deriving shear correction factor for thick laminated plates using the energy equivalence method. Struct Durab Health Monit. 2006;2(4):197. [Google Scholar]

60. Buljak V, Cocchetti G, Cornaggia A, Maier G. Parameter identification in elastoplastic material models by Small Punch Tests and inverse analysis with model reduction. Meccanica. 2018;53(15):3815–29. doi:10.1007/s11012-018-0914-3. [Google Scholar] [CrossRef]

61. Orbulov IN, Szlancsik A, Kemény A, Kincses D. Low-cost light-weight composite metal foams for transportation applications. J Mater Eng Perform. 2022;31(9):6954–61. doi:10.1007/s11665-022-06644-4. [Google Scholar] [CrossRef]

62. Celik S, Family R, Menguc MP. Analysis of perlite and pumice based building insulation materials. J Build Eng. 2016;6:105–11. doi:10.1016/j.jobe.2016.02.015. [Google Scholar] [CrossRef]

63. Villasmil W, Fischer LJ, Worlitschek J. A review and evaluation of thermal insulation materials and methods for thermal energy storage systems. Renew Sustain Energy Rev. 2019;103:71–84. doi:10.1016/j.rser.2018.12.040. [Google Scholar] [CrossRef]

64. Ayfari F, Hesami S. Development and evaluation of high-temperature resistant bituminous composites: a study on the performance of expanded perlite and polytetrafluoroethylene in asphalt mixtures. Case Stud Constr Mater. 2024;21:e03955. doi:10.1016/j.cscm.2024.e03955. [Google Scholar] [CrossRef]

65. Cornaggia A, Cocchetti G, Maier G, Buljak V. Inverse structural analyses on small punch tests, with model reduction and stochastic approach. In: Proceedings of the 2018 IEEE International Conference on Environment and Electrical Engineering and 2018 IEEE Industrial and Commercial Power Systems Europe (EEEIC/I&CPS Europe); 2018 Jun 12–15; Palermo, Italy. p. 1–5. doi:10.1109/EEEIC.2018.8494215. [Google Scholar] [PubMed] [CrossRef]

66. Pozorski Z, Pozorska J, Murčinková Z, Cekus D. On local instability of deep-profiled facings in sandwich panels. Materials. 2025;18(22):5162. doi:10.3390/ma18225162. [Google Scholar] [PubMed] [CrossRef]

67. Garbowski T, Graczyk J. Shear correction factor in corrugated board layered plates. Polish Paper Rev. 2025;81:686–90. doi:10.15199/54.2025.12.3. [Google Scholar] [CrossRef]

68. Graczyk J, Tworzydło J, Garbowski T. Parametric sensitivity of shear correction factors for multiwall corrugated structures. Materials. 2026;19(5):863. doi:10.3390/ma19050863. [Google Scholar] [PubMed] [CrossRef]

69. Graczyk J, Gajewski T, Garbowski T. Numerical determination of the shear correction factor for thin-walled steel sections using shell-based finite element modeling. doi:10.2139/ssrn.6144980. [Google Scholar] [CrossRef]

70. Pozorski Z, Lange J, Sabik A. Experimental and numerical analysis of sandwich structures. Materials. 2026;19(2):386. doi:10.3390/ma19020386. [Google Scholar] [PubMed] [CrossRef]