An Analytical Analysis of the Hydrostatic Bending to Design a Wastewater Treatment Plant by a New Advanced Composite Material

Climate change has taken a prominent toll on Algeria in recent times, culminating in severe drought conditions, particularly accentuated in the last decade. This alarming issue has significantly impacted the nation's agricultural sector [1]. The ramifications of climate change are manifest through escalated air temperatures, the melting of glaciers, rising sea levels, aggravated desertification, and a surge in extreme weather events encompassing heat waves, droughts, floods, and storms. Notably, the impact of climate change varies across regions, imposing disproportionate effects on different areas [2]. To elucidate the transformations observed within the Algerian locale spanning the last forty years, we present in Figure 1 a graphical depiction that relies on ERA5 data, sourced from the fifth-generation ECMWF Atmospheric Analysis of Global Climate. Encompassing the temporal span from 1979 to 2021 and boasting a spatial resolution of 30 km, this dataset furnishes a holistic vantage point pertaining to the dynamic evolution of the climatic milieu [3].

The profound ramifications of this climate-induced metamorphosis have triggered a surge in comprehensive investigations, seeking to unravel its multifaceted effects on Algerian agricultural practices. The in-depth econometric analysis covering the years 1980 to 2020, entitled "Impact of Climate Change on Agricultural Production in Algeria", represents a major scientific effort in this field. This study, aspired to proffer a comprehensive assessment of the intricate repercussions within Algerian agriculture, notably directs attention to the preceding two decades a span during which a staggering 30% reduction in precipitation levels has unfurled across the nation, in turn precipitating a consequential and disquieting decline in agricultural productivity [4].

Compelling statistics furnished by Algeria's national bureau serve to underscore the gravity of this situation, portraying a disconcerting 30% downturn in agricultural yield within the western region. More alarmingly, the eastern region bears witness to a pronounced 40% decrement over this same pivotal time frame [5, 6], as visually elucidated in Figure 2. This distressing trajectory places the agricultural sector on a precipitous course, grappling with the dual challenges of diminishing rainfall and localized production.

The confluence of these challenges has galvanized the need for effective and expedient remedial measures. In response, the Ministry of Hydraulics has taken resolute steps, adopting a strategic policy that pivots around the innovative reuse of wastewater. The trajectory of this wastewater treatment initiative is artfully delineated in Figure 3, embodying the overarching aim of establishing wastewater treatment protocols that synergistically dovetail with sustainable agricultural practices [7].

In the context of wastewater treatment, a wide variety of approaches come into play, encompassing diverse methods such as biological treatment utilizing activated sludge, treatment through trickling filters, lagooning, membrane filtration, and UV disinfection [8]. These methods can be implemented using an array of materials, ranging from steel, reinforced concrete, and PVC, to cut stones and compact backfilled substances. Among the numerous designs for wastewater treatment plants, one notable innovation takes the form of the "Continuous Station," distinguished by its monolithic structure. While upholding the foundational principles of conventional treatment plants, this design stands out by virtue of its compact footprint, providing an efficient solution tailored to environments with spatial limitations.

As illustrated in Figure 4, the operational rhythm of the monobloc station unveils a process underpinned by several key zones [9]:

1.png

Figure 1. Precipitation and temperature change (Climate change in Algeria 1979-2021)

2.png

Figure 2. Statistics of the production of agricultural products in Algeria (2018-2020)

3.png

Figure 3. The water system in Algeria (drinkingwater-wasre water) with purification cycle

4.png

Figure 4. The system of purification by the contumely station or monobloc system

(1): An unventilated area housing a vertical flow labyrinth.

(2): A ventilated zone.

(3): A sedimentation zone.

(4): An integrated retention space.

(5): Internal recirculation.

(6): Recirculation of impurities.

(7): Aeration through the introduction of fine bubbles.

(8): Implementation of a flow regulator.

This specific design has found practical realization within Algeria, notably gaining prominence within the 'BABA-ALI' region nestled within the Wilaya of AIN DEFLA. Initially, this installation was executed using conventional steel, a choice that, while structurally robust, exposes the framework to potential corrosion due to the chemical constituents present in wastewater. Faced with this acknowledged vulnerability, our research endeavors to both address and circumvent these risks through the advocacy of an alternative material strategy—namely, the integration of a steel alloy fortified with tungsten nanoparticles.

The burgeoning field of nanotechnology has led to revolutionary advances in a number of sectors, construction engineering being no exception [10]. Nanoparticles, owing to their exceptional properties at the nanoscale, have ignited immense interest in their potential applications to enhance conventional construction materials [11]. These nanoparticles, including silicon oxide [12, 13], ferric oxide [14], alumina oxide [15], zircon [16], and titanium oxide [17], may have inherent limitations, but their potential can be unlocked through the strategic integration of nanoparticles. Nanoparticles exhibit unique behaviors due to their high surface area-to-volume ratio and quantum effects, making them versatile candidates for enhancing thermal, electrical, chemical, and mechanical properties of construction materials [18]. In civil engineering, nanoparticles have been extensively explored to bolster the performance of foundational materials like concrete and steel. This avenue of research has led to the development of materials with enhanced strength, durability, and multifunctionality [19].

In the context of thermal enhancement, nanoparticles have demonstrated remarkable potential by improving the thermal conductivity of materials. This attribute is crucial for maintaining energy efficiency in construction, thereby contributing to sustainable infrastructure. For instance, Harrat et al. [20], investigated the thermoelastic bending behavior of concrete infused with ferric oxide nanoparticles. Similarly, Benfrid et al. [21], conducted an analytical study exploring the impact of glass powder additives on concrete beams exposed to thermal loading. Both studies concluded that the addition of these entities enhances the mechanical properties of concrete while concurrently affecting its thermal characteristics due to the nanoparticles' high thermal conductivity. In a separate study, Chatbi et al. [22] conducted a mechanical analysis of polymer structures reinforced with either aligned or randomly aligned carbon nanotubes using the MORI-TANAKA approach. The findings highlighted a significant enhancement in the mechanical properties of polymers due to carbon nanotubes reinforcement. The same authors also performed an analytical investigation on the mechanical response of concrete beams reinforced with various types of clay nanoparticles, emphasizing the enhancement capabilities of these additives through a strategic integration [23]. Still in the realm of construction materials, nanocomposites offer the promise of greater strength, enhanced ductility, improved and corrosion resistance. For example, research by Long et al. [24] has shown that the addition of nano-sized reinforcements, such as graphene oxide nanosheets, can significantly increase the dynamic mechanical properties of cement paste. Similarly, studies by Behzadian and Shahrajabian [25], Wang et al. [26], Zhang et al. [27] and Bhatta et al. [28] have explored the effects of incorporating silica nanoparticles on the mechanical and durability properties of cementitious materials, showing improvements in both strength and resistance to environmental degradation.

Additionally, nanoparticles have proven invaluable in electrically conductive composites, opening avenues for smart structures and sensors that can monitor and adapt to changing environmental conditions. McCall [29] and Magnani et al. [30], have discussed the piezoelectric properties of nano-infused polymers. Furthermore, new synthesis and characterization techniques have been developed as a result of recent advances in nanotechnology, giving researchers more precise control over the characteristics of nanocomposites. To improve dispersion and interfacial bonding, for instance, functionalized nanoparticles can be used to increase their compatibility with the matrix material [31-33].

The focus of this study centers on tungsten nanoparticles, a particular class of nanoparticles that hold promise as a formidable agent against corrosion in construction materials [34]. Corrosion poses a significant challenge in civil engineering, often resulting in deterioration and reduced lifespan of structures [35]. Tungsten, renowned for its exceptional anti-corrosion properties, has been investigated for its potential to mitigate the corrosive effects of harsh environments on construction materials. Burkov and Chigrin [36], performed an experimental study on the effect of tungsten, molybdenum, nickel and cobalt on the corrosion and wear performance of Fe-based metallic glass coatings. They concluded that doping metallic glass coatings with tungsten and nickel improves anticorrosion properties in both high-temperature oxidation and electrochemical corrosion.

Several other studies have confirmed the protective effect of tungsten against corrosion, for instant, Nishimura et al. [37] investigated the corrosion behavior of tungsten-bearing steel in a wet/dry environment containing chloride ions. This investigation aimed to address the challenges posed by force shortage, necessitating the use of weathering steels for minimal maintenance. Their findings indicated that utilizing tungsten coating and surface treatment in coastal areas had a beneficial effect on corrosion resistance. Liu et al. [38] explored the effects of a tungsten pre-implanted layer on the corrosion and electrochemical characteristics of amorphous carbon films on stainless steel. Their experimental findings demonstrated that the pre-implemented tungsten coating layer enhances the corrosion resistance of steel. Additionally, Xavier [39] delved into the impact of superior surface protection, mechanical strength, and hydrophobic properties of salinized tungsten carbide nanoparticles encapsulated within epoxy nanocomposite coatings on steel structures in marine environments. The results highlighted that the newly developed mercapto-propyltrimethoxysilane (MPTMS)/WC nanoparticles within the neat epoxy (EP) coated mild steel nanocomposite exhibited superior corrosion protection and enhanced hydrophobic behaviors (with a water contact angle of 144°). Furthermore, the investigated coatings displayed significant improvements in mechanical properties. Consequently, the compatibility of EP, MPTMS, and WC nanoparticles was established through their superior hydrophobicity, improved corrosion protection, and enhanced mechanical properties

Furthermore, beyond its anti-corrosion prowess, tungsten nanoparticles exhibit a dual role by reinforcing the mechanical integrity of materials [40]. The incorporation of these nanoparticles into steel alloys has demonstrated the capacity to enhance strength, ductility, and overall structural performance [41]. Increased tungsten particle content in tungsten-chromium metal matrix composite coatings increases wear resistance, according to Bartkowski et al. [42]. Such attributes are paramount in environments where construction materials are subjected to substantial loads, pressures, and potentially corrosive elements. Dong et al. [43], suggested the addition of tungsten carbide to enhance the mechanical properties of a graphene/copper matrix, replacing the low-dimensional nano-carbon materials as the interfaces between the nano-carbon materials and the metal matrix always present a significant challenge for achieving the best enhancement effects. They concluded that the addition of small amounts of tungsten can lead to a significant improvement in the tensile strength of the metal composite matrix. Research by Zhang et al. [44] investigated the use of tungsten fibers as reinforcement in polymer composites, highlighting their superior stiffness, strength, and thermal stability compared to conventional fibers such as glass or carbon. Moreover, studies by Liu et al. [45] and Chen et al. [46] focused on tungsten-reinforced cementitious composites, demonstrating improvements in flexural and impact resistance, as well as radiation shielding properties.

In a broader context, water's status as a precious and limited resource accentuates the imperative to develop efficient strategies for anticipating and addressing water loss. Wastewater treatment plants emerge as exceptional tools for curbing water waste and recycling utilized water. However, the corrosive tendencies of wastewater posed a challenge, demanding a solution to mitigate the impact of corrosion. Because of its unique properties, including high density and excellent corrosion and radiation protection, tungsten is particularly interesting for applications in nuclear facilities and infrastructure exposed to extreme environmental conditions. Tungsten is a very promising candidate for the design of wastewater treatment plant structures, providing protection against the ravages of aggressive water.

Within the purview of this specific inquiry, our primary focus is directed towards an in-depth exploration of the mechanical implications stemming from the introduction of tungsten nanoparticles into the components comprising wastewater treatment plant structures. This comprehensive analysis focuses on the response of these structures to external bending loads, encompassing both static and hydrostatic forces. Utilizing the refined plate theory (RPT), we simulate the WWTP component made of stainless steel analytically. We employ Hamilton's principle to construct the equations of motion for the steel plate, while Navier's analytical solutions provide equilibrium solutions for a simply supported steel panel. Our investigation encompasses critical parameters, including the volume percentage of tungsten nanoparticles seamlessly integrated into the steel matrix. The foundation of the wastewater treatment plant (WWTP) is envisaged as a plate structure, subjected to a uniformly distributed static load originating from the stored water. In tandem, the WWTP structure's walls are postulated as plates, subjected to a linearly varying hydrostatic load. At the surface area, the load is null, whereas at the bottom, it attains its maximum magnitude. Moreover, this research unveils previously unreported outcomes, casting a fresh spotlight on the role of tungsten nanoparticles as enhancers for steel alloy elements.

In essence, by elucidating the mechanical implications of incorporating tungsten nano-sized particles into wastewater treatment plant structures (WWTP), this study breaks new ground by presenting novel results that advance our understanding of materials science in the context of sustainable engineering. By uncovering novel results and exploring the complex interplay between water preservation, structural integrity and innovative materials, our research highlights the potential of tungsten nanoparticles to serve as effective improvers for steel alloying elements. These ideas not only contribute to the current discourse on sustainable engineering practices, but also pave the way for the development of a resilient infrastructure capable of meeting the challenges of wastewater treatment operations.

In conducting this analysis, we embrace the well-established refined plate theory (RPT), a framework that intricately captures both bending and shear elements within the plate thickness [49]. This theory has garnered a track record of precision and reliability, having previously demonstrated its prowess in effectively prognosticating the intricate mechanical responses of composite plates. As an illustration, Zenkour [50] used the refined plate theory to examine the bending behavior of a functionally graded material plate under varying bending loads. Ahmed et al. [51] and Bouamama et al. [52] used it for the dynamic analysis of composite plates and beams respectively. Their findings concluded that the refined plate theory aligns seamlessly with observations and yields notably more accurate outcomes when juxtaposed against the generalized higher-order deformation theory of Reddy [53]. This underscores the efficacy and precision of the refined plate theory in capturing the intricate mechanical nuances of complex plate structures.

3.1 Kinematics

The displacement field of a material point situated at coordinates (x, y, z) within the plate, as governed by the Refined Plate Theory (RPT), can be delineated in the ensuing manner:

$\begin{gathered}U(x, y, z)=u_0(x, y)-z \frac{\partial w_b(x, t)}{\partial x}+f(z) \frac{\partial w_s(x, y)}{\partial x} \\ V(x, y, z)=v_0(x, y)-z \frac{\partial w_b(x, t)}{\partial y}+f(z) \frac{\partial w_s(x, y)}{\partial y} \\ W(x, y, z)=w_b(x, y)+w_s(x, y)\end{gathered}$          (6)

The shape function expressing shear across the thickness takes the following form [54]:

$f(z)=\sinh ^{-1}\left(\frac{3 z}{h}\right)-z \frac{6}{h \sqrt{13}}$       (7)

The deformation components associated with the utilized refined plate theory are acquired through the following procedure:

$\begin{gathered}\varepsilon_x=\frac{\partial U}{\partial x} ; \varepsilon_y=\frac{\partial V}{\partial y} ; 2 \gamma_{x y}=\frac{\partial U}{\partial y}+\frac{\partial V}{\partial x} \\ 2 \gamma_{x z}=\frac{\partial U}{\partial z}+\frac{\partial W}{\partial x} ; 2 \gamma_{y z}=\frac{\partial V}{\partial z}+\frac{\partial W}{\partial y}\end{gathered}$           (8)

The constitutive stress-strain relationships for the steel matrix infused with tungsten nanoparticles, are delineated as follows:

$\left\{\begin{array}{c}\sigma_{\mathrm{x}} \\ \sigma_{\mathrm{y}} \\ \tau_{\mathrm{xy}} \\ \tau_{\mathrm{xz}} \\ \tau_{\mathrm{yz}}\end{array}\right\}=\left[\begin{array}{ccccc}C_{11}^T & C_{12}^T & 0 & 0 & 0 \\ C_{12}^T & C_{22}^T & 0 & 0 & 0 \\ 0 & 0 & C_{66}^T & 0 & 0 \\ 0 & 0 & 0 & C_{44}^T & 0 \\ 0 & 0 & 0 & 0 & C_{55}^T\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{x}} \\ \varepsilon_{\mathrm{y}} \\ \gamma_{\mathrm{xy}} \\ \gamma_{\mathrm{xz}} \\ \gamma_{\mathrm{yz}}\end{array}\right\}$         (9)

In this context, $C_{i j}^T$ symbolizes the diminished elastic constants of the amalgamated system, encompassing both the tungsten reinforcement and steel matrix. These constants are established through the application of Mori-Tanaka's homogenization methodology in which:

$\begin{aligned} & C_{11}^T=C_{22}^T=\frac{\left(1-v_{\text {hom }}\right) E_{\text {hom }}}{\left(1+v_{\text {hom }}\right)\left(1-2 v_{\text {hom }}\right)} ; \\ & C_{12}^T=\frac{v \text { Ehom }}{\left(1+v_{\text {hom }}\right)\left(1-2 v_{\text {hom }}\right)} ; \\ & C_{66}^T=C_{55}^T=C_{44}^T=\frac{E_{h o m}}{\left(1+v_{h o m}\right)} . \\ & \end{aligned}$          (10)

3.2 Equations of motion

Hamilton's principle is invoked to formulate the equations of motion for the nano-composite plate:

$\int_0^t(\delta U+\delta V) d t=0$          (11)

where, δU and δV correspond to the virtual variations of the internal strain energy within the plate and the external energy induced by the external bending loads, respectively.

The expression for the virtual strain energy imparted by the plate is as follows:

$\begin{gathered}\delta U=\int_0^A \int_{-h / 2}^{h / 2}\left(\sigma_x \delta \varepsilon_x+\sigma_y \delta \varepsilon_y+\tau_{x y} \delta \gamma_{x y}+\right. \left.\tau_{y z} \delta \gamma_{y z}+\tau_{x z} \delta \gamma_{x z}\right) d A d z\end{gathered}$        (12)

substituting the strain expression from Eqs. (8)-(9) into Eq. (12), the resultant outcome is as follows:

$\begin{gathered}\delta U=\int_A\left(N_x \frac{\partial \delta u_0}{\partial x}-M_x^b \frac{\partial^2 \delta w_b}{\partial^2 x}+M_x^s \frac{\partial^2 \delta w_s}{\partial^2 x}+N_y \frac{\partial \delta v_0}{\partial y}-\right. \\ M_y^b \frac{\partial^2 \delta w_b}{\partial^2 y}+M_y^s \frac{\partial^2 \delta w_s}{\partial^2 y}+N_{x y}\left(\frac{\partial \delta u_0}{\partial y}+\frac{\partial \delta v_0}{\partial x}\right)+2 M_x^b \frac{\partial^2 \delta w_b}{\partial x \partial y}+ \\ \left.2 M_x^s \frac{\partial^2 \delta w_s}{\partial x \partial y}+Q_{y z}\left(\frac{\partial \delta w_s}{\partial y}\right)+Q_{x z}\left(\frac{\partial \delta w_s}{\partial x}\right)\right)\end{gathered}$     (13)

In Eq. (13), the stress and moment resultants are expressed in the following form:

$\begin{gathered}N_{i j}=\int_{-h / 2}^{h / 2} \sigma_{i j} b d z ; i, j=x, y \\ M_{i j}^b=\int_{-h / 2}^{h / 2} z \sigma_{i j} b d z ; i, j=x, y \\ M_{i j}^s=\int_{-h / 2}^{h / 2} f(z) \sigma_{i j} b d z ; i, j=x, y \\ Q_{i z}=\int_{-h / 2}^{h / 2} g(z) \tau_{i z} b d z ; i=x, y\end{gathered}$          (14)

In the case of the nano-reinforced steel plate experiencing mechanical bending loads denoted as 'q', the virtual external work engendered by these loads is defined by Equation.

$\delta V=-\int_A \int_{-h / 2}^{h / 2} q\left(\delta w_b+\delta w_s\right) d A d z$        (15)

By employing Eqs. (13) and (15), within Eq. (11), and subsequently performing integration by parts while accounting for the coefficients of δu₀, δv₀, δw_b, and δw_s, the resulting equations of motion pertinent to the refined plate theory are derived as follows:

$\begin{gathered}\delta u_0: \frac{\partial N_x}{\partial x}+\frac{\partial N_{x y}}{\partial y}=0 \\ \delta v_0: \frac{\partial N_y}{\partial y}+\frac{\partial N_{x y}}{\partial x}=0 \\ \delta w_b:\left(\frac{\partial^2 M_x^b}{\partial x^2}-2 \frac{\partial^2 M_{x y}^b}{\partial x \partial y}+\frac{\partial^2 M_y^b}{\partial y^2}\right)+q=0 \\ \delta w_s:\left(\frac{\partial^2 M_x^S}{\partial x^2}-2 \frac{\partial^2 M_x^S}{\partial x \partial y}+\frac{\partial^2 M_y^S}{\partial y^2}\right)+\left(\frac{\partial Q_{x z}}{\partial x}+\frac{\partial Q_{y z}}{\partial y}\right)+ q=0\end{gathered}$            (16)

3.3 Navier’s solutions for simply supported plates

Contemplate a rectangular steel plate (as depicted in Figure 5), which is simply supported. The boundary conditions for a simply supported steel plate reinforced with spherical shaped tungsten nanoparticles are defined as follows:

$x=0$ and $x=a: v_0=w_b=w_s=N_x=M_x^b=M_x^s=0$

$y=0$ and $y=b: u_0=w_b=w_s=N_y=M_y^b=M_y^s=0$        (17)

The admissible displacement functions according to Navier, expressed as trigonometric series and fulfilling the boundary conditions outlined in Eq. (17), are presented as follows:

$\begin{aligned} u(x, y) & =\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} U_{m n} \cos (\lambda x) \sin (\mu y) \\ v(x, y) & =\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} V_{m n} \sin (\lambda x) \cos (\mu y) \\ w_b(x, y) & =\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} W_{b m n} \sin (\lambda x) \sin (\mu y) \\ w_s(x, y) & =\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} W_{s m n} \sin (\lambda x) \sin (\mu y)\end{aligned}$       (18)

here, (U_mn, V_mn, W_bmn, W_smn) are the arbitrary parameters to be determined. where, λ=mπ/a and μ=nπ/b. For the ensuing outcomes, an increased number of iterations (n=201, m=201) was employed to achieve greater precision and accuracy in the results. This decision was based on the available computational resources, enabling more comprehensive calculations and enhanced precision.

In our analysis, we posit that the transverse bending load "q" manifests as hydrostatical forms, as illustrated in Figure 6. These bending loads can be effectively represented through the utilization of the Double-Fourier series, as depicted below:

$q(x, y)=\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} Q_{m n} \sin (\lambda x) \sin (\mu x)$       (19)

where,

$Q_{m n}=\left\{\frac{8 q_0}{m n \pi^2} \cos (m \pi)\right.$       (20)

6.png

Figure 6. Simply supported steel plate subjected to external bending linear hydrostatic load

Ultimately, to attain the analytical solutions, the outcomes of the substitution can be organized into the subsequent matrix format:

$\left[\begin{array}{llll}k_{11} & k_{12} & k_{13} & k_{14} \\ k_{21} & k_{22} & k_{23} & k_{24} \\ k_{31} & k_{32} & k_{33} & k_{34} \\ k_{41} & k_{42} & k_{43} & k_{44}\end{array}\right]\left\{\begin{array}{c}U_{m n} \\ V_{m n} \\ W_{b m n} \\ W_{s m n}\end{array}\right\}=\left\{\begin{array}{c}0 \\ 0 \\ Q_{m n} \\ Q_{m n}\end{array}\right\}$           (21)

where,

$\begin{gathered}k_{11}=-A_{11} \lambda^2-A_{66} \mu^2 ; k_{12}=-\lambda \mu\left(A_{66}+A_{12}\right) ; \\ k_{13}=B_{11} \lambda^3+\left(2 B_{66}+B_{12}\right) \lambda \mu^2 ; \\ k_{14}=B_{s 11} \lambda^3+\left(2 B_{s 66}+B_{s 12}\right) \lambda \mu^2 \\ k_{21}=k_{12} ; k_{22}=-\left(A_{11} \lambda^2+A_{66} \mu^2\right) ; \\ k_{23}=B_{22} \mu^3+\left(2 B_{66}+B_{12}\right) \lambda^2 \mu ; \\ k_{24}=B_{s 22} \mu^3+\left(2 B_{s 66}+B_{s 12}\right) \lambda^2 \mu ; \\ k_{31}=k_{13} ; k_{32}=k_{23} ; \\ k_{33}=-D_{11} \lambda^4+2 \lambda^2 \mu^2\left(D_{12}+2 D_{66}\right)+D_{22} \mu^4 ; \\ k_{34}=-D_{s 11} \lambda^4+2 \lambda^2 \mu^2\left(D_{s 12}+2 D_{s 66}\right)+D_{s 22} \mu^4 ; \\ k_{41}=k_{14} ; k_{42}=k_{24} ; k_{43}=k_{34} ; \\ k_{44}=-A_{s s 11} \lambda^4-2 \lambda^2 \mu^2\left(A_{s s 12}+2 A_{s s 66}\right) \\ +A_{s s 22} \mu^4+A_{c c 55} \lambda^2+A_{c c 44} \mu^2 ;\end{gathered}$         (22)

$\begin{gathered}\left\{A, B, D, A_s, B_s, A_{s s}\right\}= \int_{-h / 2}^{\frac{h}{2}}\left\{1, z, z^2, f(z), z f(z), f(z)^2\right\} C_{i j}^T b d z e \\ \left\{A_{c c}\right\}=\int_{-h / 2}^{h / 2}\left\{g(z)^2\right\} C_{i j}^T b d z\end{gathered}$           (23)

Water is an invaluable resource, vital for human consumption and crucial in sustaining agricultural irrigation. As the need for efficient water management intensifies, Wastewater Treatment Plants (WWTPs) emerge as pivotal solutions, effectively recycling used water. However, safeguarding the integrity of these WWTP structures becomes paramount. Not only must they withstand chemical attacks to ensure water quality, but their mechanical resilience also demands attention. By fortifying WWTPs against both chemical and mechanical threats, we secure a sustainable future for water resources, fostering a resilient environment for generations to thrive.

Hence, our study casts a spotlight on the profound ramifications of introducing tungsten nanoparticles into the mechanical framework of WWPT steel components. In our comprehensive exploration, we meticulously examined an array of critical parameters. These encompassed the manipulation of tungsten reinforcement volume proportions within the steel matrix – a fundamental element of the monolithic structure. Our investigation also entailed a meticulous analysis of geometric ratios inherent in the plate configuration. Furthermore, we methodically evaluated the responses of these components under diverse bending conditions, encompassing the hydrostatic loads.

Our research has yielded a compelling array of insights, each serving to underscore the pivotal significance of our findings:

1. Introducing nano-sized tungsten into a stainless-steel matrix leads to notable enhancements in elastic characteristics of the composite, encompassing Young's modulus, bulk modulus, and shear modulus. The extent of improvement is intricately linked to the volume concentration of these additives.
2. The incorporation of an optimal volume of tungsten, precisely evaluated at 30%, into a steel matrix yields remarkable improvement. This improvement elevates the elastic constants ‘Etot’ of the composite by an impressive 31%, in stark comparison to a steel matrix infused with 30% titanium dioxide, which registers an improvement of up to 13%.
3. Wastewater treatment plant structures fortified with nano-tungsten particles manifest an exceptional capacity to optimize their response against external mechanical loads. This reinforcement markedly enhances their flexural strength and overall performance.
4. Under hydrostatic external bending loads, steel plates reinforced with tungsten nanoplatelets exhibit a striking reduction in transverse displacement. This reduction accentuates the effectiveness of nano-reinforcements in curtailing structural deformations across diverse external mechanical loading scenarios.
5. By introducing 30% of tungsten particles into the steel matrix, transverse displacement experiences a noteworthy decrease of 26%, while axial displacement diminishes by 25%.
6. Beyond their corrosion-resistant properties, tungsten particles exhibit an additional layer of mechanical advantages, further enhancing their pivotal role in structural optimization.

Whilst this paper focuses on the analytical analysis of the hydrostatic bending of the panels of a wastewater treatment plant made of a new advanced composite tungsten steel material, it offers significant theoretical contributions that will advance the field of nanotechnology in general and nanomaterials in particular, as well as structural engineering, nevertheless the authors acknowledge the limitations of plate theories, particularly regarding the linear elasticity assumption. Plate theories typically rely on the assumption of linear elasticity, which implies that materials behave linearly under loading conditions. As we strive to extend our research further, we recognize the importance of addressing non-linear deformation assumptions. Future investigations will aim to overcome these limitations by incorporating non-linear deformation models to provide more accurate predictions of plate behavior under various loading conditions.