Combined Behavior of Reinforced Concrete Out-of-Plane Parts Beams Encased with Steel Section

Abstract

This research investigated and compared the structural behavior of reinforced concrete straight beams and beams made with out-of-plane parts. This study focused on the influence of the location and number of out-of-plane parts, as well as encasing the beams with a steel section, on the ultimate strength, deflection, and rotation in addition to the ductility, energy absorption, and failure mode. A total of nine beams were modelized numerically, divided into three series. The first one included one straight beam, while the remaining two series included four beams each made with out-of-plane parts with and without steel sections. The beams with out-of-plane parts connected the two, three, four, and five concrete segments. The outcomes revealed that the beams made with out-of-plane parts showed less strength than straight beams, which increased the connected segments and reduced the ultimate strength capacity. The regular beam’s linearity was dissimilar to the zigzag beams, which showed a linearity of 32% and was reduced to 22%, 20%, 19.67%, and 16% for beam out-of-plane parts made with two, three, four, and five segments, respectively. Forming a zigzag in the plane of the beams reduced the cracking load, but the decrement depended on the number of parts, which led to more reduction in the yielding load. Concerning the deflection and deformations, the concrete straight beams failed in flexure, with maximum deflection occurring at the midspan of the beam, which was different for beams without plane parts, which showed a combined shear-torsional failure for which the maximum deformation occurred at the midspan with inclination of connected parts on the interior perpendicular axis. Encasing the beams’ out-of-plane parts with steel sections enhanced the structural behavior. The ductility and energy absorption of the out-of-plane parts beams were less than the straight ones, but encasing the beams with a steel section improved the ductility and energy absorption twice.

1. Introduction

1.1. Back Ground

The shape of the concrete member plays a significant role in the structural behavior of the structural member in addition to the architectural requirements, which may require unique textures for the aesthetics of buildings [1,2]. In many structures, such as low-rise and midrise buildings, the designer needs to change the straight path of the columns and beams to a non-straight path for architectural purposes or sometimes to fix the conflict points of the members with other equipment, such as the mechanical members (ducts, pipes, etc.). Sometimes, a change in the path of the beams is required to reduce the number of columns, as in many applications regarding this issue, such as the sky house of Tokyo, a complex structure in China, cross-bracing RC beams in metro stations in Budapest, and the balconies of theaters as seen in Figure 1 [3,4,5]. The change of path of RC beams changes the structural behavior of these beams against the applied loads, which offer varied strength, shear, bending, torsion, buckling, and other mechanical properties. The change in the concrete beam’s path causes the beams to suffer from combined behavior. Therefore, many researchers investigated the structural behavior of RC beams under combined loading (bending, shear, and torsion).

1.2. Literature Review

Owenite [6] investigated the effect of combined loading on the behavior of the RC beams. The studies focused on studying the behavior of variable steel reinforcement ratios (primary and transverse steel reinforcement). The results showed that the failure mode and crack shape were influenced by load type. Furthermore, the increase in steel ratio was more effective in enhancing the torsional strength and cracking resistance. Ali and Anis [7] investigated theoretically and experimentally the RC floor to spandrel beam to explore the effect of the loading configuration on the shear, flexural, and torsional behavior. The results showed that the ductility was reduced when the member was exposed to combined loads. Also, torsional stresses developed when the angle of the load was increased. Kamiński and Pawlak [8] presented an experimental and numerical investigation to study the effect of combined loading on the behavior of RC beams. The results showed that combining the torsion with a bending moment and shearing force reduced the stiffness of the RC beams. Elsayed et al. [9] investigated the influence of the change in the inclination angle of the cantilever’s reinforcement ratio of RC beams. The outcomes demonstrated that the increase in the steel reinforcement inclination angle had little effect on the cracking and ultimate load in addition to the stiffness but developed a torsional impact on the RC beams. Rafeeq [10] exposed the fixed-end RC beams to combined loading. The study results demonstrated that the subjection to bending in addition to the torsional shear decreased the torsional and bending strength. Amulu and Ezeagu [11] investigated the effects of combined loadings on the behavior of normal-strength reinforced concrete beams, including torsional moments, bending moments, and shear forces. This study, which utilized standard design codes and experimental work, concluded that the beam failures were due to combined torsion, shear, and bending moments. The longitudinal and transverse reinforcement ratios influence directly the beams’ capacity to resist the applied combined loads. The study also demonstrated that the beams’ capacity to withstand the effects of combined loads could be increased by providing the reinforcement obtained from torsional design calculations and the total amount of bending and shear reinforcement required at ultimate loads.

1.3. Gaps in Research

The structural behavior of reinforced concrete (RC) beams under combined loading conditions has been the subject of several studies; however, the majority of these investigations have concentrated on straight beams. Prior studies have looked at how RC beam strength, ductility, and failure modes are affected by reinforcement ratios; load configurations; and combined shear, torsion, and bending. The combined structural behavior of RC beams with out-of-plane sections (WPP), which reflect non-linear geometries frequently required by functional or architectural requirements, is still not well understood, nevertheless. The impact of the quantity and arrangement of out-of-plane components on crucial elements including ultimate strength, deformation properties, and ductility has not yet been examined in previous research. Furthermore, there has not been a thorough investigation into how encasing such beams in steel sections can enhance their structural performance under challenging stress situations.

1.4. Aims of Study

By meticulously studying the combined structural behavior of reinforced concrete beams with out-of-plane components, this work aimed to fill these gaps. The main goals were as follows:

• Examine how the quantity and arrangement of out-of-plane components affect the RC beams’ ultimate strength, deflection, rotation, ductility, energy absorption, and failure modes.
• Examine the differences in performance between straight RC beams and RC beams with out-of-plane sections under static loads.
• Assess how well RC beams with out-of-plane portions may be encased in steel sections to improve their structural performance, especially in terms of strength, ductility, and energy absorption.

In order to increase performance in architectural and structural applications, this work intended to fulfill these goals and offer crucial insights into the design and optimization of RC beams with complicated geometry.

2. Finite Element Modeling

A dimensional FE model using ANSYS APDL 2021 R1 software was employed to analyze the concrete beams’ behavior. Employing the FE offered the advantage of understanding the behavior of beams’ out-of-plane parts. This technique provided a wide range of complex results. The comparison between the experimental and numerical analysis by FE demonstrated that the FE slightly overestimated stresses due to the concrete being more homogeneous than it in fact was. However, the FE results, such as the cracking load, ultimate load, deflection, etc., were comparable and sufficiently accurate compared with the experimental work. In this section, the WPP beam analyses are carried out by utilizing FEM, as discussed in the previous section. The investigations were divided into three series, which present the influences of different parameters on the behavior of WPP beams. The first series was conducted on straight RC beams, while the second and third series included eight specimens (four models for each series) made with out-of-plane parts of concrete beams investigating the location and number of connecting segments. From this point of view, concrete is a non-homogeneous material with different behaviors regarding compression and tension; linearity and nonlinearity are different in compression and tension [12,13].

Over the years, many research studies have explained the behavior of concrete and clarified its mechanical properties. FE ANSYS can model the same behavior concretely and with high accuracy. However, concrete modeling in the form of beams with different configurations is still challenging due to the sensitivity of the concrete element in the FE packages [14,15,16,17]. The brittle behavior of this element is the most critical status in the behavior of the concrete. This research presents an analysis of a beam with a novel configuration (beam without plane parts) considering the modeling of this type of beam. When modeling the WPP beam, it was necessary to define the behavior and material of the concrete beam component; the concrete, steel reinforcement, and section behavior depended on the failure theory for each material. A plastic-damage constitutive law was used to represent the concrete and steel behavior. The efficiency of the damage constitutive rule in precisely describing the nonlinear behavior of concrete and capturing the specific failure mode was initially confirmed by experimental results conducted on concrete beams [18,19,20]. This was performed to ensure that the rule was effective. To use finite element analysis (FEA) to model the WPP beam, it was essential to consider a wide variety of material properties in addition to the geometry of the beam. In the preprocessing stage, it was necessary to select the appropriate material components with great care and precisely specify the behavior of those components. The loading and boundary conditions must first be determined before the experiment was conducted to reproduce experimental testing in the same conditions. It was possible to gain a better understanding of the behavior of materials by using constitutive models. It was possible to use the concrete plasticity failure (CPF) model for concrete, which incorporated smearing and brittle yield models to accurately represent the behavior of concrete. The material’s response to external forces could be determined with the assistance of these models, which also helped to forecast the material’s failure characteristics [20].

Performing a finite element analysis (FEA) on the WPP beams was necessary to validate their structural capabilities. The software ANSYS APDL was employed for this purpose. As part of the verification process, the load–displacement relationships, ultimate maximum loads, deflections, and failure modes generated by the FEA were compared with the results of experimental testing carried out by investigators from different institutions. As shown in Figure 2, the research frequently used a finite element model composed of many elements. Enhancements were made to the mesh to more precisely depict the delicate aspects of the joint geometry, and modifications were made to the input values to ensure that convergence conditions were consistently met. The analysis’s accuracy and reliability were evaluated through the convergence criteria. In this particular instance, non-convergence was seen when the open and close shear transfer coefficient values dropped below 0.2, indicating that adjustments were required [21]. On the other hand, optimal alignment was successfully achieved when the number of elements exceeded 8638.

3. Material Modeling

In ANSYS, modeling concrete involves defining the material properties that capture its quasi-brittle behavior and differentiate between compression and tension. The stress–strain curve of concrete typically exhibits linear behavior in both compression and tension until the yielding point is reached. After yielding, the concrete develops cracks and gradually loses strength, leading to failure through crushing. The cracks continue to propagate until they reach their maximum extent. Once the maximum strain is reached, the curve descends into the softening region and ultimately fails. To define concrete behavior in ANSYS, material properties such as the ultimate uniaxial compressive strength, ultimate uniaxial tensile strength, elastic modulus (E), modulus of rupture, Poisson’s ratio, and a reduction factor of stiffness are utilized. The constitutive model for standard concrete in ANSYS is based on damage plasticity, which simulates the gradual loss of strength and stiffness due to cracking and crushing. In ANSYS, the SOLID65 element with eight nodes and three degrees of freedom per node is commonly used to simulate concrete. This element can accommodate plastic deformation, yield, and crushing, making it suitable for capturing the behavior of concrete under various loading conditions [22].

Regarding the steel section, a shell element (membrane41) with four nodes was used, and LINK180 was used to represent the steel rebar with variation in the stress–strain behavior for both elements as seen in

Table 1. Properties of materials.

Concrete type	Element	Adopted stress–strain curve
Concrete	SOLID65	Multilinear stress–strain curve
Steel section	Membrane41	Bilinear stress–strain curve
Steel rebar	LINK180	Bilinear stress–strain curve
Steel plate	SOLID185	Linear stress–strain curve

. A bilinear isotropic material model in ANSYS represents the steel section and rebar. This material model assumes the same behavior in both compression and tension, and the Von Mises failure criterion is typically employed to evaluate failure in the steel components. By properly defining the material properties and using appropriate elements in ANSYS 2021 R1, it is possible to accurately simulate the behavior of concrete, steel sections, and steel reinforcement and analyze the structural response of WPP beams under different loading conditions [22]. In the finite element modeling of WPP beams, the connection between the concrete and steel beams is commonly represented using perfect bond theory. Using this proposed modeling technique, the finite element analysis can effectively describe the nonlinear behavior of WPP beams, including the behavior of concrete beams encased with steel sections. This allows for a more realistic simulation of the overall response and performance of WPP beams. It is important to note that this proposed modeling technique’s specific details and approaches may vary depending on the study and the researchers involved.

4. Verification Results

During a bending test, concrete usually exhibits nonlinear characteristics until it reaches the point of failure, when the material collapses, typically due to the rupture of its tensile fibers. When subjected to static stresses, the main characteristics of concrete beams are their strength, which refers to the most significant load they can bear, and stiffness, which measures the deflection they experience when a force is applied. The four-point bending test is widely regarded as the most reliable method in the literature for assessing the behavior of a beam.

Table 2. Dimensions and hardened properties of the experimental specimens.

Beam ID	Section, mm	Overall Span, mm	Shear Span, mm	Compressive Strength, MPa	Modulus of Elasticity, GPa	Poisson’s Ratio
NSC-1OP	150 × 200			35	27.6	0.22
HSC-1OP		1500	450	75	40.7	0.2
HSC-1OPT				75	40.7	0.2

details three specimens selected from the available literature data; the table also includes the specimens’ dimensions, material qualities, and test parameters. Figure 3 shows the finite element model shows the mesh of experimental beams in ANSYS.

Table 3. Verification results include the failure load.

Beam ID	Vu, Exp. (kN)	Vu, ANS. (kN)	$\frac{\mathbf{V}\mathbf{u}, \mathbf{E}\mathbf{x}\mathbf{p}}{\mathbf{V}\mathbf{u}, \mathbf{A}\mathbf{N}\mathbf{S}.}$	∆, Exp. (mm)	∆, ANS. (mm)	$\frac{∆, \mathbf{E}\mathbf{x}\mathbf{p}}{∆, \mathbf{A}\mathbf{N}\mathbf{S}.}$
NSC-1OP	369.2	364.77	98.80%	61.21	54.97	89.80%
HSC-1OP	385.99	363.99	94.30%	51.22	51.15	99.86%
HSC-1OPT	460.49	467.40	101.50%	49.62	48.63	98%

presents the calculated and observed outcomes for the load–deflection curve, ultimate load, and maximum deflection of the specimens (NSC-1OP, HSC-1OP, and HSC-1OPT) utilized for validation, as reported by Mohsin et al. [23]. The expected and observed behaviors regarding the load–deflection curves were almost indiscernible. Table 3 and Figure 4 exhibit the verification results in terms of the load–deflection relationship. The experimental to numerical ultimate load ratio varied between 98.8% and 101.5%. The experimental and numerical figures for the maximum deflection were also included in Table 3. As a result of the fragile characteristics of concrete, the experimental to theoretical deflection ratios were deemed acceptable, although they were lower than the corresponding load values. Experimental testing of crack propagation was not feasible due to the simplicity and assumptions made in the utilized finite element method (FEM). The qualities of the concrete explained the difference between the conditions of the theoretical and experimental tests. The actual and theoretical evaluations of failure load, deflection, and fracture pattern were compared, providing assurance for the application of the parameters. The final load analysis of the modeled beams revealed that due to ANSYS’s inability to converge, the beams were incapable of sustaining any further loads. In order to perform an analytical inspection of the proposed shapes of WWP, it was feasible to rely on previous approaches for modeling geometric materials due to the good convergence between the theoretical and practical results using the ANSYS program.

5. Analytical Study

This study examined the behavior of beams’ out-of-plane components by analyzing nine specimens. These specimens were constructed using regular concrete and then reinforced using a steel box positioned at the midpoint of the reinforced concrete beams. The investigation involved tests on beams with cross-sectional dimensions of 150 × 200 mm throughout 2000 mm. The beams were developed following the 2019 ACI code [24], as specified in a previous study by Mohsin et al. [23,25]. The design involved using six steel rebars, with a diameter of 12 mm, for longitudinal reinforcement and rebars, with a diameter of 8 mm, for transverse reinforcement. To prevent rapid failure in shear and torsion, stirrups were positioned at a distance of 81 mm due to the material’s eccentricity and non-homogeneous behavior. This was done since the concrete was expected to experience shear-torsional failure, as depicted in Figure 5. The compressive strength of the concrete and steel reinforcement for these beams was 50 MPa. The yield strength of the D8 mm steel reinforcement was 559 MPa, while the yield strength of the D12 mm steel reinforcement was 413 MPa. The variables in this study were testing multiple configurations of WPP beams, including a straight beam (SP); beams with a connection of two segments (WPP1); and beams with connections of three, four, and five segments (WPP2, WPP3, and WPP4).

The beams with the same configurations were reinforced with a steel section (a box of 50 × 200 mm) placed at the midsection to enhance the composite behavior of such beams, as revealed in

Table 4. Details of specimens (series 1).

ID	f’c (MPa)	EC (GPa)	Plane Parts Number	Box Steel Section
SP	50	29.60	NSC beam with straight segment	-
WPP1	50	29.60	NSC beam with two segments	-
WPP2	50	29.60	NSC beam with three segments	-
WPP3	50	29.60	NSC beam with four segments	-
WPP4	50	29.60	NSC beam with five segments	-
WPP1-C	50	29.60	NSC beam with two segments	50 × 100 mm
WPP2-C	50	29.60	NSC beam with three segments	50 × 100 mm
WPP3-C	50	29.60	NSC beam with four segments	50 × 100 mm
WPP4-C	50	29.60	NSC beam with five segments	50 × 100 mm

and Figure 5. The purpose of these variables was to compare the structural behavior of out-of-plane parts beams with the corresponding straight beams control beam; inspect the combined flexural, shear, and torsional behavior; and investigate the possibility of the steel section to reduce the effect of the eccentricity material and non-homogeneous behavior on the torsional behavior of the beams. Steel plates with thicknesses of 25 mm were placed at the loading and support zones. Table 4 displays the variables examined in this study. This analysis examined the behavior of WPP beams under fixed-end supports. The aim was to study the combined effects of flexure, shear, and torsion by applying a two-point load. The objective was to evaluate the structural performance of the tested beams. In ANSYS, the load was distributed evenly throughout the nodes of the steel plate by dividing the total load. A 25 mm thick steel plate was used to reduce the stress concentration on the loading zone’s concrete elements and protect the SOLID65 from unexpected crushing.

5.1. Test Setup and Boundary Conditions

In the analysis of the structural model, appropriate boundary conditions were essential for accurate simulations to obtain the actual behavior. The model incorporated fixed support conditions at designated nodes to prevent any translational or rotational movement, specifically restraining the base in all three translational directions (X, Y, and Z), as well as rotational degrees of freedom (RX, RY, and RZ). This setup simulated a rigid connection, ensuring stability during loading conditions. External loads were applied at specific locations as shown in Figure 6 below on the model to simulate realistic operational scenarios, including vertical static forces, with magnitudes. Defined contact surfaces between different components enabled the simulation of interaction effects, with contact conditions (e.g., bonded and frictionless) specified to capture mechanical behavior accurately during loading. Additionally, a finer mesh was applied in areas of high-stress concentration to ensure accurate results, with the mesh quality assessed to ensure sufficient convergence of the simulation results. These boundary conditions collectively provided a comprehensive framework for accurately simulating the structural behavior of the model under various loading scenarios, to investigate the combined behavior of flexural, shear, and torsional behavior of WPP beams with the application of two-point load. In ANSYS, the load was applied as the total load divided by the number of nodes of the steel plate. The steel plate was modeled with a thickness of 25 mm to avoid the stress concentration on the concrete elements under the loading zone, which prevented the SOLID65 from sudden crushing as revealed in Figure 6. Modeling of the support in this model was implemented by connecting the supports directly using the perfect bond theory. The perfect bond theory in ANSYS modeling simplifies the interaction between bonded materials by assuming that they act as a single entity without any relative motion. This approach can be beneficial for certain applications where the interface behavior is not critical, allowing for straightforward simulations with reduced computational complexity. However, this theory had several drawbacks that could significantly affect the accuracy of the results. One major drawback was that the perfect bond theory did not account for any slip or separation that may occur at the interface under load, but this drawback did not affect the structural behavior that the stresses initiated and moved toward the connection zones without the initiation of stresses that caused slip.

5.2. General Behavior

As shown by the load-midspan deflection response, as revealed in

Table 5. Results of OPP in series 1.

ID	Yield Load (kN)	Yield Displacement (mm)	Failure Load (kN)	δ Corr. to	Linearity	Ductility	Energy Absorption
				Pu (mm)	%
SP	189.70	8.03	592.80	29.53	0.32	3.676	1160.69
OPP1	77.97	10.17	354.40	31.88	0.22	3.135	1044.62
OPP2	54.36	6.59	267.30	22.36	0.20	3.391	940.16
OPP3	41.77	6.17	210.36	21.44	0.20	3.473	846.14
OPP4	24.56	3.82	156.46	16.77	0.16	2.393	761.53
OPP1-C	146.38	13.94	563.23	36.98	0.26	6.144	1734.07
OPP2-C	118.00	8.63	496.66	25.04	0.24	6.138	1419.64
OPP3-C	105.00	8.38	428.44	23.58	0.25	5.696	1201.52
OPP4-C	74.00	5.24	366.79	17.91	0.20	3.088	1012.83

and Figure 7, the SP beam had a higher ultimate load than the WPP beams. This was demonstrated by the SP beam having a more significant ultimate load. When the SP beam was subjected to the imposed stress, it displayed a reduced center deflection, a remarkable feature. The findings showed that the SP beam possessed more excellent beam ductility and stiffness than the other investigated beams. Concerning this particular instance, the SP beam exhibited a higher ultimate load, which indicated that it could withstand larger weights before it failed compared with the other beams. A greater rigidity could be seen in the SP beam than in the other beams. This indicated that the SP beam had fewer deflections or displacements when subjected to an equivalent applied load, suggesting a stronger resistance to deformation. Several factors could contribute to the ultimate load and deflection response variations among the beams. These factors included variations in the material qualities, such as compressive strength and reinforcement details, as well as variations in the geometry and design of the beams. These factors could potentially affect the beams’ capacity to carry loads, as well as their ductility and stiffness. Implementing the primary parameter was decided upon to improve the performance of the beam. This was accomplished by enclosing the concrete beam with a steel box section, which covered the area out of the plane. Increasing the load-bearing capacity and the overall strength of the beam was accomplished by incorporating steel pieces into the concrete, which increased the strength of the concrete. Additional reinforcement against shear, flexural, and torsional forces was provided by the steel section of the structure. Compared with typical RC beams that were bent, the load–deflection correlation of the RC beams tested with the out-of-plane component is depicted in Figure 7.

Both the shape and the direction of the force applied to the beam were altered due to the presence of the out-of-plane component, which ultimately resulted in a combined load of bending and torsion. The beams’ loading capacity and ductility were both reduced due to the combination of loading circumstances that were considered. Because of the discovered decrease in ductility and load-carrying capacity, the impact of out-of-plane components must be considered when designing beams. The behavior and performance of beams with out-of-plane components could be optimized by using strategies such as using high-strength steel to supply additional torsional reinforcement. This resulted in increased ductility and more excellent load resistance. It is seen in Figure 8 that the structural response of the RC beams that were tested with the out-of-plane component exhibited reduced ductility when compared with the typical RC beams that were bent. Both the shape and the direction of the force given to the beam were altered due to the presence of the out-of-plane comment, which ultimately resulted in a combined load of bending and torsion. As a result of this combined loading condition, the beams’ load-bearing capacity and ductility were reduced more than they would have been otherwise.

For the displacement distribution, the displacement was recorded at each of 250 mm along the span, which showed that WPP without steel section reduced deflection along the span due to the developing of torsion along the middle connected out-of-plane parts. The deflection decreased with the increase in out-of-plane concrete parts, and less deflection occurred for the WPP4. Encasing the steel section into the WPP beams reduced the deflection along the span for the beams, which revealed higher deflection for the concrete straight beams and less for the WPP4-C, as seen in Figure 8.

5.3. Twisting Angle

The twisting angle of the WPP beams was recorded at the midspan of the ultimate load. The rotation angles were measured by recording the reading at the connection point at the first part and the end of the midsegment, then dividing the distance from the point. The results showed that the eccentricity of the material caused torsion at the midsegment of the beam, where this segment tends to rotate inside. The analysis of these beams showed that the rotation of the beam with two segments, WPP1, was 2.344 degrees, which increased when the number of the out-of-plane parts increased, where the beams WPP2, WPP3, and WPP4 offered rotation angles of 2.763, 3.122, and 3.778, equal to 17.9%, 33.2%, and 61.2% when compared with WPP1, as seen in Figure 9a. Reinforcing the WPP beams with steel sections showed that the angle of twist reduced significantly, reinforcing the beam with two segments (WPP1-C) and revealing an angle of twist less than the corresponding beam (WPP) of 16.6%. When the number of the segment increased, the ability of the steel section to reduce the twisting angle decreased, and the beams WPP2-C, WPP3-C, and WPP4-C showed reductions in the twisting angle of 11.4%, 16.4%, and 13.5%, respectively, when compared with the corresponding beams WPP2, WPP3, and WPP4, as demonstrated in Figure 9b.

5.4. Effect of Directed Segments

The regular beam’s linearity was dissimilar to the zigzag beams, which showed a linearity of 32% and reduced to 22%, 20%, 19.67%, and 16% for beams WPP1, WPP2, WPP3, and WPP4, respectively. A comparison between the concrete beam without plane parts and the corresponding regular one showed that the cracking load of the WPP beams decreased significantly, which was 189.7 kN, and reduced to average cracking loads of 44.67 kN and 110.85 kN for the WPP beams with and without strengthening by a steel section. Creating the zigzag in the plane of the beams reduced the cracking load, but the decrement depended on the number of parts, which led to more reduction in the yielding load. The beam WPP1 fabricated as two parts showed that the cracking load was reduced by 58.9%, as revealed in Figure 10a. Also, beams WPP2, WPP3, and WPP4, fabricated with three, four, and five segments, showed that yield load decreased by 71.34%, 74%, and 87.05%, respectively, as seen in Figure 10a.

Regarding the ultimate load for a beam without plane parts, the beam SP showed an ultimate load of 421.8 kN, which was reduced to 354.4, 267.3, 210.36, and 156.46 kN for beams WPP1, WPP2, WPP3, and WPP4, respectively, as seen in Figure 11a. The reductions in the ultimate load were 16%, 36.4%, 50.1%, and 62.9%. The ultimate load of these beams refers to the load that caused torsional failure for these beams, although if the geometrical conditions (loading and dimensions) were eccentric, the WPP beams showed torsional failure. The concrete material was a non-homogeneous material that failed at the first cracked segment, making the force and stresses concentrated at these regions, considered the weakest zones in the beam. Concerning the deflection and deformations, the concrete beams with regular shape failed in flexure, with maximum deflection occurring at the midspan of the beam, which was different for beams without plane parts, which for the WPP1 showed a combined flexural–torsional failure, for which the maximum displacement occurred at the midspan with an inclination of connected parts to in the interior perpendicular axis, as seen in Figure 11. For beams WPP2, WPP3, and WPP4, combined shear–flexural failure occurred at the connection zones. The displacement of the regular beams was 29.53 mm, which was reduced by 2.2%, 14.3%,17.4%, and 43.2%.

5.5. Effect of Steel Section

Regarding the strengthened beams with steel box sections, the steel section enhanced the behavior of the beams without plane parts, which increased the cracking and ultimate load of these beams. It should be noted that the steel increased the deflection of the concrete beams, but this increment was reduced with the increase in directed segments. The cracking load of the strengthened beam WPP1-C showed higher resistance against cracking when compared with the reference beam (SP) and WPP1. The cracking load of WPP1-C revealed a value of 146.38 kN, which was higher than the corresponding beam by 87.74%, as demonstrated in Figure 10b. Comparing the same beam with the straight one revealed that the existence of the steel section reduced the reduction in the cracking load from 68.9% to 23% only, which offered an index to the efficiency of the steel section in enhancing the resistance to cracking. The comparison between the corresponding beams WPP2 and WPP2-C showed that the cracking load was enhanced by 117.1%. The comparison between the corresponding beams WPP3 and WPP3-C showed that the cracking load was enhanced by 151%. The comparison between the corresponding beams WPP4 and WPP4-C showed that the cracking load was enhanced by 201%.

Concerning the ultimate load-carrying capacity, the strengthening of WPP beams showed that the enhancement in the ultimate strength between the corresponding beams WPP and WPP-C was significant. The steel section reduced the ultimate load from 16% to zero percent, gaining an additional strength of 33.5%, as demonstrated in Figure 11b. Comparing the corresponding beams WPP1 and WPP1-C showed that the ultimate load carrying capacity increased by 58.9%. Comparing the corresponding beams WPP2 and WPP2-C showed that the ultimate load carrying capacity increased by 85.8%. Comparing the corresponding beams WPP3 and WPP3-C showed that the ultimate load carrying capacity increased by 103.7%. Comparing the corresponding beams WPP4 and WPP4-C showed that the ultimate load carrying capacity increased by 134.4%. Concerning the deflection, the variation for such types of concrete beams was not as massive as the cracking and ultimate loads except with the control straight b, which increased 73.6% for the WPP beams. The comparison between the corresponding beams WPP and WPP-C showed that the deflection capacity increased by 28%, 12%, 10%, and 6.8% for beams WPP1-C, WPP2-C, WPP3-C, and WPP4-C, respectively.

5.6. Displacement Ductility Index

Ductility can be defined as the ability of the material to undergo large deformations without rupture before failure. In concrete, it can be obtained by the percentage of steel reinforcement within it because mild steel is a ductile material that can be bent and twisted without rupture (Punmia et al. and Khamees et al. [26,27]). Ductility can be measured in terms of the ductility index. According to Kim et al. and Maghsoudi et al. [28,29], the ductility index is defined as the ratio of deflection at ultimate load (ΔU) to the yield point deflection (Δy). Authors have proposed different procedures to estimate this term, but Park [30] was based on the equivalent elastoplastic yield point. A comparison between the concrete beam without plane parts and the corresponding regular one showed that the ductility of the WPP beams decreased significantly, which was 3.676 and reduced to an average value of 3.098 for the WPP beams and increased to 5.266 for the WPP beams encased with steel section. Creating the zigzag in the plane of the beams reduced the ductility, but the decrement depended on the number of parts, which increased the number of parts and led to more reduction in the ductility. The beam WPP1, fabricated of two parts, showed that the ductility was reduced by 14.7%, as revealed in Figure 12. Also, for beams WPP2, WPP3, and WPP4 fabricated with three, four, and five segments, it was shown that ductility decreased by 7.25%, 5.5%, and 35%, respectively, as seen in Figure 12. Regarding the encased beams, the WPP-C beams showed a significant enhancement in the ductility, while the steel section for WPP1-C beam showed a ductility index of approximately two times, which increased by 96% when compared with the corresponding beam without the steel section (WPP1). With the increase in the connected parts, the ductility index increased reduced due to the torsional effect.

5.7. Energy Absorption

In this research study, an alternative approach was adopted to assess the ductility enhancement of reinforced concrete beams, focusing on the energy absorption capacity. The energy absorbed by the beams was determined by calculating the area under the load-shortening curve using numerical integration. Using energy absorption capacity as an evaluation parameter provided valuable insights into the beam’s ability to dissipate energy and undergo plastic deformations before failure. The observed improvements in energy absorption capacities signified the enhanced performance and resilience of the WPP beams, which are crucial for withstanding seismic events and other extreme loading conditions. According to the obtained results, a comparison between the concrete beam without plane parts and the corresponding regular one showed that the energy absorption of the WPP beams decreased significantly; it was 1160.69 kN.mm and reduced to an average value of 898.1 kN.mm for the WPP beams and increased to 1342 kN.mm for the WPP beams encased with a steel section. Introducing a zigzag pattern in the plane of the beams resulted in a decrease in energy absorption. However, the extent of this decrease was contingent upon the number of parts. Increasing the number of parts led to a greater reduction in energy absorption. According to Figure 13a, the beam WPP1 that was made up of two sections exhibited a 10% drop in energy absorption. Furthermore, the energy absorption of beams WPP2, WPP3, and WPP4 made up of three, four, and five segments, respectively, exhibited reductions of 19%, 26.9%, and 34.4%, as depicted in Figure 13a.

Regarding the encased beams, WPP-C beams showed a significant enhancement in energy absorption, and the steel section for the WPP1-C beam showed an increased energy absorption, which increased by 66% compared with the corresponding beam without a steel section (WPP1). With the increase in the connected parts, the energy absorption was reduced due to the torsional effect in addition to the shear, which reduced the energy absorption enhancements. The enhancement of the beam WPP2-C was reduced to 51% when the number of connected segments increased to three. Also, the beams WPP3-C and WPP4-C showed less improvement in the energy absorption, which reduced to 42% and 33% when the number of the connected segments increased to four and five, as seen in Figure 13b.

5.8. Cracking Pattern and Stress Distribution

Figure 14 and Figure 15 exhibit the stress distribution of various configurations of RC beams. The stress distribution confirmed that the results were related to the ductility improvement with slight strength hardening. The stress distribution in all beams exhibited varying patterns, with the intensity levels represented by different colors. The colors indicate the stress intensity, ranging from blue to green, yellow, brown, and red. Blue signifies the minimum stress intensity, while red corresponds to the maximum. The stress concentration is visually depicted through the gradual transition of colors from blue to green, yellow, brown, and ultimately red. These figures explain the path of stresses and the zones of their concentration. Regarding the stress distribution of reinforced concrete (RC) beams, the straight beam failed in flexure, and the shear stress developed near the support and moved at an inclination of 45 degrees toward the applied load zone. For the WPP beams, the shear stress developed near the supports and moved away toward the stepped section that connected the concrete segments, with the highest shear stress concentration occurring at the top fiber of the segments. Regarding the shear and torsional stresses along the beam span, the shear stresses were typically the highest at the supports due to the concentrated reactions. Moving away from the supports toward the midspan, the shear stresses gradually decreased as the torsional stresses increased. At the bottom fiber of the beam at midspan, the shear stresses were typically the lowest as the torsional stresses increased. The distribution of shear stresses along the beam span followed a parabolic trend, with the peak values occurring at the supports and the minimum values at the midspan. In the case of zigzag beams, the behavior was dissimilar to that of the straight beam. The shear stresses developed near the support and combined with the torsional stresses, which moved spirally along the beam, reaching the connection zone. The combined shear–torsional stresses dominated the failure in the concrete at the connection zone, causing rotation of the connected segments and crushing at this zone.

6. Conclusions

In this study, WPP beams were analyzed using three-dimensional FEM. The results obtained by FEM were compared with the experimental results in terms of load–deflection curves. Also, the study was conducted to determine the ultimate load’s maximum deflection. Based on the results provided, the following conclusions can be drawn:

• Changing the configuration of the RC beams from straight to beam out-of-plane parts reduced the ultimate load-carrying capacity by an average value of 41.4% compared with the consecutive beams. For the deflection, the reduction was by 24.3%.
• Testing the WPP beams under static loading with variable segments number showed that increasing the number of segments increased the torsional–shear stresses along the beam.
• Encasing the beams’ out-of-plane parts with steel sections enhanced the ultimate load-carrying capacity and deflection of the WPP beams, compensating for the loss the occurred due to the discontinuities of the concrete segments by gaining an additional strength of 9.9%. The deflection reduction was only 24.3% to 12.4%, which was considered an index in enhancing the ductility of these beams.
• Regarding the stress distribution of reinforced concrete (RC) beams, the straight beam failed in flexure, and the shear stress developed near the support and moved at an inclination of 45 degrees toward the applied load zone. For the WPP beams, the shear stress developed near the supports and moved away toward the stepped section that connected the concrete segments, with the highest shear stress concentration occurring at the top fiber of the segments. For the shear and torsional stresses along the beam span, the shear stresses were typically the highest at the supports due to the concentrated reactions. At the bottom fiber of the beam at midspan, the shear stresses generally were the lowest as the torsional stresses increased.
• The straight beams showed higher ductility than WPP beams, which connected the concrete parts and developed more shear stresses in the concrete design, reducing the ductility significantly. The reduction in the ductility increased with the increase in the connection zone and the concrete segment of the WPP beams, which revealed a decrement of 8–35% when the out-of-plane parts rose from two to five.
• Encasing the WPP beams with a steel box enhanced the ductility, compensating for the loss in the ductility and gaining an additional ductility of 45.3% when compared with the straight beam. Comparing the ductility index between the WPP beams with and without the steel section proved the efficiency of the steel section in improving the ductility, which showed an increment of 63–67%.
• The straight beams showed higher energy absorption than the WPP beams. In the WPP beams, the connections between the concrete parts developed more torsional and shear stresses in the concrete design. This reduction in the concrete design caused the WPP beams to have significantly reduced energy absorption compared with the straight beams.
• As the number of out-of-plane parts increased from two to five, the reduction in energy absorption increased. This increase in the out-of-plane parts and concrete segments of the WPP beams led to a 10–34.6% decrement in energy absorption. The analysis results indicated that the straight beam configuration was more favorable for maximizing energy absorption than the WPP beam design, which suffered from increased torsional and shear stresses in the concrete connections. Engineers need to carefully consider this trade-off when designing concrete beams with out-of-plane elements as the increase in these structural elements can significantly compromise the overall energy absorption capabilities of the system.
• Encasing the beams with steel boxes enhanced their energy absorption. This steel encasement compensated for the 22.6% loss in energy absorption compared with the straight beam and an additional 15.6% increase in energy absorption compared with the straight beam. Further comparisons of the energy absorption between the WPO beams with and without the steel section demonstrated the efficiency of the steel section in improving energy absorption. Including the steel section significantly increased the energy absorption, ranging from 2.2% to 49.4%.