Strength and Stiffness of Corrugated Plates Subjected to Bending

Abstract

When a beam twists, localized flexural deformation in the plate occurs at the connection between the corrugated plate and the beam, resulting in a reduction in the bracing stiffness of the corrugated plate. To accurately assess the bracing stiffness of a beam with a connected corrugated plate, it is essential to independently determine the bending stiffness of the corrugated plate and the rotational stiffness of the plate–beam joint. This study conducts bending experiments on corrugated plates subjected to in-plane bending to elucidate the stress mechanisms within the plates. The influence of the width-to-thickness ratio on the bending load and bending stiffness of the corrugated plates is examined using the width-to-thickness ratio regulations of various countries. The findings revealed that when the width-to-thickness ratio of the corrugated plates used in this study was lower than the threshold specified in the Eurocode, the bending load at the onset of stiffness degradation was approximately 80% of the maximum bending load, and the initial bending stiffness corresponded closely to the theoretical value. Conversely, when the web width-to-thickness ratio of the corrugated plates exceeded the limit prescribed in the Eurocode, it was demonstrated that the maximum bending load decreased to approximately 50% of the yield load, and the initial bending stiffness was reduced to about 95% of the theoretical value.

1. Introduction

Corrugated plates, commonly employed as roofing and wall cladding members, are often used for the roofs of large-span structures such as gymnasiums. In these applications, corrugated plates, serving as non-structural members, are connected to main beams; however, stress transfer between the plates and beams is not considered in the design phase. Consequently, cases have been documented where non-structural members, such as the roofs and ceilings of gymnasiums designated as evacuation facilities, render these buildings unusable as evacuation centers, due to the lateral buckling of beams during earthquakes [1,2,3,4].

1.1. Research on Lateral Buckling of Beams

The phenomenon of lateral buckling in beams was initially investigated by Timoshenko [5], Bleich [6], and Nethercot et al. [7]. Subsequent theoretical analyses and experimental studies revealed that lateral buckling behavior varies depending on end restraints and loading conditions [8,9,10,11,12,13]. Furthermore, it has been demonstrated that when main beams, to which non-structural members are connected, undergo lateral buckling, these non-structural members can effectively restrain the lateral deformation of the beams [14,15,16,17]. Design guidelines from various countries specify the stiffness and load requirements for bracing members, such as small beams, to counteract the lateral buckling of beams [18,19,20,21,22]. Later studies validated these design loads and bracing stiffness using moment gradients and bracing positions as variables [23,24,25,26,27,28]. However, these guidelines and studies address partial or discrete stiffeners, which are not applicable to scenarios where stiffeners, such as corrugated plates, are continuously connected to beam flanges.

1.2. Research on Continuous Stiffening Effect on Lateral Buckling of Beams

Non-structural members such as corrugated plates [29,30] (referred to as profile sheets in Eurocode), shown in Figure 1, are connected continuously to the upper flange of beams. In steel structures, studies have also been conducted on the lateral buckling behavior of composite beams with attached concrete slabs [31,32,33]. While concrete slabs offer both in-plane and out-of-plane stiffness [34,35], corrugated plates provide only out-of-plane stiffness, serving solely as continuous stiffeners to resist the lateral buckling of beams [36,37].

Kimura et al. elucidated the relationship between the lateral buckling deformation of beams and the stresses (M_β, F_u) induced in corrugated plates due to this deformation, as illustrated in Figure 2 [38]. Furthermore, the corrugated plate, rigidly connected to the beam, was modeled as a spring with bracing stiffness to restrain the lateral buckling of the beam. Through numerical analysis, they examined the influence of the corrugated plate’s bracing stiffness on the lateral buckling load of H-shaped steel beams subjected to varying gradient moments and degrees of end restraint and determined the horizontal forces and torsional moments required to restrain the beam’s deformation [39,40,41].

Yoshino et al. [42] conducted torsional experiments on beams to evaluate the bracing stiffness of corrugated plates, as depicted in Figure 1. Their study elucidated the relationship between the torsional moment in the beam that damages the corrugated plate and the torsional moment generated in the corrugated plate during the beam’s lateral buckling. These experiments revealed that, when the beam twists, localized deformation at the joint reduces the rotational stiffness of the connection between the corrugated plate and the beam. Consequently, the bracing stiffness of the corrugated plate, derived from the joint’s rotational stiffness and the plate’s bending stiffness, diminishes compared to the value assuming rigid connections. To precisely assess the bracing stiffness of the corrugated plate connected to the beam, it is essential to clarify both the joint’s rotational stiffness and the bending stiffness of the corrugated plate.

1.3. Studies on Width-to-Thickness Ratios of Cold-Formed Members

Corrugated plates, typically fabricated through pressing or rolling, are thin and prone to distortion prior to installation. Consequently, they may not achieve their designed bending strength or stiffness, creating a discrepancy between actual performance and design assumptions. Additionally, thin-walled, cold-formed members are susceptible to local buckling under shear or compressive loads [43,44,45]. Research on cold-formed members under bending has established relationships between buckling load and width-to-thickness ratios [46,47,48,49,50]. However, these studies primarily focus on C-, Z-, and I-shaped steel members; no explicit guidelines currently exist for the width-to-thickness ratio of corrugated plates. Moreover, the inclined webs of corrugated plates may not be applicable to the width-to-thickness ratio regulations [18,19,20,21,22] typically applied to beams and columns in structural design across various countries [51].

1.4. Studies on Bending Performance of Corrugated Plates

Johansson [52] and Degtyarev et al. [53,54,55,56] proposed analytical models to calculate the moment and deflection of corrugated plates; however, these models are significantly influenced by the presence of perforations and missing components. Bahr [57] performed bending experiments on corrugated plates combined with plywood, which provided insights into the strength and behavior of the composite system, but did not isolate the performance of the corrugated plate itself. Jandera et al. [58] conducted bending experiments and numerical analyses on corrugated plates subjected to hanging loads, focusing on the strength of the plates when the load was applied exclusively to the web. These studies fail to fully capture the intrinsic cross-sectional performance of corrugated plates. Although predictive methods for corrugated plate bending strength have been proposed [59,60] and incorporated into design and construction manuals [61,62,63,64,65,66], concerns remain about the accuracy of these predictions [67,68,69]. Moreover, these studies and manuals do not address the correlation between strength and stiffness when initial bending stiffness diminishes due to initial shape imperfection in the rolled corrugated plates, nor do they clarify the relationship between bending stiffness and width-to-thickness ratio.

In this paper, bending experiments on corrugated plates under in-plane bending are conducted to elucidate the stress mechanisms within the plates. Subsequently, a parametric study using numerical analysis is performed to examine the bending load and bending stiffness of corrugated plates. Additionally, the influence of the width-to-thickness ratio on the bending load and the bending stiffness is examined using the width-to-thickness ratio standards from various countries [18,19,20,21,22]. Finally, this study demonstrates the potential applicability of corrugated plates as structural members by establishing the relationship between bending performance and width-to-thickness ratios.

2. Outline of Bending Experiment on Corrugated Plates

2.1. Outline of Bending Experiment Apparatus

Figure 3 illustrates the experimental setup for conducting the three-point bending experiment on a corrugated plate. As depicted in Figure 3a, the boundary conditions at the ends of the corrugated plate replicate pin-roller support along the z-direction to impose a symmetric bending moment. The supports at both ends are composed of plates with tight frames welded atop the pins, as illustrated in Figure 3b. The tight frames and corrugated plates are bolted at the top flange and positioned 30 mm from the ends of the corrugated plate, as shown in Figure 3b,c. Consequently, the distance between the edges of the corrugated plate is L_r = L − 60 mm, as shown in Figure 3c. In this study, the flanges of the specimen illustrated in Figure 3d are designated as the top flange and bottom flange, respectively. Moreover, two types of corrugated plate cross-sections commonly employed in Japanese steel structures, as presented in Figure 4, were selected based on a previous survey [70].

Figure 4a depicts a cross-section where the web width is three times the flange width, designated as Type A. Figure 4b illustrates a cross-section where the flange width is twice the web width, designated as Type B. In this study, each cross-section shown in Figure 4 is referred to as a “unit”.

2.2. Loading Protocols

The specimen depicted in Figure 4 is mounted on the Amsler-type testing machine (Tokyo Koki Testing Machine Inc., Aichi, Japan) illustrated in Figure 3a, with a forced vertical downward displacement (y direction) applied at the central loading point at a constant rate of δ = 0.02 mm/s. Consequently, a bending moment is induced within the specimen.

2.3. Specimen Configuration

Figure 5 provides a detailed illustration of the loading beam. Figure 5a denotes the loading beam for Type A specimens. Figure 5b represents the loading beam for Type B specimens. The connector consists of a steel block, machined into a trapezoidal shape to ensure high stiffness, with a width equivalent to that of the tight frame (30 mm). The connector contacts the top surface of the specimen at the loading point marked by the B-B′ line in Figure 3c.

2.4. Material Properties

The results of the material tests for the corrugated plates are presented in

Table 1. Material properties of the corrugated plates.

Cross-Section	t (mm)	E (×103 N/mm2)	σy (N/mm2)	σu (N/mm2)
	Thickness	Young’s Modulus	Yield Strength	Ultimate Strength
Type A	0.6	165	303	361
	0.8	179	332	384
	1.0	177	318	388
Type B	0.6	185	352	401

.

The galvanized steel plate [71] was selected as the material for the corrugated plates due to its corrosion resistance and suitability for roofing members. The tensile strength tests for the steel plates were conducted according to JIS Z 2241 (JSA2011), following the Japanese Industrial Standards [72]. The yield strength and ultimate strength of the corrugated plates were 303 to 352 N/mm² and 361 to 402 N/mm², respectively.

Table 2. Details of specimens.

Specimen	Cross-Section	Number of Units	Thickness of Specimen	Length of Specimen
		Length of Unit B (mm)	(mm)	L (mm)
A1-0.6-18	Type A	One unit 200 mm	0.6	1800
A1-0.8-18			0.8
A1-1.0-18			1.0
A3-0.6-18		Three units 600 mm	0.6
A3-0.8-18			0.8
A3-1.0-18			1.0
B1-0.6-8	Type B	One unit 140 mm	0.6	800
B1-0.6-12				1200
B3-0.6-8		Three units 420 mm		800
B3-0.6-12				1200

provides a summary of the specimens. The ten test pieces were assessed based on the following four parameters in the experiment: (1) cross-sectional profile (Type A and B), (2) number of units, (3) thickness of specimen, and (4) length of specimen. The specimen’s name is provided in the sequence of the cross-sectional shape, number of units, corrugated plate thickness, and material length of the corrugated plate.

2.5. Measurement Methods

In this experiment, the bending stiffness is evaluated based on the relationship between the applied load and the displacement at the loading point. The load is recorded using an Amsler testing machine. The vertical displacement at the loading point is measured at either a single location (Disp-1) or at three locations (Disp-1–3), as indicated by the ⯅ markers in Figure 5, utilizing a contact-type displacement sensor.

Figure 6 illustrates the locations for strain measurement on the corrugated plate. The strain gauges are affixed to sections experiencing high stress concentrations during loading. Strain gauges are applied on both sides of the corrugated plate to capture the localized flexural deformation in the plate of the corrugated plate.

As illustrated in Figure 7, the imperfection deflection before experimentation of the bottom flange of each Type A specimen at ±100 mm (±L/18) prior to loading was measured. This deformation is defined as the initial shape imperfection. When the corrugated plate was positioned on a flat surface, the bottom flanges of the central units at the left and right central flanges were elevated relative to those at both ends, resulting in a y-axis curvature in the test specimen.

The initial shape imperfection exceeded twice the plate thickness, due to the thinness of the specimen and the roll-forming inaccuracies encountered during manufacturing.

3. Results of Bending Experiment on Corrugated Plates

This section provides an experimental elucidation of the bending load and bending stiffness of corrugated plates, varying by cross-sectional shape and number of units.

3.1. Bending Moment of Corrugated Plates

Figure 8 illustrates the relationship between the vertical load and the vertical displacement at the loading point. The ratio P/P_y,r represents the vertical load P applied at the center of the test specimen, as depicted in Figure 3a, relative to the vertical yield load P_y,r corresponding to the yield bending moment M_y,r of the corrugated plate. This ratio P/P_y,r is referred to as the load ratio. The ratio δ/δ_y,r denotes the vertical displacement δ at the center of the specimen relative to the vertical displacement δ_y,r at the center when the yield load is applied, as determined from the displacement gauges (Disp-1~3) shown in Figure 5.

Figure 8a illustrates the effect of the number of units, while Figure 8b demonstrates the impact of the type of corrugated plate cross-section (Type A and B). The gray straight line represents the theoretical bending stiffness for a simply supported member at both ends subjected to a concentrated load, as shown in the figure above. The theoretical bending stiffness P/δ corresponds to the stiffness of the corrugated plate in its elastic state and is calculated using the following equation:

$$P={\displaystyle \frac{48{E_{r}I}_r}{L_r^3}}\delta$$

(1)

E_r represents the Young’s modulus of the corrugated plate, I_r is the second moment of area of the cross-section, and L_r (=L − 60 mm) denotes the span between the end support points along the plate’s longitudinal axis.

The ▽ dashed marker in Figure 8 indicates the point at the initial stress range of 0.2 P_y,r, within which no reduction in stiffness was observed during the experiments conducted in this study, and is designated as Point I. The solid ▽ marker indicates the point where the slope of the experimental data declines by more than 5% from the initial bending stiffness slope P_y,r/δ_y,r derived from Equation (1), approximately at 0.4 P_y,r to 0.8 P_y,r, and is designated as Point II. Additionally, the ⯆ plot marks the maximum load P_max, designated as Point III. These plots correspond to the single-unit specimens A1-0.6, A1-1.0, and B1-0.6.

From Figure 8a it can be observed that, irrespective of the number of units, the maximum load ratio P_max/P_y,r for the specimen with a thicker plate (A1-1.0, A3-1.0) at Point III, ⯆ is greater than that for the specimen with a thinner plate (A1-0.6, A3-0.6), reaching nearly up to the yield load P_y,r. For specimens of identical plate thickness, the maximum load ratio of the one-unit specimen (A1-0.6, 1.0) is lower than that of the three-unit specimen (A3-0.6, 1.0), with a reduction of 10–12%. The load ratio at which the Type A specimen reaches Point II is higher for specimens with greater plate thickness (A1-1.0, A3-1.0) compared to those with thinner plate thickness (A1-0.6, A3-0.6).

From Figure 8b, the load ratio on the Type A specimen declines significantly after attaining the maximum load ratio. The Type B specimen reached Point II at P/P_y,r = 0.4, exhibiting a load ratio at Point II, where the initial bending stiffness decreases, lower than that of the Type A specimen. The load ratio on the Type B specimen increased progressively from Point II to the maximum load and subsequently decreased gradually after reaching the maximum load ratio.

3.2. Stress State of Corrugated Plates

Figure 9 illustrates the relationship between the axial strain ε_N in the z-axis direction near the loading point, where the bending moment reaches its peak, and the vertical displacement at the loading point. The strain measurement point for the specimen is positioned at +100 mm from the center of the specimen’s length, as depicted in Figure 6.

The axial strain represents the average of the strains ε₁ and ε₂ measured on the front and rear surfaces of the plate, as shown in Figure 9a, and is determined using the following equation:

$${\epsilon }_N={\displaystyle \frac{{\epsilon }_1+{\epsilon }_2}{2}}$$

(2)

From Equation (2), the positive values depicted in Figure 9 correspond to tensile axial strain, while the negative values represent compressive axial strain.

The red line in Figure 9 represents the elastic theory strain of the corrugated plate, calculated using the following equation:

$${\epsilon }_N^{\prime }={\displaystyle \frac{\sigma }{E_r}}={\displaystyle \frac{M}{E_r{Z}_r}}={\displaystyle \frac{{PL}_r}{{4E}_r{Z}_r}}$$

(3)

M (=PL_r/4) represents the bending moment at the loading point, E_r denotes the Young’s modulus of the corrugated plate, and Z_r signifies the sectional modulus of the entire cross-section of the corrugated plate. The plots in the figure indicate the displacements at Points I, II, and III, as identified in Figure 8.

In the Type A specimen illustrated in Figure 9a, the axial strain of the CF located at the top flange is lower than the theoretical axial strain represented by the red line in Equation (3) and is nearly equivalent to the axial strain values observed in CW3 and CW4, located on the webs adjacent to both sides of the top flange. The axial strain of CW1, positioned at the center of the web, remains nearly zero up to Point III, corresponding to the maximum load, as it lies along the neutral axis of the corrugated plate.

In the Type A specimen comprising three units, as depicted in Figure 9b, the axial strain of the CF in the central unit exceeds that of the LF and RF in the end units, indicating that the central unit bears a greater proportion of the load compared to the end units up to Point II. In the multi-unit specimen, as shown in Figure 7, the initial shape imperfection of the central unit is substantial due to the roll forming process. Consequently, the loading beam first contacts the central unit before engaging the outer units, inducing higher stress concentrations in the central unit. This stress distribution leads to uneven loading across the entire corrugated plate, and the load ratio of the three-unit A3 specimen is inferred to be lower than that of the single-unit A1 specimen. The axial strain of CF in the upper flange is below the theoretical axial strain, represented by the red line in Equation (3). However, the axial strain of CF is higher than that of CW3 located in the web. This difference arises from the variation in height (distance) from the neutral axis to the measured strain points of CF and CW3, which corresponds to the distribution of bending stress across the cross-section of the corrugated plate.

In the Type B specimen illustrated in Figure 9c, beyond Point II, the axial strain values of LF, CF, and RF at the center of the top flange decrease markedly, while the axial strain values of LCF and RCF at the center of the bottom flange continue to increase. This behavior occurs because of the top flange, subjected to compressive stress, experiences localized bending deformation, leading to a reduction in its axial strain.

Figure 10 depicts the relationship between the plate bending strain ε_M near the loading point, where the bending moment reaches its maximum, and the vertical displacement at the loading point. The measurement positions are consistent with those shown in Figure 9. The plate bending strain is derived from the differential strains measured on the front and back surfaces of the flange and web plate at the measurement positions, as illustrated in Figure 9a, and is determined using the following equation:

$${\epsilon }_M={\displaystyle \frac{{\epsilon }_1-{\epsilon }_2}{t}}$$

(4)

From Equation (4), the positive values shown in Figure 10 indicate positive plate bending, whereas the negative values denote negative plate bending.

In the Type A specimen with a single unit, as illustrated in Figure 10a, the plate bending strain of the CF located at the center of the top flange increases progressively from the onset of loading. Conversely, the plate bending strain of CW3 and CW4, positioned in the webs adjacent to both sides of the top flange, exhibits a rapid increase starting from Point II. This behavior suggests that the initial bending stiffness diminished due to the plate bending deformation of the web. Furthermore, the plate bending strain of CW3 and CW4 occurs in the direction opposite to that of CF.

The Type A specimen comprising multiple units illustrated in Figure 10b exhibits a greater initial shape imperfection compared to the single-unit specimen illustrated in Figure 10a. Consequently, it differs from the occur points of the plate bending strains for LF, CF, and RF, located at the center of the top flange. Additionally, the plate bending strain of CF, situated at the center of the top flange, and CW3, positioned in the web adjacent to the left side of the top flange, both begin at Point I. However, the plate bending strain of CW1, located at the center of the web in the central unit, increases sharply from Point II, where the initial stiffness begins to decline. This indicates that, for specimens with a greater number of units, the reduction in initial bending stiffness can be attributed to plate bending deformation across the entire web, rather than being confined to the regions adjacent to the flanges.

The Type A specimen depicted in Figure 10b has a web width greater than the flange width, leading to a decrease in initial bending stiffness due to localized deformation of the web. Conversely, the Type B specimen shown in Figure 10c features a flange width exceeding the web width. As a result, while the plate bending strain of the CF located at the center of the top flange begins to increase from Point I, the initial bending stiffness diminishes due to the rapid increase in the plate bending strains of LF and RF, positioned at the center of the top flange in the side unit, from Point II onward.

Figure 11 illustrates the axial strain distribution in the x–y plane near the loading point of the test specimen. The measurement positions are consistent with those shown in Figure 9 and Figure 10. Each line represents the elastic theory strain of the corrugated plate, calculated using Equation (3).

The axial strain in the A1-0.6 specimen, shown in Figure 11a, is nearly identical to the theoretical value up to Point III, representing the maximum load.

The axial strain in the A3-0.6 specimen, depicted in Figure 11b, falls below the theoretical value at Point II, where the initial bending stiffness diminishes across the top flanges of all units, resulting in an uneven load distribution across the entire plate. Furthermore, the axial strain at CW1 and LW2, located at the center of the web in the central and left-side units, has similarly declined, attributable to the neutral axis shift caused by localized flexural deformation in the web at Point II.

In Figure 11c, the axial strain in all top flanges of the B3-0.6 specimen is lower than the theoretical value at Point II. The axial strain in the web at Point III exhibits a value on the compressive side relative to the theoretical value.

3.3. Deformation Mechanism of Corrugated Plates

Figure 12 illustrates the final deformation state of the test specimen post loading. Regardless of the corrugated plate type, localized flexural deformation in the plate is concentrated around ±30 mm from the member’s center. This concentration arises because the bending moment generated at the edge of the connector attached to the loading beam reaches its maximum at this location. The plate bending deformation at CF and LF, situated at the center of the top flange in the central and left-side units, occurs at a position of +30 mm indicated by the circle. While the plate bending deformation at RF, located at the center of the top flange in the right-side unit, occurs at a position of −30 mm indicated by the circle. In the Type A specimen, the plate bending deformation at CW3, situated in the web adjacent to the right side of the central unit’s top flange, occurs at a position of +30 mm indicated by the square.

4. Bending Performance of Corrugated Plates

In the bending experiments presented in Section 3, it was established that both the cross-sectional shape of the corrugated plate and the number of units significantly influence the bending load and stiffness of the corrugated plate. In this chapter, a numerical analysis model is developed under conditions identical to those of the bending experiments, enabling parametric study. Subsequently, the effect of the width-to-thickness ratio on the bending load of the corrugated plate is examined using the width-to-thickness ratio standards of various countries. Furthermore, the relationship between the bending stiffness of corrugated plates and the width-to-thickness ratio is elucidated to apply corrugated plates as buckling stiffeners for beams.

4.1. Outline of Finite Element Analysis

This section describes the elasto-plastic large deformation analysis [73], commonly referred to as Pushover analysis. This analysis was conducted using the FEA software package ABAQUS ver. 2024 [74].

Figure 13 illustrates the numerical analysis model. The corrugated plate was modeled utilizing a four-node shell element (S4R) with six degrees of freedom per node and a three-node shell element (S3) [74]. In the analysis, the shell thickness was defined to correspond to the plate thickness of the test specimen. To precisely capture the stress distribution around the loading point, the elements within 100 mm of the member’s center on either side are refined to a mesh twice as fine as that of the surrounding regions. A mesh sensitivity analysis was conducted to ensure the precision and stability of the simulation outcomes. The chosen mesh size was optimized to achieve an appropriate balance between computational efficiency and solution accuracy, with convergence of the results verified within an acceptable range. The boundary conditions incorporate pin rollers at the ends of the members. At the loading point, movement is constrained exclusively to the vertical direction, with no displacement permitted along the member’s length of the corrugated plate.

The numerical analysis model employed in this study incorporates a bilinear isotropic hardening law, as illustrated in Figure 14 [75]. The isotropic hardening parameters defined are presented in

Table 3. List of calibrated isotropic hardening parameters.

Cross-Section	t (mm)	E (×103 N/mm2)	σy (N/mm2)	σy,t (N/mm2)	εy,t (×10−6)
	Thickness	Young’s Modulus	Yield Strength	True Stress	True Strain
Type A	0.6	165	303	361	0.0363
	0.8	179	332	384	0.0238
	1.0	177	318	388	0.0395
Type B	0.6	185	352	401	0.0411

Here, σ_y represents the yield stress, σ_y,t denotes the true stress, and ε_y,t signifies the true strain, with a Poisson’s ratio of 0.3. The true stress and true strain values were derived in accordance with Appendix C.6 of EN 1993-1-5: 2006 [76]. The plate thickness, Young’s modulus, and yield stress were determined from the material test results summarized in Table 1.

.

For this study, a nonlinear static analysis, commonly referred to as Pushover analysis, was conducted. The analysis was executed using the static Riks method within the ABAQUS ver. 2024 program, where the arc length increment Δ for the load–displacement curve illustrated in Figure 15 was set to 0.025 per step [74]. The corrugated plates used in the experiments were rolled, and as shown in Figure 7, displayed initial shape imperfections prior to testing. These measured initial shape imperfections were modeled as structural imperfections in numerical analysis.

Figure 16 presents a comparison of the experimental results with numerical analysis outcomes. The numerical analysis of the load–displacement relationship in Figure 16a aligns with the experimental results up to the point of maximum load, irrespective of the cross-sectional configuration. For the single-unit model, the axial strain distribution within the cross-section, as shown in Figure 16(b-1),(b-2), corresponds closely with the experimental results up to Point III, representing the maximum load. In the three-unit configuration, the axial strain distribution across the cross-section, depicted in Figure 16(b-3),(b-4), aligns with the experimental data up to Point II, where the initial bending stiffness begins to decline.

Figure 17 illustrates a comparison of the deformation observed in the corrugated plates following loading. In the Type A specimen, pronounced local deformation is evident in both the web and flange at the loading point. Conversely, in the Type B specimen, local deformation is primarily concentrated in the top flange. The deformation patterns observed in the experimental and numerical analysis are broadly consistent. The comparison between the experimental results and numerical analysis presented in Figure 16 and Figure 17 confirms the validity of the analytical model depicted in Figure 13.

4.2. Bendig Strength and Stiffness of Corrugated Plates

In this section, the bending load and bending stiffness of corrugated plates are investigated using the numerical analysis model depicted in Figure 13. The parameters in the numerical analysis include the number of units, the shear span ratio, and the width-to-thickness ratio.

Figure 18 illustrates the influence of the number of units on the load ratio. The load ratio at the point of initial bending stiffness reduction in Point II remains consistent regardless of the number of units. The load ratio at maximum load decreases with an increase in the number of units. This phenomenon occurs because, as demonstrated in Section 3.2, a greater number of units induces uneven loading across the entire corrugated plate.

Figure 19 illustrates the impact of initial bending stiffness on the shear span ratio. The shear span ratio is defined as the proportion of the corrugated plate’s length L_r, as depicted in Figure 3c, to its height d, as shown in Figure 4. The bending stiffness k_θ,i is defined as the ratio of the vertical displacement δ of the loading point to the vertical load P_i applied at the center of the member, as depicted in Figure 19a.

The value of k_θ,i is computed using the following equation:

$$k_{\theta ,i}={\displaystyle \frac{P_i}{\delta }}$$

(5)

The subscripts i indicate Point II and Point III in Figure 8.

The initial bending stiffness is denoted as k_θ,II, while the bending stiffness at the maximum load is represented as k_θ,III.

k_θ,0 refers to the theoretical bending stiffness when a concentrated load is applied at the center of the member, which is derived from the equation that incorporates Equation (1) into Equation (5).

$$k_{\theta ,0}={\displaystyle \frac{48E_r{I}_r}{L_r^3}}$$

(6)

The initial bending stiffness corresponding to Point II decreases as the shear span ratio diminishes.

The subsequent equation represents the bending stiffness k′_θ,0, which accounts for the shear stiffness obtained from both the bending stiffness k_θ,0 as calculated in Equation (6), and the shear stiffness k_s,0.

$${\displaystyle \frac{1}{{k\prime }_{\theta ,0}}}={\displaystyle \frac{1}{k_{\theta ,0}}}+{\displaystyle \frac{1}{k_{s,0}}}$$

(7)

The shear stiffness k_s,0 is determined using the following equation:

$$k_{s,0}={\displaystyle \frac{2GA}{L_r\kappa }}$$

(8)

G represents the shear modulus, A is the total cross-sectional area of the corrugated plate, and κ is the shear correction factor [77].

Here, the flanges and webs of the corrugated plate cross-section function as plate elements supported on both sides. Consequently, the stiffness of box-shaped cross-section members is presented as a reference value.

The red line in Figure 19b represents the ratio of bending stiffness k_θ,0 to k′_θ,0, which incorporates the shear stiffness of the box-shaped cross-section. In this context, the box-shaped cross-section corresponds to the flange width and web height of the corrugated plate from the Type A specimen used in the experiment, depicted in Figure 4a.

As the shear span ratio decreases, the bending stiffness k′_θ,0, which accounts for shear stiffness (illustrated by the red line), becomes smaller than k_θ,0, demonstrating a trend consistent with the experimental results. Based on the above results, it can be inferred that a smaller shear span ratio increases the contribution of the shear stiffness k_s,0 at the loading point, leading to a reduction in the bending stiffness of the corrugated plate.

Subsequently, the influence of the width-to-thickness ratio on the bending load and bending stiffness of corrugated plates will be examined in detail. Currently, there are no specific width-to-thickness ratio regulations for corrugated plates, as these are considered non-structural members and are not governed by structural design standards.

Accordingly, this study investigates the applicability of width-to-thickness ratio regulations for plate elements supported on both sides, which are typically applied to beams classified as structural members in various countries [20,21,22], to corrugated plates.

The width-to-thickness ratio for a plate element supported on both sides in AIJ [20] is determined by the following equation:

$${\displaystyle \frac{d_f}{t}}\le 1.6\sqrt{{\displaystyle \frac{E_r}{F}}}$$

(9)

$${\displaystyle \frac{d_w}{t}}\le 2.4\sqrt{{\displaystyle \frac{E_r}{F}}}$$

(10)

The d represents the width of the plate element supported on both sides, d_f denotes the flange width, and d_w refers to the web width. t represents the thickness of the plate, F is the allowable stress (N/mm²).

The width-to-thickness ratio for a plate element supported on both sides in Eurocode [21] is determined by the following equation:

$${\displaystyle \frac{d_f}{t}}\le 42\sqrt{{\displaystyle \frac{235}{F}}}$$

(11)

$${\displaystyle \frac{d_w}{t}}\le 124\sqrt{{\displaystyle \frac{235}{F}}}$$

(12)

The width-to-thickness ratio for a plate element supported on both sides in AISC [22] is determined by the following equation:

$${\displaystyle \frac{d_f}{t}}\le 1.49\sqrt{{\displaystyle \frac{E_r}{F}}}$$

(13)

$${\displaystyle \frac{d_w}{t}}\le 5.71\sqrt{{\displaystyle \frac{E_r}{F}}}$$

(14)

In these cases, the yield stress F, as obtained from the material test results presented in Table 1, is applied in Equations (9)–(14).

Figure 20 illustrates the relationship between the load ratio and the width-to-thickness ratio of corrugated plates. Figure 20a represents the flange width-to-thickness ratio, while Figure 20b represents the web width-to-thickness ratio. In Figure 20a, the red line indicates the prescribed width-to-thickness ratio obtained from Equation (9), the blue line represents the ratio obtained from Equation (11), and the green line corresponds to the ratio from Equation (13). In Figure 20b, the red line represents the prescribed value obtained from Equation (10), the blue line from Equation (12), and the green line from Equation (14).

When the web width-to-thickness ratio exceeds the specified value in Equation (12) of Eurocode [21], the disparity between the two load ratios, P_II and P_max, diminishes, with reductions reaching up to approximately 50% of the yield load. The larger width-to-thickness ratio increases the early likelihood of localized flexural deformation in the corrugated plate, which in turn lowers the load ratio as initial bending stiffness diminishes within the elastic range. Moreover, the deformation performance of the corrugated plate diminishes due to localized flexural deformation in the plate within the elastic range, leading to a diminished load increment from the point of initial bending stiffness reduction to the attainment of the maximum load.

When the web width-to-thickness ratio is below the limit specified in Equation (12) of Eurocode [21], the load ratio remains nearly constant, with the bending load ratio at the point of initial stiffness reduction reaching approximately 80% of the maximum bending load ratio.

The load ratio behavior of the corrugated plate in this study is applicable to the width-to-thickness ratio specified in Eurocode [21].

Figure 21 illustrates the relationship between the bending stiffness ratio of corrugated plates and the width-to-thickness ratio. The bending stiffness ratio k_θ,i/k_θ,0 is defined as the proportion of the bending stiffness k_θ,i of the corrugated plate to the theoretical bending stiffness k_θ,0 in Equation (6). As shown by the red line in Figure 21a, in the experiment, the slope of the line connecting the maximum load to the initial loading point is defined as the secant stiffness k_θ,III. Here, the solid ▽ marker and ▼ marker in Figure 21a represent the same Points II and III as depicted in Figure 8.

From Figure 21b,c, when the web thickness ratio of the corrugated plate is less than the value specified in Equation (12) of Eurocode [21], the initial bending stiffness k_θ,II of the FEA plot closely approximates the theoretical bending stiffness k_θ,0. The secant stiffness k_θ,III diminishes as the width-to-thickness ratio becomes small, owing to increased plastic deformation from the point of initial bending stiffness loss up to the point of maximum load, with the rate of decrease reaching approximately 20% of the initial bending stiffness. As the web width-to-thickness ratio of the corrugated plate exceeds the Eurocode [21] specification in Equation (12), the initial bending stiffness declines. Within the range considered in this study (flange width-to-thickness ratio (20 ≤ d_f/t ≤ 70) and web width-to-thickness ratio (50 ≤ d_w/t ≤ 220)), the initial bending stiffness k_θ,II reduced to around 95% of the theoretical bending stiffness in Equation (6). This is because the corrugated plates, composed of multiple units, experience uneven stress distribution within each unit due to the initial shape imperfections, as demonstrated in Section 3.2.

5. Conclusions

In the initial part of this study, bending experiments were performed on corrugated plates to elucidate the stress mechanism within these plates, and the results were analyzed and compared between two types of corrugated plates. In the latter part of the paper, the influence of the width-to-thickness ratio on the bending load and bending stiffness of corrugated plates was examined using the width-to-thickness ratio standards of various countries, through a combination of experiments and numerical analysis.

The key findings of this investigation are summarized as follows:

(1)

Through the stress distribution within the corrugated plate from the corrugated plate bending experiment, it was determined that localized deformation of the flange and web on one side contributes to a reduction in both load capacity and initial stiffness.

For the width-to-thickness ratio of the web plate exceeding that of the flange plate, the reduction in load after reaching the maximum load becomes more pronounced due to the localized bending deformation of the web. For the width-to-thickness ratio of the web plate that is smaller than that of the flange plate, the initial bending stiffness of the corrugated plate diminishes because of the localized bending deformation of the flange.

(2)

By investigating the correlation between the bending performance of corrugated plates, as determined through numerical analysis and the width-to-thickness ratio standards of various countries, it was concluded that the bending performance of these plates can be accurately assessed using the values stipulated in the Eurocode.

For corrugated plates with a width-to-thickness ratio below the Eurocode-specified threshold, the bending load corresponding to the initial reduction in stiffness was approximately 80% of the maximum bending load, and the initial bending stiffness matched the theoretical bending stiffness. Furthermore, the secant stiffness of the corrugated plate at maximum load decreased as the width-to-thickness ratio declined, with reductions reaching up to 20% of the initial bending stiffness. For corrugated plates with a width-to-thickness ratio exceeding the Eurocode-specified threshold, the maximum bending load was reduced to approximately 50% of the yield load, while the initial bending stiffness was reduced to around 95% of the theoretical bending stiffness.

The parameters investigated in this study encompass a flange width-to-thickness ratio range of 20 ≤ d_f/t ≤ 70 and a web width-to-thickness ratio range of 50 ≤ d_w/t ≤ 220.

Bending experiments were performed under monotonic loading conditions; however, the results are limited to cases where the upper and lower flange widths are identical. In practical structural applications, the widths of the upper and lower flanges of corrugated plates may differ. Consequently, it is imperative to examine the bending behavior of corrugated plates under alternating positive and negative loading conditions.