Effect of Cracks on the Compressive Ultimate Strength of Plate and Stiffened Panel Under Biaxial Loads: A Finite Element Analysis

Abstract

Crack damage can significantly reduce the ultimate strength of marine structures, potentially leading to progressive collapse. This study employs finite element analysis to investigate how cracks affect the strength of plates and stiffened panels under uniaxial and biaxial compression, providing insights essential for robust structural design. The effects of crack size and orientation are explored through a systematic evaluation of longitudinal, transverse, and bidirectional cracks—sized at 10%, 25%, and 50% of structural dimensions (plate length and plate breadth/web height)—in both plates and unstiffened panels. The analysis identifies key parameters governing strength degradation and reveals that stiffened panels are more resistant to cracking, whereas plates are more sensitive to crack orientation and loading direction. These findings underscore the role of crack characteristics and structural reinforcement in maintaining residual strength and provide guidance for improving the accuracy and reliability of ultimate strength predictions.

1. Introduction

The ultimate strength of plates and stiffened panels with cracks under compressive loads has been extensively studied by researchers due to its critical significance in various engineering applications, particularly in civil, mechanical, and marine/ocean engineering. Over the years, numerous experimental [1,2,3,4], analytical [5,6], and numerical [2,7,8] investigations have been conducted to understand the complex behavior of these structural components and enhance their performance.

The effect of cracks in plates and stiffened panels on the ultimate strength behavior has also been widely explored in the fields of ships and offshore engineering. Cracks progressively weaken structural integrity, with their severity increasing over time. Ships and ship-shaped offshore structures are continuously exposed to vertical bending moments during their operation. In particular, cracks may propagate over time due to the cyclic tensile and compressive loads induced by vertical bending moments, such as hogging and sagging during operation [1,7,9].

Various studies have been conducted to obtain the ultimate compressive strength or residual ultimate compressive strength of plates [9,10,11,12,13] and stiffened panels [4,8,10,14,15,16,17]. Recently, the ultimate limit state of a cracked ship hull girder was examined by Ashkan and Khedmati [18]. The residual ultimate strength characteristics of plates and stiffened panels with cracks may be influenced by the following key parameters: (1) material properties, (2) crack geometry and location, (3) loading conditions, (4) stiffening techniques, and (5) analysis methods.

Material properties, such as yield strength, elastic modulus, and ductility, significantly affect the ultimate strength of plates and stiffened panels. Researchers have explored different materials, including steel, aluminum, and advanced composites, to assess the influence of material properties on panels’ performance under compressive loads. Attia et al. [19] recently conducted axial compression tests on cracked aluminum stiffened panels and found that material selection significantly influences a panel’s ultimate strength, resistance to crack propagation, and overall structural stability.

The geometry and location of cracks in plates and stiffened panels are critical factors influencing their ultimate strength [10]. The crack length, orientation, and position can significantly impact a panel’s load-carrying capacity and failure mechanisms. For instance, cracks near a panel’s edges or stiffeners often lead to high-stress concentrations, reducing ultimate strength.

The effect of cracks is highly dependent on loading conditions. Ships experience cyclic loading due to waves, repeatedly undergoing hogging and sagging moments. While tensile loads promote crack propagation, compressive loads, on the other hand, can suppress crack growth by closing the crack surfaces. Researchers have investigated the effects of various loads on panels’ performance, including compression [10,16] and shear [1,7,20,21]. Recently, Hu et al. [22,23,24] numerically predicted the residual ultimate strength of plates, stiffened panels, and box girders under cyclic loading conditions. Khedmati et al. [25] conducted a sensitivity analysis on a cracked plate under axial compression via elastic buckling analysis. While crack closure can mitigate local crack-tip stresses under cyclic or bending-dominated conditions, its role under monotonic compressive loading remains limited. Several studies [26,27,28] have shown that in thin-walled structures under dominant compression, crack-face contact may occur but contributes little to global load transfer due to slippage or partial interaction. Moreover, stress redistribution around closed cracks can still trigger local buckling or failure, meaning that closure does not guarantee structural integrity. Therefore, this study models cracks as open discontinuities without contact to ensure conservative and computationally efficient estimation of ultimate compressive strength.

Stiffened panels generally exhibit superior ultimate strength compared to unstiffened plates due to the additional support provided by stiffening elements. Various stiffening techniques, including longitudinal and transverse stiffeners, corrugated panels, and sandwich structures, have been explored to enhance load-carrying capacity and resistance to crack propagation. The optimization of the design and arrangement of stiffening elements is a major focus in the ongoing endeavor of improving the ultimate strength of stiffened panels.

The development of numerical and analytical methods is crucial for advancing the understanding of the ultimate strength of plates and stiffened panels with cracks under uniaxial compressive loads. Numerical models, such as finite element (FE) and analytical models, have been used to predict the structural performance of cracked panels and identify the key factors influencing their residual ultimate strength. These methods have also contributed to the development of accurate, efficient design guidelines and standards. Recently, Li et al. and Liu et al. [29,30] assessed the ultimate strength of cracked structures using artificial neural networks (ANNs).

While substantial research has been conducted on the residual ultimate strength of cracked hull structures, most studies focus on the effects of cracks under uniaxially applied compression. However, the combined effects of crack size, orientation, and biaxial loading conditions remain insufficiently explored.

The aims of this study are as follows:

• To identify key factors affecting the ultimate compressive strength of cracked plates and stiffened panels under uniaxial and biaxial loading conditions;
• To perform detailed nonlinear finite element analyses that capture the effects of various crack sizes, orientations, types, and loading directions;
• To assess the strengths and limitations of existing predictive models and identify areas for improvement;
• To provide recommendations for enhancing the accuracy and reliability of ultimate strength prediction in cracked marine structures.

2. Target Structures and Crack Scenarios

Most hull girder collapse accidents caused by cracks occur in container ships, with these cracks typically propagating from the side shells. The target structure in this study is the outer side shell of a 9300 TEU container ship, composed of high-tensile steel (AH32) and located in the midship section. The analysis considers two structural components: (1) a plate, located between longitudinal stiffeners and transverse frames and (2) a stiffened panel, located between longitudinal girders and transverse frames.

Table 1. Material properties of target structure.

Description	Dimension
Yield strength, σY	315 MPa
Elastic modulus, E	205,800 MPa

and

Table 2. Structural dimensions of target structure.

Description	Dimension
Plate length, L	4200 mm
Plate breadth, b	860 mm
Plate thickness, tp	16 mm
Panel breadth, B	6880 mm
Number of stiffeners	7
Web height, hw	300 mm
Web thickness, tw	11 mm
Flange breadth, bf	90 mm
Flange thickness, tf	16 mm
Plate aspect ratio	4.884
Plate slenderness ratio	2.103
Radius of gyration	98.268 mm
Column slenderness ratio	0.532

present the material properties and dimensions of the target structure.

Cracks are generally classified as longitudinal (horizontal), transverse, vertical, or oblique (angular) cracks [9,31,32]. For angular cracks, the use of different layouts of geometry according to crack size and direction may introduce uncertainties by the element shape and size in FE modeling. This study examines longitudinal, transverse, and combined (bidirectional) cracks. To analyze the effect of crack size relative to the structural dimensions, three levels of crack sizes (minor, moderate, and major) were selected randomly. In all cases, the cracks were modeled as through-thickness defects, meaning that the crack depth was equal to the full thickness of the plate or web where the crack was located.

Crack types were as follows:

• A longitudinal crack is a crack in the plate length direction.
• A transverse crack is a crack in the breadth (for plates) or web height (for webs) direction.
• A combined crack is a crack in the longitudinal and transverse directions.

Crack sizes were as follows:

• A longitudinal crack extends 10%, 25%, and 50% of the plate length.
• A transverse crack extends 10%, 25%, and 50% of the plate breadth (or web height).

As illustrated in Figure 1, longitudinal and transverse cracks were assumed to be located at the center of the structure, while a combined crack was considered to propagate from the midpoint of one side of the structure. For transverse and combined cracks in the stiffened panels, both the plate and the web stiffener developed cracks in the transverse and vertical directions. In the case of a longitudinal crack, the plate and the web were completely disconnected at the crack region by assigning independent nodes to each part, ensuring that the structural discontinuity caused by the crack was properly represented.

3. FE Analysis

3.1. Modeling of Crack

In nonlinear FE analysis, accurately representing cracks is crucial for assessing their influence on structural behavior. Each crack was modeled by disconnecting the elements along the crack path to simulate the discontinuity in the geometry. In other words, the elements along the crack line were not merged or connected to adjacent nodes, ensuring that stress redistribution and local failure mechanisms were properly captured.

Since the accuracy of the analysis depends on the proper representation of crack-induced stress concentration, special attention was given to the mesh refinement in the crack region. In the FE analysis of a cracked structure under tensile loads, the elements around the crack zone should be finer than those in the uncracked area due to the crack propagation and stress concentration [33]. By contrast, the influence of the element size in the crack zone on the structural behavior is minimal in structures subjected to compressive loading [10].

A case study was conducted to Investigate the effect of the element size In the FE model of a crack zone on the structural behavior. To compare structural responses, longitudinal compression was applied to the plate containing a transverse crack extending across 50% of the plate breadth, perpendicular to loading direction. Fine elements (2.4 mm) were used around the crack zone, as shown in Figure 2a; a coarser mesh size of 21.5 mm was used for the surrounding area where finer elements were not required.

Figure 2b,c show a comparison of the structural behavior and von-Mises stress distribution of plate with and without fine elements in the crack zone. Figure 2 indicates that the element size had no significant impact on structural behavior or stress distribution, including the stress concentration. Therefore, a uniform element size was sufficient for the entire structure in the ultimate compressive strength analysis.

3.2. FE Modeling

The commercial software Ansys Mechanical APDL (Version 2022 R2) was used to conduct nonlinear FE analysis using an elastic–perfectly plastic (EPP) material curve [34]. This study employed the FE modeling technique validated through numerical benchmark studies [35,36].

The ISSC and Kim et al. [36,37] recommended the use of 10 elements in the plate breadth direction for ultimate compressive strength calculation. However, a finer mesh with more elements is necessary to achieve a more detailed stress distribution and accurately capture the ultimate strength behavior. In this study, quadrilateral shell elements with a size of 21.5 mm were assigned as follows, considering the aspect ratio of the plate, web, and flange: 40 elements in the plate breadth (b), 200 elements in the plate length (L), 20 elements in the web height (h_w), and 6 elements in the flange breadth (b_f).

The structures in this study were assumed to be simply supported because they were surrounded by longitudinal stiffeners (or girders) or transverse frames, which are strong structural members. Figure 3 illustrates the boundary conditions applied to the stiffened panels.

For ultimate strength calculation, applied loads must be sufficiently high to induce structural collapse. In this study, the applied line pressure was at 1.1 times the yield strength multiplied by thickness (plate, web and flange, respectively), and five loading ratios (σ_y:σ_x = 1:0, 3:1, 1:1, 1:3, 0:1) derived from experience [36] were considered to calculate the strength under uniaxial and biaxial compressive loads.

Due to welding, initial deflection and residual stress are inevitable initial imperfections in ship structures. This study applied buckling initial deflection, column-type initial deflection, and sideways distortion to the plates, stiffened panels, and stiffeners, respectively, as shown in Figure 4 and determined using Equations (1)–(3) [38]. However, residual stress from welding was not considered. This is because the stress may be relieved after cracking and depends on crack size and type.

$$w_{opl}=A_o\mathit{sin}\left({\displaystyle \frac{m\pi x}{L}}\right)\mathit{sin}\left({\displaystyle \frac{\pi y}{b}}\right) \mathrm{for} \mathrm{plate},$$

(1)

$$w_{oc}=B_o\mathit{sin}\left({\displaystyle \frac{\pi x}{L}}\right)\mathit{sin}\left({\displaystyle \frac{\pi y}{B}}\right) \mathrm{for} \mathrm{entire} \mathrm{stiffened} \mathrm{panel},$$

(2)

$$w_{os}=C_o{\displaystyle \frac{z}{h_w}}\mathit{sin}\left({\displaystyle \frac{\pi x}{L}}\right) \mathrm{for} \mathrm{stiffener} (\mathrm{sideways}),$$

(3)

where $A_o=0.1{\beta }^2{t}_p$, $B_o=0.0015L$, $C_o=0.0015L$, $\beta$ is the plate slenderness ratio, $t_p$ is the plate thickness, $L$ is the plate length, b is the plate breadth, B is the panel breadth, and $m$ is the buckling half-wave number in the length direction, which is the minimum integer satisfying Equation (4) [38].

$${\displaystyle \frac{(m^2/L^2+1/b^2{)}^2}{m^2/L^2+c/b^2}}\le {\displaystyle \frac{\left[\right(m+1{)}^2/L^2+1/b^2{]}^2}{(m+1{)}^2/L^2+c/b^2}}$$

(4)

where $c$ is the loading ratio (${\mathrm{\sigma }}_y/{\mathrm{\sigma }}_x$).

4. Results and Discussions

4.1. Ultimate Compressive Strength of Cracked Plates

Paik et al. [9] and Babazadeh and Khedmati [39] presented an approximate empirical formula to calculate the ultimate strength of cracked structures as expressed in Equation (5).

$${\sigma }_u={\displaystyle \frac{A_{intact}-A_c}{A_{intact}}}{\sigma }_{u\_intact}$$

(5)

where ${\sigma }_{u\_intact}$ and ${\sigma }_u$ are the ultimate strength of the intact and cracked structures, and $A_{intact}$ and $A_c$ are the cross-section area of intact structure and crack, respectively.

The concept of the formula is to determine strength reduction by considering the ratio between an effective and intact cross-section area. This approach is applicable when the crack is oriented perpendicular to the applied load. Li et al. [29] proposed an empirical formula for the ultimate strength of one-directionally cracked plates under longitudinal compressions as Equations (6a) and (6b) considering directions and locations of crack.

$${\sigma }_u=\left[1-a_1\right]·\left[1-a_3{\left({\displaystyle \frac{s}{b}}\right)}^2\right]·{\sigma }_{u\_intact} \mathrm{for} \mathrm{longitudinal} \mathrm{crack},$$

(6a)

$${\sigma }_u=\left[1-a_2\right]·\left[1-a_4{\left({\displaystyle \frac{s}{b}}\right)}^2\right]·{\sigma }_{u_{intact}}+0.045 \mathrm{for} \mathrm{transverse} \mathrm{and} \mathrm{oblique} \mathrm{crack},$$

(6b)

where ${\sigma }_{u\_intact}$ and ${\sigma }_u$ are the ultimate strength of the intact and cracked structures, $a_1$ and $a_2$ are coefficients related to the plate slenderness ratio and direction of crack, $a_3$ and $a_4$ are coefficients for the location of crack, $s$ is the location of crack center in breadth direction, and $b$ is the plate breadth.

Figure 5 and

Table 3. Comparison of ultimate strength (σ_u/σ_Y) of cracked plates under uniaxial compression with the existing formula (in Mpa).

Crack Size			Intact	10%	25%	50%
Longi. strength	Longi. crack	Present	0.688	0.664	0.641	0.623
		Equation (6a)	0.688	0.655	0.607	0.525
		Difference	0.000	0.014	0.053	0.157
	Trans. crack	Present	0.688	0.686	0.625	0.532
		Equation (5)	0.688	0.619	0.516	0.344
		Equation (6b)	0.688	0.676	0.612	0.505
		Difference with Equation (5)	0.000	0.098	0.174	0.353
		Difference with Equation (6b)	0.000	0.015	0.021	0.051
Trans. strength	Longi. crack	Present	0.270	0.262	0.241	0.204
		Equation (5)	0.270	0.243	0.203	0.135
		Difference	0.000	0.073	0.158	0.338

Difference: (1 − Present/equation).

compare the ultimate strength of plates with perpendicular cracks under uniaxial compressions, as obtained by Equations (5), (6a) and (6b) and the results of this study. It shows that Equation (5) underestimates the ultimate strength, because it assumes a uniform stress distribution across the cross-section and a linear strength reduction based solely on crack size. However, in practical scenarios, stress redistribution is not perfectly uniform, and localized stress concentrations around the crack can cause nonlinear strength degradation rather than a simple linear reduction. This observation indicates the necessity of revising the equation to account for the effects of nonuniformly distributed stress across the cross-section. Otherwise, the results from Equations (6a) and (6b) and this study showed good agreement for small cracks, as Equations (6a) and (6b) was derived from FE simulations, which account for the structural response more accurately than simplified analytical approaches.

Figure 6, Figure 7 and Figure 8 present the effect of the crack type and size on the structural behavior of the plates. The ultimate compressive strength of the intact and cracked plates under uniaxial and biaxial compressive loads is summarized and compared.

As shown in Figure 6 and Figure 7a, cracks reduced both the longitudinal strength and stiffness of the plates, regardless of crack size and type. The impact of transverse and combined cracks on the ultimate compressive strength in the longitudinal direction was more pronounced than that of longitudinal cracks, as shown in Figure 5. However, cracks smaller than 10% of the plate dimensions had a negligible effect, as they did not significantly alter the stress distribution, as illustrated in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9 and Figure A10 (Appendix A). In contrast, larger combined cracks led to a substantial reduction in both strength and stiffness, indicating their critical influence on structural integrity.

Figure 7 shows the effect of cracks on the structural response under transverse uniaxial compression. Similar to the longitudinal strength, the transverse load capacity of the structure was affected by the cracks. Interestingly, the transverse cracks did not significantly influence the transverse response (Figure 7b) because, after buckling, most of the applied load was carried by the plate edges (Figure A10). The stress distribution confirms that under transverse loading, the plate edges primarily resist the applied load, regardless of crack size. As a result, cracks that form are situated in regions with inherently lower stress, minimizing their impact on the overall structural response. Thus, no significant difference was identified between the effects of the longitudinal and combined cracks whose length and breadth were smaller than 50% of the structural dimensions. Combined cracks sized 50% of the structural dimensions made a quarter of the plate separate from the main structure (Figure A15).

When the load is applied along the crack direction, strength deterioration progresses more rapidly, as stress concentration around the crack increases, making propagation easier. Conversely, if the loading and crack directions differ, the crack may close, or the stress may be redistributed to other regions, slowing the reduction in strength. Also, it is observed that under combined loading, the structural load-carrying capacity diminishes more rapidly than in cases with a single-direction crack. This is because the interaction between the two crack directions disrupts the stress redistribution mechanism, leading to localized instability.

Figure 8 and

Table A1. Ultimate strength of cracked plates with longitudinal cracks.

Load Ratio (σx:σy)	Intact		Longi. 10%		Longi. 25%		Longi. 50%
	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY
1:0	0.688	0.000	0.664	0.000	0.641	0.000	0.623	0.000
			(3.49) *	(-)	(6.83)	(-)	(9.45)	(-)
3:1	0.625	0.208	0.617	0.206	0.582	0.194	0.546	0.182
			(1.28)	(0.96)	(6.88)	(6.73)	(12.64)	(12.50)
1:1	0.262	0.262	0.244	0.244	0.220	0.220	0.180	0.180
			(6.87)	(6.87)	(16.03)	(16.03)	(31.30)	(31.30)
1:3	0.085	0.265	0.085	0.256	0.079	0.237	0.066	0.200
			(0.00)	(3.40)	(7.06)	(10.57)	(22.35)	(24.53)
0:1	0.000	0.270	0.000	0.262	0.000	0.241	0.000	0.204
			(-)	(2.96)	(-)	(10.74)	(-)	(24.44)

* Percentage reduction in ultimate strength in %.

,

Table A2. Ultimate strength of cracked plates with transverse cracks.

Load Ratio (σx:σy)	Intact		Trans. 10%		Trans. 25%		Trans. 50%
	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY
1:0	0.688	0.000	0.686	0.000	0.625	0.000	0.532	0.000
			(0.29)	(-)	(9.16)	(-)	(22.67)	(-)
3:1	0.625	0.208	0.624	0.208	0.603	0.201	0.515	0.171
			(0.16)	(0.00)	(3.52)	(3.37)	(17.60)	(17.79)
1:1	0.262	0.262	0.262	0.262	0.247	0.247	0.225	0.225
			(0.00)	(0.00)	(5.73)	(5.73)	(14.12)	(14.12)
1:3	0.085	0.265	0.085	0.264	0.085	0.264	0.085	0.264
			(0.00)	(0.38)	(0.00)	(0.38)	(0.00)	(0.38)
0:1	0.000	0.270	0.000	0.270	0.000	0.270	0.000	0.270
			(-)	(0.00)	(-)	(0.00)	(-)	(0.00)

and

Table A3. Ultimate strength of cracked plates with combined cracks.

Load Ratio (σx:σy)	Intact		Combined 10%		Combined 25%		Combined 50%
	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY
1:0	0.688	0.000	0.668	0.000	0.596	0.000	0.367	0.000
			(2.91)	(-)	(13.37)	(-)	(46.66)	(-)
3:1	0.625	0.208	0.615	0.206	0.571	0.190	0.330	0.110
			(1.60)	(0.96)	(8.64)	(8.65)	(47.20)	(47.12)
1:1	0.262	0.262	0.258	0.258	0.209	0.209	0.131	0.131
			(1.53)	(1.53)	(20.23)	(20.23)	(50.00)	(50.00)
1:3	0.085	0.265	0.085	0.256	0.078	0.233	0.047	0.141
			(0.00)	(3.40)	(8.24)	(12.08)	(44.71)	(46.79)
0:1	0.000	0.270	0.000	0.261	0.000	0.240	0.000	0.142
			(-)	(3.33)	(-)	(11.11)	(-)	(47.41)

present the results of biaxial loading on cracked plates, highlighting the influence of crack size and type on their ultimate compressive strength. As shown in Figure 8d, small cracks had a negligible effect on the ultimate strength under biaxial compressive loads, regardless of crack orientation. Among the different crack types, the combined crack had the most significant impact, followed by the longitudinal crack. In contrast, the effect of the transverse crack varied with the loading ratio (Figure 8b) and had no influence on the ultimate strength in the transverse direction.

Figure 8f further demonstrates that a large combined crack (50% of the structural dimension) led to an approximately 50% reduction in the ultimate strength of the intact panel across all loading conditions. This substantial strength degradation occurred because the crack effectively detached one-quarter of the plate, significantly reducing its ability to resist compressive loads. The detached region no longer participated in load transfer, leading to stress redistribution and increased vulnerability to local instability (Figure A11, Figure A12, Figure A13, Figure A14 and Figure A15).

4.2. Ultimate Compressive Strength of Cracked Stiffened Panels

Figure 9 and

Table 4. Comparison of ultimate strength (σ_u/σ_Y) of cracked stiffened panels under uniaxial compressions with the existing formula.

Crack Size			Intact	10%	25%	50%
Longi. strength	Trans. crack	Present	0.773	0.77	0.761	0.738
		Equation (5)	0.773	0.761	0.743	0.713
		Difference	0.000	0.012	0.024	0.034
Trans. strength	Longi. crack	Present	0.293	0.291	0.29	0.286
		Equation (5)	0.293	0.263	0.22	0.146
		Difference	0.000	0.096	0.241	0.490

Difference: (1 − Present/equation).

compare the ultimate strength of stiffened panels with perpendicular cracks under uniaxial compression, as calculated by Equation (5). The existing formula underestimates the strength of these stiffened panels because it assumes a uniformly reduced cross-sectional area, neglecting the load redistribution effect provided by the stiffening elements. For longitudinal strength with a transverse crack, the results of this study showed little deviation from the equation, as the cracked area was relatively small compared to the original cross-section, and its impact on the overall structure was minimal. However, a significant gap appears when the structure is subjected to transverse compression with longitudinal cracks. Equation (5) assumes a uniformly reduced cross-sectional area based on the ratio of the cracked area to the total area. This leads to an underestimation, as it applies the original strength without considering the fact that, at the ultimate limit state, most of the load is carried by the edges. These edges are where the stiffening elements are concentrated (see Figure A20).

Figure 10 and Figure 11 compare the structural response of intact and cracked stiffened panels under longitudinal and transverse uniaxial compression, while Figure 12 illustrates the ultimate compressive strength of the panels under both uniaxial and biaxial compressions. The influence of longitudinal and transverse cracks on the ultimate strength of stiffened panels was less significant compared to that of plates. This is because stiffened panels have a larger load-bearing structural area and improved load redistribution capacity, reducing the localized stress concentration near the crack region. Additionally, the overall stiffness, characterized by the slope of the stress–strain curve, was only marginally affected by cracks. This is attributed to the stiffeners, which act as load-carrying elements, enhancing the global rigidity of the panel and delaying local instability by redistributing stresses away from the cracked region.

Figure 12 presents the ultimate strength results for both longitudinal and transverse directions under varying crack sizes. In the case of transverse cracks, the ultimate strength decreased linearly with increasing crack size, showing minimal differences in strength reduction between transverse and combined cracks. However, when the crack size reached 50% of the panel’s length (L) and breadth (b), the strength dropped suddenly. On the other hand, the longitudinal strength was more significantly influenced by the crack direction, with combined cracks showing a sharper decline. For crack sizes up to 25%, the longitudinal strength exhibited a near-linear reduction, with combined cracks starting at a value between the longitudinal and transverse results but showing a rapid decrease from 25%.

Figure 12 and

Table A4. Ultimate strength of cracked stiffened panels with longitudinal cracks.

Load Ratio (σx:σy)	Intact		Longi. 10%		Longi. 25%		Longi. 50%
	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY
1:0	0.773	0.000	0.739	0.000	0.708	0.000	0.681	0.000
			(4.40)	(-)	(8.41)	(-)	(11.90)	(-)
3:1	0.715	0.235	0.691	0.227	0.673	0.220	0.642	0.211
			(3.36)	(3.40)	(5.87)	(6.38)	(10.21)	(10.21)
1:1	0.305	0.300	0.306	0.301	0.298	0.294	0.282	0.278
			(−0.33)	(1.31)	(2.30)	(3.61)	(7.54)	(8.85)
1:3	0.097	0.288	0.097	0.288	0.098	0.288	0.097	0.286
			(0.00)	(0.00)	(−1.03)	(0.00)	(0.00)	(0.69)
0:1	0.000	0.293	0.000	0.291	0.000	0.290	0.000	0.286
			(-)	(0.68)	(-)	(1.02)	(-)	(2.39)

,

Table A5. Ultimate strength of cracked stiffened panels with transverse cracks.

Load Ratio (σx:σy)	Intact		Trans. 10%		Trans. 25%		Trans. 50%
	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY
1:0	0.773	0.000	0.770	0.000	0.761	0.000	0.738	0.000
			(0.39)	(-)	(1.55)	(-)	(4.53)	(-)
3:1	0.715	0.235	0.709	0.233	0.709	0.233	0.683	0.224
			(0.84)	(0.85)	(0.84)	(0.85)	(4.48)	(4.68)
1:1	0.305	0.300	0.301	0.297	0.288	0.284	0.278	0.274
			(1.31)	(2.62)	(5.57)	(6.89)	(8.85)	(10.16)
1:3	0.097	0.288	0.097	0.287	0.097	0.287	0.097	0.287
			(0.00)	(0.35)	(0.00)	(0.35)	(0.00)	(0.35)
0:1	0.000	0.293	0.000	0.292	0.000	0.292	0.000	0.293
			(-)	(0.34)	(-)	(0.34)	(-)	(0.00)

and

Table A6. Ultimate strength of cracked stiffened panels with combined cracks.

Load Ratio (σx:σy)	Intact		Combined 10%		Combined 25%		Combined 50%
	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY	σxu/σY	σyu/σY
1:0	0.773	0.000	0.757	0.000	0.648	0.000	0.538	0.000
			(2.07)	(-)	(16.17)	(-)	(30.40)	(-)
3:1	0.715	0.235	0.695	0.229	0.678	0.223	0.635	0.209
			(2.80)	(2.55)	(5.17)	(5.11)	(11.19)	(11.06)
1:1	0.305	0.300	0.305	0.301	0.293	0.289	0.255	0.251
			(0.00)	(1.31)	(3.93)	(5.25)	(16.39)	(17.70)
1:3	0.097	0.288	0.097	0.288	0.098	0.288	0.089	0.262
			(0.00)	(0.00)	(−1.03)	(0.00)	(8.25)	(9.03)
0:1	0.000	0.293	0.000	0.291	0.000	0.290	0.000	0.275
			(-)	(0.68)	(-)	(1.02)	(-)	(6.14)

present the results of biaxial loading on cracked stiffened panel, emphasizing the effect of crack size and type on the ultimate compressive strength. As shown in Figure 12d, a small crack (10% of the structural dimensions) did not significantly affect the ultimate strength of the panel under biaxial compression, similar to the behavior observed in the plate (Figure 8d). This is because the stress distribution remained largely unaffected by the crack, as illustrated in Figure A16, Figure A17, Figure A18, Figure A19, Figure A20, Figure A21, Figure A22, Figure A23, Figure A24, Figure A25, Figure A26, Figure A27, Figure A28, Figure A29 and Figure A30.

In contrast, the combined cracks, each with a size of 50% of the plate’s length and breadth (or web height), led to a longitudinal disconnection between the plate and stiffener (Figure A27d). As a result, both the strength and stiffness of the panel decreased significantly, as the structural components were no longer able to transmit loads effectively. The impact of the cracks on the panel’s performance was found to diminish as the loading ratio (${\mathrm{\sigma }}_y/{\mathrm{\sigma }}_x$) increased (Figure 12). In other words, when the longitudinal load was much smaller than the transverse force, the cracks had a negligible effect on the ultimate strength of the stiffened panel.

5. Concluding Remarks

This study investigated the ultimate compressive strength of cracked plates and stiffened panels under uniaxial and biaxial loading conditions using nonlinear finite element analysis. The effects of crack size, orientation, and location were systematically examined across various structural configurations. The key findings are as follows:

• Crack orientation had a significant influence on strength degradation. A 50% transverse crack reduced strength by 22.67%, compared to 9.45% for a longitudinal crack under the same loading,
• Stiffened panels exhibited strong resistance to cracking, with minimal loss even under severe damage scenarios. For example, a 50% longitudinal crack caused only a 2.39% reduction, while the corresponding plate suffered a 24.44% loss,
• Combined (bidirectional) cracks resulted in the most severe reduction in load-carrying capacity among all configurations,
• Structural reinforcement via stiffeners effectively redistributed stress and preserved global integrity, as clearly demonstrated by the superior performance of stiffened panels.

These findings provide practical insights that can be applied in the shipbuilding and maintenance industry to estimate residual strength in cracked structures and to plan appropriate repair strategies based on crack size and orientation. Future work will extend this framework to include crack propagation and cumulative damage under realistic loading conditions.