Experimental and Numerical Investigation on the Ultimate Strength of Stiffened Plates with Scanned Initial Geometrical Imperfection

The study of the ultimate strength of stiffened plates is a hot topic in ocean engineering. The ultimate strength and behavior of collapse of stiffened plates were investigated using experimental and numerical methods. Two stiffened plates, with one and two half-bays in both longitudinal and transverse directions, were tested under the uniaxial compression. There were clamped boundaries at both ends of the stiffened panels and a restrained boundary on the transverse frames. The novel three-dimensional laser scanning technology was used to measure the initial geometric imperfections and the ultimate deformation of the stiffened panels after the test. The initial geometric deformation was imported into the finite element model, and the ultimate strength and behavior of collapse of the stiffened plates were calculated using the finite element analysis. FE analysis results based on the measured initial geometric imperfections were compared with the test results. It is concluded that structural deformation can be well measured by three-dimensional laser scanning technology, and can be conveniently imported into the finite element analysis. With the measured initial geometric imperfections considered, the FE analysis results agree well with the experimental results in ultimate strength, behavior of collapse, and the ultimate displacement distribution of the stiffened panels.


Introduction
Stiffened plates are the fundamental building form used for ship and offshore structures. Great efforts have been made to develop the effective method for the local and global strength assessment of ship and offshore structures. So, it is significant to further understand the ultimate strength of the stiffened panel. This is especially important as the limit state practices are continually used in the structural design (ISSC, 2015). The structural test is a direct and effective method to evaluate the ultimate strength of the stiffened panel. Gordo and Guedes Soares (2008a, 2008b, 2011 carried out the compressive tests on the short, intermediate and long stiffened panels by changing the distance between transverse frames. The specimens were three-bay stiffened panels with the associated plate made of very high tensile steel S690. Ghavami (1994) carried out two series of tests on the longitudinally stiffened steel plates with and without the transverse stiffeners under the uniform axial in-plane compression until they collapsed. Tanaka and Endo (1988) conducted a series of tests on the longitudinally stiffened panels with three flat bar stiffeners by three bays. The fail-ure of the stiffened plates in the tests intended to be caused by the local buckling or tripping of the longitudinal stiffeners. The stiffened plates of two half bays plus one full bay in the longitudinal direction were used to investigate the ultimate strength and behavior of collapse by Guedes Soares (2013a, 2013b) and the effect of the width on the ultimate strength of the stiffened panels was analyzed.
The nonlinear finite element (FE) analysis is another widely used method to evaluate the strength of stiffened and unstiffened panels. This method can predict the complex collapse behavior of structures in detail by considering the geometry, boundaries and material nonlinearity. The ultimate load bearing capacity of unstiffened rectangular steel plates under a uniaxial compression was studied by Silva et al. (2013). Shanmugam et al. (2014) discussed the ultimate strength of the stiffened plates subjected to the combined action of in-plane loads and lateral pressure using the software ABAQUS. Feng et al. (2017) investigated the ultimate strength of ship stiffened panel subjected to random corrosion degradation and the influence of corrosion on the ultimate strength of the stiffened panel was quantified as a function of the corrosion volume.
Initial imperfections are unavoidably caused by the manufacture and welding of the structures and there are significant uncertainties regarding the magnitude and the deformation pattern (Guedes Soares, 1988a). It has been widely accepted that the initial imperfections reduce the rigidity and the ultimate strength of plates. Furthermore, plates under the buckling loads tend to deform in way corresponding to their initial defects (Guedes Soares, 1988b). In a word, initial imperfections have significant impact on the ultimate strength as well as the behavior of collapse of stiffened plates. Therefore, it is an important issue to accurately describe the initial geometric imperfections when assessing the ultimate strength of stiffened plates.
In recent studies (Fujikubo et al., 2005;Paik et al., 2008), the initial deformation was assumed to have the same shape as the trigonometric functions, i.e., the equivalent initial deformation. Xu and Guedes Soares (2013c) calculated the ultimate strength of the stiffened plates with measured initial geometric imperfections using a caliper ruler and the equivalent geometric imperfections, and the numerical results were compared with the experimental data. Although the equivalent geometric imperfections do not correspond accurately to real structures, they are still widely used because the measurement of the geometric imperfections is not convenient in engineering application. With the development of three-dimensional (3D) scanning technology and the availability of portable devices, the geometric imperfections measurement becomes more convenient. Hence, it is feasible that the geometrical initial imperfections of the stiffened panel for the ultimate strength assessment are measured by 3D scanning technology.
The objective of this paper is to investigate the ultimate strength and collapse behavior of stiffened plates of two half bays plus one full bay under the uniaxial compression, with the clamped boundaries at both ends of the panel and the restrained boundaries on the transverse frames using both experimental and numerical methods. The novelty lies in the measurement of the deformation of the stiffened panel applying 3D scanning technology. The initial geometrical imperfections for both plate and stiffener of the stiffened panel were measured through 3D scanning and then were restored by the inverse processing technology. The restored initial deformation of the stiffened panel was imported into the numerical analysis. The numerical results regarding the ultimate strength and behavior of collapse were compared with the experimental ones. The deformation distribution of the stiffened panel after collapse was quantified using 3D scanning measurement and was compared using the finite element analysis.

Experimental arrangement
The test aimed at the ultimate strength assessment of the stiffened panels under the uniaxial compression, with β λ clamped boundaries at both ends of the panel and restrained boundaries on the transverse frames. The stiffened panel with two bays (1/2+1+1/2) made of steel AH36 was used in this test. The currently widely used principle to assess the ultimate strength of the stiffened panel is based on the plate slenderness ratio, , and the column slenderness ratio, (Xu and Guedes Soares, 2012a;Saad-Eldeen et al., 2015).
where, is the width of the plate of one bay, is the thickness of the panel, is the yielding strength of the material and is the Young's modulus, a is the length of the plate of one bay, r is the radius of inertia of the stiffener. The plate slenderness ratio is generally from 1 to 5 (Smith, 1975;Guedes Soares, 1992) and the column slenderness ratio is from 2 to 7.5 for ship stiffened panels (Xu and Guedes Soares, 2013c). So without any loss of applicability, the stiffened plate is designed according to the plate slenderness ratio of 1.77, and the column slenderness ratio of 4.35. The thickness of the plate is 5 mm, the frame space is 400 mm and the space between stiffeners is 190 mm. The frame of the stiffened panel is a tee section bar with a web of 40 mm×4 mm and a face plate of 25 mm×5 mm respectively, and the stiffener is a flat bar of 20 mm×4 mm, as shown in Fig. 1.
A 1000 kN hydraulic actuator was used in the experiments on the stiffened panels. The uniaxial compression was accomplished by using displacement control. For each loading step, the displacement was exerted slowly enough to obtain the static structural response and the load-shortening curve was recorded. The end edges of the stiffened plates were tightly clamped by the steel blocks to produce fixed boundaries at the ends of the panel. The ends of the stiffened plate could only move with the steel blocks in an axial direction following the round bar guide rails. Four greased clevises embraced the guide rails of the square bar. YU Yang-zhe et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 446-458 447 The clevis was designed to support the frame with contact connection at ends of the frame and with gaps running perpendicular to the panel, as shown in Fig. 2. Consequently, the rotation and translation of the frame ends were respectively constrained around the long edge of the panel (x axis) and along the transverse of the panel (y axis). Due to the gaps between the clevis and the stiffened plate, no additional contacts occurred between the clevis and the stiffened panel except for the support of the clevis at the ends of the frame. So, the deformation of the stiffened plate was allowed at the frame ends. Strain gauges (s1-s7) were evenly arranged at the interval of 47.5 mm along the centerline of the back side of the panel. The strain gauges on the outside of the midpoint of the stiffeners were numbered s8 and s9, as shown in Fig. 3.

Measurement of the initial geometric imperfection
Initial geometrical imperfections caused by machining and welding are unavoidable and have significant effects on the ultimate strength and the failure mode of the stiffened panels. Therefore, it is necessary to measure the geometrical initial imperfections. As discussed in the recent study (Xu and Guedes Soares, 2012b), the displacement transducer had been used to measure the distance between a reference base plane and the plate surface. The measured initial imperfections of the plate were used directly in the FE analysis. However, the initial imperfections of the plate were only measured at very limited points in the plate and no initial geometrical imperfections were measured on the stiffeners. The device and the measured initial imperfections using that device (Xu and Guedes Soares, 2012b) were illustrated in Figs. 4a and 4b.
Herein, the initial geometric imperfections of the speci-mens were measured using three-dimensional (3D) laser scanning technology and the 3D scanning device is given in Fig. 5a. Based on the principle of laser ranging, a large number of points in the specimen surfaces were recorded with the information of three dimensional coordinates, re-  flectivity and texture. By using the software Geomagic Spark, the recorded points in surfaces of the specimens were analyzed and digitally generated (Two specimens were tested and named A and B respectively). The initial deformations of the panel and the stiffeners described by the recorded points of Specimens A and B were respectively given in Figs. 5b and 5c. The results of 3D scanning measurement obviously have a higher precision than those in Fig. 4b. Moreover, 3D scanning is especially suitable for the measurement in situ because there is no need to deploy displacement transducers or to build a reference plane. It can be seen that the initial geometrical imperfections of the two specimens are irregular. The maximum initial deformation for panel of Specimen A is smaller than that of Specimen B. The maximum initial deformation for the panel occurred in the middle bay for both specimens. Manufacturing caused the initial side-way deformation in the stiffeners. The maximum side-way deformation of Specimen A is a little greater than that of Specimen B. For Specimen A, the side-way deformation of one stiffener is obviously greater than that of the other. However, the side-way deformation of both stiffeners is similar to each other for Specimen B. On the whole, Specimen B has relatively larger initial deformation than Specimen A, which causes an adverse effect on the ultimate strength of Specimen B.
The initial deformation results by 3D scanning are easy to be imported to the finite element analysis software. The inverse processing is first carried out by using the software Geomagic Control. Geomagic Control will create the geometrical surface of the specimen through the points in the surface of the specimen recorded by the software Geomagic Spark. And then, the geometrical surface generated by the software Geomagic Control can be converted into the finite element model with the software Hypermesh. In this way, the initial geometrical imperfections of both the plate and the stiffener of the specimens were restored and were further imported into the finite element analysis software for numerical analysis. Details about mesh in FEM analysis can be found in Section 3.2. The basic idea of the 3D scanning technology and how to make the model available in the FEM analysis is given in Fig. 6.

Experimental results
The ultimate strength test was carried out in three loading cycles. The first two cycles were designed to release the residual welding stress of the specimen, and eliminate any possible gap between the specimen and the test setup. The last loading cycle aimed to get the ultimate strength, as well as the buckling and post-buckling behavior of the stiffened plate. It should be noted that the first two loading cycles should be adequately large so as to effectively release the residual stress and eliminate the gap, but keep the stiffened  YU Yang-zhe et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 446-458 449 plates in an elastic structural response state. Such experimental procedure has been widely used, as can be found in Xu and Guedes Soares (2013a, 2013b, 2013c.
In the experiment, the first loading cycle was 0 kN→100 kN→0 kN and the second loading cycle was 0 kN→150 kN→0 kN. The load-displacement curves of Specimen A during the experiment were obtained, as shown in Fig. 7. At the end of the first cycle, the axial displacement did not return to zero, which indicates that there are gaps between the specimen and the loading device. However, the load curves in the 2nd and 3rd cycles were almost the same at the initial loading stages, which illustrates that the gaps between the specimen and the loading device are eliminated and the response of the specimen is in an elastic stage. The ultimate strength of Specimen A is 351.47 kN. It can also be found that there is a sudden decline in the load-displacement curve. This is due to the tripping buckling of the stiffener that had large initial side-way deformation.
The strain-displacement curves in the final loading cycle are given in Fig. 8a for strain gauges on the panel and Fig. 8b for those on the stiffeners. All the strain gauges used in the test are unidirectional gauges and the measured strains represent the axial strains. All strains started from zero in the final cycle, which indicates that the first two loading cycles are in the elastic range. The strains of s1-s7 shared the same pattern. At the beginning of the loading, the strains of s1-s7 were negative due to the compression, until the stiffened plate reached the limit state. The strains of s8-s9 changed rapidly compared with those of the panel and the values were significantly greater than those on the panel. The strain of s8 achieved the record limit at the imposed displacement of 3.85 mm before the ultimate state of specimen A. The strain of s9 also achieved the record limit as soon as the specimen reached the limit state. As can be seen in Fig. 8b, the strain of s8 changes more severely than that of s9, and it is attributed to the stiffener with s8 having greater initial deformation than that of s9, as noted in the initial deformation measurement. The stiffener with the greater initial deformation entered a buckling state earlier, which leaded to a sudden falling of the load-bearing capacity for the stiffened plate, as shown in Fig. 7.
The load-displacement curves for Specimen B were experimentally obtained, as shown in Fig. 9. The first loading cycle was 0 kN→100 kN→0 kN and the second loading cycle was 0 kN→150 kN→0 kN. During the first two loading cycles, the trends of the curves were almost the same, which indicates that the deformation of the specimen is within the elastic range, and little gap exists in the test setup. The ultimate force achieved was 337.04 kN when the displacement reached 4.18 mm. For Specimen B, no sudden buckling of local structure occurred, such as the tripping failure of the stiffener, as found in Specimen A. As Specimen B having relatively larger general deformation than Specimen A, the ultimate strength of Specimen B is

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YU Yang-zhe et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 446-458 smaller than that of Specimen A. The strain displacement curves in the final loading cycle of Specimen B were experimentally obtained, as shown in Fig. 10. All the strains in the third loading cycle started from zero, which means that the loads in the previous loading cycles were big enough to release the residual stress and no plastic deformation was induced. The strains (s1-s7) in the middle of the panel shared the same trends, which were first negative and then positive. The negative values indicate that the panel was in compression, while with the increase of the axial load, the panel began to deform. The deformation induced the back of the panel in tension and therefore the strain values became positive. After collapse, the strains increased rapidly until the end of the test. The trends of the strain-displacement curves are similar for s8 and s9, especially at the beginning of the curves, because the initial deformation of the stiffeners is alike. The variations in the initial geometrical imperfections between the two stiffeners eventually lead to a difference in the latter part of the strain displacement curves. It is obvious that the strain on the stiffener was much larger than that of the panel before the specimen failed. The strains of the stiffeners achieved the record limit at the imposed displacement of 2.8 mm and 3.6 mm, respectively.
The 3D laser scanning method was used again after the panels collapsed and the measured deformation was compared with the tested ones, as shown in Fig. 11 and Fig. 12. It is obvious that the 3D scanning technology can record and represent the shape of the specimen regardless whether the specimen deformed or not. It is found that the failure mode of the two specimens can be classified as the stiffenerinduced failure. The scanned deformation of the stiffened panels is compared with that attained by the finite element analysis in Section 4.

Nonlinear finite element analysis for ultimate strength calculation
3.1 Finite element model Based on the ABAQUS software package, the four-node shell element S4R with six degrees of freedom at each node and five integration points along the thickness was applied to build the FE model. It accounts for both linearity and nonlinearity of large rotations and strains. Both full and reduced integration schemes are supported Wang, 2012a, 2012b;AbuBakar and Dow, 2013). Riks method was employed to investigate the ultimate strength of the stiffened plates. This method is based on the solution of the structural nonlinear static equilibrium equation and tracks the load-displacement equilibrium path of the structure during loading and unloading (ABAQUS, 2004).
The mesh of the FE model should be fine enough to give accurate and reliable results. Convergence analysis of mesh size was carried out (Ghavami and Khedmati, 2006). The element sizes chosen were 15 mm, 10 mm and 5 mm in the convergence study, and the ultimate strength of the stiffened plate were 352.2 kN, 351.6 kN and 350.7 kN, respectively.   The ultimate strength of the stiffened plate was convergent when the mesh size was 10 mm, as shown in Fig. 13. In this paper, to restore the measured initial geometric imperfection in detail, a mesh size of 5 mm was chosen.

Initial geometric imperfection
The initial geometric imperfections determine the final deformation and the ultimate strength of the stiffened plates. Thus, it is of great importance to consider the initial geometric imperfections in the FE analysis. In this paper, two methods were adopted to account for the initial geometric imperfections.
One commonly used method in the simulation of the initial geometric deformation is to present the initial deformation as a trigonometric function. The equivalent initial deformation of the stiffened plates as expressed from Eq. (3) to Eq. (5) (Zhang and Khan, 2009). All the assumed equivalent amplitudes in the following equations were on the average level.
Column-type initial deformation of the panel: Deformation of local panel plate: Side-way deformation of the stiffeners: where a is the length of the plate of one bay, is the width of the plate of one bay, m is the number of buckling half waves which can be predicted as a minimum integer satisfying Eq. (6), which is 2 in this paper, is the height of the stiffener.
In addition, the 3D scanning technology was used to measure the initial deformation of the stiffened plate. The initial surfaces of the specimens were created by Geomagic Control and then was imported into Hypermesh to generate the model mesh used in the FE analysis. The model of the plate and stiffeners with the measured initial imperfections was amplified 8 times, as can be seen in Fig. 14, where the mesh was set to be invisible and the deformation pattern of the specimens was irregular.

Material properties
The influences of the material hardening on the ultimate strength and behavior of collapse of the stiffened panel were investigated by ISSC (2009). The ideal elastic-plastic material model has also been widely used in the FE analysis, especially for complicated structures (Yu et al., 2017). To consider the effect of the material nonlinearity and accurately replicate the behavior of collapse of the stiffened plates in the FE analysis, a group of tension tests were conducted to evaluate the material properties of steel AH36, as shown in Fig. 15. The velocity of the imposed displacement was 0.15 mm/s and the force-displacement data were recorded during the process. The material properties obtained were as follows: the yielding stress 418.15 MPa, the Young's modulus N/mm 2 and Poisson's ratio . The obtained true stress-strain curve was used in the FE analysis.

Boundary conditions
The boundary conditions in the finite element analysis should reproduce the actual constraint of the test. Before analysis, the enforced displacement was applied to the loading edges to make up for the deformation caused by the loading device during assembly. In the analysis, all degrees of freedom except the axial displacement were constrained for the loading edge. Enforced displacement was applied at the reference point to simulate the axial load. For the constraint of the frames, the clevis was modeled and tied to the

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YU Yang-zhe et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 446-458 stiffened plate at the end of the frames. Tie is a kind of friction free contact connection in which the contact algorithm is activated if the contact occurs and is otherwise kept inactive (ABAQUS, 2004). As in the test, the clevis was set to constrain the rotation around the x axis and the translation along the y axis in the FEA. Because of the gaps between the stiffened panel and the clevis, the translation along the z axis and rotation around the y axis were free. All the boundary conditions in the FEA were defined as shown in Fig.16, where U x , U y and U z indicate the translation along coordinate axes x, y and z, and R x , R y and R z denote the rotation around the corresponding coordinate axis.

Results of Specimen
The normalized stress-strain curves obtained in the FE analysis and the test for Specimen A are shown in Fig. 17.
( ) is defined as the ratio of the simultaneous panel strain ( ) to the panel strain of the limit state ( ), where the panel strain is defined as the axial displacement by the length of the stiffened panel.
is defined as the ratio of the mean stress in the stiffened panel to the yielding stress of the material. A more gradual growth of at the beginning of loading in the test is due to the rearrangement of the test setup until every part of the panel, support and hydraulic machine were in full contact. The phenomenon also explains the reason that obtained in the test did not start σ a /σ y from zero. The error of obtained in the test and the FE analysis is only 0.23%. Similar to the sudden decline of the normalized stress-strain curve for the test, the finite element analysis result illustrates a rapid decline as the stiffened panel reaches the ultimate state. It demonstrates that the stiffener buckling is identified in the FEA.
After testing, the 3D scanning technology was used again to monitor the deformation of the specimen. Thus, the displacement distribution of the specimen can be obtained through the comparison of the surfaces created by 3D scanning before and after the test, as shown in Fig. 18. It can be seen that the middle bay of Specimen A deformed away from the stiffeners during test. This failure mode is classified as stiffener-induced failure (SIF) (Paik and Thayamballi, 2003).    YU Yang-zhe et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 446-458 453 The relative deformation, defined as the deformation relative to the maximum deformation of the stiffened panel, for the panel of Specimen A was obtained by 3D scanning and FEA respectively, as shown in Fig. 19a for test and Fig. 19b for FE analysis. The displacement distribution of the panel of Specimen A during the test and FE analysis kept the same pattern, as can be seen in Fig. 19. Although the specimen was symmetrically designed, the displacement distribution did not keep symmetry due to the irregular initial deformation. Compared with Fig. 5a, it can be seen that the maximum displacement of the panel occurred at the same place where the maximum initial geometric imperfection was.
The relative deformation for d1-d3 at the final load step of the test was extracted and compared with those in the FE analysis, as shown in Fig. 20. The errors of d1 and d3 are smaller than 1% while the error of d2 is 5.70%, which indicates that the results of the FE analysis are credible and highly accurate.
The side-way deformation of the stiffeners was attained by 3D scanning and FEA as well, as shown in Fig. 21. It can be found that the results of the FEA are quite close to the ones obtained via 3D scanning. The buckling of the stiffener with the greater initial deformation can be identified both in the FEA and in 3D scanning, as noted in the initial deformation measurement. However, the maximum side-way deformation of the stiffener occurred in the middle bay of the stiffened panel for both the test and FEA, instead of the position where the initial maximum imperfection occurred. This can be explained as the deformation is governed by the stiffener induced failure of the stiffened panel, where the maximum deformation will appear in the middle of the stiffened panel.

Results of Specimen B ε N
The normalized stress-strain curve for Specimen B was obtained by FE analysis and compared with the one from the test, as shown in Fig. 22. In the test, the reaction load first increased almost linearly with the increase of the axial displacement. Then, the reaction load rose slowly as exceeded 0.49. Different from that in the test, the curve of the FE analysis grew linearly with the increase of the displacement before it reached its maximum. The ultimate stress ratio of the stiffened plate was 0.38 in the FE analysis, 1.93% smaller than that in test. The curves declined gradually as the stiffened panel reached its ultimate state. This is due to   the similar initial deformation of the stiffeners, and the facts that the external loads could be carried evenly by the two stiffeners without sudden collapse of any one stiffener, as was found for Specimen A. It can be seen from Fig. 5 that the initial geometric imperfections of Specimen B are larger than that of Specimen A, and the ultimate strength of Specimen B is smaller than that of Specimen A.
The displacement distribution of Specimen B was measured by 3D scanning, as shown in Fig. 23. Similar to Specimen A, the middle bay of Specimen B deformed away from the stiffeners during test. The failure mode can also be regarded as SIF.
The comparison of the relative deformation between the test and the FEA is illustrated in Fig. 24. The measured deformation of the stiffened plate almost had the same pattern as the calculated one, except in the small areas on the upper right and left bottom of the stiffened plate. The ratios of the displacements of d1-d3 to the maximum displacement at the final load step of test were compared with those in the FE analysis, as shown in Fig. 25. The errors in deformation for d1-d3 are 0.81%, 2.43% and 4.23%, respectively. The displacement distributions and the deformation at the measuring points are in agreement with the experimental ones.
The normalized side-way deformation in the test was compared with those in FEA, as shown in Fig. 26. It can be seen that the FEA result agrees with the test one. The pat-tern of the side-way deformation in the stiffener after the test is similar to that in the initial state except for those in the area of the end bay. Because of the constrained boundary conditions, the stiffeners in the end bay have barely deformed in the test or the FEA.

Results of Specimens C and D
The finite element model with the initial deformation described by Eqs. (3)-(5) is named as Specimen C, as shown in Fig. 27a. The maximum initial displacement of this stiffened plate is 2.01 mm, which is smaller than that of Specimens A and B. Compared with Fig. 5, the initial deformation of Specimen C is quite different from those of the other two. Specimen D is the finite element model with the average initial deformation of Specimen A and Specimen B, as shown in Fig. 27b. It can be seen that the maximum initial deformation of Specimen D is a little larger than that of Specimen C.    The normalized stress-strain curves of Specimens C and D are similar to the other two specimens, as shown in Fig. 28. Although the maximum initial imperfection in Specimen C is smaller than Specimens A, B and D, the ultimate strength of Specimen C is slightly smaller than the others, as shown in Fig. 29. Because the initial imperfections of Specimen C are simulated as a trigonometric function, there are several buckling half waves in both longitudinal and transversal directions. This kind of initial deformation is more regular in comparison with the measured ones and obviously causes a decrease in the structural stiffness according to the theory of the structural stability. The ultimate strength of Specimen D, with a mean statistical value generated from the initial deformation of Specimens A and B, is at the same level as those of Specimens A and B. Although only two specimens were measured, it has to be noted that the statistical values of the initial deformation are of significance in the ultimate strength assessment if a large number of stiffened panels are measured. 3D scanning becomes a powerful tool in making the detailed measurement of the initial deformation in the stiffened panel. is the result by the empirical formula (Paik and Thayamballi, 2003) as shown in Eq. (8).

Comparisons of the results
where, is the edge function, is the critical stress, is the net sectional area of stiffener without attached plating, is the net sectional area of attached plating of effective width, is the net sectional area of attached plating.
λ e β where, is the effective column slenderness ratio, is the plate slenderness ratio.
It can be seen that the ultimate strength, with the equivalent initial geometrical imperfections in FEA, is the smallest one, while the ultimate stress calculated by IACS is the biggest one. By calculating the ultimate strength using the IACS regulations, the one bay length is used to describe the column bucking deformation of the stiffened plate. However, in the test, the final bucking wave in the axial direction is longer than the one bay length. The initial imperfection is not considered in the regulations. This may explain why the ultimate strength using the IACS calculations is the largest. The smallest ultimate strength with the equivalent initial imperfection can be attributed to relatively small structural stiffness induced by the equivalent initial imperfection described in the trigonometric model. The ultimate strength obtained from the FEA with the measured initial imperfections is close to the ones in the tests and can be further verified by way of the empirical formula.

Conclusions
The ultimate strength tests of the stiffened panels made of AH36 under axial load, with clamped boundaries at both ends of the panel and a restrained boundary on the transverse frames, were performed. The FE analysis using the measured initial imperfections and the equivalent initial imperfections were carried out and compared with the test results. The following conclusions can be derived from this study.
The initial geometric imperfection, which has great influence on the ultimate strength of the stiffened panel, can be easily measured in detail using 3D scanning technology, in contrast to the conventionally used deformation measuring transducers. The 3D scanning is especially suitable for the measurement of the structural deformation in situ. The measurement results can also be incorporated into the finite element model of the stiffened panel, convenient for the subsequent nonlinear finite element analysis. A kind of test setup for the ultimate strength of the stiffened panel was developed and the tests were successfully carried out with it. The numerical model was built based on the finite element analysis. With the initial deformation considered, the numerical results of the ultimate strength of the stiffened panel, based on the nonlinear finite element analysis, agree with those obtained by testing. The different deformation patterns shown by Specimens A and B can be attributed to the different initial geometrical imperfections that may be quantified using the 3D scanning measurement. Owing to the large tripping deformation of one stiffener, Specimen A suffers a sudden buckling of the stiffener in the ultimate state, which is identified in the finite element analysis. The ultimate load-bearing capacity descends gradually after reaching the ultimate state for Specimen B as the two stiffeners have similar initial deformation. Both Specimens A and B present stiffener induced failure of the stiffened panel. The deformation after collapse, achieved by FEA, is consistent with the measured one. Although, the magnitude of the equivalent initial imperfections is smaller than that of the measured, the ultimate strength of Specimen C is reasonably small as the structural stiffness obviously decreases due to the assumed initial geometric imperfection of the trigonometric function. With the help of 3D scanning, the statistical values, for example the mean value, of the initial geometrical imperfections are attained conveniently and transferred to the FE model for the ultimate strength analysis. Although only two specimens were tested, it can be assumed that the statistical analysis of the initial geometric imperfections can be accomplished if sufficient numbers of stiffened panels are measured using 3D scanning technology and can be applied in the ultimate strength assessment.