Mechanical buckling analysis of explosive welded joints used in shipbuilding

Abstract

Aluminum superstructures and steel hull connections are of fundamental importance in ships. This study regards the buckling analysis of explosion welded joints, made of three layers (ASTM A516 low carbon steel, pure aluminium, A5086 aluminium alloy) and used in ship structures. Tensile and compressive tests were carried out on explosion welded specimens. The Infrared Thermography was used to detect the superficial temperature of the specimen during the tensile test and three phases of the temperature evolution were observed. The Digital Image Correlation technique was applied during the tests for the detection of displacement and strain fields. A theoretical analysis, considering the different materials was also performed for the analysis of buckling, which occurred during the compressive tests. Furthermore, a non-linear finite element analysis, considering the different mechanical properties of the explosive welded joint, was performed and was validated by means of the experimental results, obtained by the compressive tests.

Keywords

Explosive welded joints buckling digital image correlation infrared thermography non-linear finite element analysis ship structures

1. Introduction

Aluminium/steel welded joints are used in shipbuilding [7,8,13,18]. Figure 1 reports a typical application of connection of the steel structure to the Al superstructure using explosive welded joints.

Fig. 1.

Connection of the steel structure to the Al superstructure using explosive welded joints.

Generally, an intermediate layer can be placed in between aluminium and steel in order minimize a brittle interfacial zone and to improve the weldability, as reported in [11,14]. The literature on the recent developments in explosive welding was reviewed in [10]. Experimental investigations of explosive welding of aluminium to steel were reported in literature [1,4,7,8,13,16].

The aim of this research activity was to study the non-linear behavior of aluminium/steel explosive welded joints used in shipbuilding, when they are subjected to compression loading and buckling takes place. The influence of the different materials was considered. The tensile properties of the materials were calculated from hardness measurements, according to the procedure reported in [6,15,17], in order to perform a non-linear FE analysis, and the FE results were validated by means of the experimental data obtained through the Digital Image Correlation technique. Two full-field techniques were applied: the Digital Image Correlation technique (DIC) for the displacement field detection and the Infrared Thermography (IRT) to monitor the surface temperature of the specimen.

The authors have already applied full-field and non-destructive techniques (Infrared Thermography, Digital Image Correlation, Computed Tomography) for the analysis of different materials and welded joints used for marine structures: explosive welded joints under static and fatigue loadings [7,8], structural steel under static and fatigue loading [5,9], and Iroko wood [3].

2. Materials

The investigated explosive welded specimens, produced by TRICLAD, consist of ASTM A516 Gr55 structural steel, clad by explosion welding with AA5086 aluminum alloy and provided with an intermediate layer of AA1050 commercial pure aluminum. The heights of the layers are: 19 mm for the ASTM A516 Gr.55, 9.5 mm for the pure Aluminium 1050A, and 6 mm for the Aluminium alloy 5086; the thickness in the transversal direction is 3 mm. The measured hardness values for the three metals are: 175 HV for ASTM A516 gr 55, 47 HV for AA 1050 interlayer, 109 HV for AA 5086 [8]. The mechanical properties could be affected from the hardening due to the impact during the explosion welding process, so the non-linear properties were calculated from hardness measurements. The authors have reported in [8] the results of the yielding and ultimate stress as a function of the materials hardness. The obtained values are reported in Table 1.

Table 1
Tensile properties obtained from hardness measurements [8]

Steel AA 5086 A 1050 A

$σ_{Y}$ [MPa] 380 285 90

$σ_{ut}$ [MPa] 590 335 155

	Steel	AA 5086	A 1050 A
$σ_{Y}$ [MPa]	380	285	90
$σ_{ut}$ [MPa]	590	335	155

A method, which uses only the values of the ultimate and yielding stresses, was applied [12] in order to obtain the true stress–strain curves of the different materials. The method is based on the Ramberg–Osgood equation type: $\begin{matrix} (1) & \frac{E ε}{σ_{Y}} = \frac{σ}{σ_{Y}} + α {(\frac{σ}{σ_{Y}})}^{m} \end{matrix}$

Assuming the plastic strain at yielding $ε_{p Y} = 0.002$ , α is calculated as follows: $\begin{matrix} (2) & α = \frac{E ε_{p Y}}{σ_{Y}} \end{matrix}$ The relationship between m and the ultimate and yielding stresses, was found equal to: $\begin{matrix} (3) & m = 3.93 {[ln (\frac{σ_{ut}}{σ_{Y}})]}^{- 0.754} \end{matrix}$ The values of the R–O parameters are reported in Table 2.

Table 2

R–O parameters

	Steel	AA 5086	A 1050 A
$σ_{Y}$ [MPa]	380	285	90
$σ_{ut}$ [MPa]	590	335	155
m ( $ε_{p Y} = 0.002$ )	7.30	15.53	6.20
α ( $ε_{p Y} = 0.002$ )	1.08	0.50	1.57

The curves, obtained for the three materials using Eq. (1), are shown in Fig. 2.

Fig. 2.

Ramberg–Osgood curves of the different materials.

3. Theoretical analysis

For a tri-material specimen subjected to tension or compression loading, in the elastic phase, the stress will not be the same for each material but the strain will be the same, so the following relations can be assumed: $\begin{matrix} (4) & σ_{1} = E_{1} ε; σ_{2} = E_{2} ε; σ_{3} = E_{3} ε; \end{matrix}$ where subscripts 1, 2 and 3 refer to steel, pure aluminium and aluminium alloy respectively. Multiplying the stress by each area (width w and height h), the total force P can be expressed as: $\begin{matrix} (5) & σ_{1} h_{1} w + σ_{2} h_{2} w + σ_{3} h_{3} w = P \end{matrix}$ which means: $\begin{matrix} (6) & E_{1} ε h_{1} w + E_{2} ε h_{2} w + E_{3} ε h_{3} w = P \end{matrix}$ obtaining that the axial strain is: $\begin{matrix} (7) & ε = \frac{P}{w (E_{1} h_{1} + E_{2} h_{2} + E_{3} h_{3})} \end{matrix}$ allowing to evaluate the axial stresses as: $\begin{array}{l} (8) & σ_{1} = E_{1} \frac{P}{w (E_{1} h_{1} + E_{2} h_{2} + E_{3} h_{3})} \\ (9) & σ_{2} = E_{2} \frac{P}{w (E_{1} h_{1} + E_{2} h_{2} + E_{3} h_{3})} \\ (10) & σ_{3} = E_{3} \frac{P}{w (E_{1} h_{1} + E_{2} h_{2} + E_{3} h_{3})} \end{array}$

Buckling is defined as the tendency to deflect a beam if it is subjected to a certain compressive stress. In order for this phenomenon to occur, in reality it is necessary that the compression load is greater than the so-called critical load $P_{c}$ [N]. To be able to calculate it Euler’s formula shown below is usually applied: $\begin{matrix} (11) & P_{c} = \frac{π^{2} E J_{min}}{l_{0}^{2}} \end{matrix}$

From a theoretical point of view we can assume that when the three materials reach the critical stress, the whole specimen will undergo buckling phenomenon. The total critical load can be calculated as the sum of the critical load of each material. Considering that the specimen is clamped from both ends ( $l_{0} = 0.5 l$ ), the critical load is: $\begin{array}{l} (12) & \begin{array}{l} P_{cth} = \frac{4 π^{2} E_{1} J_{1 min}}{l^{2}} + \frac{4 π^{2} E_{2} J_{2 min}}{l^{2}} + \frac{4 π^{2} E_{3} J_{3 min}}{l^{2}} \\ P_{cth} = 4 π^{2} \frac{(E_{1} J_{1 min} + E_{2} J_{2 min} + E_{3} J_{3 min})}{l^{2}} \end{array} \end{array}$

The total critical load and the critical stress of aluminium are plotted vs the specimen length in Fig. 3.

Fig. 3.

Total critical load and critical stress of aluminium vs specimen length.

Considering a length $l = 350 mm$ and that the specimen is clamped from both ends, the following values reported in Table 3 are obtained.

Table 3

Critical load and stress calculated for each material and total critical load for all the welded joint

$P_{c 1 th}$ (N)	$P_{c 2 th}$ (N)	$P_{c 3 th}$ (N)	$σ_{c 1 th}$ (MPa)	$σ_{c 2 th}$ (MPa)	$σ_{c 3 th}$ (MPa)	$P_{cth}$ (N)
2852	475	300	50	16	16	3627

4. Experimental test

One preliminary tensile test and five compressive tests were performed at a displacement rate of 41.6 N/s. The ARAMIS 3D 12M system was used for the 3D Digital Image Correlation analysis in order to detect full-field displacements and strains of the external surface of the specimen in two directions: vertical (longitudinal) and transverse (out of plane). For a 3D measurement setup, two cameras are used (stereo setup) that are calibrated prior to measuring. The specimen surfaces need to be prepared by means of suitable methods, for example applying a speckle pattern. After creating, images are recorded in various load stages of the specimen. After the area to be evaluated is defined (computation mask) and a start point is determined, the measuring project is computed. During computation, ARAMIS observes the deformation of the specimen through the images by means of various square or rectangular image details called facets. As shown in Fig. 4, the specimens were coated with a black/white speckle pattern for the strain measurements. The same figure shows also a scheme of the three materials involved.

Fig. 4.

Test specimen, indicating the three materials.

4.1. Tensile test and thermographic analysis

One preliminary tensile test was performed and the obtained load-displacement and stress-displacement curves (calculated as F/A) are shown in Fig. 5.

Fig. 5.

(a) Load-displacement curve, (b) stress-displacement curve.

It is possible to evaluate how load distribution inside the different metals using Eqs (8), (9), (10) and the results are reported in Fig. 6. It is easy to note, from Fig. 6, that steel is the most stressed material, while the pure Al and Al alloy are stressed in the same manner (a third of the steel), being the formulae dependent on the elastic moduli.

Fig. 6.

Load-displacement curve.

An infrared camera (model FLIR Systems SC640), with a resolution of 640 × 480 pixels, was used to detect the specimen temperature during the test. The specimen surface was black painted in order to increase its emissivity. It is well known that a metallic specimen changes its temperature when it is subjected to stresses. The temperature change is produced by two main effects: elastoplasticity and thermoelasticity [2]. Figure 7 shows the IR images of the specimen during a test.

Fig. 7.

Infrared images during a static test.

Figure 8 reports the temperature variation during the test showing the high temperature increase at rupture. Figure 9 reports the initial 300 seconds of the test and shows the temperature increment variation on the specimen surface, which is characterized by three phases: an initial approximately linear decrease due to the thermoelastic effect (phase I), then the temperature deviates from linearity until a minimum (phase II) and a very high further temperature increase until failure (phase III). The temperature evolution is similar to the thermal response of steel welded joints during static tests [9].

Fig. 8.

Temperature increment of the specimen during a tensile test.

Fig. 9.

Three phases of the temperature increment during a tensile test.

4.2. Compressive tests and buckling analysis

The test performed and the obtained experimental maximum critical loads ( $P_{c_\max_\exp}$ ) are reported in Table 4.

Table 4
Test performed and maximum critical load

Test n° L [mm] $P_{c_\max_\exp}$ [kN]

1 350 3.2

2 180 11.8

3 160 10.8

4 130 13.5

5 110 18.2

Test n°	L [mm]	$P_{c_\max_\exp}$ [kN]
1	350	3.2
2	180	11.8
3	160	10.8
4	130	13.5
5	110	18.2

The specimen was clamped from both ends and the two DIC cameras were placed in correspondence of the middle length of the specimen, as shown in the experimental setup of Fig. 10, and the monitored length of each image was about 150 mm. Figure 11 shows the deformed specimens at the end of the tests.

Fig. 10.

Experimental setup.

Fig. 11.

Specimen after the tests.

Figure 11 shows the evolution of the transverse displacement (out of plane displacement) of the external surface at different loads for test n°1 ( $l = 350 mm$ ). It is easy to see that at the beginning of the test, the external surface of the specimen, lays on the origin of the coordinate axis (left side of Fig. 12) with very small values of transverse displacement: When buckling takes place, the specimen tends to deflect, (i.e. at 3200 N the specimen surface does not lay on the coordinate axis) and the transverse displacement increases with a loss of the load (bending is predominant in the right side of Fig. 12).

Fig. 12.

Transverse displacement (out of plane displacement) evolution at different loads for test n°1.

Figure 13 plots the load versus the vertical and transverse displacements curves respectively (test n°1). It has to be noted that buckling phenomenon, considering the transverse displacement measured by the DIC technique, starts to occur before the maximum registered load due to the deviation from linearity of the curve. A first deviation from linearity is observed around 2100 N, as shown also by Fig. 12.

Fig. 13.

Load-displacement curve for test n°1: (a) vertical displacement, (b) transverse displacement.

This behaviour is further confirmed by Figs 14 and 15. Figure 14 shows the evolution of the transverse displacement (out of plane displacement) of the external surface at different loads for test n°3 ( $l = 160 mm$ ). In the uppers side of Fig. 14 the longitudinal views are reported, while in the bottom side the respective 3D view; the voids in the mask are due to the absence of correlation of the computation mask (defined in Section 4) for those points (facets) due to reflection, but they did not influence the results. Figure 15 plots the load versus the vertical and transverse displacements curves respectively, of all the tests performed.

Fig. 14.

Transverse displacement (out of plane displacement) evolution at different loads for test n°3: Up longitudinal view, bottom 3D view.

Fig. 15.

Load-displacement curves of the compressive tests: (a) vertical displacement, (b) transverse.

5. Finite element analysis

The buckling analysis can be performed using the eigenvalue method; with this type of analysis is possible to predict the theoretical deformation strength of an ideal elastic structure. It allows calculating the structural eigenvalues for a given load. Obviously, in the actual conditions, the structural imperfections and the non-linearity of the material are such that the result obtained could not coincide with the real one.

Nonlinear buckling analysis is generally more accurate than eigenvalue analysis because it employs non-linear effects, large-deflection, to predict buckling loads. Furthermore the non-linear material properties of the different materials (see Fig. 2) were considered in this study. The chosen element for the FE analyses was the solid186, having 20 nodes and three degrees of freedom: translation along the x, y and z directions; this element is indicated for the analysis of buckling and elastoplastic materials. Figure 16 shows the loading and boundary conditions; the specimen is clamped at both ends and compressive load is applied at the upper end.

Fig. 16.

Loading and boundary conditions; (a) scheme, (b) FE model.

5.1. Finite element eigenvalues analysis

As regards the analysis of eigenvalues for a given load, a distributed load is applied at one end of the sample. This load was applied to generate a compressive stress, and with a unitary module in order to determine the critical load. The useful length of the specimen does not correspond to the total length of the same; in fact, the specimen is clamped at both ends by the grips, each of which has a length of 25 mm. This means that, for example, considering a total length is 400 mm, the useful length l is equal to 350 mm.

Figure 17 shows the results of the displaced specimen and the predicted value of the $P_{cFE}$ which is 3637 N, being in good agreement with the calculated theoretical value ( $P_{cth} = 3627 N$ ) and a little different than the experimental test results ( $P_{c_\exp} \approx 3260 N$ ).

Fig. 17.

Results of the displaced specimen and predicted value of the $P_{cFE}$ .

5.2. Nonlinear finite element analysis

The true non-linear nature of this analysis permits the modeling of load perturbations, material nonlinearities. For this type of analysis, small off-axis loads are necessary to initiate the desired buckling mode (usually 5% of the expected critical load). The applied load is gradually increased until a load level is found whereby the structure becomes unstable; a very small increase in the load will cause suddenly very large deflections. Five non-linear 3D FEA analyses, of the investigated critical lengths, were carried out using a gradual increase of the applied load. The mechanical properties of each material, detected from hardness measurements and previously reported in Fig. 2, were considered as input of the FE model using the Ansys software. Multi-linear kinematic hardening models were used in this study for the non-linear analyses. A sensitivity analysis of the mesh size was performed and the element size for the convergence of the solution was find equal to 2 mm. The transverse (out plane) displacement map at the maximum load is reported in Fig. 18.

Fig. 18.

Transverse (out plane) displacement at the maximum load.

The non-linear FE analysis does not give a value of the critical load but, as the experimental test, is able to give the history of the test without the load loss, for this reason a comparison between the non-linear FEA and the experimental test is reported in Fig. 19, being in good agreement and confirming that the choice of calculating the non-linear material properties, used for the non-linear FEA, from hardness measurements, leads to reliable results.

Fig. 19.

Non-linear FEA and experimental load-transverse displacement curve.

Figure 20 illustrates a comparison between experimental data and FE results of the critical load the investigated critical lengths ( $l = 350 – 180 – 160 – 110 mm$ ).

Fig. 20.

Maximum load for different critical lengths: Experimental data and non-linear FE results comparison.

Furthermore Fig. 21 illustrates a comparison of the transverse displacement at 3200 N, reporting that both, FE and DIC results show that the Al alloy side has a bigger displacement.

Fig. 21.

Transverse displacement at 3200 N: DIC and FE images.

6. Conclusion

From static tensile tests was possible to characterize the explosion welded joints and see that the most stressed material is the steel, which carries almost all the load, while the pure Al and Al alloy carry the same load (a third of the steel). The infrared investigations allowed detecting the temperature variation on the specimen surface which is characterized by three phases similar to common metals.

The linear FE buckling analysis was in good agreement with the theoretical values but both give an overestimation of the critical load. Experimental tests showed that the buckling phenomenon, considering the transverse displacement measured by the DIC technique, starts to occur before the maximum registered load, due to the deviation from linearity of the curve. The non-linear FE analysis permits the modeling load perturbations, and material nonlinearities, and was in good agreement with the experimental investigation. Furthermore non-linear FE and DIC results show that, as expected, the Al alloy side has a bigger displacement.

Footnotes

Acknowledgements

The research reported in this paper was conducted with the facilities of the Research Project “CERISI” (“Research and Innovation Centre of Excellence for Structure and Infrastructure of large dimensions”), funded by the PON (National Operative Programme) 2007–2013. This study is part of the research activities of the Research Project PRIN (Announcement 2015) “CLEBJOINT”, project funded by the Italian Ministry of Scientific and Technological Research.

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Mechanical buckling analysis of explosive welded joints used in shipbuilding

Abstract

Keywords

1. Introduction

Table 1 Tensile properties obtained from hardness measurements [8] Steel AA 5086 A 1050 A σ Y [MPa] 380 285 90 σ ut [MPa] 590 335 155

Table 4 Test performed and maximum critical load Test n° L [mm] P c _ max _ exp [kN] 1 350 3.2 2 180 11.8 3 160 10.8 4 130 13.5 5 110 18.2

Footnotes

Acknowledgements

References

Table 1
Tensile properties obtained from hardness measurements [8]

Steel AA 5086 A 1050 A

$σ_{Y}$ [MPa] 380 285 90

$σ_{ut}$ [MPa] 590 335 155

Table 4
Test performed and maximum critical load

Test n° L [mm] $P_{c_\max_\exp}$ [kN]

1 350 3.2

2 180 11.8

3 160 10.8

4 130 13.5

5 110 18.2