I. INTRODUCTION
A. OBJECTIVE
The objective of this work is to put forward an analysis and comparison of buckling behaviour of two stiffened plate of different geometrical property with one plate having square junction to the stiffener and the other having filleted junction to the stiffener. Eigen buckling analysis through ANSYS mechanical apdl software was chosen be used. The deflection and different stress values of the stiffened plate were compared for conclusion. Additionally, the concept of parameters influencing the behaviour and strength of the stiffened plates were taken into consideration and hence the influence of the aspect ratio parameter on the ultimate strength of the stiffened plate and the influence of the slenderness ratio parameter on the failure modes were analysed for conclusion.
B. BACKGROUND
A structure which is safe in deflection and normal stress, fails in buckling. Plates are widely used in the structure which is subjected to continues loading throughout their life span. To withstand such a load, stiffeners are introduced which increases the strength, the overall stiffness and provide resistance to buckling. Plates are available in various forms such as rectangular, skew, and curve. These are different types of stiffener system, namely transverse, longitudinal and both transverse and longitudinal system. In this work, rectangular plates with longitudinal stiffener system was used.
The stiffened plate is one of the most basic component among all which is identified in bridges, buildings, airplanes, offshore structures. The main difference between the offshore structures and other structure is that the former one experiences In-plane loads in addition to the other loads. Due to their complexity in the structure, the concept of elastic instability has to studied well to proceed further. To understand the behaviour of the stiffened plate in such loading, buckling analysis is done. Basically, three forms of procedure namely theoretical and/or numerical computer-aided and/or experimental can be adopted. In this work, numerical computer aided approach along with theoretical validation was adopted. The most common computer aided programming of Finite Element Analysis- ANSYS was used.
According to the load and the behaviour, the modes of buckling differs namely local buckling, overall buckling, and tripping. The main criteria to prevent the failure was to reduce the overall buckling by confining the plate within local buckling which will increase the stiffness of the plate.
The buckling performance can be influenced by variation in cross section, which suits the case of stiffened plate with filleted junction. Larger the radius of the fillet, the weight of the whole plate structure increases. Hence the geometry of the plate influences the buckling. In such a case, the fillet radii are an important parameter.
Numerical results for both the plate are obtained through Finite Element Program software ANSYS and compared for the conclusion.
C. NEED FOR RESEARCH
The plates in the ship deck are subjected to continuous loading and are prone to buckling. In such a case, the concept of increasing the thickness or providing the stiffeners to the plate are implemented which will increase the stiffness. On the other hand, stiffened plate on a particular zone can also experience buckling due to heavy loading by which the introduction of additional structure such a filleted junction to the stiffened plate can increase the stiffness. This work is important in analysing, comparing both the plates and the previous works identified rarely used ANSYS software for simulation by which this work become unique. Additionally, it is important to understand the behaviour of the stiffened plate on different environmental condition which lies in proportion with two important parameters such as aspect ratio and slenderness ratio. Hence, this work proves to be important for all the structural engineers.
D. LITERATURE REVIEW
Razzaque and M. D. Mathers (A. Razzaque & M. D. Mathers, 1983), conducted various experiments on buckling analysis of stiffened plates and shells. Ali Rezo Poladkhan et al (Ali Rezo Pouladkhan, et al., 2011), presented a paper in which they used equilibrium equation to analyse the buckling of stiffened plates. Archie Wilmer III (Archie Wilmer III, 2003), presented a paper on buckling analysis of plates with bulb flat flange stiffener and concluded that the former is four percent less than the T-flange stiffener. Ashutosh kumar, Rachayya R. Arakerimath (Ashutosh Kumar & Rachayya R. Arakerimath, 2015), analytically derived the buckling mode shapes, the nature of buckling using equilibrium method and compared it with finite element approach which was found to be matching. Dr. Alice Mathai et al (Dr. Alice Mathai, et al., 2014), presented a non-linear buckling analysis of stiffened plate and compared it with finite element software NISA which was found to be convincing. Ghania Ikhenazen, Messaoud Saidani (Ghania Ikhazen & Messaoud Saidani, 2010), used total energy concept to analyse the chart of buckling factor versus aspect ratio and width ratio of an isotropic plate subjected to in-plane loading. J.Rhodes (J. Rhodes, 2002), presented paper on buckling analysis of stiffened plates subjected to local buckling. He also concluded that the confining the plate to maximum local buckling wuld prevent the overall buckling. Mudhujit Mukhopadhayay, Abjijit Mukherjee (Mudhujit Mukhopadhay & Abhijit mukherjee, 1990), carried out numerical analysis on stiffened iso-parametric and skew plate. He concluded that the stiffener can accommodate irregular boundaries and the buckling load reduces with eccentricity of the stiffeners. P. Seide (P. Seide, 1953), derived the effective moment of inertia from the effect of eccentricity of the stiffeners. Patrick E. Fenner, Andrew Watson (Patrick E. Fenner & Andrew Watson, 2012), presented a work on buckling analysis of stiffened plate with filleted junction. He came up with a conclusion that there is limit of adding mass at junctions to increase the local buckling stability. The fillet increases the stiffness however it also increases the critical buckling stress. He also suggested that a higher fillet size increases the value of K than the upper bound value.
II. CHARACTERISTICS OF FINITE ELEMENT METHODS
A. BUCKLING MODES OF STIFFENED PLATES
The failure of the stiffened plate can be in different forms namely local buckling, plate induced failure or stiffener induced failure overall buckling and tripping. Theoretically, the local and overall failure can be identified in all wavelengths but practically it is difficult due to the material strength at that point.
In the local mode of failure, the web line junction does not project to the out-of-plane direction but in some cases there exists where the line junction exits the out-of-plane area. This displacement will be small such like (1/10)^{ th} of the overall displacement which strengthens the fact that it does not project to the out-of-plane area. In this work, the stiffened plate which is subjected to uniaxial edge compression is adopted which allows to perform theoretical calculation and analysis around the overall or tripping failure.
In the case of overall buckling of plates, the web line junction exceeds the out-of-plane point which means the plate does not lie straight. The projection of the line junction will be in the form of half wave which looks similar to half sine wave. The overall buckling which is also referred as Euler type buckling is simultaneous failure of both plate and stiffener. If the failure takes place with stiffener on the tension side, then it is referred as plate induced overall buckling. On the other hand, if the failure is due to stiffener being on the compression side then it is referred as stiffener induced overall buckling.
Tripping failure of stiffened plate is lateral torsional buckling which takes place about the stiffener to plate junction (Imtiaz A. Sheikh, et al., 2001). It is actually sudden drop in load carrying capacity. This type of failure can be serious at times due to the drop in load carrying capacity. It has to be noted that the tripping stiffener experience tension but it may also be subjected to compression due to bending.
B. WELDING CHARACTERISTICS AND TECHNIQUES IN SHIP PLATES
Welding of ships plates and the stiffener are carried out in two different ways. Continuous welding is used as a procedure to weld the plate and the stiffener which requires huge quantity of weld. It may also add to the weight of the ship and increases the construction cost. Despite increasing the welding quantity, this procedure becomes faster and reduces the labour cost.
Alternatively, the longitudinal stiffeners are attached to plating using intermittent fillet welds. This process reduces the construction cost and also the amount of residual stresses when compared to the former welding process.
Welding the stiffener plates affects both initial distortion and the residual stresses. These two parameter can vary and number of statistical work has been done in understanding this concept by experts.
Faulkner (1975), Carlsen and Czujko (1978) and Smith et al. (1991), conducted surveys on different structures to assess the above two parameters. Faulkner (1975) assessed 300 built in stiffened plates specimens and measured the initial plate distortion. He proposed a relationship for
table shows the dimensions of the square stiffened plate, load applied to solve the same.
Table dimensions of square stiffened plate and load applied
a (mm) | b (mm) | b_{w} (mm) | t_{w} (mm) | T_{k} (mm) | stiffener spacing (mm) | load applied (KN) |
2500 | 800 | 15 | 200 | 15 | 267 | 500 |
Figure Geometry of square stiffened plate
Figure: Nodal Geometry of the square stiffened plate with boundary condition
To compare the behaviour of square stiffened plate, stiffened plate with filleted junctions was solved. The process of solving this plate involved designing of the stiffened plate which was as same as square stiffened plate but the only difference was the addition of the fillet junction between the plate and the stiffener. As mentioned in chapter I.B, fillet radii is an important criteria in these cases, it was decided to use four different radii’s of 2mm, 3mm, 4mm, 5mm respectively. Since welding technique for fillet was adopted for the connection of plate and stiffener, the influence of residual stresses was taken into consideration. The residual stresses were calculated from the equation. The residual stresses were applied initially on the fillet junction to allow for the initial deflection. Furthermore, external load was applied in the longitudinal direction of the plate to calculate the deflection, von mises stress and load factor values for respective fillet radii stiffened plates and finally the ultimate strength of the plate was calculated. The table shows the dimensions of the stiffened plate along with the fillet radii and the residual stresses calculated from the equation.
Table Dimension of the stiffened plate with fillet radii and residual stresses
a (mm) | b (mm) | b_{w} (mm) | t_{w} (mm) | t_{k} (mm) | stiffener spacing (mm) | Fillet radii (mm) | Residual stress (N/mm^{2}) | load applied (KN) |
2500 | 800 | 15 | 200 | 15 | 267 | 2 | 27.93 | 500 |
2500 | 800 | 15 | 200 | 15 | 267 | 3 | 28.61 | 500 |
2500 | 800 | 15 | 200 | 15 | 267 | 4 | 29.56 | 500 |
2500 | 800 | 15 | 200 | 15 | 267 | 5 | 30.78 | 500 |
Fillet junction
Figure View of part of stiffened plate with filleted junction
Figure Full view of stiffened plate with filleted junction
Figure Nodal geometry of the filleted junction stiffened plate with boundary conditions
Figure Mapped meshing of the stiffened plate with filleted junction
IV. RESULTS
A. EFFECT OF SLENDERNESS PARAMETER ON FAILURE MODE
The simulation of different slenderness parameter stiffened plate carried out through ansys were observed. The progressive mode of the failure of the stiffened plate subjected to unit compressive load was also observed and three different modes of failure was identified. Those modes were plate induced failure, plate buckling and dual failure. All the three different parameter β_{1}, β_{2}, β_{3 }had their respective influences on the strength and the behaviour of the plate. For the different slenderness ratio stiffened plate, the load factor value is obtained through the software and the ultimate strength is calculated by multiplying the factor with the length of the plate. From the obtained ultimate strength, plots were created for better understanding. The plots were created with the slenderness parameter and the ultimate strength of the plate. Also, the relation between the slenderness parameters and the failure mode was also understood with the help of the table. The pictures help to come forward with the future work and a conclusion of the influence of this parameter on the ultimate strength.
Table () Effect of β_{1} with β_{3 }= 0.3 on plates with uniaxial compression
β_{1} | β_{2} | β_{3} | Failure Mode |
0.2 | 0.5 | 0.3 | Plate induced overall |
0.75 | 0.5 | 0.3 | Plate induced overall |
1.5 | 0.5 | 0.3 | Plate Buckling |
2 | 0.5 | 0.3 | Plate Buckling |
2.7 | 0.5 | 0.3 | Plate Buckling |
0.2 | 1 | 0.3 | Plate induced overall |
0.75 | 1 | 0.3 | Plate induced overall |
1.5 | 1 | 0.3 | Plate Buckling |
2 | 1 | 0.3 | Plate Buckling |
2.7 | 1 | 0.3 | Plate Buckling |
0.2 | 1.5 | 0.3 | Plate induced overall |
0.75 | 1.5 | 0.3 | Plate Buckling |
1.5 | 1.5 | 0.3 | Plate Buckling |
2 | 1.5 | 0.3 | Plate Buckling |
2.7 | 1.5 | 0.3 | Dual failure |
0.2 | 2 | 0.3 | Plate Buckling |
0.75 | 2 | 0.3 | Plate Buckling |
1.5 | 2 | 0.3 | Plate Buckling |
2 | 2 | 0.3 | Dual failure |
2.7 | 2 | 0.3 | Dual failure |
Table () Effect of β2 with β_{3 }= 0.3 on plates with uniaxial compression
β_{1} | β_{2} | β_{3} | Failure Mode |
0.75 | 0.25 | 0.3 | Plate induced buckling |
1.25 | 0.5 | 0.3 | Plate buckling |
2 | 0.75 | 0.3 | Plate buckling |
2.7 | 1 | 0.3 | Plate buckling |
0.75 | 1 | 0.3 | Plate buckling |
1.25 | 1.25 | 0.3 | Plate buckling |
2 | 1.5 | 0.3 | Dual failure |
2.7 | 1.75 | 0.3 | Dual failure |
0.75 | 2 | 0.3 | Dual failure |
1.25 | 2.25 | 0.3 | Dual failure |
2 | 2.5 | 0.3 | Dual failure |
2.7 | 2.75 | 0.3 | Dual failure |
0.75 | 3 | 0.3 | Dual failure |
1.25 | 3.25 | 0.3 | Dual failure |
2 | 3.5 | 0.3 | Dual failure |
2.7 | 4 | 0.3 | Dual failure |
Table () Effect of β_{3} on plates with uniaxial compression
β_{1} | β_{2} | β_{3} | Failure Mode |
0.75 | 0.5 | 0.075 | Plate induced buckling |
1.25 | 0.5 | 0.15 | Dual failure |
2 | 0.5 | 0.225 | Dual failure |
2.7 | 0.5 | 0.3 | Dual failure |
0.75 | 1 | 0.375 | Plate buckling |
1.25 | 1 | 0.45 | Dual failure |
2 | 1 | 0.525 | Dual failure |
2.7 | 1 | 0.6 | Dual failure |
0.75 | 1.5 | 0.675 | Dual failure |
1.25 | 1.5 | 0.75 | Dual failure |
2 | 1.5 | 0.825 | Dual failure |
2.7 | 1.5 | 0.9 | Dual failure |
0.75 | 2 | 0.975 | Dual failure |
1.25 | 2 | 1.05 | Dual failure |
2 | 2 | 1.2 | Dual failure |
2.7 | 2 | 1.275 | Dual failure |
Effect of β_{1 }on P when β_{2 }=0.5
Effect of β_{1 }on P when β_{2 }=1
Effect of β_{1 }on P when β_{2 }=1.5
Effect of β_{1 }on P when β_{2 }=2
Effect of β_{2}_{ }on P when β_{3}_{ }=0.3
Effect of β_{3}_{ }on P
B. EFFECT OF ASPECT RATIO PARAMETER
A number of aspect ratio plates were studied which is subjected to uniaxial compression along with the residual stress and the initial imperfection. The below table shows the plate details along with the deflection, stress, out of plane deflection (w_{op}, w_{os}) and the residual stresses.
The below results are obtained through modelling of the plate in ANSYS, applying the loading and simulated. The effect of initial deflection and the residual stresses are obtained through
σrcσo=2ɳtpb-2ɳtw
Where, tp – thickness of the plate,
tw- thickness of the web,
σrc= residual stress.
The initial out of plane deflection of the plate is obtained by the following table.
Residual stress | Wop = min (Wop1,Wop2) | |
Wop1 | Wop2 | |
0≤(σ_{rc}/σ_{0})≤0.05 | 0.025β_{p}^{2}t_{p} | b/100 |
0.05≤(σ_{rc}/σ_{0})≤0.15 | 0.1 β_{p}^{2}t_{p} | b/100 |
0.15≤(σ_{rc}/σ_{0})≤0.20 | 0.2 β_{p}^{2}t_{p} | b/100 |
σ_{rc}/σ_{0}≥0.2 | 0.3 β_{p}^{2}t_{p} | b/100 |
The initial out of plane deflection of the web is calculated as W_{os} = 0.00075a.
The below table shows the results obtained through software simulation and the theoretical calculation. From the results, it is predicted that the more tensile residual stresses present in the web it will be proportional to the average stress stain response. Although the above reason does not have effect on the buckling strength, the residual stress plays a significant role in the load carrying capacity of the initially distorted plate.
a
mm |
b
mm |
α | t_{p}
mm |
d
mm |
t_{w}
_{mm} |
y
mm |
P
(KN) |
W_{op}
mm |
W_{os}
mm |
σ_{acr}
N/mm^{2} |
800 | 800 | 1 | 11.43 | 139.5 | 13.9 | 1.13 | 500 | 5.32 | 0.6 | 27.93 |
800 | 600 | 1.3 | 8.57 | 122.39 | 12.2 | 1.62 | 412.8 | 4.31 | 0.6 | 27.93 |
800 | 450 | 1.7 | 6.43 | 100.87 | 10.09 | 1.9 | 389.6 | 3.45 | 0.6 | 27.93 |
800 | 350 | 2.3 | 5 | 81.56 | 8.2 | 2.1 | 373.6 | 2.89 | 0.6 | 27.93 |
800 | 250 | 3.2 | 3.57 | 65.26 | 6.5 | 2.31 | 345.6 | 2.14 | 0.6 | 27.93 |
800 | 225 | 3.5 | 3.21 | 56.71 | 5.7 | 2.43 | 304.8 | 1.78 | 0.6 | 27.93 |
Table Plate dimensions for aspect ratio simulation
The figure below shows the relation between the aspect ratio and ultimate strength from which it is understood that when the aspect ratio of the plate changes there will be change in ultimate strength also. From the figure, it is understood that the aspect ratio has significant influence on the deflection range of the stiffened plate. It is also understood that the plate with increase in aspect ratio, the ultimate strength decreases by considerable percentage following a large deflection range.
INFLUENCE OF ASPECT RATIO ON ULTIMATE STRENGTH
Influence of Aspect ratio on deflection
C. ANALYSIS OF SQUARE AND FILLET JUNCTION STIFFENED PLATE
The simulation of the square stiffened plate with the dimensions mentioned in table in ansys helped out with obtaining the results consisting of deflection, buckling pattern, stress, and ultimate strength which is shown in table. The buckling pattern of the square stiffened plate is shown in figure. From this buckling pattern it is evident that this particular square stiffened plate has dual failure mode of buckling. Finally, the ultimate strength of the plate is calculated with the load factor. These results are then compared with the results of the stiffened plate with filleted junction obtained through ansys.
Table Geometry and results of square stiffened plate
a (mm) | b (mm) | b_{w} (mm) | t_{w }(mm) | t_{k }(mm) | Stiffener spacing (mm) | Load applied (KN) | Deflection (mm) | Load factor | Ultimate strength (KN) |
2500 | 800 | 15 | 200 | 15 | 267 | 500 | 1.16 | 0.17 | 427 |
Figure Buckling pattern of square stiffened plate
The stiffened plate with filleted junction was simulated in ansys by designing the plate and stiffener with fillet radii of 2, 3, 4, 5mm respectively. The loads were applied and solved for result. The solutions obtained were buckling pattern, stress and load factor. The buckling pattern of different fillet radii are shown in the figures. It is evident from these different pattern that the buckling is due to the stiffener which is stiffener induced buckling. The deflection values changes due to the varying residual stress. From the load factor, the ultimate strength of the plate was found. Two different plots namely deflection versus fillet radii and ultimate strength versus fillet radii was formed to have a better understanding to derive a conclusion between the square and fillet junction stiffened plate.
a b
c d
Figure Buckling pattern of stiffened plate with fillet radii (a)r= 2 (b) r= 3 (c) r= 4 (d) r= 5
The table shows the results obtained through solving which was used to create plots to conclude with the results
Table Geometry and results of stiffened plate with different fillet radii
a (mm) | b (mm) | bw (mm) | tw (mm) | tk (mm) | Stiffener spacing (mm) | Fillet radii (mm) | Residual stress (N/mm^{2}) | Load applied (KN) | Deflection (mm) | Load factor | Ultimate strength (KN) |
2500 | 800 | 15 | 200 | 15 | 267 | 2 | 27.93 | 500 | 6.42 | 0.279 | 697.5 |
2500 | 800 | 15 | 200 | 15 | 267 | 3 | 28.61 | 500 | 6.1 | 0.343 | 857.5 |
2500 | 800 | 15 | 200 | 15 | 267 | 4 | 29.56 | 500 | 4.5 | 0.486 | 1215 |
2500 | 800 | 15 | 200 | 15 | 267 | 5 | 30.78 | 500 | 2.3 | 0.568 | 1420 |
CONCLUSION AND FUTURE WORK
Influence of slenderness parameter on ultimate strength of the stiffened plate: The plate transverse slenderness ratio β_{1}, was observes to be the most important parameter in terms of the stiffened plate analysis as it had major effect on the strength of the stiffened plate. It can be seen from the table. Higher the ratio of β_{1},the failure of the plate is more independent rather than induced failure and the higher the ratio will reduce the ultimate strength of the plate. The above conclusion is drawn from the figures which shows that when the plate transverse slenderness parameter increases the ultimate strength of the plate decreases. The ratio of the plate to stiffener slenderness β_{2 }also have more influence on the strength of the stiffened plate. The figure shows that as value of β_{2 }increases the ultimate strength of the plate decreases linearly and decreases considerably when the ratio reaches highest value. The ratio of the plate and stiffener ratio β_{3 }played a role on the mode of failure of the plate which is dual failure, as the area of the plate and the stiffener ratio influences the plate buckling. Higher the ratio of β_{3, }the mode of failure slowly transfers from plate buckling to dual failure. It is also to be noted from the figure, as the value of β_{3 }increases the ultimate strength of the plate decreases considerably. Finally, from the above study on the slenderness parameters it is understood that these three parameter have significant influence on the ultimate strength of the stiffened plate as these values increases, the strength decreases which may affect the stiffened plate. Since in this work the slenderness parameter involved only in the compressive buckling of the plate was studied, it is recommended to carry out a study on the influence of other parameter in different structural configuration and loading pattern.
Influence of aspect ratio parameter: The objective of this work to study the influence of the aspect ratio on stiffened plate was to understand the behaviour of ultimate strength of the stiffened plate with change in aspect ratio. For realistic assessment of this parameter, initial imperfection and residual stresses were allowed in the simulation. All other parameters and the non-dimensional parameters were kept constant for all different aspect ratio’s. Through the simulation, the load factor was obtained which multiplied by length of the plate gave the ultimate strength of the plate. The ultimate strength of the plate was also influenced by the residual stresses. The table showsthe ultimate strength and the deflection values due to unit compressive load for different aspect ratio’s. From the results obtained plot were made to understand the influence on the ultimate strength of the plate. It is evident from the figure, that the ultimate strength of the plate reduces with increase in the aspect ratio and it is also seen form the figure, higher the aspect ratio the plate deflects in large values. Hence it is important to choose the aspect ratio for the plate appropriately. Ideally it is recommended to choose aspect ratio between 2 to 4. The above study was conducted keeping the length of the plate (a) constant and varying the breadth of the plate (b). Hence it is recommended to conduct a study in future with different values of ‘a’ and ‘b’ with different loading pattern.
Buckling of square and fillet junctions: The analysis of buckling of two plates with different properties through ansys brought few important points which would be novel in the ship structural design. The buckling of square stiffened plate resulted in small deflection when compared to the fillet junction plates. The line junction of the plate has been the efficient modal in the ship structural design which is evident from this simulation where the deflection of the plate is small than the fillet junction. When the fillet junction stiffened plate results are compared, it is understood that the ultimate strength of the fillet junction plate increases with increase in fillet radii. In the case of deflection, lower the fillet radii will result in large deflection and vice versa by which it can be concluded that the use of large fillet radii is efficient but there is limit to increase the radii because it could add mass to the structure resulting in higher manufacturing cost. In this part of the work, the plate with only longitudinal stiffener and the loading in uniaxial direction was examined and it is recommended to consider both transverse and longitudinal stiffened plate loaded in biaxial direction which could to be very novel for ships exposed to heavy weather.