Marine_Structural_Design Episode 6 doc

In the present study, the simplified elasto-plastic large deflection analysis described in 9.3.1 is incorporated in the Idealized Structural Unit element in order to accurately evaluate

178 Part II Ultimate Strength

Integrating AF, and dM,respectively, the force F, and the bending moment M , acting at the

bottom of a dent are obtained as:

Applying this model, the stress distributions after local buckling may be represented as shown

in Figure 9.19 In this figure, the case with one dent is indicated as case A" distribution, and

that with three dents is a case B" distribution For a case A" stress distribution, Eqs (9.22) and (9.23) are replaced with:

where,

f,"= Fbi

(9.58) (9.59)

Procedure of Numerical Analysis

Until initial yielding is detected, Eq (9.3) gives the relationship between axial compressive loads and lateral deflection The mean compressive axial strain is evaluated by Eq (9.8) After plastification has started, the analysis is performed in an incremental manner using the plastic component of deflection shown in Figure 9.13 This deflection mode expressed by Eqs (9.10) thru (9.12) gives a constant plastic curvature increment in the region Z, If the actual plastic region length ld in Figure 9.20 (a) is taken as 1, , it reduces to prescribe excess plastic curvature especially near the ends of the plastic region To avoid this, a bi-linear distribution

of plastic curvature increments is assumed in the region I d , as indicated in Figure 9.20 (b) Then, the change of the plastic slope increment along the plastic region I d , may be expressed

as:

Chapter 9 Buckling and Local of Tubular Members 179

Figure 9.19 Elasto-plastic Stress Distribution Accompanied

by Local Buckling (DENT Model)

( a ) Actual p l a s t i c zone under combined thrust and bending

r i v I

I t , - - /

(b) Distrfbution o f increment o f p l a s t i c curva t u r e

(c) Oisribution o f increment o f p l a s t i c axial

s t r a i n

Figure 9.20 Equivalent Length of the Plastic Zone

180 Part II Ultimate Strength

where d K P , is the increment of plastic curvature at the center of a plastic region

On the other hand, if dK, is assumed to be uniformly distributed along the plastic region 1, , as indicated by Eqs (9.10) thru (9.12), the change of plastic slope increment along the plastic region ld may be expressed as:

(9.65) Here, 1, is determined so that de; =de, This is equivalent to the condition that the integrated values of plastic curvature in the plastic regions are the same for both cases, which reduces to:

The above-mentioned procedure used to estimate 1, , is only an approximation In Section 9.3.2, a more accurate procedure is described To evaluate the actual plastic region size I,, for the calculated deflection, the stress is analyzed at 100 points along a span, with equal spacing and the bending moment at each point is evaluated After local buckling has occurred, plastic deformation will be concentrated at the locally buckled part For this case, 1,is considered equal to the tube's outer diameter, which may approximately be the size of the plastically deformed region after local buckling

9.3.2 Idealized Structural Unit Analysis

Pre-Ultimate-Strength Analysis

Throughout the analysis of a beam-column using the ordinary Idealized Structural Unit Method, an element is regarded to be elastic until the fully pIastic condition and/or the buckling criterion is satisfied When the axial force is in tension, a relatively accurate ultimate strength may be evaluated with the former condition along with the post-yielding calculation However, when the axial force is in compression, the ultimate strength evaluated by the latter criterion is not so accurate, since the latter criterion is based on a semi-empirical formula In the present study, the simplified elasto-plastic large deflection analysis described in 9.3.1 is incorporated in the Idealized Structural Unit (element) in order to accurately evaluate the ultimate strength under the influence of compressive axial forces

The Idealized Structural Unit Method uses the incremental analyses The ordinary increment calculation is performed until the initial yielding is detected The initial yielding is checked by evaluating the bending moment along the span of an element and the deflection expressed by

Eq (9.9) After the yielding has been detected, the simplified method described in 9.3.1 is introduced

Here, it is assumed that calculation of the (n+l)-th step has ended Therefore, the following equilibrium equation is derived similar to Eq (9.19):

Chapter 9 Buckling and Local Buckling of Tubular Members 181

= Bending moment at nodal point i at the end of the n-th step

= Bending moment due to distributed lateral load

= Bending moment given by Eq (9.1 8)

= Axial force at the end of the n-th step

= Increment of axial force during the (n+l)-th step

= Increment of bending moment at nodal point i during the (n+l)-th step

= Bending moment increment due to distributed lateral load during (n+l)-th

step and

X i , M i , M i , M i , Q, and AQ are known variables after the (n+l)-th step has ended

Considering the equilibrium condition of forces in the axial direction, geometrical conditions regarding the slope, and Eq (9.77), the following equations are obtained:

for Case A Stress Distribution:

(9.69) (9.70) (9.71)

After the initial yielding, elasto-plastic analysis by the simplified method is performed using

Eqs (9.69) thru (9.71) or Eqs (9.72) thru (9.77) at each step of the Idealized Structural Unit analysis until the ultimate strength is attained at a certain step

Here, a more accurate method is introduced to determine the length of plastic zoneZ, If the axial force P and bending moment M are given, the parameters 17 and a, (and a*), which determine the axial strain€ and curvature 4(x) are obtained from Eqs (9.17) and (9.18)

Then, the increment of the curvature d&c) from the former step is evaluated With this increment, the length of plastic zone is given as

where d4, represents the maximum plastic curvature increment in the plastic region

182 Part II Ultimate Strength

System Analysis

The procedure used for the system analysis using the proposed Idealized Structural Unit is as

follows:

- At each step of the incremental calculation, moment distributions are evaluated in elements

in which axial force is in compression

Based on the moment and axial force distribution, the stress is calculated and the yielding

of the element is checked

If yielding is detected in an element at a certain step, the initial yielding load of this element is evaluated Then, the elasto-plastic analysis is performed using Eqs (9.69) thru (9.71) or Eqs (9.72) thru (9.75) until AP becomes AX,

-

-

In the following steps, the same calculation is performed at each element where plastification takes place If dp shows its maximum value dp,, in a certain element before it reaches AX,

at a certain step, this element is regarded to have attained its ultimate strength

Pu (= Xi + dp,,) Then, all the increments at this step are multiplied by dP,,/MTi

For the element that has attained its ultimate strength, its deflection is increased by keeping the axial force constant until the fully plastic condition is satisfied at the cross-section where the

bending moment is maximum Then, this element is divided into two elements and a plastic

node is inserted at this cross-section

The results of such analyses are schematically illustrated in terms of the axial forces and bending moments in Figure 9.21 (0) represents the results of the Idealized Structural Unit Method, and the dashed line represents the results of the simplified method Up to point 4, no plastification occurs Between points 4 and 5 , yielding takes place, and the analysis using simplified methods starts where the yielding occurs No decrease is observed in this step At

the next step between points 5 and 6, the ultimate strength is attained Then, the increment of

this step is multiplied by b5/56 While keeping the axial force constant, the bending moment is increased up to point c, and a plastic node is introduced After this, the Plastic Node Method (Veda and Yao, 1982) is applied

Evaluation of Strain at Plastic Node

In the Plastic Node Method (Ueda and Yao, 1982), the yield function is defined in terms of

nodal forces or plastic potentials Therefore, plastic deformation occurs in the form of plastic

components of nodal displacements, and only the elastic deformation is produced in an element Physically, these plastic components of nodal displacements are equivalent to the integrated plastic strain distribution near the nodal point If the plastic work done by the nodal forces and plastic nodal displacements is equal to those evaluated by distributed stresses and plastic strains, the plastic nodal displacements are equivalent to the plastic strain field in the evaluation of the element stiffness matrix Veda and Fujikabo, 1986) However, there is no mathematical relationship between plastic nodal displacements and plastic strains at the nodal point Therefore, some approximate method is needed to evaluate plastic strain at a nodal points based on the results of Plastic Node Method analysis

Here, the internal forces move along the fully plastic interaction curve after the plastic node is introduced as indicated by a solid line in Figure 9.22 On the other hand, the result of accurate elasto-plastic analysis using the finite element methods may be represented by a dashed line in

Chapter 9 Buckling and Local Buckling of Tubular Members 183

the same Figure The chain line with one dot represents the results obtained from the simplified method

1 .o

P/Pp

0

Figure 9.21 Schematic Representation of Internal Forces

Figure 9.22 Determination of an Approximate Relationship Between

Axial Forces and Bending Moments

Part II Ultimate Strength

axial force is plotted by a chain line with two dots as shown in Figure 9.22

Substituting the axial force P and the evaluated bending moment from Eq (9.79) into Eqs (9.17) and (9.18), respectively, strain may be evaluated If the maximum strain (sum of the

axial strain and maximum bending strain) reaches the critical strain expressed by Eq (9.36), the post-local buckling analysis starts

Post-Local Buckling Analysis

The filly plastic interaction relationship after local buckling takes place may be expressed as

In the above expressions, d and S are given by Eqs.(9,43) and (9.44), and ei and Mbi are equal to 4 and Mb and given by Eqs (9.56) and (9.57) of the i-th dent

Here, the angle a represents the size of a locally buckled part and is a function of the axial strain e and the curvature K of a cross-section, and is expressed as:

At the same time, 4 and Md are functions of e and K through a Consequently, the fully

plastic interaction relationship is rewritten in the following form:

Chapter 9 Buckling and Local Buckling of Tubular Members 185

As described in 9.3.2.3, there exists no one-to-one correspondence between plastic nodal displacements and plastic strains at a nodal point However, plastic strains may be concentrated near the cross-section where local buckling occurs So, the axial strain and curvature at this cross-section are approximated by:

lp in the above equations represents the length of plastic zone, and is taken to be equal to the diameter D(=2R ) as in the case of a simplified method Considering Eqs (9.87) and (9.88),

the filly plastic interaction relationship reduces to:

The elasto-plastic stiffness matrix after local buckling occurs, is derived based on the filly

plastic interaction relationship expressed by Eq (9.89) The condition to maintain the plastic

state is written as:

On the other hand, the increments of nodal forces are expressed in terms of the elastic stiffness

matrix and the elastic components of nodal displacement increments as follows:

186 Part II Ultimate Strength

(9.95)

where {dh)represents the increments of nodal displacements

Substituting Eqs (9.94) and (9.95) into Eq (9.92), dAiand dAjare expressed in terms of {dh} Substituting them into Eq (9.954, the elasto-plastic stiffness matrix after local buckling is derived as:

(9.96)

For the case in which local buckling is not considered, the elasto-plastic stifbess matrix is given in a concrete form in Veda et al, 1969) When local buckling is considered, the terms

4; K, 4i and 4; K, 4j in the denominators in Ueda and Yao (1982) are replaced by

4; K, +i -'y,?y, and 4; K, q5j -'yryj, respectively

9.4 Calculation Results

9.4.1

In order to check the validity of the proposed method of analysis, a series of calculations are performed on test specimens, summarized in Table 9.4, in which a comparison is made between calculated and measured results Three types of analyses are performed a simplified elasto-plastic large deflection analysis combined with a COS model and a DENT model, respectively, for all specimens; and an elasto-plastic large deflection analysis without considering local buckling by the finite element method The calculated results applying COS model and DENT model are plotted in the following figures, along with those analyzed using the finite element method The experimental results are plotted by the solid lines

H series

This series is newly tested The measured and calculated load -deflection curves are plotted in Figure 9.7 First, the results from the simplified method have a very good correlation with those obtained from the finite element method until the ultimate strength is attained However, they begin to show a little difference as lateral deflection increases This may be attributed to the overestimation of the plastic region size at this stage

The calculated ultimate strengths are 7-10% lower than the experimental ones This may be due to a poor simulation of the simply supported end condition and the strain hardening effect

of the material Contrary to this, the onset points of local buckling calculated using Eq (9.33) agree quite well with the measured ones The post - local buckling behavior is also well simulated by the COS model, but not so well simulated by the DENT model Such difference

between the measured and the calculated behaviors applying DENT model is observed in all analyzed test specimens except for the D series This may be due to the underestimation of forces and moments acting at the bottom of a dent, and fiuther consideration may be necessary for the DENT model

Simplified Elasto-Plastic Large Deflection Analysis

Chapter 9 Buckling and Local Buckling of Tubular Members 187

C Series

C series experiments are carried out by Smith et al (1979) Specimens C1 and C2 which are not accompanied by a denting damage are analyzed The calculated results for Specimen C2

are plotted together with the measured result in Figure 9.23 Smith wrote in his paper that local

buckling took place when the end-shortening strain reached 2.5 times the yield strain E,,,

while it occurred in the analysis when the strain reached 1.4 E,, However, the behavior up to

the onset of local buckling is well simulated by the proposed method of simplified elasto- plastic large deflection analysis On the other hand, in the case of Specimen C1, local buckling takes place just after the ultimate strength is attained both in the experiment and in the

analysis However, the calculated ultimate strength is far below the measured one as indicated

in Table 9.4 This may be attributed to some trouble in the experiment, since the measured ultimate strength is 1.1 times the fully plastic strength

D Series

This series is also tested by Smith et al (1979) The analysis is performed on Specimens D1

and D2 Here, the results for Specimen D1 are plotted in Figure 9.24 It may be said that a good correlation is observed between the calculated and measured results in the ultimate strength and in the onset of local buckling However, the behavior occurring just after the local buckling is somewhat different between the experiment and the analysis This may be because the experimental behavior at this stage is a dynamic one, which is a kind of a snap-through phenomenon as Smith mentioned As for the load carrying capacity after the dynamic

behavior, the DENT model gives a better estimate than the COS model

A similar result is observed in Specimen D2 However, in this case, the predicted onset of local buckling is later than the measured one

S Series

This series is a part of the experiments carried out by Bouwkamp (1975) The calculated and

measured results for Specimen S3 are shown in Figure 9.25 First, the measured ultimate strength is far above the elastic Eulerian buckling strength This must be due to a difficulty in simulating the simply supported end condition Consequently, instability took place just after the ultimate strength was attained, and a dynamic unloading behavior may occur After this, a stable equilibrium path was obtained, which coincides well with the calculated results

188 Part I1 Ultimate Strength

Table 9.4 Specimen size, material properties and results of experiment and

2.1 1 2.12

1.74 1.71

1.66 1.73

1.02 1.01

5.56 5.56 5.56 5.56

0.0

63.50 21180.0 34.55 0.68 0.63 Present 127.00 21180.0 34.55 0.55 0.49 Present 190.50 21180.0 34.55 0.44 0.41 Preaent

0.00 20496.3 23.25 0.84 0.76 11 9.84 21210.1 23.25 0.49 0.43 11

0.00 20802.2 19.88 1.00 0.94 11 10.11 23351.5 20.29 0.60 0.59 11

0.00 20496.3 21.52 1.10 0.95 11 9.99 21006.2 28.95 0.58 0.63 11

0.00 22535.7 49.46 0.75 0.83 11 15.13 26002.8 47.52 0.50 0.47 11

0.00 20256.1 41.69 0.84 0.82 5 0.00 20256.1 41.69 0.72 0.59 5 0.00 20256.1 41.69 0.54 0.41 5 0.00 20256.1 41.69 0.32 0.29 5

Chapter 9 Buckling and Locul Buckling of Tubular Members 189

C f Cy

Figure 9.23 Comparison of Measured and Calculated Results (C2)

Figure 9.24 Comparison of Calculated and Measured Results @1) The same features are observed in Specimens S1, S2, and S3 Bouwkamp wrote in his paper that local buckling took place after the ultimate strength was attained However, no local buckling occurred for this series analysis

A Series and B Series

A and B series by Smith et al (1 979), show no local buckling in either one of the experiments and analyses The calculated ultimate strengths show good agreement with the measured ones, with the exception of Specimen AI

Members with Constraints against Rotation at Both Ends

An end rotation of a structural member in a structural system is constrained by other members This effect of constraint may be equivalent to placing springs, which resist rotation at both ends of a member when one member is isolated from the system For such a member with springs at both ends, a series of analyses are performed by changing the spring constant between 0 and 00 The wall thickness and outer diameter are taken as 20 mm and 2,000 mm,

respectively The initial deflection of magnitude M O O times the length is imposed to know the characteristics of the proposed Idealized Structural Unit model The yield stress of the material

is chosen as 30 kgf7mm2, and the magnitudes of springs at both ends are the same Local buckling is not considered in this analysis The calculation results for r/m =loo are shown

in Figures 9.26 and 9.27 Figure 9.26 represents the load vs lateral deflection relationships, and Figure 9.27 represents the change of internal forces at a mid-span point and end In these figures, the solid lines and chain lines represent the results obtained by using the present method and the finite element method, respectively On the other hand, the dashed lines

represent the analytical solutions expressed as follows:

Perfectly elastic solution

where,

(9.98) and k represents the magnitude of springs placed at both ends, and PE is given in Eq (9.6) Rigid plastic solution

Chapter 9 Buckling and Local Buckling of Tubular Members 191

where k/ko is taken as 0.0,O 1, 1.0 and 00, where ko = 4EI/1

The ultimate strength evaluated by the proposed method is slightly lower than the ultimate strength proposed by the finite element method when the constraint is weak, but it becomes higher proportionally, as the constraint is increased However, the proposed method gives a very accurate ultimate strength

In the case of K = 00, the axial load still increases after a plastic node is introduced at a mid-span point where the ultimate strength is attained according to a simplified method It begins to decrease after the hlly plastic condition is satisfied at both ends However, the load increment after a plastic node has been introduced at a mid-span point is very small

Therefore, an alternative analysis is performed, in which three plastic nodes are

simultaneously introduced at a mid-span and both ends when the ultimate strength is attained

by a simplified method The curves for K =Q) in Figures 9.26 and 9.27 are the results of the latter analysis Further considerations should be taken when regarding this procedure

i a PIP,

Figure 9.26 Load - lateral Deflection Curves of Simply Supported Tube

with End Constraint Against Rotation

H Series

A series of analyses are performed on H series specimens in order to check the accuracy of post - local buckling behavior predicted by the present method The coefficient, n, in Eq (9.78) is interchanged between 8 and 16 when using the COS model

The load vs lateral deflection relationships and the interaction relationships of internal forces are plotted in Figures 9.28 and 9.29, respectively The solid and dashed lines represent the results obtained from the present method and experiment, respectively, and the chain lines

192 Part II Ultimate Strength

represent the results obtained from the finite element method without considering local buckling

Until local buckling takes place, both results obtained from the present method and the finite

element method, show good correlation's including the ultimate strength The comparison of

these results using the FEM to the results of other experiments shows little differences among them, which may be attributed to the reasons described in 9.4.1 However, judging from the interaction relationships shown in Figure 9.29, these differences may be attributed to the material properties of the actual material and assumed material used for the analysis The yield stress used in the analysis is determined, based on the results of the tensile test, and may be

very accurate as long as the stress is in tension It is not completely clear, but there may be

some differences in the material properties in a tensile and a compressive range

Figure 9.27 Axial Force Bending Moment Relationships

Post-local buckling behavior is simulated quite well although the calculated starting points of local buckling are a little different from the measured ones The difference in the onset point

of local buckling may be due to inaccuracies of the critical buckling strain evaluated by Eq.(9.31) and the estimated strain using Eq.(9.67) At present, the value to be employed as n remains unknown Although larger values may give good results as indicated in Figures 9.28 and 9.29

The curves changing the value of n may be regarded as the results of the numerical

experiment, changing the onset point of local buckling A greater reduction is observed in the

load canying capacity (axial load) as the critical load for buckling increases

The same analysis is performed on small-scale test specimens Relatively good correlation's are observed between the calculated and experimental results for the ultimate strength in all specimens However, the calculated post-ultimate strength behavior is slightly different from the observed behavior This may be attributed to a difference in the assumed stress-strain relationship used during the analysis and the actual one An elastic-perfectly plastic stress- strain relationship is assumed in the analysis Contrary to this, the actual material showed relatively high strain hardening In order to analyze such cases, the influence of strain hardening must be taken into account The strain hardening effect may be easily incorporated

Chapter 9 Buckling and Local Buckling of Tubular Members

in the simplified analysis Applying the Plastic Node Method for the post-ultimate strength analysis is a basic idea that is presented in Ueda and Fujikubo (1986) These remain as ideas in

194 Part II Ultimate Strength

Figure 9.29 Measured and Calculated Relationship Between Axial Force

and Bending Moment

If pure bending is obtained, the axial force is zero and the proposed method does not need to

be applied In this case, the filly plastic condition will give an accurate ultimate strength

Furthermore, this method is not necessary when the axial force is in tension

9.5 Conclusions

Local buckling of tubular members is investigated in this chapter both theoretically and experimentally First, a series of experiments are carried out on large and small-scale tubular specimens Large-scale test specimens are 1/33 scale model of a bracing member in an

Chapter 9 Buckling and Local Buckling of Tubular Members 195

existing semi-submersible drilling unit, and their diameter to thickness ratio, D/t, is 78 The D/t ratio of small-scale specimens varies between 40 and 97 Axial compression tests with load eccentricity are carried out on both specimens, and pure bending tests on small-scale specimens only These experiments have shown that after the ultimate strength has been attained, local buckling takes place at the area of maximum compressive strain Two types of

buckling mode are observed, which are denoted as a cosine mode and a dent mode The

buckling wave of a cosine mode spreads about a half circle in the circumferential direction, and that of a dent mode about a quarter circle in the circumferential direction Nevertheless, it has a short wavelength in the axial direction in both modes

The load canying capacity suddenly decreases due to the initiation of local buckling

In the case of a cosine mode, the formation of local denting deformation follows at the foot of the initial cosine-buckling wave Other local denting deformations are formed adjacent to the initial dent and in the case of dent mode buckling

A simplified method is proposed to analyze the elasto-plastic behavior of a tubular member subjected to axial compression, end moments, and distributed lateral loads Two models are proposed which simulate the post-local buckling behavior of a tubular member based on the observed results of experiments They are the COS and the DENT model

Combining these models with the simplified method, a series of analyses have been performed

on the newly tested specimens and on those previously reported The analyses results are

compared with experimental results, and the validity and usefulness of the proposed simplified

methods of analysis are demonstrated

Furthermore, the Idealized Structural Unit model (element) is developed by incorporating the proposed simplified method Using this model, the ultimate strength is automatically evaluated under axial compression After the local buckling has started, its influence is reflected upon the fully plastic strength interaction relationship through plastic nodal displacements of the element Some example calculations are performed by applying the newly developed element The calculated results are compared with those obtained using the finite element method and the validity and usefulness of this element is demonstrated

Research remaining for future work is:

Accurate estimates of plastic strain and curvature at a plastic node

Accurate evaluation of critical buckling strain

System analysis using the proposed Idealized Structural Unit model

The Plastic Node Methods, as described in Part I1 Chapter 12, is a generalization of the plastic

hinge methods that have been popular for plastic analysis of beams and framed structures The

196 Part II Ultimate Strength

generalization makes it possible to effectively conduct analysis of plated structures and shell structures (see Ueda and Yao, 1982) It is also possible to include the effect of strain hardening

in the formulation, see Ueda and Fujikubo (1986) However, geometrical nonlinearity is not a subject discussed in the plastic node methods

The Idealized Structural Unit Methods (Ueda and Rashed, 1984) make use of the Plastic Node

Methods to deal with the plasticity, and utilize empirical formulae (such as those in design

codes) for ultimate strength analysis of individual components In this Chapter, however, an attempt has been made to predict the ultimate strength of the components using simplified inelastic analysis instead of empirical formulae The advantage of using the simplified inelastic analysis is its ability to account for more complex imperfection and boundary conditions that are not covered in the empirical formulae However, the disadvantage is its demand for computing effort and its complexity that may lead to loss of convergence in a complex engineering analysis

Batterman, C.S (1 9 6 3 , ‘Tlastic Buckling of Axially Compressed Cylindrical Shells”,

Bouwkamp, J.G (1979, “Buckling and Post-Suckling Strength of Circular Tubular Section”, OTC, No-2204, PP.583-592

Chen, W.F and Han, D.J (1985),”Tubular Members in offshore Structures”, Pitman Publishing Ltd, (1985)

Det norske Veritas (1981), Rules for Classification of Mobile Offshore Units (1981) Gerard, G (1 962), “Introduction to Structural Stability Theory”, McGraw-Hill International Book Company, New York

Rashed, S.M.H (1980), “Behaviour to Ultimate Strength of Tubular Offshore Structures by the Idealized Structural Unit Method”, Report SK/R 51, Division of

Marine Structure, Norwegian Institute of Technology, Trondheim, Norway

Reddy, B.D (1979), “An Experimental Study of the Plastic Buckling of Circular Cylinder in Pure Bending”, Int J Solid and Structures, Vol 15, PP 669-683

Smith, C.S., Somerville, W.L and Swan, J.W (1979), “Buckling Strength and Post-Collapse Behaviour of Tubular Bracing Members Including Damage Effects”,

Toi, Y and Kawai, T (1983), “Discrete Limit Analysis of Thin-Walled Structures (Part 5) - Non-axisymmetric Plastic Buckling Mode of Axially Compressed Circular Shells”, J Society of Naval Arch of Japan, Na 154, pp.337-247 (in Japanese)

AIAA J., V01.3 (1965), pp.316-325

BOSS, PP.303-325

Chapter 9 Buckling and Local Buckling of Tubular Members 197

Ueda, Y and Fujikubo, M (1986), “Plastic Collocation Method Considering Strain-Hardening Effects”, J Society of Naval Arch of Japan, Val.160, pp.306-317 (in Japanese)

Ueda, Y., Rashed, S.M.H and Nakacho, K (1984), “New Efficient and Accurate Method of Nonlinear Analysis of Offshore Structures”, Proceeding of OMAE,

Ueda, Y and Yao, T (1982), “The Plastic Node Method: A New Method of Plastic Analysis”, Computer Methods in Appl Mech and Eng., Va1.34, pp.1089-1104

Yao, T., Fujikubo, M., Bai, Y., Nawata, T and Tamehiro, M (1986), “Local Buckling

of Bracing Members (1st Report)”, Journal of Society of Naval Architects of Japan, Vol 160

Yao, T., Fujikubo, M and Bai, Y., Nawata, T and Tamehiro, M (1988), “Local Buckling of Bracing Members (2nd Report)”, Journal of Society of Naval Architects of Japan, Vol 164

PP.528-536