Chapter 2 Buckling of Free Pipe under External Pressure
In separated-type water tunnel structure, when the point supported steel liner is subjected to the uniform external pressure, the contact between tunnel lining and liner is difficult to happen because the developed compressive hoop thrust in pipe only shortens the circumference of liner and then enlarges the gap between liner and host. Therefore, the buckling of the uniformly point supported liner can be considered a rotary symmetric buckling likely the buckling of free pipe under external pressure. In this chapter, the buckling of free pipe is investigated.
2.1 Introduction
2.1.1 Plain Pipe vs. Stiffened Pipe
The choice of plain versus stiffened steel pipe for a tunnel liner is primarily not only a cost issue, but also have to consider other issues such as the safety, constructability, and the inspection/ maintenance. For instance, steel pipe with diameter to thickness ratios of more than about 300 or thickness more than 45mm are usually not practical due to handling limitations.
On the other hand, the steel pipe must have the sufficient capacity to resist the internal pressure and external pressure, while the external pressure is vulnerable to cause the buckling of pipe.
Generally, the steel pipe thickness designed for external pressure is usually much thicker than that for internal pressure. In engineering practice, plain steel pipe has many advantages as shown in followings,
• Design and analysis are relatively simple and well understood.
• Outer diameter of a plain steel pipe is less than a stiffened steel pipe; hence the excavated tunnel diameter can be reduced.
• Manufacture of a plain steel pipe is simpler, particularly for wall thickness less than about 20mm. The manufacture cost per unit length of stiffened steel pipe is generally higher than a plain pipe with same thickness, considering the required stiffener welding work.
However, when the water tunnel is built in deep underground, the groundwater pressure is relatively high and the structure design has to consider the bucking of steel pipe under external pressure principally. In such case, the stiffened pipe is required considering its following advantages,
• Stiffened steel pipe can be designed lighter than a plain pipe.
• Buckling mode of a stiffened pipe can be controlled, and the long steel pipe collapse can be avoided if appropriately design.
• Thinner pipe can be used, hence not only quantities of steel materials but also the welding works are possibly reduced. In addition, the construction of deep water tunnel can become possibility when the steel pipe rolling process is limited in thickness due to available fabrication machinery and other restrained conditions.
Otherwise, the application of the external circular stiffeners usually called ring stiffener should be considered when the thickness of a plain pipe designed for external pressure exceeds the thickness required by internal pressure. Finally, the design should be carried out based on economic considerations of the following three available options: a) increasing the thickness of the pipe, b) adding external stiffeners to a pipe with thickness required for internal pressure, and c) increasing the thickness of pipe and adding external stiffeners, and satisfy the design requirement for the external pressure. In addition, the economic comparison between plain and stiffened pipe must consider the extra cost of welding, tunnel excavation and backfill.
In the current study, the stiffened pipe is considered as the tunnel liner in principle considering the safety and cost of the water tunnel built in urban deep underground. The stiffeners are installed with a constant spacing on the steel pipe, and are welded with fillet welding around and at the exterior of pipe. The stiffener is assumed having sufficient second moment of inertia to avoid buckling of itself, since it is principally used to holding the pipe in a circular shape. As the stiffener types, there are 4 common types as shown in Fig. 2.1, called tee bar, rolled channel stiffener, rolled plate stiffener, and flat bar, respectively. Tee bar is theoretically the most efficient as a stiffener, however the complicated fabrication is the vital shortcoming. Moreover, the available reduction of cross section is limited because the failure of flange or web has to be avoided. Rolled channel stiffener is also structural efficient, by which a greater spacing of stiffener becomes possible because of its larger inertia moment and two connection points with steel pipe. The problem of rolled channel stiffener is that there are no effective means to treat the void created by stiffener. A rolled plate oriented parallel to the steel pipe cannot add the second moment of inertia of steel pipe to resist buckling. It is therefore that the rolled plate is not used as main stiffener, just used for reinforcing the connection of steel pipe. A flat bar (Rolled or cut plate placed perpendicular to pipe) are the most commonly used stiffener due to not only its easily manufacturing but having efficient second moment of inertia to improve the capacity of buckling resistance. Accordingly, in this study, the flat bar is adopted as stiffener in terms of the design and construction of stiffened pipe. The cross-section profile of a pipe stiffened with flat bar is shown in Fig. 2.2, where the related notations about stiffened pipe are shown1).
a) Rolled tee b) Rolled channel c) Rolled plate d) Rolled /Cut plate (Tee Bar) (Parallel to pipe) (Perpendicular to pipe) (Flat Bar) Fig. 2.1 Typical stiffeners
Steel pipe
Fig. 2.2 Wall cross-section of stiffened pipe
2.1.2 Existing Buckling Theories Review
In the past decades, many researchers have studied the buckling problem, for instance, Von Mises, Donnel, Southwell, Timoshenko, Flügger, Tokugawa , etc.., investigated the free pipe with two ends held in circular shape and presented analytical solutions respectively; Kendrick, Bryant etc. investigated the stiffened pipe and presented analytical equations taking into account the flexural rigidity of stiffener, Koter, Yamaki and others investigated the nonlinear theory of thin shells and the influence of imperfection.
Plain Pipe
When a free pipe is subjected to uniform external pressure, the tangential compressive stress will be developed in pipe and increases with the external pressure increases. When the tangential compressive stress reaches a limit value, the pipe is not able to maintain its initial circular shape, and distorts unstably in buckling.
For a finite long free pipe with radius R, thickness t, the buckling has been discussed by Von.
Mises2), Donnel3), Southwell4), Timoshenko5), Flügger6), Tokugawa7), etc.. The corresponding buckling equations will be referred in following section for stiffened pipe. However for an infinite long free pipe, the buckling equation can be derived using the Euler buckling theory, by assumed as a ring with the second moment of inertia (I).The buckling equation is expressed as follows,
2 3 3
2
12 ) 1 ( ) 1
( ⎟
⎠
⎜ ⎞
⎝
− ⎛
− =
= R
t E n R
EI
Pcr n (2.1a)
However, if the pipe is longitudinal restrained, Eq. (2.1a) should be modified by considering the Poisson’s effect. The equation of critical pressure is then given by the following,
合 成 断 面 中 立 軸
R 2S
t
tr
hr
合 成 断 面
Rc
Rt Be=1.56
R0
Effective cross-section Neutral axis
' 3 3 2
2 2
12 ) 1 ( )
1 ( 12
) 1
( ⎟
⎠
⎜ ⎞
⎝
− ⎛
⎟ =
⎠
⎜ ⎞
⎝
⎛
−
= −
R t E n R
t v
E
Pcr n (2.1b)
Where, the E’=E/(1-v2) is often used in practice, commonly called effective elastic modulus.
As for a long free pipe, since the lowest critical pressure is always produced when the number of wave n is equal to 2, the number of waves is commonly given 2. Thus the buckling equation of a long pipe can be expressed by Eq. (2.2).
' 3
4 ⎟
⎠
⎜ ⎞
⎝
= ⎛ R
t
Pcr E (2.2)
Moreover, the analytical solution using Eq. (2.2) is only valid for hydrostatic pressure with the acting direction normal to pipe surface. For a conservative load maintaining their direction, the denominator 4 in Eq. (2.2) should be replaced by 3.
Stiffened Pipe
As the ring-stiffened pipe is generally considered a stiffened cylindrical shell, the following studies should be mentioned. The buckling of ring-stiffened cylindrical shells under external pressure has been studied by the following researchers. Reis and Walker8) analyzed the local buckling strength of ring-stiffened cylindrical shells by non-linear buckling analysis. Y.
Yamamoto9) studied the general instability of ring-stiffened cylindrical shells using experiments.
S. S. Seleim et al. 10), systemically studied the buckling behavior of ring-stiffened cylinders.
From the studies mentioned above, it is confirmed that the buckling behavior of ring-stiffened cylindrical shells involves three types of failure: inter-ring shell buckling, general buckling, and ring-stiffener stripping. Regarding buckling design, Charles P. Ellinas and William J. Supple11) conducted a comprehensive investigation on buckling design for ring-stiffened cylinders.
Generally, the studies on buckling of stiffened pipe can be divided into two groups. One group focused the buckling of the shell between adjacent stiffeners, while another group focused the buckling of overall pipe. The two buckling forms are usually called general buckling and local buckling, and their buckling forms are shown in Fig. 2.3, respectively. The representatives of former group include Von Mises, Donnel, Southwell, Timoshenko, Flügger, Tokugawa, etc, while the latter are represented by Kendrick12), Bryant13), etc..
In the research of former group, the stiffened pipe is simply assumed that the two ends of a plain pipe are held in circular shape, and buckling occurs in a rotary-symmetric buckling with sinusoidal wave. The related equations were derived by Southwell in 1913, Von Mises in 1914, Timoshenko in 1938, Flügger in 1960, Tokugawa in 1961, Donnel in 1976. However, since the Timoshenko’s equation obtained by Von Mises firstly are usually used in engineering practice.
a) Initial shape b) bucking of inter stiffener shell c) buckling of overall pipe (Local buckling) (General buckling)
Fig. 2.3 Buckling forms of stiffened pipe
The Timoshenko and Von Mises’ buckling equation for a stiffened pipe with radius R, thickness t and spacing S is given in Eq. (2.3). Where, the approximate wave number can be determined by following Eq. (2.4).
( )
⎥⎥
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢⎢
⎢
⎣
⎡
⎟ +
⎠
⎜ ⎞
⎝
⎛
− + −
− − +
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ⎟ +
⎠
⎜ ⎞
⎝
− ⎛
=
1 1 1 2
) 1 ( 1 12 )
1 (
1
2 2
2 2 3 2 3 2 2
2
2 S
n R n n
R Et
S n R R n
Pcr Et
π ν π ν
(2.3)
( )
8
2 2 2
4
) 1 ( 12
3
R t
S R
n
ν
π
−
⎟⎠
⎜ ⎞
⎝
⎛
= (2.4)
When the spacing S in Eq. (2.3) is replaced by the length of a plain pipe L, the Von Mises’
buckling equation can also be used for a finite long pipe.
In Japan, Tokugawa equation is also used in buckling design as shown following,
( ) ( ) ( )
( )
⎥⎥⎦⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
+
− −
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ + −
+ +
−
= 2
2 2
2 2 4
2 2 2
0 2 2 2
2 4 2 2
0 2 1
) 1 ( 3
2 2
1 2
α α
α μ α
α
nn n n
D t n n
D E t
Pcr (2.5)
Where,
S D 2 π 0
α =
25 . 0 0 5 . 0
63 0
.
1 ⎟
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
= ⎛
t D S
n D
In the latter group, for the buckling of stiffened pipe, Kendrick and Bryant presented their theoretical solutions using the energy approach in 1953, 1954 respectively, based on the phenomenon of buckling deformation that stiffeners and shell distorts simultaneously when buckling occurs. Kendrick and Bryant’s theoretical equations are given in Eq. (2.6) and Eq.
(2.7), respectively.
S R n I n
n R
Pcr Et e 3
2 2 2 2 2 2
2
4 ( 1)
) )(
1 2 (
+ − +
−
= +
α α
α
(2.6)S R R
n I n
n R P Et
c
cr e 2
0 2 2 2 2 2 2
2
4 ( 1)
) )(
1 2 (
+ − +
−
= +
α α
α
(2.7)Kendrick’s theoretical solutions was derived by assuming the displacement variations as sinusoidal functions and applying the Rayleigh-Ritz approach, and was experimentally verified by Galletly et al.. Bryant used the same means in derivation of theoretical equation, however, the effect of stiffeners’ eccentricity and the thickness of shell were taken into account. Moreover, Kendrick’s equation has been adopted in BS 5500 1994, while Bryant’s formula has been recommended by the Structural Stability Research Council (SSRC) in America and Associate of Civil Engineer in Japan.
Nonlinear and Imperfection Theory
When a cylindrical shell is subjected to small external pressure, the small deformation with rotations and strain will occur. In theoretical analysis, the liner expressions for both displacement-strain relation and stress-strain relations can be applied, and the basic governing equations in terms of deformation of shells results in the classical linear theory of elasticity.
However, if shells deforms largely, the geometric nonlinear and material nonlinear in terms of displacement-strain relation and stress-strain relations will happen. In such case, the governing equations have to be expressed by adding many nonlinear terms, which results in the nonlinear theory of elasticity.
The famous research was carried out by Donnell14) and Yamaki15). The first nonlinear theory of cylindrical shells was established during analysis of torsion buckling of thin-walled pipe by Donnell in 1933. Due to its relative simplicity and practical accuracy, the theory has been most used for analyzing both buckling and post-buckling problems. However, Donnell’s nonlinear theory was established based on many assumptions as follows,
a) shells is sufficiently thin,
b) strains are sufficiently small and Hooke’s law holds, c) there is no deformation in the middle plane,
d) points of shells lying initially normal to middle plane remains the normal to middle
surface of shell after deformation, and
e) normal stress in the direction transverse to shells can be disregarded.
Where, the assumptions d) and e) consists the so-called Kirchhoff-love hypotheses. From assumptions above, Donnell’s theory is only suitable for the shallow shell with small deformation, while for the analysis of larger deformation of cylindrical shell it is not applicable.
Yamaki found out the problems and presented a modified nonlinear theory based on Flügge theory. Using modified nonlinear theory, the buckling and post-buckling problems were investigated, otherwise the influence of initial imperfection was investigated and corresponding solution was presented.
As the imperfection problem of thin shells, the Koitor’s research should be mentioned. In 1945, Koiter16) clarified the bifurcation stability with the asymptotic analysis of total potential energy of system, by which the initial post-buckling behavior and the influence of small initial imperfections on the critical pressure are reasonably predicted. Moreover, in Koitor’s research, the general nonlinear theory of thin shells and various simplifications of energy functions were also discussed. Based on the general nonlinear theory, the equations of equilibrium were described in both fundamental state I and an infinitesimally adjacent state II. It was clarified that the energy function for both dead and pressure loading are identical. As the imperfection-sensitivity analysis, the imperfection of a structure is closely related to its initial post-buckling behavior and the theory is exact in the asymptotic sense. As shown in Fig. 2.4, the shape of the secondary initial equilibrium path plays a vital role in determining the influence of the initial imperfections. When the initial portion of the secondary path has a positive curvature (see Fig. 2.4 a)), the structure can develop considerable post-buckling strength and loss of stability of primary path does not result in structural collapse. However, when the initial portion of the secondary path has a negative curvature (see Fig. 2.4 b)) then in most cases buckling will occur violently and the magnitude of the critical load is subject to the degrading influence of initial imperfections. For those cases where as shown in Fig. 2.4, the bifurcation point is symmetric with respect to the buckling deflection, the initial post-buckling behavior is governed by
λ/λc=1+bη2 (2.8a) Where, λ is the applied load (axial load P or external pressure p) and λc is the classical buckling load. The amplitude of the buckling displacement normal to the shell w have been normalized with respect to the shell thickness t, thus η=w/t. Accordingly, if the post-buckling coefficient b is negative, the equilibrium load falls following buckling and the buckling load of the real structure λs is expected to be imperfection-sensitive. In this case, the asymptotic relationship between the buckling load of the imperfect shell and the imperfection amplitude η* is
(1-ρs)3/2= 1.5(-3b) 0.5|η*|ρs (2.8b)
a) Stable post-buckling b) Unstable post-buckling Fig. 2.4 Equilibrium paths for perfect and imperfect shells
Where, ρs=λs/λc and η* is the normalized amplitude of the initial imperfection in the form of the classical asymmetric buckling mode17). In the latter the theory was developed and refined by Thompson and Hunt18), etc..
However, as the buckling of stiffened pipe under external pressure, that the imperfection has far less influence on the critical pressure, and in most cases the experimental buckling pressure is higher than the analytical value by as much as 15% has been verified by the Tennyson19).
2.1.3 Numerical Analysis Method and FEM Software20)
The numerical analysis is carried out using FEM software MSC.Marc, which has been used to analyze numerous problems successfully in various fields. The buckling analysis provided can estimate the collapse loading / buckling load of a structure in three means: 1) linear buckling analysis in which the eigenvalue analysis is extracted in a linear problem, 2) nonlinear buckling analysis in which eigenvalue analysis is performed in a nonlinear problem based on the incremental stiffness matrices, and 3) Arc-length method which is usually applied for both geometric (large deformation) and material (elasto-plastic material) nonlinear problem. The buckling load can be estimated directly when apply 1) and 2) analysis method. However, the estimation of buckling load requires to investigate the behavior of load and displacement when use the arc-length method.
In MSC.Marc, to activate the buckling option in the program, the parameter BUCKLE should be used. If a nonlinear buckling analysis is performed, also use the parameter LARGE DISP.
Otherwise, the history definition option BUCKLE can be used to input control tolerances for buckling load estimation (eigenvalue extraction by a power sweep or Lanczos method). The
Perfect shell
Imperfect shell
Imperfect shell Perfect shell
Limit point Bifurcation point
η λ/λc λ/λc
λ/λc
η
buckling load can be estimated after every load increment. However, the BUCKLE INCREMENT option should be used if a collapse load calculation is required at multiple increments. The linear buckling load analysis is correct when one takes a very small load step in increment zero, or makes sure the solution has converged before buckling load analysis (if multiple increments are taken). Although the linear buckling (after increment zero) can be done without using the LARGE DISP parameter, in which case the restriction on the load step size no longer applies, the estimated bucking load should be used with caution, as it is not conservative in predicting the actual collapse of structures. Generally, for a buckling problem that involves material nonlinearity (for example, plasticity), the nonlinear problem must be solved incrementally because a failure to converge in the iteration process or non-positive definite stiffness can signal the plastic collapse during the analysis. Moreover, for extremely nonlinear problems, since the BUCKLE option cannot produce accurate results, the history definition option AUTO INCREMENT should be used to allow automatic load stepping in a quasi-static fashion for both geometric large displacement and material (elastic-plastic) nonlinear problems.
By the option the elastic-plastic snap-through phenomena can be handled and the post-buckling behavior of structures can be analyzed.
In the liner buckling analysis, the buckle option solves the following eigenvalue problem by the inverse power sweep or Lanczos method using the following matrix formula.
(
, ,)
] 0[K+
λ
iΔKG Δu u Δσ φ
i = (2.9a) Where, K is the stiffness matrix of structure, ΔKG is assumed a linear function of the load increment ΔP to cause buckling.The geometric stiffness ΔKG used for the buckling load calculation is based on the stress and displacement state change at the start of the last increment. However, the stress and strain states are not updated during the buckling analysis. The buckling load Pcr is therefore estimated by:
P P
Pcr = 0+
λ
iΔ (2.9b) Where, for increments greater than 1, P0 is the load applied at the beginning of the increment prior to the buckling analyses, and λi is the ith mode value obtained by the power sweep or Lanczos method. As the control tolerances, the maximum number of iterations and the convergence tolerance can be inputted. For the inverse power sweep method, the power sweep terminates when the difference between the eigenvalues in two consecutive sweeps divided by the eigenvalue is less than the tolerance. The Lanczos method concludes when the normalized difference between all eigenvalues satisfies the tolerance.In the current study, the numerical analysis for investigation of buckling behavior is conducted applying linear buckling analysis, considering the non-linear characteristics has little
influence on the elastic buckling of a perfect cylindrical under uniformly external pressure. As the analysis result, the estimated critical load can be used to evaluate whether material nonlinearities occurred before buckling. Moreover, the buckling mode should be plotted and studied, by which whether the modeled mesh size is sufficient to describe the collapse mode can be checked.
2.2 Buckling Behavior of Ring-Stiffened Pipe
2.2.1 Generalization on Buckling of Stiffened Pipe
In general, the buckling of stiffened pipe is much more complicated than that of plain pipe, the bucking form (type) include a) buckling between stiffeners (local buckling ), b) buckling of general pipe (general buckling), and, c) tripping of stiffeners. As shown in Fig.2.5, the local buckling occurs in the cylinder between stiffeners and distorts into several waves, while the stiffeners remain circular shape; the general buckling stands for the buckling of overall stiffened pipe accompanied with stiffeners and pipe deformed in same waves. The tripping of stiffener is same as the buckling of a plate with a clamped edge twist about its point of attachment to shell, and can be avoided if the aspect of cross section is ensured21).
Fig. 2.5 Local buckling and general buckling
However, in practice, an un-appropriate conception is prevailing until present that the buckling between stiffeners will occur in a stiffened pipe with relatively heavy stiffeners, while general buckling failure will occur in a stiffened pipe with the light stiffener. Actually, the buckling failure type of stiffened pipe is determined not only by the flexural rigidity of stiffeners but the spacing of stiffeners and the geometries of pipe. Buckling behavior of stiffened pipe with respect to buckling type has been investigated by author using numerical analysis method, where the uniformly stiffened pipe is focused, and the second moment of inertia referring the flexural rigidity of stiffener is briefly named stiffness, since Young’s modulus is a constant22) 23).
Stiffener Pipe
Pipe
Stiffener Pipe
Stiffener
Stiffener
Pipe
General buckling Local buckling
2.2.2 Investigation through Numerical Analysis
The buckling behavior of stiffened pipes was investigated using numerical analysis. Since the un-stiffened cylindrical shells always occurs general buckling, while stiffened cylindrical shells have three types of buckling types, the decisive element can be considered the ring stiffeners.
Therefore the effects of stiffeners in terms of its stiffness and spacing should be investigated first. Meanwhile, the influence of the pipe
dimensions is examined. Pipe dimensions of the analysis models and their material properties are given in Table 2.1 and Table 2.2 respectively.
However, where to prevent ring stiffener from stripping failure, the ratio of height and thickness hr/ht is set less than 17 based on Eq.(2.10).
y r
r f
t E
h ≤0.4 (2.10)
The buckling behavior is discussed from the two aspects of: a) effects of stiffness and spacing of stiffeners b) effect of the pipe dimensions.
Effects of Stiffness and Spacing of Stiffeners The buckling behavior is investigated in terms of buckling wave and critical pressure, as well as the effects of stiffness and spacing of stiffener, as shown in Fig.2.6
Table 2.1 Analysis Models Pipe geometries Stiffener
variations model
R(m) t(mm) L(m) S(m) Ir(m4)
1 1.5 2 1.0 3
10 3 0.5
4 10 6 1.0
5 10 9 1.0
6 5.0 3 1.0
7 5.0 6 1.0
8 5.0 9 1.0
9
1.50
7.5 3 1.0 hr/tr
≤17
Table 2.2 Material properties Yong's
Modulus
Poisson’s Ratio
Yield Stress E(N/mm2) ν σy (N/mm2)
2.1+E5 0.3 325
a) Buckling waves b) Critical pressure Fig.2.6 Effects of stiffness and spacing on buckling of uniformly stiffened pipe
Ir
0 500 1000 1500 2000 2500 3000 3500
0 20 40 60 80 100 120
Model1 Model2 Model3
Buckling Pressure Pcr (kN/m2)
Second moment of area of ring stiffeners Ir (cm4)
I I
I II
II
II
I : General buckling II: Local buckling
0 2 4 6 8 10 12 14
0 20 40 60 80 100
Model1 Model2 Model3
Number of lobes n
Second moment of area of ring stiffeners Ir（cm4） General buckling lobes
(Model1)
Local buckling lobes (Model1)
I I
II
II
II
I : General buckling
II: Local buckling Critic
al pressure Pcr (kN/m2 )
Second moment of inertia of stiffeners Ir(cm4) Second moment of inertia of stiffeners Ir(cm4)
Fig.2.6 a) and b) show the buckling behavior with changing of the stiffness and spacing of ring stiffeners. Where, the curves of the stiffness of ring stiffeners, Ir, versus the number of lobes, n, and the critical external pressure, Pcr, are presented for different spacing of ring stiffeners.
Fig.2.7 a) indicates that all models experience two buckling types with the increasing of stiffeners’ stiffness. The first one is the general buckling with deformation of pipe and ring stiffeners as one cylinder in the form of sinusoid waves. The second one is the local buckling with deformation of the inter-stiffener shell in a form of diamond waves. The number of buckling waves is not successive: it is less when in general buckling, and then becomes a greater constant value when in local bucking. As the effect of spacing, with the spacing of ring stiffeners decreasing, the number of
waves decrease in general buckling and increase in local buckling. Fig.2.7 b) indicates that the buckling pressure continuously increases in general buckling and then maintains the maximum value of general buckling when in local buckling.
Where the limit stiffness is defined as the second moment of inertia of stiffener at which the buckling behavior changes from general buckling to local buckling.
However, the buckling pressure changes with the change of the spacing: in general the smaller the spacing is, the larger the buckling pressure is for both general buckling and local buckling, with respect to same stiffness of ring stiffeners.
Effects of Pipe Geometries
The effect of the geometries including length and thickness of pipe on buckling behavior is shown in Fig.2.7. Figure 2.7 indicates that the buckling behavior are all identical in terms of alteration from general buckling to local buckling with increasing of the stiffness of ring stiffeners, and the critical pressure increasing in general buckling while
a) Wall thickness forms
b) Length and thickness
Fig.2.7 Effects of pipe dimensions on buckling of uniformly stiffened pipe
0 300 600 900 1200 1500
0 5 10 15 20 25 30 35 40
MODEL6 MODEL9 MODEL1
座屈圧力 Pcr(kN/m2)
補 剛リ ブの 断 面二 次 モ ーメ ン ト Ir(cm4)
t=10mm
t=7.5mm
t=5mm
Model6 Model9 Model2
Critical pressure Pcr (kN/m2 )
Second moment of inertia of stiffeners I 4)
0 100 200 300 400 500
0 500 1000 1500 2000
20 40 60 80 100 120
Model6 Model7 Model8
Model2 Model4 Model5
Buckling Pressure Pcr (kN/m2)
Second moment of area of ring stiffeners Ir (cm4)
Buckling Pressure Pcr (kN/m 2)
Second moment of inertia of stiffeners Ir(cm4)
Critical pressure Pcr (kN/m2 ) Critical pressure Pcr (kN/m2 )
t=5mm t=10mm L=3.0 m
L=6.0 m L=9.0 m
almost maintaining the maximum critical pressure of general buckling in local buckling.
However, from Fig.2.7 a), that the critical pressure and limit stiffness increase with the increase of wall thickness can be observed. In terms of the length of stiffened pipe, as shown in Fig.2.7 b), the limit stiffness of ring stiffeners, is required to be larger with an increase of pipe length, while the local buckling pressures are almost the same values, which are determined only by inter-ring shells. Moreover, the influence of length change on the limit stiffness and critical pressure becomes greater with the thickness increase.
Summary
From this study, the buckling behavior was clarified as shown in Fig.2.8. For an individual stiffened pipe, with the increase of flexural rigidity of stiffener briefly named stiffness the critical pressure increases in first phase of general buckling, while almost maintains constant in second phase of local buckling. Furthermore, the buckling type is always general buckling before the stiffener stiffness reaches the limit value, then turns to local buckling once exceeds the limit value. Also, from Fig.2.8, a similar behavior is shown for pipe stiffened with a different spacing. However, the changes in terms of limit stiffness of stiffeners and the critical pressure due to the spacing change are expressed significantly. Generally the wider the spacing is the smaller the limit stiffness of stiffeners, and the smaller the critical pressure. As the effects of pipe dimensions, the length, radius, and wall-thickness of pipe affect the buckling behavior in terms of the critical pressure and the limit stiffness of stiffeners. Generally, the limit stiffness decreases with the increasing of the ratio of radius to thickness (R/t), as well as the critical pressure. Similarly, with the increasing of length, the limit stiffness and critical pressure decreases.
Fig.2.8 Buckling behavior of uniformly stiffened pipe (spacing S1>S2)
General buckling
Local buckling
L
P1cr
Pcr
0
Pcr
L
I2r
:Critical pressure :Critical pressure of pipe
:Critical pressure of local buckling : Second moment of inertia of stiffener :Limit second moment of inertia
L
Pcr 0
Pcr
Pcr
L
Ir
Ir
S2 S1
L
P2cr
L
I1r Ir
2.2.3 Consideration and Subject
From the results mentioned above, it is clear that the buckling behavior of uniformly stiffened pipe is more complicated. For a stiffened pipe, there are two potential buckling types, and the buckling type is determined not only by stiffness and spacing of stiffeners but also by the pipe dimensions. The complex buckling behavior makes the solution of buckling problem more difficult in terms of the critical pressure and buckling mode, because the mechanisms of two buckling types are far distinct and there is not an existing method to estimate the buckling types previously. The obtained bucking behavior brings a light to solve the buckling problem of stiffened pipe. If the limit stiffness of stiffeners is known previously, the buckling type can be predicated by comparing the stiffness of existing stiffeners with limit stiffness, since the local buckling happens only when the stiffness of stiffeners exceeds the limit value. As the limit stiffness of stiffener, its magnitude can be obtained considering that the maximum critical pressure of general buckling is identical to that of local buckling at the limit stiffness.
However, the theoretical analysis for buckling of stiffened pipe requires the corresponding buckling equations to calculate the critical pressure. For local buckling, the theoretical equation may be simple because only the inter-stiffener shell is concerned. Whereas the general buckling, that the theoretical equation is much complicated because the stiffener has to be taken into account, and the interaction between stiffener and pipe shell still remains a difficult question.
The theoretical equations will be introduced as well as its utilization method in the next section.
2.3 Derivation of Theoretical Buckling Equations for Ring-stiffened Pipe24)
The buckling behaviors of uniformly stiffened pipe have been clarified. In particular, the buckling type that will eventually change from general buckling to local buckling with the increasing of stiffness of ring stiffeners was identified. Since the general buckling is far different from the local buckling, and the pipe and stiffeners is required to consider simultaneously, the accurate critical pressure of an arbitrary stiffened pipe can be obtained only if the buckling type is known previously. In flowing paragraphs, the theoretical equations of critical pressure are derived taking into account characteristics of general buckling and local buckling first. The solution for buckling of stiffened pipe is then presented based on the theoretical equations. Finally, the verification of the theoretical equations and the presented solution are carried out using numerical analysis and existing experiments.
2.3.1 Introduction
As discussed above, since the buckling of stiffened pipe involves two buckling types of general buckling and local buckling, it is necessary to derive the theoretical equations in terms of general buckling and local buckling, respectively. As for the initial imperfection25), its influence on critical pressure is smaller in the case that Baterdf parameter Z（Zb b=(1-v2) 1/2L2/R/t）
is greater than 103, while can be reduced due to supporting of stiffeners in the case that Z is smaller than 102. Furthermore, since the imperfection of stiffened pipe is still in the research stage and there is not a practical solution until present, in the current study, the imperfection is disregarded in theoretical analysis, however will be taken into account in stiffened pipe design using safe factor. Otherwise, the material stress state is another important issue for buckling analysis, because the elastic or inelastic buckling is determined by whether the stress level of material is beyond the proportional limit or not. In [Buckling Design Specification]26) published by JSCE, a simple formula as shown in Eq. (2.11) has been given to estimate the instability type for a cylindrical shell. If Kc＞1.2, the buckling is elastic buckling, or inelastic buckling. For an infinite long stiffened pipe, the buckling can always be considered elastic buckling generally considering of the infinite long length.
2 / 1 4
/
3 )
(2 2 )
( RE
Lf t
Kc = R y (2.11) On the other hand, for the theoretical analysis, in the case that it is difficult to determine the exact buckling load in complex structures using the Euler formula due to the difficulty in organizing the constant stiffness matrix, the buckling load is often approximated using energy conservation. This means that predicting buckling load is often referred to as the energy method in structural analysis27), 28), 29). Therefore, in the current study, the theoretical equations based on buckling behavior are discussed using the energy method disregarding the imperfection and
nonlinear of material, while the solution for buckling pressure uses the Ritz method30), which is a variational method named after Walter Ritz, and one effective method for finding approximations to the lowest energy eigenstate or ground state in mechanics.
2.3.2 Strains in Shell
The strains of pipe can be discussed using Timoshenko’s bending theory of thin shells. The arbitrary infinitely small element is presented in Fig. 2.9 a), which is obtained through cutting off from shell by two pairs of adjacent planes normal to the middle surface of shell and containing its principal curvatures. In addition, when the lateral sides of the element are displaced parallel to themselves owing to stretching of middle surface, the deformation of element is shown in Fig.
2.9 b), where x, y and z denote axial,
tangential and radial direction in terms of cylindrical coordinate. From the Fig. 2.9, the unit elongations (strain) of a thin lamina at a distance z from the middle surface can be given as,
R z
z R x x
x x x
x 0 0
'
0 1 1 )
(
ε χ
ε
ε
= − − = − (2.12)R z z R
y y
y y y
y
0 0 '
0 1 1 )
(
ε χ
ε
ε
= − − = − (2.13)Where,
ε
xandε
yare strains of the lamina,ε
x0,ε
0y and χ0x, χ0y are strains and curvature change of middle surface respectively, with respect to x and y directions．Furthermore, the shear strain of focused lamina during bending of shell should be taken into account. Considering that it is consisted of twisting of focused elementχ
0xy and the shearing strain in middle surfaceγ
xy0 , the shear strain of the lamina can be given as Eq. (2.14).0
0 2z xy
xy
xy
γ χ
γ
= − (2.14) From the equations discussed above, the strains of arbitrary point of pipe can be obtained, upon the strain and curvature change of middle surface were given.a) b)
Fig. 2.9 Variations of an infinite small element and flexural deformation
y
Middle surface
x
z
O
z
Ry Rx
Focused lamina t
0
γyx 0
γxy
γxy
εx 0
εx εy
0
εy
γyx
Rx(y)
ε0ds
R’x(y)
x(y)
O
z ds
O’
z
x(y)
Focused lamina Middle surface
2.3.3 General Buckling Equation
As discussed before, general buckling is the buckling type that the pipe and stiffeners deformed in same waves.
Accordingly, the curvature change of stiffeners can be assumed to be the same as that of pipe. Fig. 2.10 shows the model, loading as well as the cylindrical coordinate, where u, v and w are the displacements of axial, tangential, and radial direction
respectively. Also, the detailed information of longitudinal cross section of stiffened pipe can be referred to Fig. 2.2.
Stiffened pipes are reasonably considered to have orthotropic properties in most applications because of the deformation of stiffened pipe resulted from interaction of bending behaviors of ring beam and thin shells. In the current study, the stiffened pipe as orthotropic structure is discussed through investigating the pipe and stiffeners respectively. However, out-of-plane bending, torsional and warping strain of the stiffeners are disregarded because they are found that their effects can be negligible for isotropic stiffened pipe in previous studies 12), 31), and are not expected to be significant for the orthotropic case.
However, for such a complicated system, the solution of buckling of stiffened pipe need to use the potential energy theory. The potential energy of the ring-stiffened cylinder is expressed as follows:
s
p V
V
U − −
=
∏ (2.15) Where,
Π : total potential energy of stiffened pipe Vp: strain energy of pipe
Vs : strain energy of ring stiffeners
U : work done by external pressure during buckling
Strain Energy in Pipe
The general strain energy of a thin shell can be written as follows:
dxdydz V [
x x y y z z xy xy xz xz yz yz]
2
1 σ ε + σ ε + σ ε + τ γ + τ γ + τ γ
= ∫∫∫
(2.16)Where, the assumption for thin shells and the stress-strain relationships from Hock are introduced as follows:
Fig. 2.10 Schematic of model and loading
External pressure P
R y ⅴ u
w
x z R
θ t
L Pipe Plat stiffener
=0
= xz
yz
γ
γ
(2.17a)z =0
σ
(2.17b) )1 2 ( x y
x
E
ε νε
σ ν
+= − (2.18a) )
1 2 ( y x
y
E
ε νε
σ ν
+= − (2.18b)
xy xy
E
γ τ ν
) 1 ( 2 +
= (2.18c) Substituting Eqs. (2.18) into Eq. (2.16), the strain energy then can be written into Eq. (2.19),
dxdydz V E
x y x y( 1 )
xy]
2 2 1
) [ 1 ( 2
1
2 2 22
ε ε νε ε ν γ
ν + + + −
= − ∫∫∫
(2.19)Since the pipe is treated as the isotropic thin shells, substituting Eq. (2.12), Eq. (2.13) and Eq.
(2.14) into Eq. (2.19), the strain energy of pipe then can be written as
∫∫∫
+ + +( )
−= − 2 02 0 0 02 1 02
2 2 1
) { 1 ( 2 1
xy y
y x x
p
V E
ε νε ε ε ν γ
ν
( ) ( ) ( 1 ) ]
[
2 z χ
x0ε
x0+ νε
0y+ χ
0yε
0y+ νε
x0+ − ν γ
0xyχ
xy0−
(2.20)( )
dxdydzz2[
χ
x02 + 2νχ
x0χ
y0 +χ
y02 + 2 1−ν χ
xy0 2]}+
Integrating Eq. (2.20) with respect to t from -0.5t to 0.5t and rearranging the equation, the strain energy of pipe is then expressed by bending strain energy (Vp1) and extensional strain energy (Vp2) of pipe.
2
1 p
p
p
V V
V = +
(2.21)Where， Et dxdy
Vp [( x y) 2(1 )( x y xy )]
) 1 ( 24
02 0 0 2
0 0 2
3
1 χ χ ν χ χ χ
ν + − − −
= −
∫∫
Et dxdy
Vp2 =2(1−ν2)
∫∫
[εx02+2νεx0εy0+ε202+21(1−ν)γxy02]Strain Energy in Stiffener
The strain energy of stiffeners is computed using curve beam theory. The plain strain is assumed to distribute invariably. Using cross-section area Ar, second moment of inertia Ir0, the strain energy of a stiffener can be expressed as
∫∫
∫∫ +
= EI dxdy
EA dxdy
V
s r y r y22 0
2
2 ε χ
(2.22)Since the stiffeners’ deformation is identical with shell’s strain for general buckling, the curvature change is same as that of shell. In addition, the strains produced due to axial force is relatively smaller than that due to bending deflection, the strains can be expressed using the curvature change of shell. Thus, the strain energy of a stiffener can be written as Eq. (2.23), where, Ir is expressed as Eq. (2.24), is defined as effective second moment of inertia.
∫∫
∫∫
=
⎥⎦
⎢ ⎤
⎣
⎡ + +
= EI dxdy
EI dxdy t
h V EA
y r
y r r
r r s
2 0
2 0 0 2
) 2 (
) 2 (
2 ) 2 (
χ
χ
(2.23)
0
2 4
)
( r r
r
r A h t I
I = + + (2.24) Where,
r r
r dA hb
A =
∫∫
=2 12
0 r r
r t h
I =
Moreover, since the pipe is stiffened uniformly, the number of stiffeners then can be computed as L/S for a stiffened pipe. Accordingly, the strain energy of all stiffeners is obtained as shown below,
∫
= EI dy
S
Vs L r y02
2 χ (2.25)
Work done by External Pressure
The work done by external pressure is estimated from the bending deformation of infinite element (see Fig. 2.11). When pipe is subjected to external pressure, the axial force Ny develops around stiffened pipe before buckling, and the structural stability is kept.
However, once buckling occurs, the radial displacement is developed, which induces a relative rotation angle χyRdθ/2 between middle surface and axial force. The axial force then can be divided into perpendicular force Nysin(χyRdθ/2) and tangential force Nycos(χyRdθ/2) with respect to deflected middle surface. Considering the assumption of
Fig.2.11 Work mechanism of infinite small element during buckling
Ny
Ny
R Ny
dθ
Rdθ p
w Deflected
Initial
2 θ/ χyRd
) 2 / sin(χ Rdθ Ny y
) 2 / cos(χ Rdθ Ny y
2 θ/ χyRd 2
θ/ χyRd
inextensional deformation of cylindrical shell, the work done by tangential force can be disregarded, the work done by external pressure during buckling is therefore equal to that done by perpendicular force. The Eq. (2.26) expresses the work done by external pressure for a infinite element．
wdx Rd
Nysin(χy θ/2)
− (2.26) Moreover, since χyRdθ/2 is very small and relative rotation angle χy is identical to curvature change of middle surface of pipe, the total works can be written as follows,
dx wd R N
U =−2
∫∫
yχy0 θ (2.27) where，pR
Ny = (2.28) Substituting Eq. (2.28) into Eq. (2.27), the total works done by external pressure of stiffened pip can be expressed as shown in Eq. (2.29).
dx pR wd
U =− 22
∫∫ χ
y0θ
(2.29)Total Potential Energy and Solution
Based on the above discussion, the overall potential energy can be obtained by substituting Eq.
(2.20), Eq. (2.25) and Eq. (2.29) into Eq. (2.15), as shown in following equation.
dx pR wd
V V
U − p − s = −
∫∫ χ
yθ
=
∏ 2 0
2
( ) χ χ dxdy
χ Et χ
xy y x y
∫∫
x+ − − −
− − [( ) 2 1 ( )]
) 1 ( 24
02 0 0 2
0 0 2
3
ν χ
ν
( ) dxdy
Et
xy y
x
∫∫
x+ + + −
− 1 ]
2 2 1
2 [
02 02
2 0 2 0
0
νε ε ε ν γ
ε
− SL EI2r∫ χ
y02dy (2.30)As similar with other variational method, a trial wave function is required on the system in Ritz method. The trial function should be selected to meet boundary conditions (and any other physical constraints). Since the exact function is not known previously the trial function should contain one or more adjustable parameters, which are varied to find a lowest energy configuration. Where, the displacements in x, y and z direction denoted u, v and w respectively are expressed by trial wave function as shown below, taking into account the support boundary at two ends and the buckling deformations.
L x n m
A
u= sin
θ
cosπ
(2.31a)L x n m
B
v= cos
θ
sinπ
(2.31b)L x n m
C
w= sin
θ
sinπ
(2.31c) where，A，B and C：constant factors of displacement u, v and w m：number of longitudinal buckling waves
n：number of circumferential buckling waves
As for the strain-displacement relation, the equations can be given as follows according to Flügger’s cylindrical shell theory:
x u
x ∂
= ∂
ε
0 (2.32a)) 1 (
0 v w
y R −
∂
= ∂
ε θ
(2.32b)γ θ
∂ + ∂
∂
=∂ u
R x v
xy
0 1 (2.32c)
2 0 2
x w
x ∂
= ∂
χ
(2.32d)) 1 (
2 2 2
0 w w
y R +
∂
= ∂
χ θ
(2.32e) )1( 2
0
x v x
w
xy R ∂
+∂
∂
∂
= ∂
χ θ
(2.32f)Additionally dy= Rdθ, substituting it and Eqs. (2.32) into Eq. (2.30) and integrating, the total potential energy of stiffened pipe can be written as following equation.
⎥⎦
⎢ ⎤
⎣
⎡ − +
− +
⎩ −
⎨⎧
⎥⎦⎤
⎢⎣⎡ + −
− −
=
Π 2 2 2 2 2 2 2 2 2 Arn2 n2
tl 1 2
m ) 1 B ( n ) 1 2( m 1
1 A Et R 4
L
α μ μ α μ
μ π
( )
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ − +
− + +
− + +
− −
−
− ( 1) ) 1
1 1 (
12 1 ) 1
1 ( 2
2 2 2
2 2 2 2 2 2 2
2 2
R A
n A I
n tl R m
n t Et p C
r r r
α μ μ
⎭⎬
− ⎫
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ −
+
− +
− A BC m AC
n tl AB
mn
α μ
rμ α
μ
1 21 2 )
1 (
2
(2.33) Moreover, based on the assumption of in-extensional deformation of a cylindrical shell, C= Bn
