Ultimate bending capacities of steel pipelines under combined loadings

Abstract

This article derives analytical solutions for steel pipelines subjected to combined bending moments, axial forces, and internal pressure by considering the effect of ovalization on the plastic region. Both lower and upper bound solutions are proposed based on the selection of a general modulus of pipe material. The ultimate bending capacity, as well as the moment–curvature relationship, can be obtained from the analytical solutions. A comparison of the analytical solutions with the experimental and numerical results finds the bending capacities and the moment–curvature curves in good agreement. Therefore, the proposed analytical solutions can be used to study the behavior of steel pipelines subjected to combined loadings.

Keywords

analytical solutions combined loadings ovalization steel pipelines ultimate bending capacities

Introduction

Buried pipelines are typically subjected to complex loading combinations of internal pressure caused by the actions of the fluids they convey, the axial forces due to thermal and Poisson ratio effects, and the bending moments induced by differential soil movements that may result from seabed movements, seismic activities, frost heaves, and thaw settlements (Emamgheiss, 2007; Palmer and King, 2004; Sen, 2006). Consequently, the study of the bending moment capacities and deformational characteristics of steel pipes under combined loadings is highly significant.

Extensive research has been conducted on the theoretical study of the bending moment capacity of steel pipes under combined loads. Brazier (1927) derived an analytical solution for the ultimate capacities of pipes under bending moments by considering the effect of ovalization. Gerard and Becker (1957) provided a number of interactional relationships for pipes under various loading combinations, which was applicable for pipes with large diameter-to-thickness ratios. Mohareb and Murray (1999), Mohareb et al. (2001) proposed an interactional expression concerning internal pressure, axial forces, and bending moments without accounting for the effects of ovalization and local or global instability. Hauch and Bai (2000) derived an analytical solution for the ultimate capacity of pipes subjected to combined loads by adopting the interaction criterion of multi-axial stresses proposed by Galambos (1998) and considering isotropy in the longitudinal and circumferential direction. Subsequently, Mohareb (2002, 2003) developed general plastic interaction equations for pipes under axial forces, torsion, biaxial bending moments, biaxial shear forces, and internal or external pressure by the lower and upper bound theorems. Schaumann et al. (2005) derived an analytical solution for pipes subjected to combined internal pressure and bending moments based on the equilibrium differential equation of beams. Using elastic-perfectly plastic, linear hardening and power hardening models, Li et al. (2009) suggested analytical solutions for perfect pipes subjected to combined pressure, axial forces, and bending moments.

In the above studies, the effects of ovalization of pipe cross-sections and local buckling on bending capacity are ignored. In addition, fully plastic resistances are reached along the cross-section of steel pipes subjected to bending moments. These simplifications and assumptions seem inappropriate, particularly for pipes with large diameter-to-thickness ratios. Referring to the theory of Mohareb (2002, 2003) and Hauch and Bai (2000), and considering the effect of the ovalization of the plastic region, analytical solutions are thus derived for steel pipelines subjected to combined bending moments, axial forces, and internal pressure.

Basic assumptions and equations

Basic assumptions

To obtain the analytical solution of the ultimate bending capacities for pipelines under internal pressure, axial forces, and bending moments tractable, the following assumptions are introduced:

For a pipe that attains its ultimate bending capacity, a portion of the cross-section is elastic, while the remaining portion is plastic. Any effect of strain hardening in the plastic area is ignored.

The cross-section of the pipeline remains plane both before and after bending so that a plastic neutral axis exists and divides the cross-section into compressive and tensile regions.

The pipeline is considered to be a thin-walled structure. Radial stress and shear stresses are ignored.

The effect of strain localization is ignored.

Because the curvature radius $ρ$ is much greater than the radius of the pipe, the difference of the curvature radius between the outer and inner fibers along a pipe section could be negligible. The curvature radius of the pipe centerline is used during the derivation.

The yield region of the cross-section is simplified as an arched beam.

Basic equations

Based on the maximum distortional energy density yield criterion, that is, the von Mises yield criterion (Boresi and Sidebottom, 1985), a given point on the pipe section reaches yield when the following condition is satisfied

\begin{matrix} \frac{1}{2} [(σ_{r} - σ_{θ})^{2} + (σ_{θ} - σ_{z})^{2} + (σ_{z} - σ_{r})^{2} \\ + 6 (τ_{r θ}^{2} + τ_{θ z}^{2} + τ_{rz}^{2})] = σ_{y}^{2} \end{matrix}

(1)

where $σ_{r}$ , $σ_{θ}$ , $σ_{z}$ , $τ_{r θ}$ , $τ_{θ z}$ , and $τ_{rz}$ are stress tensor components expressed in the cylindrical coordinate system, as shown in Figure 1. $σ_{y}$ is the yield strength of pipe material.

Figure 1.

Stress component acting on pipe.

For pipelines subjected to combined internal pressure, axial forces, and bending moments, the radial stress $σ_{r}$ and the shear stresses $τ_{r θ}$ , $τ_{θ z}$ , and $τ_{rz}$ are low relative to the longitudinal and circumferential stress components $σ_{z}$ and $σ_{θ}$ , and they are generally ignored according to thin-walled theory. Therefore, from equation (1), the interaction relation for pipelines subjected to combined internal pressure, axial forces, and bending moments can be simplified as

σ_{θ}^{2} - σ_{θ} σ_{z} + σ_{z}^{2} = σ_{y}^{2}

(2)

Based on the first assumption in section “Basic assumptions,”equation (2) can be solved to obtain the following expressions for the longitudinal limit compressive and tensile stress $σ_{c}$ and $σ_{t}$

\begin{matrix} σ_{c} = \frac{1}{2} σ_{θ} - \sqrt{σ_{y}^{2} - \frac{3}{4} σ_{θ}^{2}} \\ σ_{t} = \frac{1}{2} σ_{θ} + \sqrt{σ_{y}^{2} - \frac{3}{4} σ_{θ}^{2}} \end{matrix}}

(3)

where $σ_{θ}$ is related to the internal pressure P, $σ_{θ} = PR / t$ ; R is the average radius; and t is the wall thickness of the pipeline.

Analytical solution for bending capacity under combined loadings

The inward stress components

The main characteristic of a pipeline under bending is the ovalization of its cross-section due to the inward stress components $q_{c}$ and $q_{t}$ that generate from the longitudinal stress components $σ_{c}$ and $σ_{t}$ , as shown in Figure 2.

Figure 2.

Ovalization mechanism and inward stress components.

The inward stress components $q_{c}$ and $q_{t}$ are the vertical components of $σ_{c}$ and $σ_{t}$ , respectively. Take example of a piece of pipe with length dl and curvature radius $ρ$ , as shown in Figure 2. According to the equilibrium and geometry equations, the following expressions can be derived

q_{i} d s_{i} dl = 2 σ_{i} td s_{i} \sin \frac{θ}{2} = σ_{i} t θ d s_{i} (i = c, t)

(4)

θ = \frac{dl}{ρ}

(5)

where the subscripts c and t represent the compressive and tensile regions, respectively; $d s_{c}$ and $d s_{t}$ are the lengths of the compressive and tensile plastic regions, respectively, on the pipe section; $θ$ is the bending angle of the piece of pipe under the bending moment; and $ρ$ is the curvature radius corresponding to the angle of $θ$ , as shown in Figure 2.

According to equations (4) and (5), the inward stress component $q_{i}$ can be expressed as

q_{i} = \frac{σ_{i} t}{ρ} (i = c, t)

(6)

The influence coefficient of ovalization on pipe bending moment

At the initial stage of pipe bending, the curvature radius $ρ$ is large and the entire pipe section is in an elastic stage. Any ovalization in the entire section is ignored during the stage. As the curvature radius decreases, the top and/or bottom segments of the pipe enter plastic yielding. Assuming that the distribution of the axial stress is as follows: the range of $[α_{c}, π - α_{t}]$ and $[π + α_{t}, 2 π - α_{c}]$ are the elastic zones and the range of $[2 π - α_{c}, α_{c}]$ and $[π - α_{t}, π + α_{t}]$ are the plastic zones, as shown in Figure 3. The ovalization of the elastic region is ignored and the ovalization of the plastic zones is considered below.

Figure 3.

Deformation of pipe cross-section under inward stress.

Under the inward stress components, the points $A_{c}$ , $B_{c}$ , $A_{t}$ , and $B_{t}$ moved to ${A'}_{c}$ , ${B'}_{c}$ , ${A'}_{t}$ , and ${B'}_{t}$ , respectively, with no vertical displacement. According to the mechanical characteristics and the geometry properties of the pipe section, the compressive and tensile plastic regions of the cross-section can be simplified as arch beams, as shown in Figure 4(a) and (b).

Figure 4.

Simplified structures of the plastic regions: (a) compressive plastic region and (b) tensile plastic region.

Based on Castigliano’s first and second theorems (Hibbeler, 2005), the vertical displacements $Δ d_{c}$ and $Δ d_{t}$ for the outer fibers of the compressive and tensile regions $G_{c}$ and $G_{t}$ , as shown in Figure 4(a) and (b), can be calculated as

\begin{matrix} Δ d_{i} = \frac{12 q_{i} R^{4}}{E' t^{3}} (\frac{1}{4} α_{i}^{2} - \frac{1}{4} α_{i} \sin 2 α_{i} - \frac{7}{8} \cos 2 α_{i} - \frac{1}{8} + \cos α_{i} + \frac{\sin 2 α_{i}}{α_{i}} - \frac{2 \sin α_{i}}{α_{i}}) (i = c, t) \end{matrix}

(7)

where $E'$ is the general modulus of pipe material that will be discussed further in section “Verification.”

The vertical displacements $Δ d_{c}$ and $Δ d_{t}$ of the plastic region due to ovalization are relatively small compared with the pipe average radius R, so it is reasonable to regard the vertical displacement of a given point as a linear change. Therefore, for arbitrary points $F_{c}$ and $F_{t}$ that lie in the compressive and tensile regions and move to points ${F'}_{c}$ and ${F'}_{t}$ after ovalization as shown in Figure 4(a) and (b), the vertical displacements $Δ d_{c}^{F}$ and $Δ d_{t}^{F}$ can be expressed as

Δ d_{i}^{F} = \frac{Δ d_{i}}{L_{i}} L_{i}^{F} (i = c, t)

(8)

where $L_{c}^{F}$ and $L_{c}$ are the vertical dimensions from points $F_{c}$ and $G_{c}$ to the straight line $A_{c} B_{c}$ and $L_{t}^{F}$ and $L_{t}$ are the vertical dimensions from points $F_{t}$ and $G_{t}$ to the straight line $A_{t} B_{t}$ , as shown in Figure 4(a) and (b).

Based on the parallel axis theorem, the bending moment of the plastic region can be divided into two parts: one is the moment of the plastic region with respect to the straight line $A_{c} B_{c}$ or $A_{t} B_{t}$ , called the demarcation line of the plastic and elastic regions; the other is the moment from the demarcation line to the pipe centerline. The first part of the bending moment is affected by ovalization, whereas the other part is independent of ovalization. Therefore, to determine the influence coefficient of ovalization on the pipe’s bending moment, the first part of the moment must be obtained first. After ovalization, the bending moments of the compressive and tensile plastic regions, $M_{c}$ and $M_{t}$ , with respect to the demarcation straight line $A_{c} B_{c}$ or $A_{t} B_{t}$ can be calculated as

\begin{matrix} M_{i} = \int_{l_{i}} σ_{i}^{F} (L_{i}^{F} - Δ d_{i}^{F}) td s_{i} \\ = \int_{l_{i}} σ_{i}^{F} L_{i}^{F} (1 - \frac{Δ d_{i}^{F}}{L_{i}^{F}}) td s_{i} (i = c, t) \end{matrix}

(9)

where $l_{c}$ and $l_{t}$ are the length of the arch beams for the compressive and tensile plastic regions; $σ_{c}^{F}$ and $σ_{t}^{F}$ are the longitudinal stresses of the micro-elements ${F'}_{c}$ and ${F'}_{t}$ ; $Δ d_{c}^{F}$ and $Δ d_{t}^{F}$ are the vertical displacements of $F_{c}$ and $F_{t}$ due to ovalization; and $d s_{c}$ and $d s_{t}$ are the lengths of the micro-elements ${F'}_{c}$ and ${F'}_{t}$ .

Substituting equation (8) into equation (9), the bending moment $M_{i}$ can be rewritten as

\begin{matrix} M_{i} = \int_{l_{i}} σ_{i}^{F} L_{i}^{F} (1 - \frac{Δ d_{i}}{L_{i}}) td s_{i} = (1 - \frac{Δ d_{i}}{L_{i}}) \\ \int_{l_{i}} σ_{i}^{F} L_{i}^{F} td s_{i} (i = c, t) \end{matrix}

(10)

In equation (10), $\int_{l_{i}} σ_{i}^{F} L_{i}^{F} td s_{i}$ represents the bending moment without accounting for ovalization, defined as $M_{oi}$ , and $(1 - (Δ d_{i} / L_{i}))$ can be seen as the influence coefficient of ovalization on the bending moment. Therefore, the bending moments of the plastic regions with respect to the straight line $A_{c} B_{c}$ and $A_{t} B_{t}$ can be expressed as

M_{i} = Q_{i} M_{oi} (i = c, t)

(11)

where $M_{oc}$ and $M_{ot}$ are the bending moments of the compressive and tensile plastic regions with respect to the straight line $A_{c} B_{c}$ and $A_{t} B_{t}$ , respectively, without accounting for ovalization; $M_{c}$ and $M_{t}$ are the bending moments considering ovalization; and $Q_{c}$ and $Q_{t}$ are the influence coefficients of ovalization for the compressive and tensile plastic regions on the bending moment, expressed as

Q_{i} = (1 - \frac{Δ d_{i}}{L_{i}}) (i = c, t)

(12)

Based on the principle of minimum potential energy (Callen, 1985), there is no concave deformation for a pipeline in a stable equilibrium state under a combined internal pressure, axial forces, and bending moments. Thus, when the influence coefficient is less than zero, that is, $Δ d_{c} > L_{c}$ or $Δ d_{t} > L_{t}$ , then $Δ d_{c} = L_{c}$ and $Δ d_{t} = L_{t}$ are given.

The relationship between the neutral axis and plastic regions

To determine the bending moment applied to a pipe section for a given curvature radius $ρ$ and angle of neutral axis $ψ$ , the angles of the compressive and tensile plastic regions $α_{c}$ and $α_{t}$ , that is, the stress distribution of the pipe cross-section, should be obtained first. Figure 5 shows a general stress distribution of a pipe cross-section, and the corresponding deformation of the pipe in the longitudinal direction is displayed in Figure 6.

Figure 5.

General stress distribution of the pipe cross-section.

Figure 6.

General pipe deformations along the axial direction.

Based on Figures 5 and 6, the following equations can be obtained

σ_{i} = E ε_{i} = E \frac{2 Δ l_{i}}{dl} (i = c, t)

(13)

\frac{θ}{2} = \frac{dl}{2 ρ} = - \frac{Δ l_{c}}{R (\cos α_{c} - \cos ψ)} = \frac{Δ l_{t}}{R (\cos ψ - \cos α_{t})}

(14)

where E is the elastic modulus of the pipeline; $ε_{c}$ and $ε_{t}$ are the strains of the outer fibers of the compressive and tensile elastic regions; and $Δ l_{c}$ and $Δ l_{t}$ are the elongations corresponding to $ε_{c}$ and $ε_{t}$ , as shown in Figure 6.

From equations (13) and (14), the relationship between the neutral axis angle $ψ$ and the angles of the compressive and tensile plastic regions $α_{c}$ and $α_{t}$ can be written as

\begin{matrix} \cos α_{c} = \cos ψ - ρ \frac{σ_{c}}{ER} \\ \cos α_{t} = - (\cos ψ - ρ \frac{σ_{t}}{ER}) \end{matrix}}

(15)

Determination of the pipe bending moment

For a given curvature radius $ρ$ , equation (15) gives the relationship between $α_{c}$ , $α_{t}$ , and $ψ$ . Depending on the location of the plastic regions, four cases must be considered when estimating the bending capacities. The bending moment of the pipe section M is divided into six parts during calculation: $M_{1}$ , the bending moment of the compressive plastic region with respect to the demarcation line; $M_{2}$ , the bending moment of the compressive plastic region from demarcation line to pipe centerline; $M_{3}$ , the compressive elastic region to the pipe centerline; $M_{4}$ , the tensile elastic region to the pipe centerline; $M_{5}$ , the tensile plastic region to the demarcation line; and $M_{6}$ , the tensile plastic region from the demarcation line to the pipe centerline. It is noteworthy that $M_{1}$ and $M_{5}$ are affected by ovalization.

Case 1

\begin{matrix} ρ \frac{σ_{c}}{ER} \leq (\cos ψ - 1) \\ ρ \frac{σ_{c}}{ER} \geq (\cos ψ + 1) \end{matrix}

This case indicates that the entire pipe section is in an elastic stage. Figure 7 shows the stress distribution of the pipe cross-section, where ${σ'}_{c}$ and ${σ'}_{t}$ are the compressive and tensile stresses on the outer fiber of the pipe section. Figure 8 displays the pipe deformation along the longitudinal direction.

Figure 7.

Stress distributions for Case 1.

Figure 8.

Pipe deformations for Case 1.

From Figures 7 and 8, the following formulas can be obtained

{σ'}_{i} = E {ε'}_{i} = E \frac{2 Δ l_{i}}{dl} (i = c, t)

(16)

\frac{θ}{2} = \frac{dl}{2 ρ} = - \frac{Δ l_{c}}{R (1 - \cos ψ)} = \frac{Δ l_{t}}{R (1 + \cos ψ)}

(17)

Therefore, the longitudinal compressive and tensile stresses on the outer fiber of the pipe section, ${σ'}_{c}$ and ${σ'}_{t}$ , can be expressed as

\begin{matrix} {σ'}_{c} = - \frac{ER (1 - \cos ψ)}{ρ} \\ {σ'}_{t} = \frac{ER (1 + \cos ψ)}{ρ} \end{matrix}}

(18)

For Case 1, the axial force can be expressed as

\begin{matrix} F = 2 [\int_{0}^{ψ} \frac{{σ'}_{c} (\cos θ - \cos ψ)}{(1 - \cos ψ)} Rtd θ + \int_{ψ}^{π} \frac{{σ'}_{c} (\cos ψ - \cos θ)}{(\cos ψ + 1)} Rtd θ] \\ = 2 [\frac{{σ'}_{c} Rt}{(1 - \cos ψ)} (\sin ψ - ψ \cos ψ) \\ + \frac{{σ'}_{t} Rt}{(\cos ψ + 1)} (π \cos ψ - ψ \cos ψ + \sin ψ)] \end{matrix}

(19)

A dichotomy method (David and Trefethen, 1997) is employed to obtain highly accurate solutions of the neutral axis $ψ$ . For a given $ψ$ , if the deviation of the calculated axial force F from equation (19) and the actual value $F'$ is greater than a predefined small value, the value range of $ψ$ is halved until the deviation is less than the small value. It is worth mentioning that the initial range of $ψ$ is (0, π). The dichotomy method is also employed to determine $ψ$ for the following three cases.

For Case 1, the influence of ovalization on the bending moment is ignored in the elastic region of the pipe section. The bending moment M of the pipeline can be calculated from the moment equilibrium equation with respect to the pipe centerline

\begin{matrix} M = M_{3} + M_{4} \\ = - 2 [\int_{0}^{ψ} \frac{{σ'}_{c} (\cos θ - \cos ψ)}{(1 - \cos ψ)} RtR \cos θ d θ \\ + \int_{ψ}^{π} \frac{{σ'}_{t} (\cos ψ - \cos θ)}{(\cos ψ + 1)} RtR \cos θ d θ] \\ = - 2 [\frac{{σ'}_{c} R^{2} t}{(1 - \cos ψ)} (\frac{1}{2} ψ + \frac{1}{4} \sin 2 ψ - \sin ψ \cos ψ) \\ + \frac{{σ'}_{t} R^{2} t}{(\cos ψ + 1)} (- \cos ψ \sin ψ - \frac{1}{2} π + \frac{1}{2} ψ + \frac{1}{4} \sin 2 ψ)] \end{matrix}

(20)

Case 2

\begin{matrix} \cos ψ - 1 < ρ \frac{σ_{c}}{ER} < 0 \\ ρ \frac{σ_{t}}{ER} \geq \cos ψ + 1 \end{matrix}

This case signifies only a portion of the compressive region reaching yield. Figure 9 shows the stress distribution of the pipe cross-section of Case 2. ${σ'}_{t}$ and $σ_{c}$ , the tensile stress and the limit compressive stress on the outer fiber of pipe section, respectively, are determined from equations (18) and (3).

Figure 9.

Stress distribution for Case 2.

The axial force can be expressed as equation (21) for a pipe with the stress distribution as shown in Figure 9, in which $α_{c}$ is expressed by equation (15). The location of the neutral axis, that is, the angle of $ψ$ , could be obtained similar to Case 1, and the plastic compressive region, that is, the angle of $α_{c}$ , can then be determined

\begin{matrix} F = 2 [\int_{0}^{α_{c}} σ_{c} Rtd θ + \int_{α_{c}}^{ψ} \frac{σ_{c} (\cos θ - \cos ψ)}{(\cos α_{c} - \cos ψ)} Rtd θ \\ + \int_{ψ}^{π} \frac{{σ'}_{t} (\cos ψ - \cos θ)}{(\cos ψ + 1)} Rtd θ] \\ = 2 [σ_{c} Rt α_{c} + \frac{σ_{c} Rt}{(\cos α_{c} - \cos ψ)} \\ (\sin ψ - ψ \cos ψ - \sin α_{c} + α_{c} \cos ψ) \\ + \frac{{σ'}_{t} Rt}{(\cos ψ + 1)} (π \cos ψ - ψ \cos ψ + \sin ψ)] \end{matrix}

(21)

The influence of the ovalization of the compressive plastic region on the bending moment is determined from equation (12). For a given curvature radius $ρ$ , the bending moment M acting on the pipe can be obtained through the integral of the stress over the cross-section

\begin{matrix} M = M_{1} + M_{2} + M_{3} + M_{4} \\ = - 2 [Q_{c} \int_{0}^{α_{c}} σ_{c} R t R (\cos θ - \cos α_{c}) d θ + σ_{c} R^{2} t α_{c} \cos α_{c} + \int_{α_{c}}^{ψ} \frac{σ_{c} (c o s θ - c o s ψ)}{(c o s α_{c} - c o s ψ)} R t R \cos θ d θ + \int_{ψ}^{π} \frac{{σ^{'}}_{t} (c o s ψ - c o s θ)}{(c o s ψ + 1)} R t R \cos θ d θ] \\ = - 2 {Q_{c} σ_{c} R^{2} t (\sin α_{c} - α_{c} \cos α_{c}) + σ_{c} R^{2} t α_{c} \cos α_{c} + \frac{σ_{c} R^{2} t}{(c o s α_{c} - c o s ψ)} [\frac{1}{2} (ψ - α_{c}) + \frac{1}{4} (s i n 2 ψ - s i n 2 α_{c}) - c o s ψ (s i n ψ - s i n α_{c})] + \frac{{σ^{'}}_{t} R^{2} t}{(c o s ψ + 1)} [- c o s ψ s i n ψ - \frac{1}{2} (π - ψ) + \frac{1}{4} s i n 2 ψ]} \end{matrix}

(22)

Case 3

\begin{matrix} ρ \frac{σ_{c}}{ER} \leq \cos ψ - 1 \\ 0 < ρ \frac{σ_{t}}{ER} < \cos ψ + 1 \end{matrix}

This case represents only a portion of the tensile region reaching yield. Figure 10 shows the stress distribution of the pipe cross-section. ${σ'}_{c}$ and $σ_{t}$ , the compressive stress and the limit tensile stress on the outer fiber of pipe section, respectively, can be determined from equations (18) and (3).

Figure 10.

Stress distribution for Case 3.

The axial force for a pipe with the stress distribution shown in Figure 10 can be approximately expressed as equation (23), where the angle of $ψ$ and the plastic tensile region, that is, the angle of $α_{t}$ , can then be obtained

\begin{matrix} F = 2 [\int_{0}^{ψ} \frac{{σ^{'}}_{c} (c o s θ - c o s ψ)}{(1 - c o s ψ)} R t d θ + \int_{ψ}^{π - α_{t}} \frac{σ_{t} (c o s ψ - c o s θ)}{(c o s ψ + c o s α_{t})} R t d θ + \int_{π - α_{t}}^{π} σ_{t} R t d θ] \\ = 2 [\frac{{σ^{'}}_{c} R t}{1 - c o s ψ} (s i n ψ - ψ \cos ψ) + \frac{σ_{t} R t}{(c o s ψ + c o s α_{t})} (π \cos ψ - α_{t} \cos ψ - \sin α_{t} - ψ \cos ψ + s i n ψ) + σ_{t} R t α_{t}] \end{matrix}

(23)

For the stress distribution shown in Figure 10, the influence of ovalization on the bending moment is determined from equation (12). For a given curvature radius $ρ$ , the bending moment M of the pipeline can be calculated as follows

\begin{matrix} M = M_{3} + M_{4} + M_{5} + M_{6} \\ = - 2 [\int_{0}^{ψ} \frac{{σ^{'}}_{c} (c o s θ - c o s ψ)}{1 - c o s ψ} R t R \cos θ d θ + \int_{ψ}^{π - α_{t}} \frac{σ_{t} (c o s ψ - c o s θ)}{(c o s ψ + c o s α_{t})} R t R \cos θ d θ + Q_{t} \int_{π - α_{t}}^{π} σ_{t} R t R (\cos θ + c o s α_{t}) d θ + σ_{t} R^{2} t α_{t} \cos (π - α_{t})] \\ = - 2 {\frac{{σ^{'}}_{c} R^{2} t}{1 - c o s ψ} (\frac{1}{2} ψ + \frac{1}{4} s i n 2 ψ - s i n ψ c o s ψ) + \frac{σ_{t} R^{2} t}{c o s ψ + c o s α_{t}} [c o s ψ (s i n α_{t} - s i n ψ) - \frac{1}{2} (π - α_{t} - ψ) + \frac{1}{4} (s i n 2 α_{t} + s i n 2 ψ)] + Q_{t} σ_{t} R^{2} t [- s i n α_{t} + α_{t} \cos α_{t}] - σ_{t} R^{2} t α_{t} \cos α_{t}} \end{matrix}

(24)

Case 4

\begin{matrix} \cos ψ - 1 < ρ \frac{σ_{c}}{ER} < 0 \\ 0 < ρ \frac{σ_{t}}{ER} < \cos ψ + 1 \end{matrix}

Case 4 represents both portions of compressive and tensile regions reaching yield. Figure 11 shows the stress distribution of the pipe cross-section of Case 4.

Figure 11.

Stress distributions for Case 4.

Similarly, the axial force can be expressed as equation (25) for a pipeline with a stress distribution as shown in Figure 11. The angle of the neutral axis $ψ$ can be obtained from this equation, and the plastic compressive and tensile regions, that is, the angles of $α_{c}$ and $α_{t}$ , can then be derived from equation (15)

\begin{matrix} F = 2 [\int_{0}^{α_{c}} σ_{c} R t d θ + \int_{α_{c}}^{ψ} \frac{σ_{c} (c o s θ - c o s ψ)}{(c o s α_{c} - c o s ψ)} R t d θ + \int_{ψ}^{π - α_{t}} \frac{σ_{t} (c o s ψ - c o s θ)}{(c o s ψ + c o s α_{t})} R t d θ + \int_{π - α_{t}}^{π} σ_{t} R t d θ] \\ = 2 [σ_{c} R t α_{c} + \frac{σ_{c} R t}{(c o s α_{c} - c o s ψ)} (s i n ψ - ψ \cos ψ - \sin α_{c} + α_{c} \cos ψ_{c}) + \frac{σ_{t} R t}{(c o s ψ + c o s α_{t})} (π \cos ψ - α_{t} \cos ψ - \sin α_{t} - ψ \cos ψ + s i n ψ) + σ_{t} R t α_{t}] \end{matrix}

(25)

For the stress distribution displayed in Figure 11, the influence of ovalization on the bending moment is determined from equation (12). For a given curvature radius $ρ$ , the bending moment M applied to the pipe can be obtained through the integral of the stress over the cross-section

\begin{matrix} M = M_{1} + M_{2} + M_{3} + M_{4} + M_{5} + M_{6} \\ = - 2 [Q_{c} \int_{0}^{α_{c}} σ_{c} RtR (\cos θ - \cos α_{c}) d θ + σ_{c} R^{2} t α_{c} \cos α_{c} \\ + \int_{α_{c}}^{ψ} \frac{σ_{c} (\cos θ - \cos ψ)}{(\cos α_{c} - \cos ψ)} RtR \cos θ d θ \\ + \int_{ψ}^{π - α_{t}} \frac{σ_{t} (\cos ψ - \cos θ)}{(\cos ψ + \cos α_{t})} RtR \cos θ d θ \\ + Q_{t} \int_{π - α_{t}}^{π} σ_{t} RtR (\cos θ + \cos α_{t}) d θ + σ_{t} R^{2} t α_{t} \cos (π - α_{t})] \\ = - 2 {Q_{c} σ_{c} R^{2} t (\sin α_{c} - α_{c} \cos α_{c}) + σ_{c} R^{2} t α_{c} \cos α_{c} \\ + \frac{σ_{c} R^{2} t}{(\cos α_{c} - \cos ψ)} \\ [\frac{1}{2} (ψ - α_{c}) + \frac{1}{4} (\sin 2 ψ - \sin 2 α_{c}) - \cos ψ (\sin ψ - \sin α_{c})] \\ + \frac{σ_{t} R^{2} t}{(\cos ψ + \cos α_{t})} \\ [\cos ψ (\sin α_{t} - \sin ψ) - \frac{1}{2} (π - α_{t} - ψ) + \frac{1}{4} (\sin 2 α_{t} + \sin 2 ψ)] \\ + Q_{t} σ_{t} R^{2} t [- \sin α_{t} + α_{t} \cos α_{t}] - σ_{t} R^{2} t α_{t} \cos α_{t}} \end{matrix}

(26)

The computational procedure

It is difficult to manually obtain the analytical solution from the above derived theoretical expressions. A program is developed using C++ language to determine the ultimate bending capacity, the relationship between the moment and curvature, and the parameters characterizing the deformation behavior of the pipe cross-section such as the location of the neutral axis, the compressed plastic region, and the tensioned plastic region.

At the initial stage of bending, a pipe has a lower bending moment and greater curvature radius $ρ$ , so a large value such as 10,000 is assigned to $ρ$ as the initial condition. The curvature radius decreases with the increase in the bending moment, so a small value such as 1 is assigned to $ρ$ as the termination condition for a pipeline with a large bending deformation. Figure 12 shows the flow chart of the computational procedure.

Figure 12.

Flow chart of the computational procedure.

Verification

To obtain the bending moment of a pipe using the presented equations, the parameter $E'$ should be determined first. As mentioned earlier, $E'$ is used to determine the vertical displacements in the compressive and tensile plastic regions. The vertical displacement is composed of elastic and plastic displacements. Thus, $E'$ is assumed as the general modulus considering the comprehensive effects of elasticity and plasticity. Based on the physical meaning, the value of $E'$ is typically in the range of strain-hardening modulus $E_{t}$ , defined as the slope of the line between the yield point and the limit point on the stress–strain curve of the pipe, and Young’s modulus E. Thus, the results of $E' = E_{t}$ and $E' = E$ are calculated to represent the lower and the upper bound solutions, respectively. In the process of calculation, the strain-hardening modulus E_t is supposed as a constant, and independent of the stress state.

To verify the validity of the analytical solutions, the ultimate bending capacity and moment–curvature relationship of the steel pipes are investigated through the test and numerical methods below.

Test verification

Experimental data from Dorey (2001), Mohareb (1995), Ozkan and Mohareb (2009), and Wang et al. (2014) are used to validate the analytical solutions proposed in this article. Table 1 lists the geometric parameters, material properties, and loading conditions of these tests, where $σ_{u}$ is the ultimate strength of the pipe material; $E_{t}$ is the strain-hardening modulus; and P and $F'$ are the internal pressure and axial force, respectively.

Table 1.

Geometric parameters, material properties, and force conditions of test specimens.

Specimen	Grade	D (mm)	D/t	σ_y (MPa)	σ_u (MPa)	E (MPa)	E_t (MPa)	P (MPa)	F′ (kN)
CP20N-2	X70	762	89.7	524.2	656.9	201,835	1065	2.15	−1499
CP40N-2	X70	762	89.7	524.2	656.9	201,835	1065	4.30	−930
CP80N-2	X70	762	89.7	524.2	656.9	201,835	1065	8.61	209
T20P40N	X70	762	89.7	524.2	656.9	201,835	1065	4.30	1898
T20P80N	X70	762	89.7	524.2	656.9	201,835	1065	8.61	1898
CP0W-2	X70	762	87.6	523.0	636.7	203,442	881	0.00	−2069
CP20W	X70	762	87.6	523.0	636.7	203,442	881	2.15	−1499
CP40W	X70	762	87.6	523.0	636.7	203,442	881	4.30	−930
CP80W	X70	762	87.6	523.0	636.7	203,442	881	8.61	209
T20P40W	X70	762	87.6	523.0	636.7	203,442	881	4.30	1898
T20P80W	X70	762	87.6	523.0	636.7	203,442	881	8.61	1898
UGA508	X56	508	64.3	391.0	520.0	200,000	708	0.00	−1303
UGR508	X56	508	64.3	391.0	520.0	200,000	708	0.00	−652
DGA508	X56	508	64.3	391.0	520.0	200,000	708	9.91	−216
DLR508	X56	508	64.3	391.0	520.0	200,000	708	9.91	1820
UGA324	X52	324	51.0	378.0	547.0	200,000	927	0.00	−644
HGA324	X52	324	51.0	378.0	547.0	200,000	927	5.16	−409
DGA324	X52	324	51.0	378.0	547.0	200,000	927	10.33	−184
P00-N00-M100	X65	508	81.2	541.0	669.0	209,000	780	0.00	0
P00-N20-M95	X65	508	81.2	541.0	669.0	209,000	780	0.00	897
P40-N00-M89	X65	508	81.2	541.0	669.0	209,000	780	4.59	0
P40-N20-M94	X65	508	81.2	541.0	669.0	209,000	780	4.59	897
P40-N40-M89	X65	508	81.2	541.0	669.0	209,000	780	4.59	1793
P80-N40-M72	X65	508	81.2	541.0	669.0	209,000	780	9.19	1793
P40-N00-t12	X52	356	29.0	387.9	625.8	207,000	1974	10.38	0

Comparison of pipe ultimate bending capacities

Table 2 compares the ultimate bending capacities of the test results and the analytical results proposed by Mohareb (2002, 2003) and this article, where the ratio represents the predicted result/test result.

Table 2.

Comparison of predicted ultimate bending capacity with test results (unit: kN m).

Specimen	Test results	Mohareb’s results	Ratio	Predicted by analytical solutions for different $E'$
Specimen	Test results	Mohareb’s results	Ratio	E_t	Ratio	E	Ratio
CP20N-2	1921	2558	1.332	1661	0.865	2237	1.165
CP40N-2	1804	2405	1.333	1515	0.840	2086	1.156
CP80N-2	1548	1627	1.051	1032	0.667	1468	0.948
T20P40N	2352	2403	1.022	1985	0.844	2315	0.984
T20P80N	1865	1697	0.910	1383	0.742	1783	0.956
CP0W-2	1883	2622	1.392	1789	0.950	2369	1.258
CP20W	1911	2389	1.250	1696	0.887	2294	1.201
CP40W	1834	2236	1.219	1552	0.846	2146	1.170
CP80W	1627	1566	0.963	1080	0.664	1549	0.952
T20P40W	2316	2456	1.060	2024	0.874	2371	1.024
T20P80W	1921	1936	1.008	1427	0.743	1854	0.965
UGA508	685	694	1.013	512	0.747	690	1.007
UGR508	769	755	0.982	600	0.780	740	0.962
DGA508	416	311	0.748	228	0.548	309	0.743
DLR508	579	554	0.957	479	0.827	547	0.945
UGA324	211	208	0.986	170	0.805	217	1.031
HGA324	192	181	0.943	148	0.770	193	1.007
DGA324	146	119	0.815	102	0.699	132	0.904
P00-N00-M100	780	853	1.094	700	0.898	823	1.055
P00-N20-M95	779	826	1.060	609	0.781	794	1.019
P40-N00-M89	687	784	1.141	585	0.852	761	1.107
P40-N20-M94	794	818	1.030	672	0.847	789	0.994
P40-N40-M89	788	784	0.995	576	0.731	758	0.961
P80-N40-M72	748	690	0.922	579	0.774	678	0.907
P40-N00-t12	468	463	0.989	461	0.986	505	1.079
Average of the ratios			1.049		0.799		1.020
Standard deviation			0.153		0.094		0.109

It can be seen from Table 2 that the average ratio of Mohareb’s results is 1.049, which is better than the lower ( $E' = E_{t}$ ) bound solution 0.799 and inferior to the upper ( $E' = E$ ) bound solution 1.020. The standard deviation of the lower bound solution is minimum, indicating that these results have lower dispersion. Among all 25 tests, there are 14 cases that Mohareb’s predicted results are greater than the test results, indicating that Mohareb’s results are not conservative. The ratios of the lower bound solutions to test ones range from 0.548 to 0.986; thus, all lower bound solutions are conservative. As for the upper bound solutions by $E' = E$ , the predicted bending capacities agree quite well with experimental results; however, about half of predicted bending capacities are overestimated.

The ratio of diameter to thickness is a significant factor for failure modes of steel pipelines subjected to combined loadings. Figure 13 shows the relationship between the diameter–thickness ratio and the ultimate bending capacity, where $M_{predict}$ is the analytical result and $M_{test}$ is the experimental result. It can be seen that the diameter–thickness ratio has little effect on the predicted accuracy of the methods proposed by Mohareb and this article.

Figure 13.

Relationship between diameter–thickness ratio and the ultimate bending capacity.

The influence of initial internal pressure on failure modes of steel pipelines subjected to bending moment is significant. Figure 14 displays the interaction between pipe bending capacities and initial loading conditions, where $P_{y}$ is the yield internal pressure $(P_{y} = (2 σ_{y} t) / (D - t))$ and $F_{y}$ is the yield axial tensile force ( $F_{y} = σ_{y} A$ , A is the area of the pipe cross-section). For the initial conditions with low internal pressure, the predicted results of Mohareb and the upper bound method are greater than the experimental results; for the initial conditions with high internal pressure, the predicted results of the upper bound method are close to and lower than the experimental results. The predicted results of the lower bound method are lower than the experimental results for any initial loading conditions.

Figure 14.

Interaction between pipe ultimate capacities and loading conditions.

Moment–curvature relationship

Figure 15 shows the curves of global moment versus global curvature for four tests. The experimental curves are denoted by the solid black line; the red-dashed line represents the lower bound results with $E' = E_{t}$ , and the green dotted-dashed line indicates the upper bound results with $E' = E$ .

Figure 15.

Global moment–curvature curves comparisons between theoretical derivation and tests: (a) CP20N-2, (b) T20P40W, (c) P40-N40-M94, and (d) P40-N00-t12.

As seen in Figure 15(a)–(d), both the moment–curvature curves of $E' = E_{t}$ and $E' = E$ match the experimental results well during the elastic range for all four specimens. As the curvatures increase, the moment–curvature curves predicted by the lower and the upper bound solution depart from the test result. This difference is likely due either to geometric imperfections within the pipe wall that were not modeled in this study, the assumptions used in the pipe plastic region during the derivation or the ignoring of the buckling. It is also noted that, in general, the experimental curves lie between the lower and upper bound curves.

Numerical verification

Based on the finite element method, numerical analysis of steel pipes is conducted to compare with the analytical solutions. The finite element analysis (FEA) simulator ABAQUS (ABAQUS, 2011) was selected to predict pipe behavior. Shell element S4R is adopted because it is efficient and reliable for large displacement, large rotation, and finite membrane strain shell analysis. A piecewise linear isotropic hardening stress–strain constitutive model, the von Mises yield criterion, and the associative flow rule were applied to describe the material behavior. The automatic Riks method is used to calculate the post-buckling of the steel pipe under monotonically increasing moment.

Two types of steel pipes are investigated through a numerical method and compared with the lower and upper bound solutions. Table 3 lists the geometric parameters and the material properties of the two types of steel pipes and the details of the true stress–strain curve are given in Dorey (2001) and Mohareb (1995). Determinations of the length of the pipes such as L/D = 2, 3.35, 5.31, 7.28, and 10 for X56, and L/D = 2.23, 3.54, 4.86, 7.22, and 10 for X70 are carried out. The results indicate that L = 2700 mm can ignore the boundary effects for both the two types of specimen. A mesh convergence study is also conducted for both the X56 and X70 specimens. The results of the study show that the 48 elements along the pipe circumference and 90 elements along the pipe axis for X56, and 60 by 90 elements for X70 can provide reliable bending capacities.

Table 3.

Geometric parameters and material properties of the steel pipes.

Grade	σ_y (MPa)	σ_u (MPa)	E (MPa)	E_t (MPa)	D (mm)	t (mm)
API 5L X56	391	520	200,000	708	508	7.9
API 5L X70	524.2	656.9	210,000	1050	762	8.3

The finite element model should be validated first prior to studying the comparison of the numerical solutions with the analytical solutions. The test specimens UGA508 (Mohareb, 1995) and T20P80N (Dorey, 2001) are adopted to validate the finite element model. The finite element model of the specimen T20P80N is shown in Figure 16. The comparison of the global moment and curvature curve between the experimental and the FEA result is provided in Figure 17. From Figure 17, we can see that the finite element model can predict the ultimate capacity well, as well as the deformation behavior of pipeline. The errors of ultimate moment only −5.008% and 3.157%, respectively, where error = (FEA result − experiment result)/experiment result. Therefore, the finite element model can be used with confidence to validate the analytical solutions.

Figure 16.

Finite element model of the pipe.

Figure 17.

Comparison of global moment curvature curve: (a) UGA508 and (b) T20P80N.

Comparison of pipes ultimate bending capacities

Figures 18 and 19 display the comparison of the ultimate bending capacities between the FEA, the analytical results proposed by Mohareb (2002, 2003), and the lower and upper bound solutions of this article under different internal pressures and axial forces, where $P_{y}$ is the yield internal pressure and $F_{y}$ is the yield axial tensile force.

Figure 18.

Comparison of the ultimate bending capacities for X56 pipes: (a) F/F_y = 0, (b) F/F_y = −0.2, (c) F/F_y = −0.4, and (d) F/F_y = −0.6.

Figure 19.

Comparison of the ultimate bending capacities for X70 pipes: (a) F/F_y = 0, (b) F/F_y = −0.2, (c) F/F_y = −0.4, and (d) F/F_y = −0.6.

From Figures 18 and 19, it can be seen that for the initial conditions with low axial force, the ultimate bending capacities from FEA are close to and lower than the upper and Mohareb’s solutions for most of X56 and X70 pipes. With the increase in axial force, the ultimate bending capacities predicted by FEA move away from the upper bound results and gradually closer to the lower bound results. The predicted results of the lower bound solutions are lower than the numerical results for any initial pressure and axial force. The FEA results lie between the lower and the upper bound solutions for most pipes, except for the X70 pipes with a high pressure. It is worth mentioning that axial forces are relatively low for most test specimens mentioned in section “Comparison of pipe ultimate bending capacities,” so the predicted bending capacities of the upper bound solutions agree quite well with test results.

Moment–curvature relationship

Figures 20 and 21 show the comparison of the moment–curvature curves between the numerical results and the upper and the lower bound solutions with different internal pressures and axial forces for X56 and X70 pipes. From these two figures, it can be seen that the moment–curvature curves for the FEA lie between the lower and upper bound curves for most of the pipes. During the elastic range, the curves of the FEA are completely consistent with those obtained from the lower and upper bound method. After entering the plastic range, the FEA moment–curvature curves depart from that predicted by the lower and upper bound curves as described in section “Moment–curvature relationship.” In the plastic range, the ultimate moment capacities of the lower bound solutions initially arose, followed by an apparent descending curve, and the longest yield plateau is observed for the upper bound solutions, with a slowly decreased curve.

Figure 20.

Analytical and numerical moment–curvature curves for X56 pipes: (a) P/P_y = 0, F/F_y = 0; (b) P/P_y = 0.4, F/F_y = 0; (c) P/P_y = 0.8, F/F_y = 0; (d) P/P_y = 0.2, F/F_y = −0.2; (e) P/P_y = 0.6, F/F_y = −0.2; (f) P/P_y = 0, F/F_y = −0.4; (g) P/P_y = 0.4, F/F_y = −0.4; and (h) P/P_y = 0.2, F/F_y = −0.6.

Figure 21.

Analytical and numerical moment–curvature curves for X70 pipes: (a) P/P_y = 0, F/F_y = 0; (b) P/P_y = 0.4, F/F_y = 0; (c) P/P_y = 0.2, F/F_y = −0.2; (d) P/P_y = 0.4, F/F_y = −0.2; (e) P/P_y = 0.6, F/F_y = −0.2; (f) P/P_y = 0.2, F/F_y = −0.4; (g) P/P_y = 0.4, F/F_y = −0.4; and (h) P/P_y = 0.4, F/F_y = −0.6.

Conclusion

Considering ovalization of the plastic region, the analytical solutions were derived for steel pipelines subjected to combined bending moments, axial forces, and internal pressure. The lower and the upper bound bending capacities were obtained, along with the relationship between the moment and curvature. In comparing the analytical results with the experimental and numerical results, the following conclusions can be drawn:

For the initial conditions with low axial forces, the predicted ultimate bending capacities of the upper bound solutions were close to and greater than the test and numerical results; with increased axial force, the upper bound results move away from the test and the numerical results. In general, the predicted ultimate bending capacities of the upper bound solutions agree quite well with experimental and numerical results; however, some of the cases are overestimated.

Under the initial conditions with low axial forces, there were some errors between the lower bound solutions and the results of experiments and FEA, and the errors decreased gradually with the increase in axial force. For the lower bound solutions, the bending capacities are underestimated for almost all of the test and numerical cases.

For the global moment versus global curvature curves of steel pipelines subjected to combined bending moments, axial forces, and internal pressure, both the lower and upper bound curves nearly matched those of the experimental and numerical results in the elastic range. As the curvatures increase, the moment–curvature curves predicted by the lower and upper bound solutions depart from the test and FEA results.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Chinese Ministry of Science and Technology (Grant No. 2011CB013702) and supported by the Program for New Century Excellent Talents in University (Grant No. NCET-11-0051).

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