Casing/Tubing design manual october 2005 Chevron

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θz

θr

∂

r dr

rθ

∂

rθ

∂

r dr

∂

r dr

zθ

∂

zθ

∂

z dz

∂

z dz

∂

z dz

Figure D-4. Equilibrium of a Differential Element Centered at (r, θ, z)

−−

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

+− − −

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

+− + +

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

+− + +

⎛

⎝

⎜

⎞

⎠

⎟

σθσ

∂σ

∂

σσ

∂σ

∂θ

σσ

∂σ

∂θ

σσ

∂σ

∂

θχ θ

θθ

zr zr

ddz

dr r

ddz

drdz

dz rdrd rdrd dz

sin

cos

(D-30)

and in the

-direction, moment equilibrium about the center of the element is

expressed as:

()

[]

()

σθσ

∂σ

∂

∂σ

∂θ

θθ

ddz

dr r

ddz

drdz r

ddrdzr

−

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

−−+

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

22 22

(D-31)

Rearranging equations (D-30) and (D-31), factoring the common factor (drd

dz),

using the small angle approximations (

sin , cos ,

≅

<1for 1

), and ignoring

higher order terms, the final forms of these equations can be written:

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October 2005

∂σ

∂

∂θ

∂σ

∂

θθ

rr z r

+++

−

0 (D-32)

θθ

(D-33)

where the required result of moment equilibrium is symmetry of the shear stress

components. Force equilibrium requirements in the

and directions yield

respectively,

∂

∂θ

∂

θθθθ

rzr

rr z r

++++

(D-34)

∂σ

∂

∂σ

∂θ

∂σ

∂

zrz

rr z r

++++

(D-35)

and the moment equilibrium equations in the

and r

directions yield

respectively,

θθ

z z rz zr

, (D-36)

D.3.3 Special Cases

As was the case with the strain-displacement relations, it is convenient to

consider certain useful simplifications to equations (D-32), (D-34), and (D-35).

However, because of the inter-relationship between stress and strain via the

material properties, it will be necessary to postpone a discussion of these special

cases until the constitutive equations have been introduced.

D.4 Stress-Strain Relations

The concepts of both stress and strain are independent of the material under

consideration. In order to relate the response (strain) to the loading (stress), it is

necessary to introduce additional parameters characterizing the particular

material being studied. The resulting expressions are termed constitutive

relations since their exact form will depend on the constitution of the material.

A given set of constitutive relations will be applicable only over a certain range of

conditions. For example, the equations relating stress and strain (rate) for water

at atmospheric pressure are only valid below 100

°C (212°F). Above that

temperature, a phase change occurs (water

→steam) and a new set of stress-

strain rate relations applies.

A similar phenomenon occurs with tubular steel. Below a certain stress state,

steel may be classified as an elastic material with corresponding elastic stress-

strain relations. Above a certain stress state, the physical properties of steel

change and the steel are characterized as an elastic-plastic material. Above this

critical stress state, referred to as yield, it is necessary to formulate a new set of

equations to describe the response of the steel to external loads.

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October 2005

D.4.1 One-Dimensional Loading

Figure D-5 depicts the results of a uniaxial tension test on a sample of casing

steel. Dividing the applied force, F, by the cross-sectional area of the sample, A,

the axial stress can be determined. Similarly, the axial strain

∆

LL/ can also

be directly measured. The right-hand portion of Figure D-5 is a plot of stress as a

function of strain. Notice from the figure that as the sample is loaded along path

OA, the relation between stress and strain is linear,

mb (D-37)

where m is the slope of the line and

is the value of

0. If the sample

were loaded to any point between O and A and then unloaded, the path OA

would, within practical limits, be retraced to the origin. Such behavior is termed

elastic. Evaluating the constants in equation (D-37),

and

so that:

E (D-38)

The constant E is called Young’s modulus.

API

0.005

ε = ∆L / L

σ = F / A

∆L

Figure D-5. Typical Uniaxial Stress-Strain Curve

Before proceeding to higher loading states, it is worth mentioning that axial strain

is not the only dimensional change that will occur in the bar sample. Extension in

the axial dimension will be accompanied by contraction in the transverse

direction. This contraction in the transverse direction will also vary linearly with

the axial loading in the sense that:

µεε

transverse

−=−=

(D-39)

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October 2005

where

is Poisson’s ratio. Notice that

can also be measured directly during

the deformation experiment. The quantities

and

are independent and

completely determine isotropic linearly elastic stress-strain behavior for

isothermal loading.

Consider now an experiment where the sample is loaded along path OABC. The

following facts are pertinent:

• Above the stress state corresponding to point A, the relation between

stress and strain is no longer linear.

• Upon unloading from any point C above point A, the metal will not

retrace the original loading path, but rather will unload along path CDE

which is parallel to path OA. Unloading along path CDE occurs

elastically.

• After it is completely unloaded, the material will now possess a

permanent, inelastic strain (

= OE).

• If the bar at state E is now reloaded, the stress-strain behavior will be

elastic and follow path EDC. Continued loading beyond the stress level

corresponding to point C will follow path CFG.

• Point G represents the ultimate strength of the material. States beyond

that corresponding to point G are undefined.

Behavior of the sample beyond point A is termed inelastic. Following the first

occasion for which the stress corresponding to point A is exceeded, it will be

necessary to compile a new set of constitutive (stress-strain) relations for the

material. Because of its importance as a limit point of elastic behavior

, the stress

corresponding to point A is given a special name, the initial yield stress, denoted

. As will be emphasized later, the yield stress will change with inelastic

deformation. For example, recalling the comments of the previous paragraph,

after the material is loaded to point C on the stress-strain curve, point C becomes

the new yield stress. Loading/unloading along the line CDE is characterized by

elastic behavior. If subsequent loading goes beyond point C, the yield stress will

again change.

The description of material behavior beyond yield is much more complicated in

that now, stress-strain behavior depends not only on the current state, but also

on the previous history of loading. As an example, consider points B and D on

the uniaxial stress-strain curve. Both points correspond to the same value of

stress. However, it is obvious from the figure that a small increase in stress, ,

will produce different values of strain increment depending on whether the

current state actually corresponds to point B or to point D.

For loadings beyond the initial yield stress, two additional moduli are commonly

used to describe material behavior. The first modulus, called the tangent

modulus,

, is the instantaneous slope of the stress-strain curve,

For simplicity, the yield point, signifying the onset of inelastic behavior, and the

proportional limit, signifying the upper limit of applicability of equation (D-38) have

been assumed to coincide. However, for real materials, this is not always true.

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October 2005

≡

(D-40)

The second modulus, called the secant modulus,

is the slope of a line from the

origin to the point on the curve representing the current stress/strain state:

≡

(D-41)

Careful examination of Figure D-5 reveals that for all states of stress,

allfor

EEE ≥≥ (D-42)

and

EE Efor

≤

(D-43)

Clearly,

and are functions of both the current stress state and the entire

history of loading leading to the current stress state.

D.4.2 American Petroleum Institute (API)

Parameters

In the preceding section, the uniaxial stress-strain curve was discussed in

general terms common to the study of mechanics of materials. By convention,

special groups will introduce alternate definitions that are felt to better serve the

needs of a particular engineering community.

D.4.2.1 API Yield Stress

The yield stress,

, is a difficult, if not impossible, number to obtain

experimentally. Close examination of the character of the curve in Figure D-5 will

indicate that, experimentally, by the time one detects inelastic behavior, the yield

stress has already been exceeded. In order to circumvent this experimental

problem, it is common practice to approximate the yield stress by means of a

suitable alternate definition.

For example, probably the most popular alternate definition of yield stress is the

so-called 0.2% offset stress which is determined by the intersection of the

experimental stress-strain curve with a line parallel to the elastic portion of the

curve (i.e., slope =

) and passing through the point  = 0,  = 0.002 (such a

line would be similar to line CDE in Figure D-5). On the other hand, the API

defines the yield stress as the stress corresponding to

= 0.005

. As indicated in

Figure D-5,

API

, the API yield stress, does not in general correspond to

The value of strain corresponding to yield may vary with grade. For example, the yield

stress for grade P-110 is defined as the stress corresponding to

= 0.006, and the yield

stress for grade Q-125 is defined as the stress corresponding to

= 0.0065.

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October 2005

D.4.2.2 Ultimate Stress and Elongation

The ultimate stress,

, is the maximum stress recorded at the time the

specimen breaks. Corresponding to this stress will be a given value of strain. The

value of this ultimate strain, or elongation, is a measure of the ductility of the

steel.

The two stress-strain pairs (

API

, 0.005) and (

, Elongation) are the only

points taken from the uniaxial stress-strain curve for use in API formulas. Table

D-1 presents a list of the minimum acceptable values of

API

and

for tubular

steel grades currently recognized by the API.

Table D-1. API Minimum Tensile Requirements

Minimum Yield Minimum Ultimate

Grade Metric

(MPa)

English

(ksi)

Metric

(MPa)

English

(ksi)

H-40 276 40 414 60

J-55 379 55 517 75

K-55 379 55 655 95

N-80 552 80 689 100

L-80 552 80 655 95

C-90 621 90 689 100

T-95 655 95 724 105

P-110 758 110 862 125

Q-125 862 125 931 135

D.4.3 Needleman’s Model

The stress-strain curve illustrated in Figure D-5 is typical of oil field steels and it

is sometimes useful to have an equation to describe such a curve. The

advantages of such an equation are:

• Elimination of the need to interpolate between experimental data points,

because the equation will explicitly define a stress corresponding to any

given strain.

• The ability to perform parameter studies to determine the effects of

various stress-strain parameters on tubular performance.

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October 2005

0.5

1.5

2.5

00.511.522.53

ε/ε

n = 2

n = 1

n = 5

n = 20

n =

Figure D-6. Behavior of Needleman's Model for Various Values of Work-Hardening

Parameter

Of course, below the yield stress, the stress-strain curve is linear and described

completely by equation (D-38). The problem arises in attempting to describe the

nonlinear or inelastic portion of the curve.

A variety of equations or models have been proposed in the literature for

describing, or fitting, the stress-strain curve generated by a uniaxial test. It is

beyond the scope of this discussion to detail these various proposals. Rather,

here only the model proposed by Needleman [1975] will be discussed.

Needleman’s model may be expressed in equation form as:

εε

for

nnfor

≤

⎛

⎝

⎜

⎞

⎠

⎟

+−

⎡

⎣

⎢

⎤

⎦

⎥

≥

⎧

⎨

⎪

⎩

⎪

, (D-44)

where

is the strain corresponding to yield and is the work-

hardening parameter which determines the shape of the stress-strain curve

beyond yield. Needleman’s model is continuous and has a continuous derivative

for all finite values of . n

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October 2005

Figure D-6 is a plot of

versus

for various values of . Notice that

. If we associate work hardening with the amount of work (area under

the curve) necessary to increase the strain after the yield stress has been

exceeded, then an increase in the value of

corresponds to a decrease in the

work hardening character of the stress-strain curve beyond yield

1<<∞n

Table D-2 lists values of

and n that correspond to the minimum yield and

ultimate properties of API tubular steels for casing. Figure D-7 presents this

same information graphically. Notice that for each grade the proportional limit or

yield stress, denoted in this text as

, and the API yield stress,

API

, differ by

approximately 70 to 105 MPa (10,000-15,000 psi). This discrepancy is due to the

arbitrary definition of

API

, and is an important fact to keep in mind when the

structural limits of certain aspects of tubular behavior are defined in terms of the

onset of yield.

Table D-2. Needleman Parameters for API Grades

Yield Stress

Grade Metric

(MPa)

English

(psi)

H-40 187 27,100 9.8

J-55 287 41,600 12.6

K-55 235 34,100 6.6

L-80 477 69,200 23.0

N-80 461 66,900 17.5

C-90 567 82,300 39.1

T-95 603 87,400 41.0

P-110 678 98,400 28.8

Q-125 800 116,100 47.5

To be entirely correct, work hardening refers to the amount of permanent,

unrecoverable work associated with an increase in applied stress. However, even with

this definition,

is inversely proportional to the work hardening character of the

material being modeled.

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October 2005

100

120

140

160

0 0.005 0.01 0.015 0.02 0.025

(ksi)

H-40

K-55

T-95

P-110

Q-125

C-90

L-80

J-55

N-80

Figure D-7. Stress-Strain Curves for API Steels using Needleman's Model

Manipulation of equation (D-44) yields the following expressions for parameters

previously introduced in terms of Needleman’s model:

EE E for

=== ≤

εε

(D-45)

EE for

⎛

⎝

⎜

⎞

⎠

⎟

≥

−

εε

(D-46)

for

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

−

⎡

⎣

⎢

⎤

⎦

⎥

≥

εε

(D-47)

D.4.4 Ideal Plastic Behavior (Perfect Plasticity)

The stress-strain curve in Figure D-5 satisfactorily models the most general type

of uniaxial behavior that will be considered in this text. It happens that a number

of tubular steels exhibit stress-strain behavior that, within the limits of strain

D-20 Casing/Tubing Design Manual

October 2005

necessary to solve the majority of practical problems, possess a zero slope

beyond yield. Such behavior is termed ideally or perfectly plastic.

Recall Figure D-6 and notice that in the limit as the value of the work-hardening

parameter n becomes infinite, we may model ideal plastic behavior. In particular,

from equation (D-44),

lim lim

limexp ln

→∞ →∞

→∞

⎛

⎝

⎜

⎞

⎠

⎟

+−

⎡

⎣

⎢

⎤

⎦

⎥

⎛

⎝

⎜

⎞

⎠

⎟

+−

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

(D-48)

The limit of the expression in braces can be evaluated using l’Hospital’s rule,

lim ln

→∞

⎛

⎝

⎜

⎞

⎠

⎟

+−

⎡

⎣

⎢

⎤

⎦

⎥

(D-49)

which, upon substitution into equation (D-48), leads to:

lim

→∞

1.(D-50)

and, from equations (D-40) and (D-41),

Eforn

≥→

∞

(D-51)

Eforn

=≥

εε

, →∞

(D-52)

Notice that for

the inverse relation

→∞

(

)

εεσ

is no longer single valued.

That is, given

it is not possible to determine a unique value of strain. At

first glance, it would appear that the uncertainties associated with a multi-valued

strain-stress relation would render the limiting case

n →

∞

a circumstance to be

avoided. However in many problems of practical interest, useful solutions can be

obtained from ideal plastic behavior [Prager and Hodge, 1951]. In such

instances, inconveniences associated with the uncertainty of the stress-strain

relation are outweighed by the simplicity of the description of the plastic stress

state. Further, as might be anticipated from Figure D-6, ideal plastic behavior is

also useful for providing a lower bound or conservative estimate of the load

response of a particular steel grade inasmuch as all beneficial effects of work

hardening are ignored.

D.5 Multi-Dimensional Loading

During actual field application, any point in a tubular string will be subjected to

multiple stresses rather than the simple uniaxial stress state discussed above. As

was the case in the introductory discussion, the response of the casing steel can

be divided into an elastic response and an inelastic response, the demarcation

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October 2005