**Flexibility or Stress Analysis**

Flexibility or Stress Analysis is an engineering that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.

**Objective of Stress Analysis**

Pipe stress analysis provides the necessary techniques for engineers to design piping systems without overstressing and overloading the piping components & connected equipment.

The objective of stress analysis can be listed as follows:

- To limit the stresses in the piping system to the limiting value.
- To limit the deflection in the piping system to the limiting value.
- To limit the loads on nozzles of connected equipment.
- To limit the loads on supports.
- To check for leakage at flange joints.
- Unintentional disengagement of pipes from supports.
- Excessive displacement.

**Pipe Stress Analysis -Terminology**

**Stress: –** The resistance developed in the material per unit area against the applied force is the stress in the material. It can be simply specified as force per unit area of the material.

Stress (σ) =Force / Cross sectional area

**Strain: –** A component subjected to load undergoes deformation. The deformation is quantified by strain defined as change in length per unit length of the material.

Mathematically:ε = Δ L / L (Longitudinalstrain)

i.e. ε = 1 Inch / 100 Inch = 0.001

When a bar is in longitudinal strain it also tends to change its dimensions in the direction perpendicular to the applied load. The change in dimension in perpendicular direction to the original dimension in that direction is called Lateral strain. For example in case of bar, when axial load is applied,

Along with increase in length, its diameter is reduced.

Thus in this case, ε = ΔD / D (**Lateral strain**)

The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.

ν = Lateral strain / Longitudinal strain (Within Elastic limit)

**Modulus of elasticity ( E ):** – Up to certain limit of loading known as proportional limit the strain developed in the material is in direct proportion of the stress. This law is called Hooke’s law and the constant of proportionality, E, is called as modulus of elasticity (Young’s modulus), which is a definite property of the material.

Mathematically: E=σ / ε

**Stress-strain behaviour of material:**

A typical stress-strain behaviour of a ductile material under tensile loading is given in the figure, taking A 53 Gr B – a commonly used CS material.

**Yield Strength:** The stress at yield point is known as Yield strength of the material which is the maximum stress the material can withstand without undergoing permanent deformation. Though the material does not break immediately beyond this stress the functionality of the member gets affected and hence the stress on the member is not allowed to exceed the Yield strength under normal operating condition.

**Ultimate Tensile strength**: The maximum stress in the stress-strain curve of the material is the Ultimate Tensile Strength of the material. This is the point beyond which the material becomes unstable under load and breaks after uncontrolled yielding. This point signifies the beginning of the reduction in cross-section area (Necking).

**Allowable stress :** Due to uncertainties in the loading and behaviour of the materials, especially in complex configurations, accurate assessment of the stresses will involve exhaustive analysis and testing which will be very cumbersome, time consuming & expensive. Widely followed method of design accounts for the uncertainties in the loading and the material behaviour by introducing a factor called Factor of safety.

The yield strength / Ultimate tensile strength of the material, as obtained from standard property charts is divided by the factor of safety to reach at allowable stress of the material.

Mathematically,

Allowable stress, σ =Yield Strength (orUTS) / Factor of safety

**Loads on Piping based on Stress Analysis**

Loadings on piping systems can be broadly classified based on their nature as primary and secondary. Primary loading occurs from sustained loads like dead weight and not self-limiting in nature i.e.; deformation will not bring relaxation on stress. Secondary loads, like thermal expansion loads, are self-limiting in nature; deformation will result in redistribution of stress. Secondary loads do not cause failure of the component in a single application. They are important from fatigue consideration.

Loads in piping systems can also be classified as static and dynamic loads, based on their effect.

Static load on piping system include:

- Weight (Dead loads and Live loads)
- Thermal expansion and contraction effect (Secondary in nature)
- Effect of support, anchor and thermal movements
- Internal and external pressure loading

Live loads underweight include snow, ice loads etc. and dead loads consists of weight of pipe material, fluid, valves and other superimposed permanent loads.

Dynamic load on piping system include:

- Impact force
- Wind load
- Seismic load
- Vibration
- Relief valve discharge load

**Stresses in Piping**

There are main four types of stresses which affect a piping element from analysis point of view.

**Circumferential \ Hoop Stress:** Under the internal pressure loading this stress is developed tangential to the cross-section (This stress acts in a direction parallel to the pipe circumference).

Sc = PD / 2t

**Longitudinal Stress:** This stress is developed normal to the cross-section of pipe. Longitudinal stress because of pressure

S_{l}= PD / 4t

**Radial Stress:** Stresses in the radial direction across the wall thickness of pipe are called radial stresses. Its value is equal to internal pressure at the inside of the pipe wall and a stress equal to atmospheric pressure at the pipe’s external surface.

Note that the radial stress is zero at the outer radius of the pipe, where the bending stresses are maximized. For this reason, this stress component has traditionally been ignored during the stress calculation.

**Axial Stress:** Under external loads in the axial direction axial loads are developed in the pipe.

Sa = F / A

**Bending Stress:** Bending stresses are developed in pipe under the loads acting in a plane normal to the axis of pipe. These may be caused due to temperature, weight of pipe, weight of contents, snow and ice, wind or earthquake.

Sb = M / Z

Where, M = Bending moment

Z = Section modulus of pipe

**Shear Stress:** This stress is sum of two components i.e. torsional stress and direct shear stress. Direct shear stress is usually negligible ( These shear stresses are distributed such that they are maximum at the neutral axis of the pipe and zero at the maximum distance from the neutral axis , since these stresses are usually small, shear stresses due to forces are traditionally neglected during pipe stress analysis ) . Torsional stresses are developed when the pipe is subjected to twisting moment

St = T / 2J

Where, T = Torque lb-in

J = Polar moment of inertia

The distribution of the stresses in the pipe is as described below:

**Piping Flexibility**

Piping systems in plant during service need to accommodate their own thermal expansion as well as the thermal displacement of the connected equipment and support structure so as to avoid,

- Failure of piping system due to overstress
- Leakage of flange joints
- Distortion of connected equipment

It can be seen that

- A very high pipe reaction load is imposed on the equipment nozzle.
- The load on the pipe is beyond the critical buckling load it can sustain which implies that the pipe is unstable.
- The stress on the piping is much higher than the allowable

The piping system is not flexible, that is, the piping system is stiff.

Mathematically

Flexibility = 1/ Stiffness

**Types of flexibility:** There are two type of flexibility

- Axial flexibility
- Bending flexibility

**Axial Flexibility **

Consider member of cross sectional area A, length L and modulus of elasticity E subjected to an axial load P.

Deflection, δ = PL / AE

Stiffness, k = P/ δ = AE / L

Axial Flexibility = 1 / k = L / AE

**Bending Flexibility**

Consider the same member is subjected to bending load P,

Deflection δb = PL^{3} / CEI

C = Constant depending on the boundary condition Which turn depends upon the type of supports, loading and location of applied bending load.

Stiffness = P / δb = CEI / L^{3}

Bending flexibility = 1 / k = L^{3}/ CEI

It can be seen that the bending flexibility is proportional to the third power of the length as compared to the length itself in the case of axial flexibility. This means piping become more flexible by providing length which takes the deflection in bending as compared to length in the axial direction.

**Method of providing flexibility:**

Axial flexibility can be provided using bellows.

Bending flexibility can be provided using loops, offsets, bends etc.

**Flexibility Factor**

Early attempts to analyse the stresses in piping systems containing elbows disclosed that the established structural engineering theory and the results of experiment did not agree at all well; practical piping systems were found to be far more flexible than the theory predicted.

**Selection of Critical Lines for Flexibility or Stress Analysis.**

Generally this is based on the project specification, however following point are considered for selection of critical lines.

- Lines of NPS 2” and larger with a design temp. of 260 Deg.C
- Lines of NPS 6” and larger with a design temp. of 120 Deg.C
- All lines with a design temp. below -29 Deg C
- Non-metallic lines NPS 3” and above
- All lines NPS 16 “and above.
- All lines with very long horizontal or vertical straight runs (H > 500’, Vertical >50’)
- Lines subjected to two-phase flow, steam out, regeneration or cyclic condition.
- All piping connected to sensitive equipment (Pumps, Compressor, and Turbine)
- All fired heater, steam generator, shell & tube exchanger with bellow & Air cooler piping.
- Piping connected to vessels, tanks or equipment subjected to differential settlement or any significant external displacement.
- All systems containing expansion joint, coupling, pressure relief valve, jacketed lines.
- Large diameter tank connection.
- Lines subjected to Vacuum conditions.
- Underground piping above 65 Deg.C
- Category “ M “ piping, defined as per B31.3
- All lines with a design temp. over 400 Deg C
- All lines subjected to vibration.

**Flexibility Analysis**

Formal Stress Analysis is Not Required:

No formal analysis of adequate flexibility is required for a piping system which:

Duplicates or replaces without significant change system operating with a successful service record.

Can easily be judged adequate by comparison with previously analyzed systems.

Is of uniform size, has no more than two points of fixation, no intermediate restraints, and falls within the limitations of empirical Eq.

Dy/(L-U) 2 ≤ K1

Where

D = Outside diameter of pipe, mm (in.)

Y = Resultant of total displacement strains, mm (in.), to be absorbed by the piping system

L = Developed length of piping between anchors, m (ft).

U = Anchor distance, straight line between anchors, m (ft)

K1 = 208,000 SA /Ea, (mm/m)²

= 30 SA /Ea, (in./ft)²

SA = Allowable displacement stress range. MPa (ksi)

Ea = Reference modulus of elasticity at 21°C (70°F), Mpa (ksi)