Hooke’s law is a principle of engineering mechanics and physics related to the properties of a material. There are basically two statements for Hooke’s law; discovered by the English scientist Robert Hooke. The first statement relates to the elongation of spring subject to the application of force. And the second statement is known as the law of elasticity and relates the relation of stress and strain of materials. In this article, we will explore more details about Hooke’s law.
Hooke’s Law Statement for Springs
For Springs, Hooke’s law states that the force (F) required to extend or compress a spring by some distance (x) varies proportionately with respect to that distance. This means more force will be required to elongate a spring more and vice versa.
When elastic materials like springs are stretched using a force (F), deformation of atoms and molecules occurs until stress is applied and they get back to their initial state when the stress is removed.
Hooke’s Law Equation for Spring
Mathematically, Hooke’s law for Springs is expressed as:
F= -k.x
Where, In the SI unit
F is the force; N
x is the extension or compression of the spring; mm
k is the constant of proportionality also known as the spring constant in N/m
The negative sign in Hooke’s law is explaining the force as a restoring force that is trying to return the spring to its equilibrium position
Let us consider a spring with the load application as shown in the figure.
Fig. 1: Hooke’s Law for Springs
The figure above (Fig. 1) shows the stable condition of the spring when no load is applied to the spring at x=0; the condition of the spring when it gets elongated to the amount x1 under the load of m1; and the condition of the spring when it gets elongated to x2 under the influence of load m2.
Depending on the material, different springs will have different spring constants.
Hooke’s Law Statement relating Stress & Strain
With respect to the stress and strain of a material, Hooke’s law states that the stress of a material is proportional to its strain within the elastic limit of the material.
The strain will remain in the body as long as the stress acts and when the stress is removed the body will regain its shape. This property of materials is known as elasticity. So basically, Hooke’s law provides the basis for elasticity and so it is known as the elasticity principle or law of elasticity.
Hooke’s law Formula
Mathematically, within the elastic region of a material, Hooke’s law formula is expressed as
σ = Eε
Where, in SI units
σ is the stress; Pa
E is the modulus of elasticity or Young’s modulus, Pa (Normally GPa)
ε is the strain, dimensionless
Hooke’s Law Graph
The below graph (Fig. 2) shows the stress-strain curve for a low-carbon steel material to explain Hooke’s law
Fig. 2: Stress-Strain Curve explaining Hooke’s law
The material tends to have elastic behavior up to the yield strength point (Point B), after that the material loses its elasticity and exhibits plasticity.
From the origin until the proportional limit (Point A) near the yield point (Point B), the straight line implies that the material is following Hooke’s law. But, beyond the elastic limit that is between the proportional limit and yield strength, the material starts losing its elastic nature and exhibits plasticity. The area that is under the curve from the origin to the proportional limit falls under the elastic range. The area that is under the curve from a proportional limit to the rupture point falls under the plastic range. In the plastic region, Hooke’s law is not applicable.
In many real-life situations such as wind blowing on a tall building, and a musician plucking a string of a guitar, the deformation is proportional to the stress/Force and Hooke’s law holds good.
Applications of Hooke’s Law
The applications of Hooke’s Law are as follows:
Hooke’s Law is used in all branches of science and engineering; For understanding the behavior of elastic materials there is no substitute for Hooke’s law.
It is used as the fundamental principle behind the manometer, the balance wheel of the clock, and a spring scale.
Foundation for seismology, molecular mechanics, and acoustics.
Limitations of Hooke’s Law
Even though Hooke’s law is used extensively in Engineering, it’s not a universal principle. The law is not applicable as soon as the elastic limit of a material is exceeded. Normally for solid particles, Hooke’s law provides accurate results when the deformations are small. Many materials deviate from Hooke’s law even well before reaching the elastic limit.
Hooke’s Law calculations
In Engineering studies, Hooke’s law equation is widely used to calculate the Force required for getting the desired deformation. For example, in spring hanger supports used in piping engineering Hooke’s law is used to know how much pipe weight the spring will carry while allowing desired thermal movement.
Spring Constant or Young’s modulus is decided following Hooke’s law calculations. For Example, If 2,000 N force is applied to displace a spring by 0.5 m, it’s Spring Constant as per Hooke’s law calculation will be k=F/x=2000/0.5=4000 N/m
To explain why pipe stress-range calculations might not be correct if pipe support friction is not properly considered. The magnitude of the potential error in stress-range calculation is dependent on the geometry of the pipe layout, the type of supports used, and the coefficient of friction. Margins of potential error in the stress range could be large enough to invalidate the pipe stress analysis.
Most pipe stress software does not give any warning of the potential error discussed below.
Fundamental assumptions for pipe stress analysis
The calculation of stress range is fundamental to every pipe stress analysis. Calculated values of stress range are necessary to assess cyclic loads that could cause fatigue failure of a component in the piping system.
Cyclic loading is the repeated application and removal of temperature, displacement, pressure, or any other mechanical loads. A load cycle consists of two halves.
i) “Loading”. e.g. From “cold” to “hot” (or “unloaded” to “loaded”)
ii) “Unloading”. From “hot” to “cold” (or “loaded” to “unloaded”).
A fundamental and implicit definition of a load cycle is that each load cycle must begin and end at the same point and stress state. A particular sequence of loading and unloading is not a cycle if it does not begin and end at the same point and stress state.
The stress difference between “cold” to “hot” is not equal and the opposite to that from “hot” to “cold” is not a stress range. If a stress difference is not a stress range, then it is not valid for use in fatigue and design code calculations.
Design code requirements for stress-range
ASME B31.3 2018 section 319.2.3 Displacement Stress Range sub-section (b) states
“While stresses resulting from displacement strains diminish with time due to yielding or creep, the algebraic difference between strains in the extreme displacement condition and the original (as-installed) condition (or any anticipated condition with a greater differential effect) remains substantially constant during any one cycle of operation. This difference in strains produces a corresponding stress differential, the displacement stress range, that is used as the criterion in the design of piping for flexibility“.
The key phrase from 319.2.3 that is sometimes not considered in pipe stress calculations is “(or any anticipated condition with a greater differential effect)“.
Calculated stress-range
Most pipe stress analysis software calculates the stress difference from the “as installed position” to the first “hot position” (or “loaded position”).
Typical load cases starting from “as-installed‑position” would be
Load Case
Description
Comments
L1
W + P1 + T1 (Weight + Pressure + Temperature)
Stress and sustained loading in the operating condition. i.e. First “hot‑position” (or “loaded-position”).
L2
W + P1 (Weight + Pressure)
Stress and sustained loading in the “as-installed-position” .
L3
L1 – L2 = (W + P1 + T1) – (W + P1) = T1
The stress difference caused by temperature change T1 between the “as-installed-position” and “hot-position”.
The stress-differential L3 can only be used as a stress range for fatigue calculation for a particular load sequence if two conditions are met.
Condition 1 The stress difference between the “as-installed-position” and “hot-position”. Must be equal and opposite to The stress difference between the “hot-position” and “cold-state position”.
AND
Condition 2 All subsequent cycles based on the same loading must begin and end at the same points as the first cycle. [See Fig. 1].
Therefore, the “cold-state position” must be the same as the “as-installed position”. The pipe system must return to its “as-installed position” after the load has been removed.
The first half of a loading and unloading sequence is not a stress range if the stress difference is not equal to that of all subsequent cycles for the same sequential load. [See Fig. 2].
“Cold-state position” is defined for this discussion as the point to which the pipe returns after the removal of thermal or mechanical loads associated with the load sequence being considered.
“Hot position” means the same as “loaded position”.
“Cold-state position” means the same as “unloaded position”.
Fig. 1: Stress-range when the system returns to its “as-installed position”
Fig. 2: Stress-range when the system does not return to its “as-installed position”. (Pipe support friction creates a small stress reversal in this example)
Accounting for the adverse effects of friction in Stress-range calculations
As discussed above, pipe stress analysis based on stress-range calculations is only correct if the pipe returns to its “as-installed position” at the end of every loading and unloading sequence.
Stress ranges based on the “as-installed position” will only be correct in a real system if:
a) The pipe system is supported on hanger rods. i.e. The pipe will naturally “swing back” to its “as‑installed position.
b) The pipe system has no intermediate supports between fixed (anchor) points.
Any sliding support involving friction will prevent the pipe from returning to its “as‑installed position” by some amount.
The only way to ensure that stress-range calculations based on “as-installed positions” are reasonably valid for design is to ensure that the pipe is fully controlled.
Pipe supports must be designed to ensure that the pipe system always returns to “near enough” and its “as-installed position” to ensure errors are within acceptable limits. For example, by fitting guides or line-stops, using hangers, using low friction sliding supports, etc.
Pipe stress software
Default load cases for most pipe stress analysis applications calculate the stress range from the “as‑installed position” to the “first hot position”. Results could be in error if the user is not aware of the effect of friction on stress range.
Some pipe stress applications can calculate the stress difference for both the loading and unloading half-cycles. For example, PASS/START-PROF
The additional functionality enables the user to determine if the unloading half‑cycle is a B31.3 319.2.3 “anticipated condition with a greater differential effect”.
Example
Description:
The purpose of the simplified example pipe system below is to demonstrate the effect of friction on the calculated stress differences discussed above. Two scenarios are considered using identical geometry.
i) High friction – coefficient of pipe support friction = 0.3
ii) Low friction – coefficient of pipe support friction = 0.1
Each scenario considers the effect of friction preventing the pipe from returning to its “as‑installed position” after the first “cool-down” (unloading). Each scenario includes load cases to compare stress differences for:
a) “heat-up” (loading) from the “as-installed position” to the first “hot position”.
b) “cool-down” (unloading) from the first “hot position” to the “cold-state position”.
This example assumes that the pipe system continues to cycle between the first “hot position” and the “Cold‑state position” at the end of the first load cycle. [See Fig. 2 and Fig. 4].
The effect of “Stiction” has not been considered. However, it is noted that “stiction” will exaggerate the effect of friction discussed here.
Fig. 3: Example system geometry
Pipe 90 mm OD, 6 mm wall, E = 202713 MPa, n = 0.292, Density = 7833.413 kg/m3, g = 9.807 m/s2,
Full of water (1000 kg/m3) (Arbitrary numerical values for purpose of this example only).
Node
Description
10
Anchor
15
Rest support, with friction
25
Rest support, with friction
30
RY rotational restraint to represent the mutual rotational restraint of legs 10 – 30 and 30 – 40 Displacement DZ 100 mm “hot”, 0 mm displacement “cold”.
30 – 40
Represents a straight section of pipe. The expansion of 30 – 40 would cause 30 to expand 100 mm in Z direction. (The axial expansion of 10 to 30 is ignored for simplicity of calculation)
Fig. 4: Example system subject to deflection and friction
LC3 W + D2 + F2 “cold-state” postion Deflection DZ
mm
0
8.9
2.9
0
Friction forces F1 resisting “heat up” m = 0.3 Z direction, added as a force to simulate friction.
N
–
-252
-225
–
Friction forces F2 resisting “cool-down” m = 0.3 Z direction, added as a force to simulate friction.
N
–
253
227
–
LC4 = L1 – L2 (“Hot Position” minus “As-installed” position)
MPa
22.4
18.5
25.9
58
LC5 = L1 – L3 (“Hot Position” minus “cold-state” position)
MPa
6.6
30.7
26.3
78
LC5 – LC4 Error in “stress-range” based on “as‑installed” position
MPa
15.8 (-70%)
12.2 (66%)
0.4 (1%)
20 (34%)
Discussion of results
Table 1 – Support friction m = 0.1
Table 1 lists deflections and stress differences for load cycling with pipe support friction m = 0.1.
i) “As-installed to “hot‑position”. (Load case LC4). i.e. Warm-up (loading).
ii) “Hot position to “cold-state”. (Load case LC5). i.e. Cool-down (unloading).
Results are calculated based on the classical beam formula using Mathcad. (An identical Autodesk Nastran beam element model produced nearly identical results).
The stress difference for warm-up load case LC4 is less than the stress difference for cool‑down load case LC5. (Except for node 10). Therefore, LC4 cannot be used as the basis for the maximum system “stress range” as discussed above. [See Fig. 2].
If the system continues to cycle between “hot-position” and “cold‑state position”, then load case LC5 would be defined as the stress range. [See Fig. 2]. i.e. A maximum stress range of 51.8 MPa. (15% higher than 45.1 MPa calculated for load cycles between the “as-installed position” and “hot‑position”).
It is noted that the LC5 stress difference at node 10 is reduced by friction. The higher stress difference at node 10 for LC4 is not a stress range for the reasons discussed. However, in general, LC4 would still need to be checked as a single application.
Table 2 – Support friction m = 0.3
Table 2 results are the same as Table 1 discussed above but for m = 0.3.
Similarly, if it can be assumed that the pipe returns to the “hot position” after cooling to the “cold‑state position”, then load case LC5 would be the stress range. [See Figure 2]. i.e. A maximum stress range of 78.5 MPa. (34% higher than 58.5 MPa calculated for load cycles between the “as-installed position” and “hot‑position”).
Table 3 and Table 4 show PASS/START-PROF results equivalent to Table 1 and Table 2 respectively. PASS/START-PROF results show a close correlation with the manual calculations and general-purpose FE beam analysis (Autodesk Nastran 2021).
Summary
Stress ranges used for fatigue analysis require that the stress difference for warm‑up (loading) is the same as that for cool-down / unloading. In other words, the stress difference for warm-up/loading from the “as-installed position” is not a stress range if it is different than the stress difference for subsequent load cycles.
The stress difference during warm-up (loading) from its “as-installed position” is only valid as a stress range if the pipe system can be guaranteed to return precisely to the same position during cool‑down (unloading) for all subsequent load cycles.
This discussion used a simple example to demonstrate that pipe stress-range calculation based on the “as‑installed position” can be significantly in error if friction prevents the pipe from returning to the “as-installed position” after cool-down / unloading.
The magnitude of the error depends on the coefficient of friction, the pipe layout, and the type of pipe supports used. The use of guides, line-stops, and other methods of controlling pipe movement has a significant effect on the magnitude of any potential error.
Care must be taken when guides or other pipe supports controlling movement are removed to reduce the calculated displacement stresses and loads from the “as-installed position”. For example, removing guides from sliding supports to reduce calculated over-stress caused by seismic anchor movements.
Pipe stress theory and piping design codes both require that the most onerous stress range is used for purpose of assessing the fatigue life of a piping system for safe operation over its design lifetime.
B31.3 2018 section 319.2.3 states “While stresses resulting from displacement strains diminish with time due to yielding or creep, the algebraic difference between strains in the extreme displacement condition and the original (as-installed) condition (or any anticipated condition with a greater differential effect) remains substantially constant during any one cycle of operation. This difference in strains produces a corresponding stress differential, the displacement stress range, that is used as the criterion in the design of piping for flexibility“.
In the author’s experience, B31.3 319.2.3 is sometimes interpreted to mean only “the extreme displacement condition and the original (as-installed) condition”.
However, B31.3 actually states “the algebraic difference between strains in the extreme displacement condition and the original (as-installed) condition (or any anticipated condition with a greater differential effect) remains substantially constant during any one cycle of operation”.
The simple example above demonstrates that a failure to consider the phrase “(or any anticipated condition with a greater differential effect)” in B31.3 319.2.3 can lead to very different and potentially inaccurate calculation results.
** Future parts of “Pipe stress range calculated correctly?” will discuss how calculated stress range is affected by real changes of pipe support friction over the plant life. For example, what happens to the validity of a stress calculation if one pipe support rusts more than another?
Young’s Modulus | Modulus of Elasticity | Elastic Modulus | Young’s Modulus of Steels
Young’s modulus, also known as the modulus of elasticity or elasticity modulus is named after the British physicist Thomas Young. This is a very useful parameter in material science. Young’s modulus specifies the measure of the ability of a material to withstand length changes under tensile or compressive forces. Young’s modulus is defined mathematically as the ratio of the longitudinal stress to the strain within the elastic range of the material.
Young’s Modulus Formula
As explained in the article “Introduction to Stress-Strain Curve“; the modulus of elasticity is the slope of the straight part of the curve. Young’s modulus is a fundamental mechanical property of a solid material that quantifies the relationship between tensile (or compressive) stress and axial strain. It is denoted by the letter “E” and mathematically expressed as E=Stress/Strain
E=σ/ϵ;
Here σ=Stress=Force (F)/Cross-Sectional Area (A)=F/A ϵ=Strain=Change in Length(δl)/Original Length (l)=δl/l
So, E=(F/A)/(δl/l)=F*l/A*δl
As the strain is the ratio of two lengths, it is dimensionless. Hence, the unit of Young’s modulus, E =the unit of stress=N/m2 in the Metric system and psi (pound per square inch) in the English System.
For a specific material, the value of Young’s modulus or the modulus of elasticity is constant at a specified temperature. But with a change in temperature the value of Young’s modulus changes. With an increase in the material temperature, the modulus of elasticity of a material decreases.
The Youngs modulus does not depend on the geometry of the material. With the change in shape, length, the moment of inertia, weight, etc. the value of the modulus of elasticity does not change.
Which is more elastic: Rubber or Steel?
Young’s modulus signifies the elasticity of a material. The more the value of elastic modulus means the more elastic the material is. For example, as compared to rubber, the value of young’s modulus is more for steel material (Refer to Table 2). So, Steel material will regain its shape more easily as compared to the rubber on the application of force. Hence, Steel is more elastic than rubber. But sometimes it creates confusion when asked all of a sudden.
Use of Young’s Modulus
Young’s modulus is used
To calculate the change in length (deformation or deflection) of a material under tensile or compressive loads.
To calculate the amount of force required for a specific extension under specified stress.
Young’s modulus can be used to calculate various other moduli (for example rigidity modulus, bulk modulus, etc) of a material.
Practical Applications of Young’s Modulus
There are numerous practical examples of Young’s modulus. A few of the same as we find in piping and related engineering are provided here
The thermal stress generated in a piping system is calculated using Young’s modulus (E). Thermal Stress= Thermal Strain X E= E.α.ΔT; Here, α=co-efficient of thermal expansion and ΔT=change in length due to temperature.
In flange leakage analysis using the ASME Sec VIII method, Young’s modulus is required to calculate flange stresses. Flange modulus of elasticity data is provided as input for design and ambient temperature conditions.
For designing the load-carrying capability of steel and concrete structures, civil engineers use Young’s modulus along with allowable deflection data.
Young’s modulus of Steel
As steel is the most widely used material in industries, we will have a look at Young’s modulus of Steel for getting a rough estimate of the values. The following table provides typical values of Young’s modulus with respect to increasing temperature for ferrous materials.
Table 1: Young’s Modulus for Steel
Approximate Young’s Modulus for Some Other Materials
Table 2 below provides approximate Young’s modulus for some other well-known materials.
Table 2: Young’s Modulus of Various Materials
Factors Affecting Young’s Modulus
The main parameters that affect Young’s modulus of material are:
Temperature (With an increase in temperature, Young’s modulus decreases)
Presence of Impurity in the material like secondary phase particles, non-metallic inclusions, alloying elements, etc.
The stress analysis methodology of air cooler piping is quite different from other types of heat exchangers. This is mainly due to its different construction and supporting arrangements. The principal application of an air-cooled heat exchanger is to maintain a heat balance through the addition or removal of heat between streams of two different operating temperatures. Air fin cooler (AFC) or air cooler uses the air stream as the cooling medium. Air is circulated by multi-blade propeller-type fans as heat exchanger media. AFC unit consists of fin tube bundles with a header box attached to each end, supported horizontally by a C-shaped steel frame or structure. In the following paragraphs, we will explore the stress analysis philosophy of air cooler piping systems using Caesar II.
The latest revision of equipment GA drawing (covering all dimensions, Materials, nozzle allowable, and weight of AFC)
Allowable nozzle load (API 661 tables can be used in absence of vendor allowable loads if the air cooler is designed as per API 661)
Mechanical datasheet (Optional).
Stress Analysis with known Displacement of Air Fin Cooler From Vendor
If the vendor gives a thermal displacement value in the air cooler GA drawing then with the known value of displacements we can analyze the system. At the piping flange connecting the equipment nozzle enter the value of DX, DY, and DZ value in displacement vector -1, keeping RX, RY, and RZ values zero.
This is the most simple type of analysis and In this case, there is no need to model the air cooler in CAESAR II. However, in most situations, the displacements are not available. So, the stress analysis engineer has to model the equipment by taking data from the GA drawing.
Modeling air cooler in Caesar II
Modeling up to the top or bottom of the header
Model pipe up to matching flange of piping and AFC nozzle as per the given stress isometrics.
From the equipment GA drawing, model as per dimensions given up to the top or bottom of the header with proper thickness and size of the nozzle.
Provide a C-node anchor at the junction of the header and inlet or outlet nozzle.
Fig. 1: Modeling of Air Cooler in Caesar II
Modeling part after defining the c-node anchor
As per Fig-1, define 520 and 620 both as ANCHOR nodes with C-node as 521 and 621 respectively for checking of nozzle load.
Model equipment parts from nodes 521 to 530 and 621 to 630 as rigid elements with the same temperature of piping up to the centerline of the header without weight.
Then model header as node numbers 530 to 535, 630 to 635, and 530 to 630 with weight and temperature the same as piping (15 percent of total empty weight, total weight between 535 to 635).
Break the element 530 to 630 giving node 700 half of its length.
Model tube bundle with node 700 to 710 (rigid element) as per its length given in GA drawing with weight (70 percent of total empty weight) and give supports REST+ PTFE (0.1 FRICTION) at every one-meter distance along the bundle length.
Model 710 to 715, 715 to 720, 710 to 725, and 725 to 730 nodes as header elements with weight and the same average temperature of inlet and outlet. (15 percent of total empty weight, distribute in these two nodes).
Model 715 to 921 and 725 to 821 rigid elements without weight taking an average of inlet and outlet temperature of piping. (921 and 821 are C-nodes).
Modeling Part Of Equipment Restraint Nodes (Fixed Header)
The whole AFC has been supported at four ends on PTFE (Teflon) pad.
One header act as a fixed and the other as a floating end.
See vendor drawing for defining support nodes as fixed or floating.
As per Fig-1, the north side header (nodes 525 to 635) is fixed and hence south side header (nodes 720 to 730) is a floating end.
Define fixed-end nodes 535 and 635. Give Rest (Y) support with 0.1 friction coefficient, Axial stop (X) North-south directional stop with 2 mm gap, and Guide (Z) East-West with standard 12mm gap as per API 661 at both nodes.
Define floating end nodes 720 and 730. Give Rest (Y) support with 0.1 friction and Guide (Z) East-West with a standard 12mm gap as per API 661 at both nodes.
Note: The guide gap can be increased as per the lateral thermal movement of the Air fin cooler to avoid excessive nozzle load. Practically for that slot, the length can be increased after approval of the equipment department and vendor.
Modeling Part After Defining C-Node Anchor (Split Header)
Split header case modeling in CAESAR-II, is described below:
As per Fig-2, define 310 to 410 both as ANCHOR nodes with C-node as 311 and 411 respectively of inlet nozzle and 510 and 610 are of outlet nozzle as C-nodes 511 and 611 for checking of the nozzle load.
From C-nodes 311 and 411, model up to the centerline of the upper header in split header case as rigid elements 311 to 315 and 411 to 415 without weight but with an average of inlet and outlet temperature.
Then model half depth of top header elements as 315 to 320 and 415 to 420.
The top header will rest on the bottom header with a Teflon pad so define nodes 320 and 420 as +Y with 0.1 friction and C-node as 321 and 421.
Model 315 to 325 and 325 to 415 as a rigid element with half of the distance between two inlet nozzles with the half weight of one header and average temperature (Distribute the weight in 315-325 and 325-415).
Model 325 to 330 as a rigid tube bundle element with a length of bundle given in the drawing and provide supports along the length.
Model elements 330 to 340 as rigid elements with the distance between two header center lines without weight.
Model 340 to 345 and 340 to 350 as fixed header elements with the half weight of the total header weight.
Model elements 340 to 355 as connecting tubes to the bottom header with an average temperature of inlet and outlet and half of the weight of the tube bundle.
Model node numbers 355 to 515, 515 to 520, and 355 to 615, 615 to 620 as rigid elements indicating a lower header with the proper distance from GA and half of one header weight.
Model node number 321 (C-node) to 515 and 421 (C-node) to 615 as half of the depth of the lower header without weight.
Model elements 515 to 511 and 615 to 611 as rigid elements as half of the depth of the lower header.
Model bottom nozzle part and give 510 and 610 as ANCHOR with C-node restrain as 511 and 611.
Modeling Part of Equipment Restraint Nodes (Split Header) is as follows:
Defining of restraint for the fixed and floating end will be the same as mentioned in the fixed header case.
Here in Fig 2, nodes 345 and 350 are support nodes of the fixed end whereas nodes 520 and 620 are support nodes of the floating end.
There is only one extra support between two split headers at nodes 320 and 420 which are +Y with 0.1 friction coefficient and C-node as 321 and 421.
Fig. 2: Modeling of Air Cooler Split Header in Caesar II
Next, create the load cases in a similar way for all other equipment and check the output results for consistency with respect to codes and standards. Air cooler nozzle loads are qualified with API 661 table values in absence of vendor-given allowable loads.
Fig. 3 below shows a typical GA drawing of an air cooler.
PVC piping is one of the versatile types of plastic piping. PVC pipes have been in use for many years servicing a lot of applications in various market sectors. These pipes are made out of a material which is known as polyvinyl chloride. PVC pipes possess good strength and are highly durable with very good corrosion resistance properties. They are economical, robust, and come in various sizes with a range of pipe fittings. They can be used for both warm and cold water applications. PVC piping is easily available in the market. In water distribution and sanitary applications, PVC piping accounts for more than 60% of the market share. Fig. 1 shows a typical application of PVC Piping in Building Services.
Fig. 1: Typical PVC Piping in Building Services
What is PVC Pipe used for?
PVC pipes are used for manufacturing sewage pipes, irrigation, and water mains. PVC pipes possess very long-lasting properties, they are easy to install, strong, lightweight, durable, and easily recyclable hence making them cost-efficient and sustainable. The smooth surface of PVC pipes encourages faster water flow due to a low amount of friction as compared to metallic or concrete piping. These pipes can also be manufactured to different lengths, diameters, and wall thicknesses according to international standards such as DIN 8061 and ASTM D1785. Some other typical applications include:
Fire Sprinkler Piping
Radiant Floor Heating
Piping inside a Swimming Pool
Chilled Water Systems
Ice Melting
PVC Piping Schedule
PVC piping is available in different sizes and schedules, denoted by the abbreviation “SCH” followed by a number (e.g., SCH 40, SCH 80). The schedule indicates the wall thickness and pressure rating of the pipe. The higher the schedule number, the thicker the pipe wall and the higher the pressure it can handle. SCH 40 is the most common schedule for general-purpose applications, offering a balance between strength and cost.
How Are PVC Pipes Made?
Chemical Reaction:
These pipes have their origin in the chemical gas known as vinyl chloride. This vinyl chloride is exposed to the sunlight creating a chemical reaction which is known as polymerization which turns into a whitish solid material. In achieving the shape and solidity of a PVC pipe, chemicals are introduced to one another. The natural gas is heated to create ethylene. This process is known as cracking. Then sodium chloride is formed using electrolysis which results in chlorine and sodium hydroxide.
Molecular bonding:
Vinyl chloride monomer is made by introducing chlorine and ethylene. Molecules are then bonded from each molecule’s end resulting in a long chain of PVC polymer. Hence, plastic is created. This polymerized plastic which is known as thermoplastic PVC powder is melted and molded into the piping. This results in a tube of PVC plastic. Due to the chemical process, PVC becomes very solid and rigid and is less likely to break during earthquakes and can therefore withstand pressures that any metals cannot tolerate which is why polyvinyl chloride (PVC) is the preferred material used for plumbing and underground wiring.
Manufacturing of PVC Piping
PVC pipes are manufactured using a machine called an extruder in sizes as small as 16mm and large as 630mm. The PVC plastic is being routed through a double screw stem extruder or a conical twin screw. The wall thickness of the PVC hose is determined by molding. A PVC pipe extruder helps in making the diameter of the pipe. The standard PVC pipe extruder has a production speed of about twenty meters per minute. The hose runs through a vacuum pump. At the end of the assembly line, a ring-cutting machine is employed to divide the PVC pipes into sections of individual pipes which are then cooled and racked. The completed PVC pipes are sent to the warehouse for labeling, final inspection, and shipping.
Fig. 2: Manufacturing of PVC Piping
UV resistance of PVC piping
The PVC pipe undergoes surface oxidation and exposure to sunlight. The pipe changes its color from gray to white which becomes more susceptible to impact damage but the strength of the piping is not reduced. By applying a thick coating of latex paint, UV protection can be provided.
Heat Tracing of PVC piping
To maintain a constant elevated temperature, heat tracing of PVC piping is required. The tracing is known as electric tracing. The pressure-temperature rating of the piping system should be more than the maximum heat trace temperature.
Supporting PVC Piping Systems
Pipe saddles that are cut from PVC pipe are used at support locations. It should not rest directly on steel. The supports must be provided close to the inline items like valves. The piping connected to the vibrating equipment such as pumps should be isolated by using Teflon or rubber expansion joints. Fig. 3 below shows the typical support spacing for PVC piping systems.
Fig. 3: Support Spacing for PVC Piping Systems
PVC Piping Fittings
To help in directional change, branch connection, size change, etc. of PVC pipe routing, a long list of commonly used PVC piping fittings is available.
Tees:
These are PVC pipe fittings with three ends, one on the side at a 90-degree angle and two in a straight line. They can connect two lines into one main line. PVC tee connections are often used for PVC structures. These are extremely versatile fitting which is some of the most used parts in plumbing. Tees have slip socket ends but threaded versions are also available.
Elbows:
For the piping system to get a turn (directional change), you will need to bend the pipeline around with PVC elbows. These elbows are most commonly available at 90-degree and 45-degree angles. Most elbows have slip socket ends but threaded versions are also available.
Crosses:
This is a slightly less common type of PVC piping fitting that joins a four-pipe section. Crosses have four slip connections that meet at a 90-degree angle thereby forming a plus shape. These crosses are usually used in building frameworks out of PVC pipe. They are used to divide fluid flow in various directions.
Couplings and Unions:
Couplings are the most simple and inexpensive type of PVC pipe fitting. Couplings are small parts that connect one part to another, permanently, that is, they can connect pipe to pipe and pipe to fittings. These are available in female threaded or slip ends depending on the requirement.
Unions connect things but are not as permanent as couplings and can be taken apart. They are used in building temporary structures like tent supports and can be taken apart when the structure is not needed. It has a ring in the center separating the two ends from each other which allows easy maintenance and deconstruction.
Caps and Plugs:
The job of a PVC cap is to stop the flow. Caps are provided at the end of a pipeline that is not connecting to another pipe. The purpose of the cap is to stop a pipeline that is planned to expand later or easy access to the system when needed. Caps also add a finished look to a pipe. They go around or outside of the pipe so as to have a socket or female threaded end.
Plugs are also like caps but they stop the flow in a fitting instead of stopping the flow in a pipe. Because of this, they go inside the fitting which means they have a male threaded end.
Fig. 4: PVC Piping Fittings
Adapters (Female and male):
Adapters are also known as reducing couplings that are versatile fitting. These are designed to change the pipe’s end type which allows it to connect to fittings and pipes of many sizes. They can have slip socket ends or threaded ends to connect to different pipes and fittings. They can be male or female-threaded as well as a socket.
Bushings:
They are like adapters but focus on connecting pipes of different diameters. They are threaded fittings which makes them apart from other types of fittings which in turn makes maintenance and pipeline customization a lot easier. They are often seen in landscaping applications due to their better performance with water fittings than metal which may cause rust.
Nipples:
In some situations, two female ends in a PVC system have to be connected. The PVC pipe fitting in this job is a nipple. Fitting with two male threaded ends is a nipple. This type of fitting requires a tight fit and is mostly made with schedule 80 PVC; however, they are still compatible with 40 scheduled parts.
Flanges:
These are the type of PVC fittings that allows the attachment of accessories and other items to the pipe. Increases the strength of the pipe. It is a disc-like fitting creating a tight seal by pressing two surfaces together with bolts, edges, and clamps. Mostly bolts are used to join two surfaces. They are available in threaded or slip ends. They are made with schedule 80 PVC because of the strength requirement of the flanges.
PVC Piping Joining Methods
The following joining methods are used for joining two PVC pipes.
Solvent Welded Joints
Flanged Joints
Threaded Joints
Applicable Standards for PVC Piping
The below-listed PVC piping standards are widely used as the design guide for PVC Piping systems.
ASTM D1785 -Standard Specification for Poly Vinyl Chloride (PVC) Plastic Pipe, Schedule 40, 80, and 120
Stress-Strain Curve is a graphical plot of a material’s Stress and its Strain. Stress is plotted on the Y-Axis and Strain is plotted on the X-axis. This Stress and Strain curve provides the relation between stress and strain and the material’s stress behavior with an increase in strain. In material science and mechanical engineering, the stress-strain curve is widely used to understand the strength, deformation, and failure criteria of any material. In this article, we will explore details about the stress-strain curve.
Generation of Stress-Strain Curve
The Stress-Strain curve is plotted during the tensile test of a test specimen inside the Universal Testing Machine (UTM). In that instrument, the force on the standard specimen is increased till its failure and a plotter keeps recording the stress and strain.
Fig. 1 shows a typical stress-strain curve of Steel.
Fig. 1: Stress-Strain Curve of Steel
The stress & strain curve shown above describes various engineering parameters as listed below:
Yield Strength:
The Yield Strength of a material is the maximum stress after which the elongation becomes plastic and permanent deformation starts. Once the yield strength of a material is reached, large deformation occurs with very little increase in the applied load. The material will regain its shape once the stress is removed if the yield point is not reached.
In the stress-strain curve, yield strength is the point from where the stress deviates its proportionality to strain. For a few materials, the yield strength in the stress-strain curve is distinct but for a few others, it is not. Hence, a concept of proof stress is used to denote the yield strength in the stress & strain curve for those materials. Proof Stress is indicated by drawing a parallel line to the linear portion of the stress-strain curve at a strain value of 0.002 (or 2%).
Ultimate Tensile Strength:
The ultimate tensile strength or tensile strength of a material is the maximum stress value of the stress-strain curve. This is the maximum stress value for any material before final failure. For brittle materials, tensile strength is used as a stress basis in design calculations. In the stress-strain curve, the ultimate tensile strength can be decided accurately for all types of materials.
Young’s Modulus:
Young’s modulus is defined as the ratio of stress to strain. It is a measure of the stiffness of an elastic material. As mentioned in Fig. 1, it is the slope of the line in the straight part of the stress-strain curve.
Importance of Stress-Strain Curve
The stress-strain curve of material provides engineers with a long list of mechanical properties needed for engineering design. The capacity of a material to withstand loads prior to fracture is obtained from the stress-strain curve. The allowable material stress values are normally decided from the yield strength value for ductile materials and from the tensile strength value for brittle materials. The curve also provides a rough estimate of its deformation under loading conditions.
The stress-strain curve also helps in fabrication processes like extrusion, bending, rolling, etc. From the curve, the amount of force required for plastic deformation can be calculated.
Stress-Strain Curve of Aluminum
The stress-strain curve of the most widely used ductile material Steel is shown in Fig. 1 above. Fig. 2 below shows the typical stress-strain curve for Aluminum. For Aluminum the yield strength is not distinct; So the yield strength is decided using the proof stress method.
Fig. 2: Stress-Strain Curve for Aluminum
Stress-Strain Curve for Cast Iron
Cast Iron is a brittle material. For brittle materials, yield strength is not present as these materials fail all of a sudden. So, tensile strength is the main important parameter for brittle materials like Cast Iron, glass, and Concrete. Fig. 3 below provides a typical stress-strain curve for cast iron and concrete.
Fig. 3: Stress-Strain Curve for Cast Iron & Concrete
Stress-Strain Curve for Elastomers
Elastomers normally exhibit permanent plasticity. So, the stress-strain curve of elastomers is quite different from ductile and brittle materials. Fig. 4 below shows a typical example of the stress-strain curve of elastomers, plastic material, and brittle polymers.
Fig. 4: Stress-Strain Curve for Elastomers
Stress-Strain Curve for Perfectly Plastic Materials
Perfectly plastic or ideal plastic material will not show any work-hardening during plastic deformation. Fig. 5 shows the stress-strain curve for perfectly plastic, linear elastic, viscoelastic, and elastoplastic materials.
Fig. 5: Stress-Strain Curve for Perfectly Plastic Material
Please note that
the stress-strain curves in tension and compression for all materials are different. The stress-strain behavior (curve shape) may be similar but stress values differ considerably.
the stress-strain curve at room temperature is different from the same curve at other temperatures. Fig. 6 shows a typical example of a stress-strain curve for stainless steel and fiber-reinforced composite materials.
Fig. 6: Stress-Strain Curve at Various Temperatures
Fig. 7 below shows a comparative image showing stress-strain curves of various materials.
Fig. 7: Comparative Stress-Strain Curve for materials
