# Advanced Mechanics of Materials and Applied Elasticity: Analysis of Stress

This chapter is from the book

## 1.9 Plane-Stress Transformation

A two-dimensional state of stress exists when the stresses and body forces are independent of one of the coordinates, here taken as z. Such a state is described by stresses s x , s y , and t xy and the x and y body forces. Two-dimensional problems are of two classes: plane stress and plane strain. In the case of plane stress, as described in the previous section, the stresses s z , t xz , and t yz , and the z-directed body forces are assumed to be zero. The condition that occurs in a thin plate subjected to loading uniformly distributed over the thickness and parallel to the plane of the plate typifies the state of plane stress (Fig. 1.10). In the case of plane strain, the stresses t xz and t yz and the body force Fz are likewise taken to be zero, but s z does not vanish* and can be determined from stresses s x and sy .

We shall now determine the equations for transformation of the stress components s x , s y , and t xy at any point of a body represented by an infinitesimal element, isolated from the plate illustrated in Fig. 1.10. The z-directed normal stress s z , even if it is nonzero, need not be considered here. In the following derivations, the angle q locating the x' axis is assumed positive when measured from the x axis in a counterclockwise direction. Note that, according to our sign convention (see Sec. 1.5), the stresses are indicated as positive values.

Consider an infinitesimal wedge cut from the loaded body shown in Fig. 1.11a, b. It is required to determine the stresses s x' and t x'y' , which refer to axes x', y' making an angle q with axes x, y, as shown in the figure. Let side AB be normal to the x' axis. Note that in accordance with the sign convention, s x' and t x'y' are positive stresses, as shown in the figure. If the area of side AB is taken as unity, then sides QA and QB have area cos q and sin q, respectively.

Equilibrium of forces in the x and y directions requires that

Equation 1.16

where px and py are the components of stress resultant acting on AB in the x and y directions, respectively. The normal and shear stresses on the x' plane (AB plane) are obtained by projecting px and py in the x' and y' directions:

Equation a

From the foregoing it is clear that . Upon substitution of the stress resultants from Eq. (1.16), Eqs. (a) become

Equation 1.17a

Equation 1.17b

Note that the normal stress sy' acting on the y' face of an inclined element (Fig. 1.11c) may readily be obtained by substituting q + p/2 for q in the expression for s x' . In so doing, we have

Equation 1.17c

Equations (1.17) can be converted to a useful form by introducing the following trigonometric identities:

The transformation equations for plane stress now become

Equation 1.18a

Equation 1.18b

Equation 1.18c

The foregoing expressions permit the computation of stresses acting on all possible planes AB (the state of stress at a point) provided that three stress components on a set of orthogonal faces are known.

Stress tensor. It is important to note that addition of Eqs. (1.17a) and (1.17c) gives the relationships

s x + s y = s x' + s y' = constant

In words then, the sum of the normal stresses on two perpendicular planes is invariant—that is, independent of q. This conclusion is also valid in the case of a three-dimensional state of stress, as shown in Section 1.13. In mathematical terms, the stress whose components transform in the preceding way by rotation of axes is termed tensor. Some examples of other quantities are strain and moment of inertia. The similarities between the transformation equations for these quantities are observed in Sections 2.5 and C.4. Mohr's circle (Sec. 1.11) is a graphical representation of a stress tensor transformation.

### Polar Representations of State of Plane Stress

Consider, for example, the possible states of stress corresponding to s x = 14 MPa, s y = 4 MPa, and t xy = 10 MPa. Substituting these values into Eq. (1.18) and permitting q to vary from 0° to 360° yields the data upon which the curves shown in Fig. 1.12 are based. The plots shown, called stress trajectories, are polar representations: s x' versus q (Fig. 1.12a) and t x'y' versus q (Fig. 1.12b). It is observed that the direction of each maximum shear stress bisects the angle between the maximum and minimum normal stresses. Note that the normal stress is either a maximum or a minimum on planes at q = 31.66° and q = 31.66° + 90°, respectively, for which the shearing stress is zero. The conclusions drawn from this example are valid for any two-dimensional (or three-dimensional) state of stress and are observed in the sections to follow.

### Cartesian Representation of State of Plane Stress

Now let us examine a two-dimensional condition of stress at a point in a loaded machine component on an element illustrated in Fig. 1.13a. Introducing the given values into the first two of Eqs. (1.18), gives

 s x' = 4.5 + 2.5 cos 2q + 5 sin 2q t x'y' = –2.5 sin 2q + 5 cos 2q

In the foregoing, permitting q to vary from 0° to 180° in increments of 15° leads to the data from which the graphs illustrated in Fig. 1.13b are obtained [Ref. 1.7]. This Cartesian representation demonstrates the variation of the normal and shearing stresses versus q 180°. Observe that the direction of maximum (and minimum) shear stress bisects the angle between the maximum and minimum normal stresses. Moreover, the normal stress is either a maximum or a minimum on planes `q` = 31.7° and `q` = 31.7° + 90°, respectively, for which the shear stress is zero. Note as a check that s x + s y = s max + s min = 9 MPa, as expected.

The conclusions drawn from the foregoing polar and Cartesian representations are valid for any state of stress, as will be seen in the next section. A more convenient approach to the graphical transformation for stress is considered in Sections 1.11 and 1.15. The manner in which the three-dimensional normal and shearing stresses vary is discussed in Sections 1.12 through 1.14.

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.