Home > Articles > Engineering > General Engineering & Hardware

This chapter is from the book

1.13 Principal Stresses in Three Dimensions

For the three-dimensional case, it is now demonstrated that three planes of zero shear stress exist, that these planes are mutually perpendicular, and that on these planes the normal stresses have maximum or minimum values. As has been discussed, these normal stresses are referred to as principal stresses, usually denoted s 1, s 2, and s 3. The algebraically largest stress is represented by s 1, and the smallest by s 3: s 1 > s 2 > s 3.

We begin by again considering an oblique x' plane. The normal stress acting on this plane is given by Eq. (1.28a):

Equation a


The problem at hand is the determination of extreme or stationary values of s x' . To accomplish this, we examine the variation of s x' relative to the direction cosines. Inasmuch as l, m, and n are not independent, but connected by l 2 + m 2 + n 2 = 1, only l and m may be regarded as independent variables. Thus,

Equation b


Differentiating Eq. (a) as indicated by Eqs. (b) in terms of the quantities in Eq. (1.26), we obtain

Equation c


From n 2 = 1 – l 2m 2, we have u2202.gifn/u2202.gifl = –l/n and u2202.gifn/u2202.gifm = –m/n. Introducing these into Eq. (c), the following relationships between the components of p and n are determined:

Equation d


These proportionalities indicate that the stress resultant must be parallel to the unit normal and therefore contains no shear component. It is concluded that, on a plane for which s x' has an extreme or principal value, a principal plane, the shearing stress vanishes.

It is now shown that three principal stresses and three principal planes exist. Denoting the principal stresses by s p , Eq. (d) may be written as

Equation e


These expressions, together with Eq. (1.26), lead to

Equation 1.31


A nontrivial solution for the direction cosines requires that the characteristic determinant vanish:

Equation 1.32


Expanding Eq. (1.32) leads to

Equation 1.33



Equation 1.34a


Equation 1.34b


Equation 1.34c


The three roots of the stress cubic equation (1.33) are the principal stresses, corresponding to which are three sets of direction cosines, which establish the relationship of the principal planes to the origin of the nonprincipal axes. The principal stresses are the characteristic values or eigenvalues of the stress tensor t ij . Since the stress tensor is a symmetric tensor whose elements are all real, it has real eigenvalues. That is, the three principal stresses are real [Refs. 1.8 and 1.9]. The direction cosines l, m, and n are the eigenvectors of t ij .

It is clear that the principal stresses are independent of the orientation of the original coordinate system. It follows from Eq. (1.33) that the coefficients I 1, I 2, and I 3 must likewise be independent of x, y, and z, since otherwise the principal stresses would change. For example, we can demonstrate that adding the expressions for s x' , s y' , and s z' given by Eq. (1.28) and making use of Eq. (1.30a) leads to I 1 = s x' + s y' + s z' = s x + s y + s z . Thus, the coefficients I 1, I 2, and I 3 represent three invariants of the stress tensor in three dimensions or, briefly, the stress invariants. For plane stress, it is a simple matter to show that the following quantities are invariant (Prob. 1.27):

Equation 1.35


Equations (1.34) and (1.35) are particularly helpful in checking the results of a stress transformation, as illustrated in Example 1.7.

If now one of the principal stresses, say s 1 obtained from Eq. (1.33), is substituted into Eq. (1.31), the resulting expressions, together with l 2 + m 2 + n 2 = 1, provide enough information to solve for the direction cosines, thus specifying the orientation of s 1 relative to the xyz system. The direction cosines of s 2 and s 3 are similarly obtained. A convenient way of determining the roots of the stress cubic equation and solving for the direction cosines is presented in Appendix B, where a related computer program is also included (see Table B.1).

Example 1.6. Three-Dimensional Stress in a Hub

A steel shaft is to be force fitted into a fixed-ended cast-iron hub. The shaft is subjected to a bending moment M, a torque T, and a vertical force P, Fig. 1.20a. Suppose that at a point Q in the hub, the stress field is as shown in Fig. 1.20b, represented by the matrix


Figure 1.20

Figure 1.20 Example 1.6. (a) Hub-shaft assembly. (b) Element in three-dimensional stress.

Determine the principal stresses and their orientation with respect to the original coordinate system.


Substituting the given stresses into Eq. (1.33) we obtain from Eqs. (B.2)

s 1 = 11.618 MPa,

s 2 = –9.001 MPa,

s 3 = –25.316 MPa

Successive introduction of these values into Eq. (1.31), together with Eq. (1.30a), or application of Eqs. (B.6) yields the direction cosines that define the orientation of the planes on which s 1, s 2, and s 3 act:

l 1 = 0.0266,

l 2 = –0.6209,

l 3 = 0.7834

m 1 = –0.8638,

m 2 = 0.3802,

m 3 = 0.3306

n 1 = –0.5031,

n 2 = –0.6855,

n 3 = –0.5262

Note that the directions of the principal stresses are seldom required for purposes of predicting the behavior of structural members.

Example 1.7. Three-Dimensional Stress in a Machine Component

The stress tensor at a point in a machine element with respect to a Cartesian coordinate system is given by the following array:

Equation f


Determine the state of stress and I 1, I 2, and I 3 for an x', y', z' coordinate system defined by rotating x, y through an angle of q = 45° counterclockwise about the z axis (Fig. 1.21a).

Figure 1.21

Figure 1.21 Example 1.7. Direction cosines for q = 45°.


The direction cosines corresponding to the prescribed rotation of axes are given in Fig. 1.21b. Thus, through the use of Eq. (1.28) we obtain

Equation g


It is seen that the arrays (f) and (g), when substituted into Eq. (1.34), both yield I 1 = 100 MPa, I 2 = 1400 (MPa)2, and I 3 = –53,000 (MPa)3, and the invariance of I 1, I 2, and I 3 under the orthogonal transformation is confirmed.

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.


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.


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.

Cookies and Related Technologies

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.


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


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


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.


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.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020