Show that the two equilibrium equations in polar coordinates for plane stress (see Ar + 1...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Show that the two equilibrium equations in polar coordinates for plane stress (see Ar + 1 r 20 1000 + ar 20 + do. 0 1 + - (arr - 000) 0 re 20 20 You Press + 20 10 - 0 T Ar Get ar 201 Or Ar L Show that the two equilibrium equations in polar coordinates for plane stress (see Ar + 1 r 20 1000 + ar 20 + do. 0 1 + - (arr - 000) 0 re 20 20 You Press + 20 10 - 0 T Ar Get ar 201 Or Ar L
Expert Answer:
Answer rating: 100% (QA)
The image you have provided contains two equilibrium equations for an element under plane stress in polar coordinates and a diagram showing the stresses acting on an infinitesimal element in polar coordinates The two equations are 1 rrr 1r r 1r rr 0 2 rr 1r 2r r 0 These equations are derived from the equilibrium conditions on an infinitesimal element Lets break down the derivation For equilibrium in any system the sum of forces must be zero This must also be true for an infinitesimal element of a material undergoing deformation When dealing with a plane stress problem we consider an infinitesimal element in the shape of a sector of an annulus with internal radius r external radius rdr central angle d and thickness t into the screen The stresses on the element are rr radial stress circumferential or hoop stress r shear stress in the r direction and the opposite r shear stress in r direction To derive the first equation from your image 1 Write down the forces in the radial direction r direction Consider the stresses multiplied by the area over which they act t is the thickness into the page rr on the inside face of the sector acts over an area of r d t rr rrr dr on the outside face acts over an area of rdr d t accounting for the change in rr over dr r on the bottom face acts over an area of dr t r r d on the top face acts over an area of dr t accounting for the angular change in r on both the left and right faces will have vertical components that cancel out and horizontal components that contribute to the radial force balance 2 Consider the fact that the sum of the radial forces should equal zero meaning they are in equilibrium write an equation and divide by the differential elements and t to get the forces per unit length Incoming radial force rr r d t minus the outgoing force rr rrr dr rdr d t plus the contributions from the shear stress and the radial component of hoop stress should combine to zero 3 Expand the equation cancel out the common terms and divide through by dr d r to simplify leading to the provided differential equation rrr 1r r 1r rr 0 Similarly for the second equation performing the force balance in the direction tangential to the radius and following a similar process of considering the forces and moments acting on the infinitesimal element taking changes over dr and d into account and equilibrating them will yield rr 1r 2r r 0 This process utilizes the concepts of static equilibrium under plane stress conditions infinitesimal calculus and the assumption that the element considered is small enough for the stress ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
Students also viewed these mechanical engineering questions
-
Find the angle that maximizes the area of the isosceles triangle whose legs have length (Figure 17), using the fact the area is given by A = 1/2 2 sin . l 0 l
-
The following additional information is available for the Dr. Ivan and Irene Incisor family from Chapters 1-5. Ivan's grandfather died and left a portfolio of municipal bonds. In 2012, they pay Ivan...
-
The Home Depot, Inc. (HD) operates over 2,200 home improvement retail stores and is a competitor of Lowe's (LOW). The following data (in millions) were adapted from recent financial statements of The...
-
Write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. 1. (5 - x) / (3x2 + x - 1) 2. (4x2 - 1) / 2x (x + 1)2
-
Preparing a schedule of cash payments for selling and administrative expenses The budget director for Metro Cleaning Services prepared the following list of expected selling and administrative...
-
Defendants in a criminal proceeding have a right to the assistance of counsel. Petty Crime Pat was charged with driving with a revoked drivers license four times. Under the law, the fourth offense...
-
During the 1984 Olympics Frank and Irma rented their home in Hollywood, California and enjoyed a three-week vacation at their condo in Malibu, California. By 2000, they decided to retire to the Palm...
-
1 What is HRM? Explain the Functions of HRM? 2 What is HRP? Explain the process of HRP? 3 4 5 6 7 What is Recruitment? Explain the factors affecting Recruitment. What are the different Sources of...
-
Evaluate the integral using the indicated trigonometric substitution. V= -3x x +9 dx, x = 3 tan(0)
-
Summarize the Waterfall project management life cycle. Summarize a project manager's roles within the Waterfall project management life cycle. Explain CRM Software and how it works. Support your...
-
These items are taken from the financial statements of Cullumber Ltd. at December 31, 2024: Accounts payable $21,050 Accounts receivable 22,780 Accumulated depreciationbuildings 49,600 Accumulated...
-
3. Show the output of the following program and briefly explain your answer. public class Test { public static void main(String[] args) { } String s "Java"; StringBuilder builder new...
-
Financial analysts forecast Crestwood Equity Partners (CEQP) growth for the future to be 3.1 percent. The firm just paid a $1.22 dividend. What is the value of their stock when the required rate of...
-
Business Plan for a Doggy Daycare: personal experience, education, and training in the area/ industry of proposed business for all key stakeholders Discuss the strengths and weaknesses of primary...
-
Nyame Bekyere Wholesale Supply and Distribution Company (NBWSDC) sells parts, equipment, and supplies for recreational activities like plastic slides, bouncy castles, swings and electronic trains. In...
-
Evaluate the function at the given value(s) of the independent variable. Simplify the results. (x) = cos 2x (a) (0) (b) (- /4) (c) (/3) (d) ()
-
Professor Patricia (Patty) Pate is retired from the PalmSprings Culinary Arts Academy (PSCAA). She is a single taxpayer and is 68 years old. Patty lives at 98 Colander Street, Henderson, NV 89052....
-
Dr. George E. Beeper is a single taxpayer. He lives at 45 Mountain View Dr., Apt. 321, Spokane, WA 99210. Dr. Beeper's Social Security number is 775-88-9531. Dr. Beeper works for the Pine Medical...
-
David and Darlene Jasper have one child, Sam, who is 6 years old. The Jaspers reside at 4639 Honeysuckle Lane, Los Angeles, CA 90248. David's Social Security number is 577-11-3311, Darlene's is...
-
An amorphous material is: (a) Mica (b) Lead (c) Rubber (d) Glass
-
Discuss the application vernier caliper and its method of measurement.
-
In center lathe, cutting tool is fed in ____ directions with reference to the lathe axis.
Study smarter with the SolutionInn App