Mohr's Stress Circle

10 Key excerpts on "Mohr's Stress Circle"

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  Design Engineer's Sourcebook

  • K. L. Richards(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  14

  Introduction to Analytical Stress Analysis and the Use of Mohr’s Circle

  14.1 Introduction

  Of all the graphical methods used by an engineer for visualizing problems, there is little doubt that Mohr’s circle is the best known; in its various applications it is an aid to visualizing a stress or strain problem.

  The object of this chapter is to explore with a reasonable thoroughness three important uses of Morh’s circle. These applications confront the engineer and designer in many branches of mechanical and structural engineering.

  In the development of the circle construction, it has been considered desirable to present a concise account of the underlying theory. It is possible for the reader more interested in applications to accept these results and go straight to those sections dealing with the constructions and applications.

  Nearly all textbooks, of necessity, deal with the circle in too cursory a manner and leave many detailed questions unanswered. It is hoped that this chapter devoted solely to the circle will answer more of these questions. In illustrating the text with examples of a practical nature, other topics of strength of materials will be quoted, the background of which can be found in the most strength of materials textbooks.

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  Design Engineer's Handbook

  • Keith L. Richards(Author)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  229 9 Introduction to Analytical Stress Analysis and the Use of the Mohr Circle 9.1 INTRODUCTION Of all the graphical methods used by an engineer there is little doubt that the Mohr circle is the best known; in its various applications it is an aid to visualizing a stress or strain problem. The object of this chapter is to explore with a reasonable thoroughness three important uses of the circle. These applications confront the engineer and designer in many branches of mechanical and structural engineering. In the development of the circle construction it has been considered desirable to present a con-cise account of the underlying theory. It is possible for the reader more interested in applications to accept these results and go straight to those sections dealing with the constructions and applications. Nearly all textbooks, of necessity, deal with the circle in too cursory a manner and leave many detailed questions unanswered. It is hoped that this chapter devoted solely to the circle will answer more of these questions. In illustrating the text with examples of a practical nature other topics of strength of materials will be quoted, the background of which can be found in most strength of materials textbooks. 9.2 NOTATION A = Area a = Circle constant B = Breadth (of beam cross section) D = Depth (of beam section) E = Modulus of elasticity e = Strain (usually with suffix to indicate direction) F = Force h = Distance I = Second moment of area (with suffix of the type xx or cg) J = Polar moment (with suffix of the type xy) M = Moment (with suffix to indicate axis) n = Factor of safety O = Origin of graph P = Force, or pole point P 1,2 = Pole points R = Radius of circle, radius of curvature T = Torque (usually with suffix to indicate axis) y = Distance from neutral axis of a beam to a given point β = Angle γ = Poisson’s ratio, angle

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  Soil Mechanics

  Concepts and Applications, Third Edition

  • William Powrie(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  (The angle, measured at the centre of the Mohr circle, between diameters representing different planes is twice the physi-cal angle between those planes in reality). 2.2.6 Mohr circle of strain Strain, like stress, may be classified as either direct or shear (Figure 2.5). Direct strain is given the symbol ε . Engineering shear strain, defined in Figure 2.5b, is given the symbol γ . Mohr circle of effective stress Centre at s uni2032= ( uni03C3uni2032 1 + uni03C3uni2032 3 ) Radius t = ( uni03C3uni2032 1 – uni03C3uni2032 3 ) t t u (a) (b) O Radius t = ( uni03C3 1 – uni03C3 3 ) 1 2 1 2 Mohr circle of total stress Centre at s = ( uni03C3 1 + uni03C3 3 ) 1 2 1 2 uni03C4 O uni03C4 uni03C3uni2032 3 uni03C3uni2032 3 uni03C3uni2032 1 uni03C3uni2032 uni03C3 1 uni03C3uni2032 3 uni03C3, uni03C3uni2032 uni03C3 3 s uni2032 s uni2032= s ( uni03C3uni2032, uni03C4 ) 1 2 ( uni03C3uni2032 1 + uni03C3uni2032 3 ) 2 uni03B8 Figure 2.4 Mohr circles of stress showing (a) the circles representing total and effective stress separated by the pore water pressure, u; and (b) the stress state on an imaginary 'cut' at an angle θ anticlock-wise from the plane on which the major principal effective stress acts. Soil strength 65 © 2010 Taylor & Francis Group, LLC The engineering shear strain, γ , is the overall decrease in the angle between the positive directions of two perpendicular lines—in the case of Figure 2.5, the x and y axes.) The same systems of subscripts are used as for stresses. If a material is elastic, the directions of the principal strain increments will coincide with the directions of the principal stress incre-ments during an increment of loading. This may not be true in other cases. The state of strain in a plane containing two principal strains may be represented by the Mohr circle of strain , as shown in Figure 2.6. This is exactly analogous to, and is used in the same way as, the Mohr circle of stress.

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  Fundamentals of Machine Component Design

  • Robert C. Juvinall, Kurt M. Marshek(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  The circle provides a convenient graphical solution for the magnitude and orientation of principal stresses  1 and  2 . These stresses are shown on a principal element at point A, drawn in Figure 4.28. Note that the #1 principal plane is located by starting with the x plane and rotating counterclockwise half of the 56 ∘ measured on the circle, and so on. y y x A x τ yx = 30.6 ksi τ yx τ xy σ x = 40.8 ksi Direct view of element A y (0, +30.6) x (40.8, –30.6) τ max = 37 ksi σ 2 = –17 ksi +τ +σ τ xy 34° 56° 0 σ 1 = 57 ksi FIGURE 4.27 Mohr circle representation at point A of Figure 4.25. 4.10 Stress Equations Related to Mohr’s Circle 99 y y x A σ 1 = 57 ksi σ 2 = –17 ksi 28° x FIGURE 4.28 Principal element at A (direct view) shown in relation to x and y faces. y y x A σ = 20 ksi τ = +37 ksi τ = –37 ksi σ = 20 ksi 17° x FIGURE 4.29 Maximum shear element at A (direct view) shown in relation to x and y faces. 6. Figure 4.28 shows the magnitude and orientation of the highest normal stresses. It may also be of interest to represent similarly the highest shear stresses. This is done in Figure 4.29. Observe again the rules of a. rotating in the same direction on the element and the circle, and b. using angles on the circle that are twice those on the element. Comment: In support of neglecting the transverse shear stress in step 1, it is of interest to note that its maximum value at the neutral bending axis of the 1-in.-diameter shaft is 4V ∕3A = (4)(2000 lb)∕ [(3)()(1 in.) 2 ∕4] = 3.4 ksi 4.10 Stress Equations Related to Mohr’s Circle The derivation of the analytical expressions relating normal and shear stresses to the angle of the cutting plane is given in elementary texts on strength of materials and need not be repeated here.

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  Geosynthetic Reinforced Soil (GRS) Walls

  • Jonathan T. H. Wu(Author)
  • 2019(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  Figure 1.6 ). However, we shall continue to make use of “stress” for the design and analysis of earth structures because it has proven to be an extremely useful tool. Keep in mind, however, that stress in soil is merely a “defined” parameter. When referring to stresses in a soil, it should be viewed on a macro‐scale, and the soil is considered a continuum for the purposes of engineering analysis.

  Figure 1.6

  Stress at a point in a soil mass: (a) reality (micro‐scale) and (b) idealized as being a uniform continuum (macro‐scale)

  1.1.3 Mohr Circle of Stress

  The Mohr circle of stress, as shown in Figure 1.7 (a), is a plot of normal stress vs. shear stress of all permissible stresses at a point under two‐dimensional conditions. Every point on a Mohr circle represents the normal and shear stresses on a particular plane at that point. A Mohr circle of stress therefore can be regarded as a graphical representation of stresses at a point under two‐dimensional conditions. There are an infinite number of points on a Mohr circle, and each point corresponds to a plane passing through the point of interest. Two distinct planes exist on a Mohr circle where shear stress . These planes are called principal planes. The stresses on the principal planes are known as principal stresses, denoted by σ

  1

  and σ

  3

   , as shown in Figure 1.7 (a). The principal stress σ

  1

  is the largest normal stress (or major principal stress), whereas the principal stress σ

  3

  is the smallest normal stress (or minor principal stress).

  Figure 1.7

  (a) Two‐dimensional representation of stress at a point by a Mohr circle of stress and (b) sign conventions of normal and shear stresses for plotting Mohr circles

  Figure 1.7 (b) shows the sign conventions for plotting a Mohr circle of stress. In soil mechanics, we denote compressive normal stress as positive and tensile normal stress as negative, which is opposite to the sign convention commonly used in structural mechanics. We do so because soil has little tensile resistance, and most normal stresses in geotechnical engineering analysis are compressive. The sign convention of considering compressive stress as positive avoids having to show nearly every normal stress in geotechnical engineering analysis with a negative sign. A shear stress that makes a clockwise rotation about any point outside of the plane is considered a positive shear stress (see Figure 1.7

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  Mechanical Properties of Engineered Materials

  • Wole Soboyejo(Author)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  Note that the locus of the circle describes all the possible states of stress on the element at the point, P, for various values of 0 between 0° and 180°. It is also important to note that several combinations of the stress components (crxx, a yy, zxy) may result in yielding, as the plane angle, 0, is varied. These combinations will be discussed in Chap. 5. When a generalized state of triaxial stress occurs, three M ohr’s circles [Fig. 3.8(a)] may be drawn to describe all the possible states of stress. These circles can be constructed easily once the principal stresses, cr1? o 2 >

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  Multiaxial Notch Fatigue

  • Luca Susmel(Author)
  • 2009(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Fig. 1.2 ).

  Mohr’s circle sketched in Fig. 1.2b then allows the stress state relative to any plane parallel to the z -axis and passing through point O to be easily determined. For instance, as sketched in Fig. 1.3 , the maximum shear stress, τ max , is equal to R c , whereas the stress perpendicular to such a plane is equal to σc . On the contrary, the principal stresses σ1 and σ2 can be calculated as: σ1 = σc + R c and σ2 = σc – R c . Figure 1.4 shows the procedure to determine these principal stresses from Mohr’s circle as well as the orientation of the principal axes.

  1.3 Orientation of the plane experiencing the maximum shear stress, τ max .

  1.4 Orientation of principal stresses σ1 and σ2 .

  Finally, it can be useful to remember that, under plane stress, the normal, σn (ϕ ), and the shear stress, τ n (ϕ ), relative to a generic material plane perpendicular to the component’s surface and having normal unit vector n at angle ϕ to the x -axis can be calculated as follows (Socie and Marquis, 2000 ):

  σ n

  ( ϕ )

  =

  σ x

  +

  σ y

  2

  +

  σ x

  -

  σ y

  2

  cos

  (

  2 ϕ

  )

  +

  τ

  x y

  sin

  (

  2 ϕ

  )

  1.9

  1.9

  τ n

  ( ϕ )

  =

  σ x

  -

  σ y

  2

  sin

  (

  2 ϕ

  )

  -

  τ

  x y

  cos

  (

  2 ϕ

  )

  1.10

  1.10

  1.4 Amplitude, mean value, range and load ratio, R , under uniaxial cyclic loading

  Assume that the body of Fig. 1.1 is subjected to an external system of forces resulting, at point O , in the following uniaxial stress state:

  [

  σ

  ( t )

  ]

  =

  [

  σ x

  ( t )

  0 0

  0 0 0

  0 0 0

  ]

  1.11

  1.11

  where

  σ x

  (t ) =

  σx

  , m +

  σx

  , a sin(ωt ). In this definition t is time, ω is the angular velocity, and

  σ x

  , m and

  σ x

  , a are the mean value and the amplitude of stress component

  σ x

  (t ), respectively. It is worth remembering here also that, by definition, a cycle is a sequence of changing stress states that, upon completion, produces a final stress state which is identical to the initial one (see the example reported in Fig. 1.5

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  Strain Patterns in Rocks

  A Selection of Papers Presented at the International Workshop, Rennes, 13-14 May 1982

  • P. R. Cobbold, W. M. Schwerdtner(Authors)
  • 2015(Publication Date)
  • Pergamon

    (Publisher)

  A line is drawn between these points and a circle is drawn about the fine as diameter. The circle has its center at (a, b) and diameter D, where INTRODUCTION M O H R circles centered on the horizontal axis of Mohr diagrams have been used for many years to provide a geometric representation of symmetric second-order tensor quantities, like the state of stress at a point or the state of strain. Recently Robin (1977), De Paor (1979) and Lister & Williams (1983) have employed Mohr circles that are not centered on the horizontal axis of Mohr diagrams, to represent asymmetric second-order tensors, and De Paor (this issue) has found an antique example of the idea in the work of De La Hire (1685). The aims of this paper are to show why Mohr circles provide vaHd representations of any second-order ten-sor, symmetric or asymmetric, and to explore some applications that arise in the study of bodies that have been inhomogeneously deformed. The discussion is (T22J12) (T11.-T21) T11 T12 2 2 • .T21T22. 1 1 Tij = Fig. 1. Mohr circle for an asymmetric second-order tensor Τ with components as given, (a, b) are the coordinates of the center of the circle and D is its diameter. 279 280 W. D. M E A N S 2a = Γπ + 2b = Tn -(2) (3) (4) (T22J12) This circle turns out to be a Mohr circle representing the tensor T. Before demonstrating this we underscore two steps in the construction. We reverse the sign of Γ21 before treating it as a coordinate value in Mohr space, and we use pairs of values in columns of matrix (1) to locate each point in Mohr space. The sign change will be familiar to those used to the difference between tensor and Mohr circle sign conventions for stresses or strains. Either Τ 12 or Γ21 can be changed. The resulting two Mohr circles are mirror images of each other across the horizontal axis.

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  Practical Rock Mechanics

  • Steve Hencher(Author)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  Commonly used methods, such as hydraulic-fracturing, only provide three principal stress measurements, and then are interpreted on the assumption that one principal stress direction is vertical. It takes very special equipment to measure the total stress state at a point, and even then the data may only be locally relevant because of geological complexity. Hudson (1989) and Hudson and Harrison (1997) clearly explain stress analysis and its measurement and are recommended reading. For those who wish to delve more deeply, then Jaeger et al. (2007) and Farmer (1983) offer fuller discussion. If 3D analysis is important because of the geometry of the situation, for example at the intersection of tunnels, then there are software programmes that can be used to calculate the stress conditions, including Examine3D (Rocscience) and FLAC3D (Itasca). Where the 3D geological conditions are important, software like 3DEC (Itasca) can be used to analyse the situation. Fortunately, for many practical engineering and structural geological analyses, 2D analysis is adequate. One of the advantages of 2D analysis is that the models are relatively easy to set up, so that many simple analyses can be conducted quickly and relatively cheaply, to explore the relative importance of various assumptions in the ground models (Starfield and Cundall, 1988). For most engineering projects, one assumes that gravity is one of the principal stress directions, usually the major one, but this is not always the case, as discussed in more detail in Chapter 3. 2.2 MOHR CIRCLE REPRESENTATION OF STRESS STATE For most projects, the situation can be adequately considered in two dimensions, chiefly in the plane containing both the maximum σ 1 and minimum σ 3 principal stresses – that is, at right angles to σ 2 . It is in this plane that failure generally occurs in nature, as in the Anderson (1951) theory of major faulting discussed in Chapter 3.

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  Mechanics of Materials

  With Applications in Excel

  • Bichara B. Muvdi, Souhail Elhouar(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  On the other hand, failure by fracture occurs if the point lies outside this area. 7.9.1.2 Mohr’s Theory The maximum principal stress theory discussed above requires that the ultimate strength in tension be numerically equal to the ultimate strength in compression. This, of course, is not true for most structural materials, and, therefore, the maximum principal stress theory does not yield satisfactory answers. Mohr’s theory, on the other hand, provides more satisfactory answers in predicting failure of materials for which the ultimate strength in tension is different from that in compression. We will limit our discussion to the simplified Mohr’s theory which is based solely on knowledge of the ultimate strength in tension, σ ut , and the ultimate strength in compression, σ uc . With this information, two Mohr’s circles can be drawn, one with diameter σ ut and center at A and the second with diameter σ uc and center at B as shown in Figure 7.31 . A plane stress state in which the two in-plane principal stresses are tensile (positive) and the out-of-plane principal stress is zero (i.e., σ 1 and σ 2 are positive and σ 3 = 0) will not lead to failure by fracture if the corresponding circle lies entirely within the confines of the circle with diameter σ ut and center at A . In other words, no failure occurs if σ 1 < σ ut and σ 2 < σ ut . Similarly, if the plane stress state is such that the two in-plane principal stresses are compressive (negative) and the out-of-plane principal stress is zero (i.e., σ 1 = 0, σ 2 and σ 3 are negative), then it will not lead to failure if the corresponding circle lies entirely within the con-fines of the circle with diameter σ uc and center at B . In other words, no failure occurs if | σ 2 | < | σ uc | B A σ u σ u σ 1 σ 3 -σ u -σ u FIGURE 7.30 A graphical representation of Equation 7.100 in which only the marble-colored area below the diagonal AB applies.

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