# CH.4. STRESS. Continuum Mechanics Course (MMC)

Size: px
Start display at page:

Transcription

1 CH.4. STRESS Continuum Mechanics Course (MMC)

2 Overview Forces Acting on a Continuum Body Cauchy s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion Properties of the Cauchy Stress Tensor Cauchy s Equation of Motion Principal Stresses and Principal Stress Directions Mean Stress and Mean Pressure Spherical and Deviatoric Parts of a Stress Tensor Stress Invariants Lecture Lecture Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7

3 Overview (cont d) Stress Tensor in Different Coordinate Systems Cylindrical Coordinate System Spherical Coordinate System Mohr s Circle Mohr s Circle for a 3D State of Stress Determination of the Mohr s Circle Mohr s Circle for a D State of Stress D State of Stress Stresses in Oblique Plane Direct Problem Inverse Problem Mohr s Circle for a D State of Stress Lecture 8 Lecture 9 Lecture 0 Lecture Lecture Lecture 3 3

4 Overview (cont d) Mohr s Circle a D State of Stress (cont d) Construction of Mohr s Circle Mohr s Circle Properties The Pole or the Origin of Planes Sign Convention in Soil Mechanics Particular Cases of Mohr s Circle Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 4

5 4.. Forces on a Continuum Body Ch.4. Stress 5

6 Forces Acting on a Continuum Body Forces acting on a continuum body: Body forces. Act on the elements of volume or mass inside the body. Action-at-a-distance force. E.g.: gravity, electrostatic forces, magnetic forces fv = ρb( x, t) dv Surface forces. V body force per unit mass (specific body forces) Contact forces acting on the body at its boundary surface. E.g.: contact forces between bodies, applied point or distributed loads on the surface of a body fs = t ( x, t) ds V (traction vector) surface force per unit surface 6

7 4.. Cauchy s Postulates Ch.4. Stress 7

8 Cauchy s Postulates. Cauchy s st postulate. The traction vector tremains unchanged for all surfaces passing through the point P and having the same normal vector n at P. ( P, ) t = t n REMARK The traction vector (generalized to internal points) is not influenced by the curvature of the internal surfaces.. Cauchy s fundamental lemma (Cauchy reciprocal theorem) The traction vectors acting at point P on opposite sides of the same surface are equal in magnitude and opposite in direction. ( P, ) = ( P, ) t n t n REMARK Cauchy s fundamental lemma is equivalent to Newton's 3 rd law (action and reaction). 8

9 4.3. Stress Tensor Ch.4. Stress 9

10 Stress Tensor The areas of the faces of the tetrahedron are: S S S = ns = ns = ns 3 3 with n { n } T,n,n3 The mean stress vectors acting on these faces are ( ) ( ) ( ) * * * * * * 3* * t = tx ( ˆ ˆ ˆ S, n), t = tx ( S, e ), t = tx ( S, e ), t = tx ( S, e 3 3) * * xs S,, 3 ; x i i i = S S mean value theorem The surface normal vectors of the planes perpendicular to the axes are n = eˆ ; n = eˆ ; n3 = eˆ3 REMARK Following Cauchy s fundamental lemma: The asterisk indicates an not i t x, eˆ = t x, eˆ = t x i,, 3 mean value over the area. ( ) ( ) i i ( ) ( ) { } 0

11 Mean Value Theorem [ ] Let f : a,b R be a continuous function on the closed interval [ a,b], and differentiable on the open interval ( a,b), where a < b. * Then, there exists some x in a,b such that: ( ) ( ) f x = f x dω Ω * Ω ( ) [ ] I.e.: f : a,b R gets its * mean value f ( x ) at the interior of a,b [ ]

12 Stress Tensor From equilibrium of forces, i.e. Newton s nd law of motion: resultant body forces R = fi = miai ρ b dv + t ds = a ρdv = ρ adv i i V V V dm V ( ) ( ) ( 3) ρ = b dv t ds t ds t ds t ds ρ a dv V S S S S V 3 Considering the mean value theorem, ( ) ( ) ( 3) ( ρb)v+ t S t S t S t S = ( ρa)v * * * * * * 3 i i 3 * * ( ) * ( ) * ( 3) * * ( ρb) h S + t S t ns t ns t n3s = ( ρa) hs 3 3 Introducing S = ns i {,, 3} and V = Sh, resultant surface forces

13 Stress Tensor If the tetrahedron shrinks to point O, * * * S O h 0 ( i) ( i ( ˆ ) ) ( O, ˆ ) { } * * * xs x lim i O t xs, e i i = t ei i,,3 h 0 ( ) S ( O ) x x lim t x, n = t, n lim ( ) h = lim ( ) h = 3 3 * * ρ b ρ a 0 h 0 h 0 The limit of the expression for the equilibrium of forces becomes, = t = t = t ( ) ( ) ( 3) * * * * * * ( ρb) h+ t t n t n t n3 = ( ρa) h 3 3 ( O, ) = t n ( ) ( ) ( 3) (, ) ( i) t O n t n i = 0 3

14 Stress Tensor ( i ) ( ) ( ) Considering the traction vector s Cartesian components : () i t P = t ( P) eˆ = σ eˆ ( i σ ) ij P = tj ( P) j j ij j σ = σ eˆ eˆ ij i j In the matrix form: T t = n σ = σ n j {,,3} T [ t ] = [ ] [ n ] j i ij ji i { } i, j,,3 Cauchy s Stress Tensor ( ) t σ ( ) ( ) t ( P, ) (, ) ( i) t n = t n () i t j P n = t j ni = niσ ij σij t( P, n) = n σ ( P) ( 3) t i P t ( ) t ( 3) t 4

15 Stress Tensor REMARK The expression t( P, n) = n σ ( P) t( P, n) = n σ ( P, ) = t n n σ is consistent with Cauchy s postulates: ( P, ) = ( P, ) t n t n REMARK The Cauchy stress tensor is constructed from the traction vectors on three coordinate planes passing through point P. σ σ σ σ3 σ σ σ 3 σ3 σ3 σ 33 Yet, this tensor contains information on the traction vectors acting on any plane (identified by its normal n) which passes through point P. 5

16 4.4.Stress Tensor Components Ch.4. Stress 6

17 Scientific Notation Cauchy s stress tensor in scientific notation σ σ σ σ σ σ σ σ σ σ Each component σ ij is characterized by its sub-indices: Index i designates the coordinate plane on which the component acts. Index j identifies the coordinate direction in which the component acts. 7

18 Engineering Notation Cauchy s stress tensor in engineering notation σ σx τxy τ xz τ yx σ y τ yz τzx τzy σ z Where: σ a τ ab is the normal stress acting on plane a. is the tangential (shear) stress acting on the plane perpendicular to the a-axis in the direction of the b-axis. 8

19 Tension and compression σ ij or σ The stress vector acting on point P of an arbitrary plane may be resolved into: a vector normal to the plane ( σ = σ n) an in-plane (shear) component which acts on the plane. ( τ ; τ = ) n n τ The sense of with respect to n defines the normal stress character: σ = σ n n σ n >0 tensile stress (tension) <0 compressive stress (compression) The sign criterion for the stress components is: a positive (+) negative ( ) τ ab positive (+) negative ( ) tensile stress compressive stress positive direction of the b-axis negative direction of the b-axis n 9

20 4.5.Properties of the Cauchy Stress Tensor Ch.4. Stress 0

21 Cauchy s Equation of Motion Consider an arbitrary material volume, Cauchy s equation of motion is: σ+ ρ b= ρa x σ ij + ρbj = ρaj j xi {,, 3} In engineering notation: σ τ x yx τzx ρbx = ρax x y z τxy σ y τzy ρby = ρay x y z τ τ xz yz σ z ρbz = ρaz x y z V ( ) ( ) b x,t x V * t x x,t V REMARK Cauchy s equation of motion is derived from the principle of balance of linear momentum.

22 Equilibrium Equations For a body in equilibrium a= 0, Cauchy s equation of motion becomes σ+ ρ b= 0 x V σ ij + ρbj = 0 j,,3 xi { } The traction vector is now known at the boundary * (, ) σ (, ) (, ) * {,, 3} n x t x t = t x t x V ni σ ij = tj j internal equilibrium equation equilibrium equation at the boundary The stress tensor symmetry is derived from the principle of balance of angular momentum: T σ = σ σ = ij σ ji i, j {,,3}

23 Cauchy s Equation of Motion Taking into account the symmetry of the Cauchy Stress Tensor, Cauchy s equation of motion σ+ ρ b= σ + ρ b= ρa x σij σ ji + ρbj = + ρbj = ρaj j xi xi Boundary conditions * n σ = σ n= t (,) x t x V {,, 3} V * ni σ ij =σ ji ni = tj t V i j ( x, ) x, {,, 3} ( ) ( ) b x,t x V * t x x,t V 3

24 Principal Stresses and Principal Stress Directions 4 Regardless of the state of stress, it is always possible to choose a special set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system. The three planes perpendicular to the principal axes are the principal planes. The normal stress components in the principal planes are the principal stresses. [ σ ] σ σ σ 3 = x σ x σ σ 3 σ 3 σ 3 σ σ σ σ x x x x 3 x x 3 σ x σ 3 σ

25 Principal Stresses and Principal Stress Directions 5 The Cauchy stress tensor is a symmetric nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers. For the eigenvalue λ and its corresponding eigenvector v: σ v =λv [ σ λ] v = 0 [ σ λ ] not det = σ λ = 0 λ σ λ σ λ σ 3 3 REMARK The invariants associated with a tensor are values which do not change with the coordinate system being used. x INVARIANTS λ I ( σ) λ I ( σ) λ I ( σ) = σ x σ σ 3 σ 3 σ 3 σ σ σ σ x x x x 3 x x 3 σ characteristic equation x σ 3 σ

26 Mean Stress and Mean Pressure Given the Cauchy stress tensor and its principal stresses, the following is defined: Mean stress σ σm = Tr ( σ ) = σii = ( σ+ σ + σ3) Mean pressure p = σm = σ+ σ + σ3 3 ( ) A spherical or hydrostatic state of stress: σ σ = σ = σ 3 σ σ 0 0 σ 0 0 = σ 0 0 σ σ σ σ σ σ σ σ σ σ REMARK In a hydrostatic state of stress, the stress tensor is isotropic and, thus, its components are the same in any Cartesian coordinate system. As a consequence, any direction is a principal direction and the stress state (traction vector) is the same in any plane. 6

27 Spherical and Deviatoric Parts of a Stress Tensor The Cauchy stress tensor σ=σ σ sph + σ The spherical stress tensor: can be split into: Also named mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body σsph : = σm= Tr ( σ) = σii REMARK 3 3 The principal directions of a stress tensor The stress deviator tensor: and its deviator stress component coincide. Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body σ = devσ = σ σ m 7

28 Stress Invariants Principal stresses are invariants of the stress state: invariant w.r.t. rotation of the coordinate axes to which the stresses are referred. The principal stresses are combined to form the stress invariants I : I = Tr ( σ) = σ = σ + σ + σ ii 3 I = I = σσ + σσ 3+ σσ 3 I 3 = det ( σ) ( σ: σ ) ( ) These invariants are combined, in turn, to obtain the invariants J : J = I = σ ii J = ( I + I) = σσ ij ji = ( σ: σ) 3 J3 = ( I + 3II + 3I3) = Tr ( σ σ σ) = σσ ij jkσki REMARK The I invariants are obtained from the characteristic equation of the eigenvalue problem. REMARK The J invariants can be expressed in the unified form: i Ji = Tr ( σ ) i {,, 3} i 8

29 Stress Invariants of the Stress Deviator Tensor The stress invariants of the stress deviator tensor: I = Tr ( σ ) = 0 I = σ : σ I = σ σ σ σ σ σ + + I 3 = det σ = σ σ σ 33 + σ σ 3σ 3 σ σ 33 σ 3σ σ 3σ = σ σ σ 3 These correspond exactly with the invariants J of the same stress deviator tensor: J ( ) = I = 0 ( ) ( ij jk ki ) J = ( I + I ) = I = ( σ : σ ) 3 J 3 = ( I + 3I I+ 3I 3) = I 3 = Tr ( σ σ σ ) = ( σσ ij jkσ ki )

30 4.6. Stress Tensor in Different Coordinate Systems Ch.4. Stress 30

31 Stress Tensor in a Cylindrical Coordinate System The cylindrical coordinate system is defined by: dv = r dθ dr dz x= r cos θ x( r, θ, z) y = r sin θ z = z The components of the stress tensor are then: σx τx y τ x z σr τr θ τrz σ = τxy σ y τ yz τr θ σθ τ = θz τx z τ y z σ z τrz τθ z σ z 3

32 Stress Tensor in a Spherical Coordinate System The cylindrical coordinate system is defined by: dv = r sen θ dr dθ dϕ The components of the stress tensor are then: σx τxy τ xz σr τr θ τr φ σ τxy σ y τ yz = τrθ σθ τθφ τxz τ yz σ z τr φ τφθ σ φ x x= r sen θ cosφ r, θϕ, y = r sen θ senφ z = r cosθ ( ) 3

33 4.7. Mohr s Circle Ch.4. Stress 33

34 Mohr s Circle Introduced by Otto Mohr in 88. Mohr s Circle is a two-dimensional graphical representation of the state of stress at a point that: will differ in form for a state of stress in D or 3D. illustrates principal stresses and maximum shear stresses as well as stress transformations. is a useful tool to rapidly grasp the relation between stresses for a given state of stress. 34

35 4.8. Mohr s Circle for a 3D State of Stress Ch.4. Stress 35

36 Determination of Mohr s Circle Consider the system of Cartesian axes linked to the principal directions of the stress tensor at an arbitrary point P of a continuous medium: x 3 The components of the stress tensor are σ 0 0 σ 0 σ 0 with 0 0 σ 3 The components of the traction vector are σ 0 0 n σ n t = σ n= 0 σ 0 n = σ n 0 0 σ 3 n 3 σ3 n 3 ê σ 3 x σ ê x ê 3 where n is the unit normal to the base associated to the principal directions 36

37 Determination of Mohr s Circle The normal component of stress σ is n σ = tn = [ σ n, σ n, σ n ] n = σ n + σ n + σ n T n 3 t The squared modulus of the traction vector is n σ n = σ n t t = σ + τ : τ : = τn = t t = σn + σn + σ3n3 σn + σn + σ3n3 = σ + τ The unit vector n = n must satisfy n n n = Mohr's 3D problem half - space Locus of all possible ( στ, ) points? 37

38 Determination of Mohr s Circle The previous system of equations can be written as a matrix equation which can be solved for any couple σ σ σ 3 n σ + τ σ σ σ3 n = σ n 3 A x b n 0 T A feasible solution for x requires that n 0 n for the, n, n 3 expression n + n + n = to hold true. 3 0 n3 Every couple of numbers ( στ, ) which leads to a solution x, will be considered a feasible point of the half-space. The feasible point is representative of the traction vector ( στ, ) on a T plane of normal n n, n, n 3 which passes through point P. The locus of all feasible points is called the feasible region. 38

39 Determination of Mohr s Circle The system σ σ σ 3 n σ + τ σ σ σ3 n = σ n 3 A x b can be re-written as A ( I) σ + τ σ + σ σ+ σσ n = 0 ( ) 3 3 ( ) ( ) ( σ σ3) A ( II) σ + τ σ + σ σ+ σσ n = ( σ σ3) A ( III) σ + τ σ + σ σ+ σσ n = 0 3 ( σ σ) with ( ) ( 3) ( 3) A = σ σ σ σ σ σ 39

40 Determination of Mohr s Circle Consider now equation ( III) : A σ + τ σ + σ σ+ σσ n = 0 It can be written as: ( ) σ a + τ = R which is the equation of a semicircle of center and radius R : C R ( ) 3 = ( σ + σ ), 0 = ( σ σ) with ( σ σ ) ( σ σ ) ( σ σ ) n with a = ( σ + σ) R= + 4 ( ) ( 3) ( 3) A = σ σ σ σ σ σ ( σ σ ) ( σ σ ) ( σ σ ) n C3 3 REMARK A set of concentric semi-circles is obtained with the different values of n 3 with center C and radius R 3 3( n3) : min n3 = 0 R3 = ( σ σ) n3 = max R3 = ( σ + σ) σ3 40

41 Determination of Mohr s Circle Following a similar procedure with ( I) and ( II), a total of three semi-annuli with the following centers and radii are obtained: C = [ ( 3),0] σ + σ C = [ ( 3),0] σ + σ a a C = 3 [ ( ),0] σ + σ a 3 R R R R R R min max max min min 3 max 3 = = σ a ( σ σ ) 3 = = σ a ( σ σ ) 3 = = σ a ( σ σ ) 3 3 4

42 Determination of Mohr s Circle Superposing the three annuli, 44 The final feasible region must be the intersection of these semi-annuli Every point of the feasible region in the Mohr s space, corresponds to the stress (traction vector) state on a certain plane at the considered point

43 4.9. Mohr s Circle for a D State of Stress Ch.4. Stress 45

44 D State of Stress 3D general state of stress D state of stress σ σx τxy τ xz τ yx σ y τ yz τzx τzy σ z σ σx τxy τ yx σ y σ z 3D problem REMARK In D state of stress problems, the principal stress in the disregarded direction is known (or assumed) a priori. σ σx τ yx τxy σ y D (plane) problem 46

45 Stresses in a oblique plane Given a plane whose unit normal n forms an angle θ with the x axis, Traction vector σx τxy cosθ σx cosθ + τxy sinθ t = σn = τxy σ y sinθ = τxy cosθ σ y sinθ + σ n Normal stress Shear stress σx σ y τ = tm = sin θ τ cos θ σx + σ y σx σ y σθ = tn = + cos( θ) + τxy sin ( θ) θ ( ) ( ) xy cosθ sinθ n = m = cosθ sinθ 47 Tangential stress θ is now endowed with sign τ ( τ 0 or τ < 0) Pay attention to the positive senses given in the figure θ θ

46 Direct and Inverse Problems Direct Problem: Find the principal stresses and principal stress directions given σ in a certain set of axes. Inverse Problem: Find the stress state on any plane, given the principal σ stresses and principal stress directions. equivalent stresses 48

47 Direct Problem 49 In the x and y axes, τ = 0 then, σx σ y τα = sin ( α) τxy cos( α) = 0 τ xy tan ( α ) = σx σ y Using known trigonometric relations, τ xy ( α ) =± =± + σx σ y tg ( α ) + τ xy sin cos ( α ) σx σ y =± =± + tg ( α ) σx σ y + τ α xy This equation has two solutions:.. α ( sign " + ") π α = α + ( sign " ") These define the principal stress directions. (The third direction is perpendicular to the plane of analysis.)

48 Direct Problem The angles and θ = α are then introduced into the equation θ α = σx + σ y σx σ y σθ = + cos( θ) + τxy sin ( θ) to obtain the principal stresses (orthogonal to the plane of analysis): σ α σx + σ y σx σ y σ = + + τ σx + σ y σx σ y σ = + τ xy xy θ α 50

49 Inverse Problem Given the directions and principal stresses σ and, to find the σ stresses in a plane characterized by the angle β : Take the equations Replace,, and to obtain: θ β σ+ σ σ σ σβ = + σ σ τβ = sin ( β) cos ( β) 5

50 Mohr s Circle for a D State of Stress Considering a reference system and characterizing the inclination of a plane by, From the inverse problem equations: σ+ σ σ σ σ = σ σ τ = sin ( β) cos Squaring both equations and adding them: + σ σ σ τ σ σ + = Eq. of a circle with center C and radius R. Mohr s Circle ( β) x y REMARK This expression is valid for any value of. C R = σ σ σ+ σ =,0 5

51 Mohr s Circle for a D State of Stress The locus of the points representative of the state of stress on any of the planes passing through a given point P is a circle. (Mohr s Circle) The inverse is also true: ( ) Given a point στ, in Mohr s Circle, there is a plane passing through P whose normal and tangential stresses are and τ, respectively. σ R = σ σ + σ σ Mohr's D problem space σ a ( β ) = = cos sin ( β ) σ σ = τ τ σ σ = R R σ+ σ C =,0 53

52 Construction of Mohr s Circle Interactive applets and animations: by M. Bergdorf: from MIT OpenCourseware: from Virginia Tech: From Pennsilvania State University: 54

53 Mohr s Circle s Properties A. To obtain the point in Mohr s Circle representative of the state of stress on a plane which forms an angle β with the principal stress direction : σ. Begin at the point on the circle (representative of the plane where acts).. Rotate twice the angle in the sense σ σ β. 3. This point represents the shear and normal stresses at the desired plane (representative of the stress state at the plane where acts). σ

54 Mohr s Circle s Properties B. The representative points of the state of stress on two orthogonal planes are aligned with the centre of Mohr s Circle: π This is a consequence of property A as β = β+. 56

55 Mohr s Circle s Properties C. If the state of stress on two orthogonal planes is known, Mohr s Circle can be easily drawn:. Following property B, the two points representative of these planes will be aligned with the centre of Mohr s Circle.. Joining the points, the intersection with the σ axis will give the centre of Mohr s Circle. 3. Mohr s Circle can be drawn

56 Mohr s Circle s Properties D. Given the components of the stress tensor in a particular orthonormal base, Mohr s Circle can be easily drawn: This is a particular case of property C in which the points representative of the state of stress on the Cartesian planes is known.. Following property B, the two points representative of these planes will be aligned with the centre of Mohr s Circle.. Joining the points, the intersection with the σ axis will give the centre of Mohr s Circle. 3. Mohr s Circle can be drawn σ σx τxy = τxy σ y 58

57 Mohr s Circle s Properties The radius and the diametric points of the circle can be obtained: σ σx τxy = τxy σ y σ σ R = + τ x y xy 59

58 Mohr s Circle s Properties Note that the application of property A for the point representative of the vertical plane implies rotating in the sense contrary to angle. σx τxy σ = τxy σ y 60

59 The Pole or the Origin of Planes The point called pole or origin of planes in Mohr s circle has the following characteristics: Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane parallel in space to that line... 6

60 The Pole or the Origin of Planes The point called pole or origin of planes in Mohr s circle has the following characteristics: If a straight line, parallel to a given plane, is drawn from the pole, the intersection point represents the state of stress on this particular plane... 6

61 Sign Convention in Soil Mechanics The sign criterion used in soil mechanics, is the inverse of the one used in continuum mechanics: In soil mechanics, σ β negative ( ) positive (+) tensile stress compressive stress τ β positive (+) negative (-) counterclockwise rotation clockwise rotation continuum mechanics But the sign criterion for angles is the same: positive angles are measured counterclockwise τ σ * β * β = τ β = σ β soil mechanics 63

62 Sign Convention in Soil Mechanics For the same stress state, the principal stresses will be inverted. continuum mechanics soil mechanics τ σ * β * β = τ β = σ β σ σ β * = σ * = σ * π = β + 64 The expressions for the normal and shear stresses are σ σ σ + σ * * * * * * σβ cos( β π) * * * * σ σ σ σ = * * cos( ) σ + σ σ σ β = + cos β* β cos = + * * * * σ σ * σ + σ * * σ sin σ β = ( ) β = sin + * β = sin σ β σ β ( ) τ β τ β π τ β sin β* The Mohr s circle construction and properties are the same in both cases ( ) ( ) like in continuum mechanics

63 4.0. Particular Cases of Mohr s Circle Ch.4. Stress 65

64 Particular Cases of Mohr s Circles Hydrostatic state of stress Mohr s circles of a stress tensor and its deviator ( σ sph = σ m) σ = σ + σ sph σ = σm + σ σ = σm + σ σ = σ + σ 3 m 3 Pure shear state of stress 66

### CHAPTER 4 Stress Transformation

CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z

### Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

### 16.20 Techniques of Structural Analysis and Design Spring Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T

16.20 Techniques of Structural Analysis and Design Spring 2013 Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T February 15, 2013 2 Contents 1 Stress and equilibrium 5 1.1 Internal forces and

### CHAPER THREE ANALYSIS OF PLANE STRESS AND STRAIN

CHAPER THREE ANALYSIS OF PLANE STRESS AND STRAIN Introduction This chapter is concerned with finding normal and shear stresses acting on inclined sections cut through a member, because these stresses may

### 3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

### Draft:S Ghorai. 1 Body and surface forces. Stress Principle

Stress Principle Body and surface forces Stress is a measure of force intensity, either within or on the bounding surface of a body subjected to loads. It should be noted that in continuum mechanics a

### PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

### Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

50 Module 4: Lecture 1 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-Coulomb failure

### Stress transformation and Mohr s circle for stresses

Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.

### 9. Stress Transformation

9.7 ABSOLUTE MAXIMUM SHEAR STRESS A pt in a body subjected to a general 3-D state of stress will have a normal stress and shear-stress components acting on each of its faces. We can develop stress-transformation

### MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

### VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.

### Lecture Notes 3

12.005 Lecture Notes 3 Tensors Most physical quantities that are important in continuum mechanics like temperature, force, and stress can be represented by a tensor. Temperature can be specified by stating

### Surface force on a volume element.

STRESS and STRAIN Reading: Section. of Stein and Wysession. In this section, we will see how Newton s second law and Generalized Hooke s law can be used to characterize the response of continuous medium

### Equilibrium of Deformable Body

Equilibrium of Deformable Body Review Static Equilibrium If a body is in static equilibrium under the action applied external forces, the Newton s Second Law provides us six scalar equations of equilibrium

### Classical Mechanics. Luis Anchordoqui

1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

### Combined Stresses and Mohr s Circle. General Case of Combined Stresses. General Case of Combined Stresses con t. Two-dimensional stress condition

Combined Stresses and Mohr s Circle Material in this lecture was taken from chapter 4 of General Case of Combined Stresses Two-dimensional stress condition General Case of Combined Stresses con t The normal

### CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics

CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models

### 3D Elasticity Theory

3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

### 7. STRESS ANALYSIS AND STRESS PATHS

7-1 7. STRESS ANALYSIS AND STRESS PATHS 7.1 THE MOHR CIRCLE The discussions in Chapters and 5 were largely concerned with vertical stresses. A more detailed examination of soil behaviour requires a knowledge

### Continuum mechanism: Stress and strain

Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

### Lecture Notes 5

1.5 Lecture Notes 5 Quantities in Different Coordinate Systems How to express quantities in different coordinate systems? x 3 x 3 P Direction Cosines Axis φ 11 φ 3 φ 1 x x x x 3 11 1 13 x 1 3 x 3 31 3

LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )

### Principal Stresses, Yielding Criteria, wall structures

Principal Stresses, Yielding Criteria, St i thi Stresses in thin wall structures Introduction The most general state of stress at a point may be represented by 6 components, x, y, z τ xy, τ yz, τ zx normal

### GG303 Lecture 17 10/25/09 1 MOHR CIRCLE FOR TRACTIONS

GG303 Lecture 17 10/5/09 1 MOHR CIRCLE FOR TRACTIONS I Main Topics A Stresses vs. tractions B Mohr circle for tractions II Stresses vs. tractions A Similarities between stresses and tractions 1 Same dimensions

### Variable Definition Notes & comments

Extended base dimension system Pi-theorem (also definition of physical quantities, ) Physical similarity Physical similarity means that all Pi-parameters are equal Galileo-number (solid mechanics) Reynolds

### L8. Basic concepts of stress and equilibrium

L8. Basic concepts of stress and equilibrium Duggafrågor 1) Show that the stress (considered as a second order tensor) can be represented in terms of the eigenbases m i n i n i. Make the geometrical representation

### Unit IV State of stress in Three Dimensions

Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength

### Introduction to Seismology Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

### Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.

Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u

### Exercises for Multivariable Differential Calculus XM521

This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done

### Mechanics of Materials Lab

Mechanics of Materials Lab Lecture 5 Stress Mechanical Behavior of Materials Sec. 6.1-6.5 Jiangyu Li Jiangyu Li, orce Vectors A force,, is a vector (also called a "1 st -order tensor") The description

### Properties of the stress tensor

Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress

### Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

51 Module 4: Lecture 2 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-coulomb failure

### both an analytical approach and the pole method, determine: (a) the direction of the

Quantitative Problems Problem 4-3 Figure 4-45 shows the state of stress at a point within a soil deposit. Using both an analytical approach and the pole method, determine: (a) the direction of the principal

### SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e

### ME 243. Lecture 10: Combined stresses

ME 243 Mechanics of Solids Lecture 10: Combined stresses Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil

### Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India AIR WALK PUBLICATIONS

Advanced Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India Dr. S. Ramachandran Prof. R. Devaraj Professors School of Mechanical

### Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9

MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)

### Two Dimensional State of Stress and Strain: examples

Lecture 1-5: Two Dimensional State of Stress and Strain: examples Principal stress. Stresses on oblique plane: Till now we have dealt with either pure normal direct stress or pure shear stress. In many

### The hitch in all of this is figuring out the two principal angles and which principal stress goes with which principal angle.

Mohr s Circle The stress basic transformation equations that we developed allowed us to determine the stresses acting on an element regardless of its orientation as long as we know the basic stresses σx,

### CH.6. LINEAR ELASTICITY. Multimedia Course on Continuum Mechanics

CH.6. LINEAR ELASTICITY Multimedia Course on Continuum Mechanics Overview Hypothesis of the Linear Elasticity Theory Linear Elastic Constitutive Equation Generalized Hooke s Law Elastic Potential Isotropic

### By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

### CONTINUUM MECHANICS. lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern

CONTINUUM MECHANICS lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern Contents Tensor calculus. Tensor algebra.................................... Vector algebra.................................

### Bone Tissue Mechanics

Bone Tissue Mechanics João Folgado Paulo R. Fernandes Instituto Superior Técnico, 2016 PART 1 and 2 Introduction The objective of this course is to study basic concepts on hard tissue mechanics. Hard tissue

### V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

### Strain analysis.

Strain analysis ecalais@purdue.edu Plates vs. continuum Gordon and Stein, 1991 Most plates are rigid at the until know we have studied a purely discontinuous approach where plates are

### Solutions for Fundamentals of Continuum Mechanics. John W. Rudnicki

Solutions for Fundamentals of Continuum Mechanics John W. Rudnicki December, 015 ii Contents I Mathematical Preliminaries 1 1 Vectors 3 Tensors 7 3 Cartesian Coordinates 9 4 Vector (Cross) Product 13 5

### ! EN! EU! NE! EE.! ij! NN! NU! UE! UN! UU

A-1 Appendix A. Equations for Translating Between Stress Matrices, Fault Parameters, and P-T Axes Coordinate Systems and Rotations We use the same right-handed coordinate system as Andy Michael s program,

### ANALYSIS OF STRAINS CONCEPT OF STRAIN

ANALYSIS OF STRAINS CONCEPT OF STRAIN Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an

### GG612 Lecture 3. Outline

GG61 Lecture 3 Strain and Stress Should complete infinitesimal strain by adding rota>on. Outline Matrix Opera+ons Strain 1 General concepts Homogeneous strain 3 Matrix representa>ons 4 Squares of line

### Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its

### (MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e

EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The

### 1 Stress and Strain. Introduction

1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may

### CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM

CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM Summary of integral theorems Material time derivative Reynolds transport theorem Principle of conservation of mass Principle of balance of linear momentum

### ENGINEERING OF NUCLEAR REACTORS NOTE L.4 INTRODUCTION TO STRUCTURAL MECHANICS. Lothar Wolf, Mujid S. Kazimi and Neil E.

.31 ENGINEERING OF NUCLEAR REACTORS NOTE L.4 INTRODUCTION TO STRUCTURAL MECHANICS Lothar Wolf, Mujid S. Kazimi and Neil E. Todreas Nominal Load Point 35 55 6569 100 σ 1-5 -45-57.7-60 Max τ theory DE theory

### Useful Formulae ( )

Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation

### MATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions

MATH 55 Applied Honors Calculus III Winter 11 Midterm 1 Review Solutions 11.1: #19 Particle starts at point ( 1,, traces out a semicircle in the counterclockwise direction, ending at the point (1,. 11.1:

### Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

### 2. Mechanics of Materials: Strain. 3. Hookes's Law

Mechanics of Materials Course: WB3413, Dredging Processes 1 Fundamental Theory Required for Sand, Clay and Rock Cutting 1. Mechanics of Materials: Stress 1. Introduction 2. Plane Stress and Coordinate

### 3D Stress Tensors. 3D Stress Tensors, Eigenvalues and Rotations

3D Stress Tensors 3D Stress Tensors, Eigenvalues and Rotations Recall that we can think of an n x n matrix Mij as a transformation matrix that transforms a vector xi to give a new vector yj (first index

### Basic concepts to start Mechanics of Materials

Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen

### Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

### Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction

### 1. Background. is usually significantly lower than it is in uniaxial tension

NOTES ON QUANTIFYING MODES OF A SECOND- ORDER TENSOR. The mechanical behavior of rocks and rock-like materials (concrete, ceramics, etc.) strongly depends on the loading mode, defined by the values and

### Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature

Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte

### 27. Folds (I) I Main Topics A What is a fold? B Curvature of a plane curve C Curvature of a surface 10/10/18 GG303 1

I Main Topics A What is a fold? B Curvature of a plane curve C Curvature of a surface 10/10/18 GG303 1 http://upload.wikimedia.org/wikipedia/commons/a/ae/caledonian_orogeny_fold_in_king_oscar_fjord.jpg

### Lecture 3 The Concept of Stress, Generalized Stresses and Equilibrium

Lecture 3 The Concept of Stress, Generalized Stresses and Equilibrium Problem 3-1: Cauchy s Stress Theorem Cauchy s stress theorem states that in a stress tensor field there is a traction vector t that

### Chapter 5. The Orientation and Stress Tensors. Introduction

Chapter 5 The Orientation and Stress Tensors Introduction The topic of tensors typically produces significant anxiety for students of structural geology. That is due, at least in part, to the fact that

### Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure

### A short review of continuum mechanics

A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material

### Chapter 3 Vectors. 3.1 Vector Analysis

Chapter 3 Vectors 3.1 Vector nalysis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Coordinate Systems... 6 3.2.1 Cartesian Coordinate System... 6 3.2.2 Cylindrical Coordinate

### Before you begin read these instructions carefully:

NATURAL SCIENCES TRIPOS Part IB & II (General Friday, 30 May, 2014 9:00 am to 12:00 pm MATHEMATICS (2 Before you begin read these instructions carefully: You may submit answers to no more than six questions.

### CURVATURE AND RADIUS OF CURVATURE

CHAPTER 5 CURVATURE AND RADIUS OF CURVATURE 5.1 Introduction: Curvature is a numerical measure of bending of the curve. At a particular point on the curve, a tangent can be drawn. Let this line makes an

### Constitutive Equations

Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module

### 19. Principal Stresses

19. Principal Stresses I Main Topics A Cauchy s formula B Principal stresses (eigenvectors and eigenvalues) C Example 10/24/18 GG303 1 19. Principal Stresses hkp://hvo.wr.usgs.gov/kilauea/update/images.html

### Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch.

HOMEWORK From Dieter -3, -4, 3-7 Module #3 Transformation of stresses in 3-D READING LIST DIETER: Ch., pp. 7-36 Ch. 3 in Roesler Ch. in McClintock and Argon Ch. 7 in Edelglass The Stress Tensor z z x O

### STRESS TENSOR 3.1. STRESSES

3 STRESS TENSOR The definitions of stress vector and stress components will be given and the equations of equilibrium will be derived. We shall then show how the stress components change when the frames

### Lecture Triaxial Stress and Yield Criteria. When does yielding occurs in multi-axial stress states?

Lecture 5.11 Triaial Stress and Yield Criteria When does ielding occurs in multi-aial stress states? Representing stress as a tensor operational stress sstem Compressive stress sstems Triaial stresses:

### Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

### In this section, mathematical description of the motion of fluid elements moving in a flow field is

Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

### Elements of Rock Mechanics

Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

### Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems

Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Bishakh Bhattacharya & Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 19 Analysis of an Orthotropic Ply References

### Chapter 3. Forces, Momentum & Stress. 3.1 Newtonian mechanics: a very brief résumé

Chapter 3 Forces, Momentum & Stress 3.1 Newtonian mechanics: a very brief résumé In classical Newtonian particle mechanics, particles (lumps of matter) only experience acceleration when acted on by external

### Is there a magnification paradox in gravitational lensing?

Is there a magnification paradox in gravitational lensing? Olaf Wucknitz wucknitz@astro.uni-bonn.de Astrophysics seminar/colloquium, Potsdam, 26 November 2007 Is there a magnification paradox in gravitational

### 7.4 The Elementary Beam Theory

7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be

### Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor

Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),

### Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

### Phys 7221 Homework # 8

Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with

### SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

### Chapter 10. Rotation of a Rigid Object about a Fixed Axis

Chapter 10 Rotation of a Rigid Object about a Fixed Axis Angular Position Axis of rotation is the center of the disc Choose a fixed reference line. Point P is at a fixed distance r from the origin. A small

### THE CAUSES: MECHANICAL ASPECTS OF DEFORMATION

17 THE CAUSES: MECHANICAL ASPECTS OF DEFORMATION Mechanics deals with the effects of forces on bodies. A solid body subjected to external forces tends to change its position or its displacement or its

### ARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13

ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem

### Week 7: Integration: Special Coordinates

Week 7: Integration: Special Coordinates Introduction Many problems naturally involve symmetry. One should exploit it where possible and this often means using coordinate systems other than Cartesian coordinates.

### Chapter 3. Load and Stress Analysis

Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3

### Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

### MATH 332: Vector Analysis Summer 2005 Homework

MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

### 14. Rotational Kinematics and Moment of Inertia

14. Rotational Kinematics and Moment of nertia A) Overview n this unit we will introduce rotational motion. n particular, we will introduce the angular kinematic variables that are used to describe the