Mechanics of Materials Lab

Transcription

1 Mechanics of Materials Lab Lecture 5 Stress Mechanical Behavior of Materials Sec Jiangyu Li Jiangyu Li,

2 orce Vectors A force,, is a vector (also called a "1 st -order tensor") The description of any vector (or any tensor) depends on the coordinate system used to describe the vector 100 lbf 100 lbf 100 lbf +y = 100 lbf (y-direction) +y' ' = 70.7 lbf (x'-direction) lbf (y'-direction) " 45 degs +y" 60 deg = 86.6 lbf (x"-direction) + 50 lbf (y"-direction) 2

3 Normal and Shear orces A "normal" force acts perpindicular to a surface A "shear" force acts tangent to a surface P = Normal orce V = Shear orce 3

4 orces Inclined to a Plane Since forces are vectors, a force inclined to a plane can always be described as a combination of normal and shear forces Inclined orce P = Normal orce V = Shear orce 4

5 Moments A moment (also called a "torque" or a "couple") is a force which tends to cause rotation of a rigid body A moment is also vectoral quantity (i.e., a 1 st -order tensor)... M M 5

6 Static Equilibrium A rigid solid body is in "static equilibrium" if it is either - at rest, or - moves with a constant velocity Static equilibrium exists if: Σ = 0 and Σ M = 0 50 lbf 50 lbf (40 lbf) +y (40 lbf) (30 lbf) (30 lbf) 60 lbf (30 lbf) (BALL ACCELERATES) (30 lbf) (NO ACCELERATION) (40 lbf) (40 lbf) 50 lbf 50 lbf 6

7 ree Body Diagrams and Internal orces An imaginary "cut" is made at plane of interest Apply Σ = 0 and Σ M = 0 to either half to determine internal forces R (= ) "cut" (or) R (= ) 7

8 ree Body Diagrams and Internal orces The imaginary cut can be made along an arbitrary plane Internal force R can be decomposed to determine the normal and shear forces acting on the arbitrary plane "cut" R (= ) P V 8

9 Stress: undamental Definitions Two "types" of stress: normal stress = σ = P/A shear stress = τ = V/A where P and V must be uniformly distributed over A P = Normal orce V = Shear orce σ = P/A τ = V/A A = Cross-Sectional Area 9

10 Distribution of Internal orces orces are distributed over the internal plane...they may or may not be uniformly distributed "cut" "cut" M σ = /A σ =? M M 10

11 Infinitesimal Elements A free-body diagram of an "infinitesimal element" is used to define "stress at a point" orces can be considered "uniform" over the infinitesimally small elemental surfaces +y +z dx dy dz 11

12 Stress Element Jiangyu Li,

13 Labeling Stress Components Two subscripts are used to identify a stress component, e.g., "σ xx " or "τ xy " (note: for convenience, we sometimes write σ x = σ xx, or σ xy = τ xy ) 1st subscript: identifies element face 2nd subscript: identifies "direction" of stress +y τ xy σ xx σ xx 12

14 Admissable Shear Stress States +y +y τ yx +y τ xy τ xy τ xy τ xy τ xy Σ = 0 ΣM = 0 (inadmissable) Σ = 0 ΣM = 0 (inadmissable) If: τ yx τ yx = τ xy Σ = 0 ΣM = 0 (admissable) 13

15 Stress Sign Conventions The "algebraic sign" of a cube face is positive if the outward unit normal of the face "points" in a positive coordinate direction A stress component is positive if: stress component acts on a positive face and "points" in a positive coordinate direction, or stress component acts on a negative face and "points" in a negative coordinate direction. 14

16 Stress Sign Conventions +y +y σxx τxy σxx σxx τxy σxx All Stresses Positive σxx and τxy Negative Positive 15

17 3-Dimensional Stress States In the most general case, six independent components of stress exist "at a point" 1 M1 2 +y +z σxx τxz τzy τxy σzz 3 M2 4 16

18 Plane Stress If all non-zero stress components exist in a single plane, the (3-D) state of stress is called "plane stress" +y +y +z τxy τxy σxx σxx σxx σxx 17

19 Uniaxial Stress If only one normal stress exists, the (3-D) state of stress is called a "uniaxial stress" +y +y +z 16

20 ree Body Diagram Defines the Coordinate System Prof. M. E. Tuttle "cut" +y +y 17

21 ree Body Diagram Defines the Coordinate System y' y' "cut" P V x' σy'y' τx'y' x' 18

22 ree Body Diagram Defines the Coordinate System Prof. M. E. Tuttle "cut" (a plane) y" Py"y" x" y" σy"y" x" Vy"z" Vy"x" τy"z" τy"x" z" z" 21

23 Stress Transformations Within a Plane Given stress components in the x-y coordinate system (σ xx, σ yy, τ xy ), what are the corresponding stress components in the x'-y' coordinate system? +y τxy σxx σy'y' τx'y'? +y' σx'x' ' θ 20

24 Stress Transformations Stress components in the x'-y' coordinate system may be related to stresses in the x-y coordinate system using a free body diagram and enforcing Σ = 0 +y +y' Σx' = 0 ' τxy τx'y' θ σxx σxx σx'x' τxy "cut" 23

25 Stress Transformation Equations By enforcing Σx' = 0, Σy' = 0, it can be shown: σ x' x' = σ xx + σ yy 2 σ y' y' = σ xx + σ yy 2 + σ xx σ yy 2 σ xx σ yy 2 cos2θ + τ xy sin2θ cos2θ τ xy sin2θ τ x'y' = σ xx σ yy 2 sin 2θ + τ xy cos2θ 22

26 Extreme Values Normal Stress 2τ xy tan 2θ p = σ x σ y σ x + σ y σ x σ y 2 σ 1,2 = ± ( ) + τ xy Shear Stress σ x σ y tan 2θ s = 2τ xy σ x σ y 2 τ 1,2 = ± ( ) + τ 2 2 xy Jiangyu Li,

27 Mohr s Circle R = σ x σ y 2 ( ) + τ 2 2 xy Shear stress tending to rotate the element clockwise are plotted above the axis Jiangyu Li,

28 Three-Dimensional Stress Jiangyu Li,

29 "Stress": Summary of Key Points Normal and shear stresses are both defined as a (force/area) Six components of stress must be known to specify the state of stress at a point (stress is a "2 nd -order tensor") Since stress is a tensoral quantity, numerical values of individual stress components depend on the coordinate system used to describe the state of stress Stress is defined strictly on the basis of static equilibrium; definition is independent of: material properties strain 27

30 Morh s Circle Jiangyu Li,

31 Mohr s Circle 2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning is a trademark used herein under license. Jiangyu Li,

32 Assignment Mechanical behavior of materials HW 6.1, 6.5 Jiangyu Li,