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 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) + 70.7 lbf (y'-direction) " 45 degs +y" 60 deg = 86.6 lbf (x"-direction) + 50 lbf (y"-direction) 2
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
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
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
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
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
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
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
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
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
Stress Element Jiangyu Li,
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
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
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
Stress Sign Conventions +y +y σxx τxy σxx σxx τxy σxx All Stresses Positive σxx and τxy Negative Positive 15
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
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
Uniaxial Stress If only one normal stress exists, the (3-D) state of stress is called a "uniaxial stress" +y +y +z 16
ree Body Diagram Defines the Coordinate System Prof. M. E. Tuttle "cut" +y +y 17
ree Body Diagram Defines the Coordinate System y' y' "cut" P V x' σy'y' τx'y' x' 18
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
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
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
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
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 2 2 2 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,
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,
Three-Dimensional Stress Jiangyu Li,
"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
Morh s Circle Jiangyu Li,
Mohr s Circle 2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning is a trademark used herein under license. Jiangyu Li,
Assignment Mechanical behavior of materials HW 6.1, 6.5 Jiangyu Li,