1' Stress and Strain

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1. Stress and Strain

• ENGR 310 Mechanics of Materials Fall, 2009
• Tomasz Arciszewski

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Hierarchy of Sciences

• Physics - focus on the relationships between the
  properties of matter and energy
• Mechanics - a sub-domain of physics, focus on the
  action of forces on bodies or fluid that are at
  rest or in motion
• Applied/Engineering Mechanics - a sub-domain of
  mechanics, focus on engineering applications
• Mechanics of Materials - a sub-domain of applied
  mechanics, focus on the relationships between the
  external loads applied to a deformable body and
  the intensity of internal forces acting within
  the body

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Mechanics of Materials

• An engineering science dealing with the modeling
  of behavior (analysis) of structural members
  considering
• External loads
• Internal forces and stresses
• Deformations and strains
• Stability

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Statics versus Mechanics of Materials

• Statics - focus on a rigid body, determination of
  forces applied to this body
• Mechanics of Materials - focus on a deformable
  body, determination of its behavior

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Rigid Versus Deformable Body
Rigid body AB distance constant Deformable body
AB distance changes when external loads are
applied
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Statics versus Mechanics of Materials

• Statics - an outside look, a global view
• A system of forces and couples applied to a given
  rigid body
• A system of forces of interactions in a given
  configuration of rigid bodies
• Mechanics - an inside look, a local view
• Internal forces (inside a given member)
• Deformations at a given point

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External Loads

• A system of forces, couples of forces, surface
  forces, temperature field, etc. applied to a
  given deformable body

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Surface Forces

• ... are forces of interaction between two bodies
  which are distributed over a contact surface

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External Forces
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Idealized Surface Forces

• Concentrated force a force applied at a point
• Linear distributed load
• a system of forces distributed along a line
• described by the distributed loading function
  (curve) w(s)
• with a resultant (resultant force) FR equivalent
  to the area under w(s) applied at the centroid of
  the loading curve

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Body Forces (Mass Forces)

• Forces acting on a given body without any direct
  contact with another body and distributed through
  the body
• Gravity forces (weights) caused by the field of
  gravity
• Earthquake forces caused by the movements of the
  entire structural system
• Forces caused by electromagnetic field
• Applied at the centroids of the individual bodies
  (structural members)

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Support Reactions

• Forces of interaction between a given body and
  its supports
• In general, they are surface forces
• Usually, they are idealized as concentrated
  forces and couples of forces
• Support translations are prevented by forces
  (reactant forces, reactions)
• Support rotations are prevented by couples of
  forces (reactant couples of forces, reactions)

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Supports Connections their Idealization
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Hinges
are connections between structural members
which do not prevent the relative rotation of
connected members and transfer only forces of
interactions
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Conditions of Equilibrium

• A body is in equilibrium when both the resultant
  force and the resultant couple are equal to zero,
  in vector terms
• ?F 0 and ?Mo0
• A balance of forces and a balance of moments
  occurs, in scalar terms
• ?Fx 0, ?Fy 0, ?Fz 0
• ?Mx0, ?My0, ?Mz0 or
• ?Fx 0, ?Fy 0, ?Mz0 for a planar system

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Free Body Diagram

• A graphical representation (visualization) of all
  necessary and sufficient information about a
  given member or a structural system to use
  conditions of equilibrium for various analytical
  purposes. It contains
• A representation of a given member
• External loads and their locations
• Reactions and their locations

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Method of Sections

• Construction of imaginary sections, or cuts,
  through the various parts of a given solid body
• Opening by cuts a given body to reveal the
  distribution of forces of interaction between two
  parts of a body, which balance external loads
• Conditions of equilibrium allow the determination
  of resultants of forces of interaction in the
  form of resultant force FR and resultant moment
  MRo at any specific point O

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Forces of Interaction Conditions of Equilibrium

• Conditions of equilibrium are necessary and
  sufficient to determine resultants of forces of
  interaction
• Conditions of equilibrium are insufficient to
  determine the distribution of forces of
  interaction

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Section, Internal Loading Internal Forces at a
Point
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Cross Section

• A concept related to the analysis of structural
  members
• A section perpendicular to the longitudinal axis
  of a given member
• It is usually a vertical section for horizontally
  positioned beams
• Usually, the point O is located at the centroid
  of a given cross section

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Components of Resultant Force Moment

• Both vectors can be resolved into components
  normal and tangent to the section
• Vector MRO is resolved into
• M - called bending moment and tangent to the
  plane
• T - called torsional moment and normal to the
  plane
• Vector FR is resolved into
• N - called normal force and normal to the plane
  
• V - called shear force and tangent to the plane

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Normal and Tangent Components
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Right-Hand Rule

• Use your right-hand curled hand
• The thumb gives the arrowhead sense of the vector
• The fingers show the tendency to rotate

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Coplanar Loading

• A body is subjected to a coplanar system of
  forces (loaded in a single, usually vertical
  plane)
• Only normal forces, shear forces and bending
  moments exist in all cross sections
• It is the main focus of our course (sorry)

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Coplanar Loading Internal Forces
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Limitation of Conditions of Equilibrium

• Conditions of equilibrium are necessary and
  sufficient to determine resultants of forces of
  interaction
• Conditions of equilibrium are insufficient to
  determine the distribution of forces of
  interaction

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Mechanics of Materials Main Focus

• Determination of distribution of forces of
  interaction over a section of a deformable body

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Material Assumptions

• Continuous - consists of continuum, or uniform
  distribution of matter with no voids
• Cohesive - all portions connected together,
  behaves as a single piece of matter (body)
• Deformable - distance between two points changes
  when loading applied

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Resultant Force and Moment
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Distribution of Forces of Interaction

• Section divided into a very large number of very
  small but finite ?A area
• A finite yet very small force ?F acts on ?A area
• ?F is resolved into ?Fz (normal) ?Fy (tangent)
  ?Fz (tangent) components

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Stress at a Point

• When ?A approaches zero, the ratio ?F to ?A
  approaches a finite limit called STRESS AT A
  POINT
• ? lim ?F/?A
• when ?A ? 0
• The stress vector acts along the line of action
  of ?F
• Sometimes called traction vector

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Stress at a Point

• Stress at a point is a measure of intensity of
  the internal forces (forces of interaction) on a
  specific plane passing through this point
• Fundamental concept of mechanic of materials, of
  structural engineering, and of life in general

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Stress Resolution

• Stress at a point is a vector
• It can be resolved into three perpendicular
  components acting along x, y, and z axes
• Z axis is normal to the section
• X and y are in the plane of the section

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Normal Stress at a Point

• ?z lim ?Fz/?A
• when ?A ? 0

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Shear Stresses

• Two components in the plane normal to z axis
• ?zx lim ?Fx/?A
• when ?A ? 0
• Vector parallel to x
• ?zy lim ?Fy/?A
• when ?A ? 0
• Vector parallel to y

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General State of Stress

• A specific point is selected
• A cubic element is cut out around the point
• Its faces are perpendicular to x, y, and z axes

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General State of Stress
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Face Considered
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Sign Convention

• Normal stress - subscript represents the axis
  normal to a given face
• Shear stress
• First subscript represents the axis to which a
  given face is normal
• Second subscript represents the axis to which
  the stress vector is parallel
• Two subscripts represent together the face (xy -
  face parallel to axes x and y)

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Units

• Stress a ratio of force to the area acted upon
• SI Units
• Newton per square meter, N/m2, Pascal, Pa (very
  small)
• MN/m2, mega Pascal, MPa
• 1 MPa 106 Pa
• 1 GPa 109 Pa
• US Customary Units
• Pound per square inch, psi
• 1 ksi 103 psi

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Average Normal Stress Assumptions

• Prismatic member
• Straight member both before and after load is
  applied
• Axial tensile force applied
• Saint-Venants Principle
• Distribution of stresses at both member ends NOT
  considered
• Homogeneous material
• Isotropic material

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Prismatic Member

• Straight (longitudinal) centroidal axis
  connecting centroids of all x-sections
• All x-sections identical in terms of
• Shape
• X-sectional area

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Saint-Venants Principle (Assumption of Flat
Sections)

• Two cases of a deformable prismatic member under
  axial loading applied at both ends
• Rigid plates at both ends, distributed loading,
  no shape change, identical deformations of all
  parts of a member, uniform distribution of normal
  stresses for all x-sections
• No plates, concentrated forces, significant shape
  changes at both ends of member, uniform
  distribution of normal stresses only in the
  central part

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Saint-Venants Principle

• At a distance equal to, or greater that the width
  of a member, the distribution of normal stresses
  at all x-sections is the same, whether the member
  is loaded by uniformly distributed forces or by
  concentrated forces. Also, the stress
  distribution is independent of the actual mode of
  application of the loads.

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Saint-Venants Principle Illustration
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Saint-Venants Principle Limitations

• The actual applied load and that used in the
  analysis must be statically equivalent
  (conditions of equilibrium are satisfied)
• The principle is incorrect for the vicinity of
  the load application points

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Axial Tensile Force
is a tensile force applied along the
longitudinal centroidal axis of a member
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Homogeneous Material
is a material which has the same physical and
mechanical properties throughout its volume, for
any point within the body
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Isotropic and Anisotropic Material

• Isotropic material has the same mechanical
  properties in all directions for any point within
  a given body (examples steel)
• Anisotropic material has different mechanical
  properties for different directions for any point
  within a given body (examples concrete, wood)

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Average Normal Stress Distribution

• Assumptions
• Constant uniform deformation
• Constant normal stress

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Average Normal Stress

• ?z P/A
• where
• ?z - average normal stress at any point on the
  x-section
• P - internal axial (centroidal) force (internal
  resultant normal force)
• A - x-sectional area of the member
• Units - psi, ksi, Pa, KPa, MPa

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Uniaxial Stress

• Conditions of equilibrium must be satisfied for
  all cubes
• Resultant forces acting on the parallel faces of
  the cube (top bottom) must be equal

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Applications of Average Normal Stress

• Both tension and compression
• When compression is considered, only short
  members (no buckling) can be properly analyzed

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Maximum Average Normal Stress

• Uniform distribution of normal stresses assumed
  for a given x-section
• Maximum average normal stress is equal to average
  normal stress
• Important from pragmatic point of view
  (dimensioning of members under axial tensile load)

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Average Normal Stress Example

• A prismatic steel member under two axial tensile
  forces F
• F 50 KN
• Circular x-section, 20 mm diameters
• Calculate normal average stress

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Average Shear Stress

• Shear stress (shear stress component) is tangent
  to the cutting section
• Planes AB and CD

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Average Shear Stress

• External force F is balanced by two internal
  resultant shear forces V (resultants of shear
  stresses)
• V force is equivalent of a stream of shear
  stresses acting on a given x-section (resultant)
• Uniform shear stress distribution is assumed
• Pure shear, simple or direct shear, occurs only
  in simple connections

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Average Shear Stress

• ?avg V/A
• where
• ?avg - average shear stress at the section
• V - internal resultant shear force at the section
  (
• A - area at the section
• Units - psi, ksi, Pa, KPa, MPa

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Single Shear

• It occurs in simple connections of two members
• Single plane of shearing
• Shear force V is equal to external force F

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Double Shear

• It occurs in simple connections of three members
• Two planes of shearing
• Shear force V is equal to half of external force
  F

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Complementary Stresses

• A stress cube is considered
• Only shear stresses in the vertical plane
  parallel to zy are show
• are a pair of equal in magnitude shear stresses
  in 2 normal planes, which are both directed to or
  from the line of intersection of their planes

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Stresses on Inclined Plane

• Prismatic member
• Axial tensile (centroidal) loading
• Cross section is considered first carrying
  average normal stress uniformly distributed
• Inclined plane is considered next carrying
  uniformly distributed s stresses
• S stresses are resolved into normal and shear
  (tangential) stresses

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Stresses on Inclined Plane

• ?n ?x cos2(?)
• ?n - (1/2)?xsin(2?)

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Structural Design Process (Designing)

• is a process which starts when needs for a
  given structural system, or for a modification of
  a given system, are realized and it ends when the
  final design, a description of a new or a
  modified system is produced. It has two major
  stages, including
• conceptual designing and
• detailed designing

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Conceptual Designing

• It is the 1st stage in the structural design
  process in which a design concept is developed.
• A design concept is an abstract description of a
  future structural system in terms of symbolic
  attributes (For example type of members, type of
  joints, type of loading)
• A design concept, an example a truss - a system
  of straight members connected by hinges and
  loaded at joints

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Detailed Designing

• It is the 2nd stage in the structural design
  process in which a design concept is converted
  into a detailed design
• A detailed design is a description of a future
  structural system in terms of numerical
  attributes (dimensions, weights, etc.)

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Detailed Designing, Major Activities

• Stress analysis
• Dimensioning, determination of x-sections of the
  individual members
• Optimization, determination of optimal x-section
  of the individual members (minimum weight, cost,
  security, etc.)

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Stress Analysis

• Determination of stresses in the individual
  structural members
• Determination if the occurring stresses are safe

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Allowable Load and F.S.

• Allowable load is the magnitude of load which can
  be safely applied to a given structural member
• Failure load is the magnitude of load which
  causes the structural failure of a given
  structural member (buckling, excessive
  deformations, collapse, fracture, etc.
• Factor of Safety, F.S. Ffail/Fallow

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Allowable Stress and F.S.

• Stresses are assumed as linearly related to loads
• Allowable stress is the magnitude of stress which
  can safely occur in a given structural member
• Failure stress is the magnitude of stress which
  causes the structural failure of a given
  structural member
• Factor of Safety
• F.S. ?fail/ ?allow or
• F.S. ?fail/ ?allow

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Simple Connections Design Assumptions

• Isotropic and homogeneous material
• Perfectly linear elastic behavior
• Small deformations

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Tension Member

• A prismatic member
• Axial tensile (centroidal) force
• Arequired P/ ?allow

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Connector subjected to Shear

• Pinned or bolted connections
• Transfer of loading through the pinn
• Friction is neglected
• Uniform distribution of shear stresses
• A P/ ?allow, A - x-section of the bolt

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Contact (Bearing) Stress

• Neglected in the book
• ?b P/(bxd)
• (b times d) is projected contact area
• Required contact area
• (bxd) P/(?b)allow

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Area to Resist Bearing

• Direct contact of two surfaces
• Uniform distribution of normal stresses
• Arequired P/(?b)allow

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Area to Resist Shear caused by Axial Load

• Shear stresses act on the shearing surface
• Contact length l to be determined
• Lrequired P/?d ?allow

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Deformation

• External loading is applied to a body
• Shape and size of a body are changed
• Deformation occurred

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Deformation

• A rubber membrane subjected to tension
• Changes in white lines
• Vertical line elongates
• Horizontal line shortens
• Inclined line changes length and rotates

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Strain

• Strain is a formal measure of deformations
• Two deformation types
• Linear deformations (elongation, contraction)
• Angular deformations (change in angle between two
  line segments originally normal)

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Normal Strain

• Consider line AD
• Initial length ?s
• Final length ?s
• Average normal strain
• ?avg (?s -?s)/?s
• Normal strain at a point
• ? lim (?s -?s)/?s
• ?s ? 0
• Normal strain is dimensionless quantity

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Shear Strain

• Consider angle CAB between lines n and t
• Initial angle ?/2, final angle ?
• Shear strain at a point (A)
• ?nt ?/2 - ?, when B ? A and C ? A

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Shear Strain Interpretation

• It measures the change in an angle
• It is measured in radians
• When ? less ?/2, positive shear strain

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Cartesian Strain Components

• A deformable body is considered
• A rectangular elements is assumed around a given
  point
• Very small initial dimensions ?x, ?y, ?z
• Initial angles ?/2

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Cartesian Strain Components

• Deformed shape a parallelepiped
• In general, for each pair of edges
• ?s ? (1 ?) ?s
• Final lengths for three sides
• ?x ? (1 ?x) ?x
• ?y ? (1 ?y) ?y
• ?z ? (1 ?z) ?z
• Final angles
• ?/2 - ?xy
• ?/2 - ?yz
• ?/2 - ?xz

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Strain Physical Interpretation

• Normal strains cause a volume change
• of a rectangular element
• Shear strains cause a change in shape

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Small (Engineering) Strain Analysis

• Strains assumed very small (deformations very
  small)
• ? ltlt 1
• ? is very small
• sin ? ?
• cos ? 1