- Stress analysis

Index

Title Page

-Overview on stress analysis 2

-Stress strain curve 4

-Mohr's circle 5

-Cylinder stress

( Thin & Thick vessel )8

-References 10

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- Stress analysis

Stress analysis

Overview:

Stress analysis is the general term used to describe analyses of the results quantities of the stresses and strains and it's related to the strength, stiffness, and life expectancy of the sample

Definitions:

Stress: is "force per unit area" or the ratio of applied force F and cross section - defined as "force per area."

tensile stress: stress that tends to stretch or lengthen the material - acts normal to the stressed area

compressive stress:  stress that tends to compress or shorten the material - acts normal to the stressed area

shearing stress:  stress that tends to shear the material - acts in plane to the stressed area

Tensile or Compressive Stress - Normal Stress

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Tensile or compressive stress normal to the plane is usually denoted "normal stress" or "direct stress" and can be expressed as

σ = Fn / A        

where

σ = normal stress ((Pa) N/m2, psi)

Fn = normal component force (N, lbf )

A = area (m2)

Shear Stress:Stress parallel to the plane is usually denoted "shear stress" and can be expressed as

τ = Fp / A        

where

τ = shear stress ((Pa) N/m2)

Fp = parallel component force (N, lbf)

A = area (m2)

Strain:Strain is defined as "deformation of a solid due to stress" and can be expressed as

ε = dl / lo

   = σ / E        

where

dl = change of length (m, in)

lo = initial length (m, in)

ε = unit less measure of engineering strain

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- Stress analysis

E = Young's modulus (Modulus of Elasticity) (N/m2 (Pa), lb/in2 (psi))

(Young's modulus can be used to predict the elongation or compression of an object)

E = stress / strain

   = σ / ε 

   = (Fn / A) / (dl / lo)      

Shear Modulus:S = stress / strain

   = τ / γ 

   = (Fp / A) / (s / d)        

Where

S = shear modulus (N/m2) (lb/in2, psi)

τ  = shear stress ((Pa) N/m2, psi)

γ = unit less measure of shear strain 

Fp = force parallel  to the faces which they act

A = area (m2, in2)

s = displacement of the faces (m, in)

d = distance between the faces displaced (m, in)

Stress Strain curve : It describes the relationship between the stress and strain that a particular material displays is known as that particular material's stress–strain curve. It is unique for each material and is found by recording the amount of deformation (strain) at distinct intervals of tensile or compressive loading (stress)

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- Stress analysis

Yield strength: the proportional limit.  For metals, it is often, but not always the same in tension and compression at which the stress-strain curve becomes non-linear

The   ultimate strength : is the maximum engineering stress on the

Sample

Mohr's circleMohr's circle is a geometric representation of the 2-D transformation of stresses. The Mohr circle is then used to determine graphically the stress components acting on the system

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- Stress analysis

Stress transformation equations

Steps of drawing Mohr's circle:

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- Stress analysis

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- Stress analysis

Cylinder stress A cylinder stress is a stress distribution with rotational symmetry;

that is, which remains unchanged if the stressed object is rotated about some fixed axis.

Cylinder stress patterns include:

Circumferential stress or hoop stress, a normal stress in the tangential direction;

Axial stress, a normal stress parallel to the axis of cylindrical symmetry;

Radial stress, a stress in directions coplanar with but perpendicular to the symmetry axis.

Thin-walled pressure vessel theory:

An important practical problem is that of a cylindrical or spherical object which is subjected to an internal pressure p. Such a component is called a pressure vessel

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- Stress analysis

 (for a cylinder)

 (for a sphere)where

P is the internal pressure t is the wall thickness r is the mean radius of the cylinder.  is the hoop stress.

When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.

Though this may be approximated to

Thick-walled vessels :

A and B are constants of integration, which may be discovered from the boundary conditions

r is the radius at the point of interest

"If   then   and a solid cylinder cannot have an internal pressure so "

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- Stress analysis

References:1. FEA Concepts J.E. Akin 2. NC state university 3. Solid Mechanics Part I Kelly4. http://inventor.grantadesign.com/ 5. http://www.engineeringtoolbox.com/ 6. https://en.wikipedia.org

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