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2-D STRESS ANALYSIS2-D STRESS ANALYSIS • Introduction to 2-D Stress Analysis• 2D and 3D Stresses and Strains• von Mises and Tresca Yield Criteria• 2-D Stress and Strain Results in

NASTRAN

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Introduction to 2-D Stress Introduction to 2-D Stress Analysis Analysis

Up until now, we have only used rod shaped structural finite elements. These elements have been used to model framed or articulated structures such as trusses and frames.

We are now going to begin working with two dimensional (planar) triangular (3 noded) or quadrilateral (4 noded) elements defined in a two dimensional plane to model 2-D dimensional elastic solid mechanisms and structures.

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Introduction to 2-D Stress AnalysisIntroduction to 2-D Stress AnalysisQuestion: Why 2-D Stress Analysis?

Answer: In complex shaped structural or machine components stress concentration effects are often important factors that must be considered in the design process

Note: A 1-D Beam or Truss (rod) analysis of the tension member shown below ignores the stress concentration in the fillet section and at the hole.

Tension member

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Introduction to 2-D Elastic Stress AnalysisIntroduction to 2-D Elastic Stress AnalysisTwo-dimensional stress analysis allows the engineer to

determine detailed information concerning deformation, stress and strain, within a complex shaped two-dimensional elastic body.

Assumptions• Deformations and strains are very small• Material behaves elastically – stress and strain

are related by Hooke’s Law.• Hooke’s Law is a matrix equation relating 3

normal stresses and one shear stress to 3 normal strains and one shear strain }{]D[}{or}]{D[}{ 1

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Introduction to 2-D Elastic Stress AnalysisIntroduction to 2-D Elastic Stress Analysis

2-D Stress analysis allows the engineer to model complex 2-D elastic bodies by discretizing the geometry with a mesh of finite elements.

Modeled as

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Introduction to 2-D Stress AnalysisIntroduction to 2-D Stress Analysis

Each finite element will deform because of the applied loading and boundary conditions on the body.

Applied loading – generates

deformation and stresses

Each node displaces in x and z direction

x

z

u2x

u2z

1

2

3

4

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Introduction to 2-D Stress AnalysisIntroduction to 2-D Stress AnalysisEvery point within the element may translate in

the x and z direction P(x,z) -> P’(x+ux, z+uz)

Note: ux and uz: x- and z-displacements are bi-linear functions of x and z.

where a0, a1,.. b3 are unknown constants

x

z

ux

uz

P

P’

xzbzbxbb)z,x(uxzazaxaa)z,x(u

3210z

3210x

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Introduction to 2-D Stress AnalysisIntroduction to 2-D Stress AnalysisIn 1-D analysis assuming small displacements

and elastic material behavior we have:

a) A strain-displacement eqn.,

and, b) a stress-strain eqn. (1D Hooke’s Law)

Question: What is the 2D or 3D form of these equations?

dxdu

LLL

0

0

E

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2-D Strain-Displacement Equations2-D Strain-Displacement Equations

xux

x

z

uzz

There are two normal strains and :x z

xu

zu zx

xz

and one shear strain :xz

x

z

x

x

z

z

x

z

xz

Below are examples of simple constant strain stateswhere only one strain component is non-zero!

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2-D Strain-displacement Equations2-D Strain-displacement Equations

Question: What are the variations of the strain components for a element where the displacement field is defined by:

?

Note: ux and uz: x- and z-displacements are bi-linear functions of x and z.

where a0, a1,.. b3 are unknown constants

x

z

ux

uz

P

P’

xzbzbxbb)z,x(uxzazaxaa)z,x(u

3210z

3210x

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3D Isotropic Stress/Strain 3D Isotropic Stress/Strain LawLaw

Three-dimensional Hooke’s Law: stress/strain relationships for an isotropic material

x

z

yσy

σy

σz

σz

σxσx

As you recall, an isotropic body can have normal stresses acting on each surface: σx, σy, σz

When the only normal stress is σx this causes a strain along the x- axis according to Hooke’s Law

Ex

x

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x

x

3D Isotropic Stress/Strain 3D Isotropic Stress/Strain LawLaw

Note, that a tensile stress in the x direction, produces a negative strains in the y and z directions This is called the Poisson effect.

These negative strains are computed via:

where:

E is Young’s Modulus ν is Poisson’s ratio

Ex

zy

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3D Isotropic Stress/Strain 3D Isotropic Stress/Strain LawLaw

Since the material is isotropic, application of normal stresses in the x, y, and z directions generates, a total normal strain in the x direction:

+ +

EEEzyx

x

x

x

yz

y z

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3D Isotropic Stress/Strain 3D Isotropic Stress/Strain LawLaw

The total normal strains in the y and z directions can be determined in a similar manner:

EEE

EEE

EEE

zyxz

zyxy

zyxx

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3D Isotropic Stress/Strain 3D Isotropic Stress/Strain LawLaw

Rearranging the above equations and yields 3 equations relating normal stresses and strains :

These equations can also be written in matrix notation: {σ}=[D]{ε}

])1([)21)(1(

])1([)21)(1(

])1([)21)(1(

zyxz

zyxy

zyxx

E

E

vE

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Shear Stress/Strain RelationshipsShear Stress/Strain RelationshipsHooke’s law also applies for shear stress and

strain: τ=Gγ where G is the shear modulus, τ is a shear stress, and γ is a shear strain. For 3-D this results in a further 3 equations.

x

z

yσy

σy

σz

σz

σxσxτxy

τyxzxzxzx

yzyzyz

xyxyxy

G2G

G2G

G2G

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Introduction to 2-D Stress Introduction to 2-D Stress Analysis Analysis

2-D planar elements are used to model complex 2-D geometries. They must connect at common nodes to form continuous structures. They are extremely important in the following analysis types:

Plane Stress Plane Strain Axisymmetric

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Plane StressPlane StressPlane Stress, in NASTRAN, is defined to be a state of stress

in which the normal and shear stresses perpendicular to the x-z plane are zero, the y-thickness is very small, and the constraints (TX,TZ) and loads act only in the x-z plane and throughout the y-thickness.

•σy = 0

• τxy= 0, τyz=0

• ‘thickness’, y dimension, is very small compared to x and z dimensions

•Loads act only in the x-z plane and throughout the y-thickness

y

x

z

F1

F3

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Plane StrainPlane StrainPlane Strain, in NASTRAN, is defined to be a state of strain

in which the normal strain in the y-direction, εy and the shear strains, γxy and γyz are zero. Note, the y-thickness of the body is very large, and constraints and loads act in x-z plane throughout thickness.

y x

z

•The ‘thickness’, y-dimension of the body is very large (“infinite”). Note, the NASTRAN finite element model analyzes only a ‘unit’ thickness

•Loads and constraints act only in the x-z plane through a unit y-thickness

• Forces are defined as force per unit y-length

•A plane stress state, where y is a very large value, does not approximate plane strain conditions!

• εy = 0, γxy= 0, γyz =0

F

F

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Axisymmetric AnalysisAxisymmetric AnalysisAxisymmetric conditions exist when the problem

geometry is such that the z-axis is an axis of symmetry, and the displacement is radially symmetric (independent of In this case the only non-zero shear strain, is γxz. Note, that loads and constraints act over a ring defined by 0 < in the x-z plane.

The only shear strain that is non-zero is γxz

A ‘thickness’of 1 radian in the direction around the z-axis is modeled by NASTRAN

z

x

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von-Mises Stressvon-Mises StressA stress quantity that is proportional to the the strain

energy density associated with a change in shape (with a zero volume change) at a material point is the von-Mises stress which is defined by:

NOTE: the von-Mises stress is a scalar measure of the stress state (the normal and shear stresses) at any point within a body

)(6)()()(2

1 2zx

2yz

2xy

2xz

2zy

2yxvm

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von Mises Yield von Mises Yield CriterionCriterion

The von Mises criterion is an experimentally based law that can be used to determine whether the stress state in a material causes plastic flow (or yielding). Note, the von Mises criterion is based on the strain energy density associated with a change in shape (with a zero volume change) at a material point.

The criterion simply states that when: the material point is

elastic the material point is

yielding

Y Y

vm

vm

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Tresca Yield CriterionTresca Yield Criterion Tresca yield criterion is another material model

which may be used to determines whether a stress state in a material causes yielding.

The criterion simply states that when: the material point is elastic the material point is yielding

YY

Tresca

Tresca

3 2 1Tresca

2

τ 31max

maxTresca 2

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2D Stress Analysis Results in 2D Stress Analysis Results in NASTRANNASTRAN

After analyzing an NASTRAN model, results in superview can be found:

• With displacement just as we did with beam and truss models

• In results von Mises• In results Stress tensor Global Based

– yy is the normal stress in y direction– zz is the normal stress in the z direction– yz is the in-plane shear stress