ECIV 520 A Structural Analysis II

Lecture 4 – Basic Relationships

Process of Matrix/FEM Analysis

Accurate Approximations of Solutions

Reliability of Solution depends on choice of Mathematical Model

Process of Matrix/FEM Analysis

Reliability of Solution depends on choice of Mathematical Model

Process of Matrix/FEM Analysis

Accurate Approximations of Solutions

Process of Matrix/FEM Analysis

Process of Matrix/FEM Analysis

Accurate Approximations of Solutions

Reliability of Solution depends on choice of Mathematical Model

StressResultant Force and Moment represent the resultant effects of the actual distribution of force acting over sectioned area

Stress

AssumptionsMaterial is continuousMaterial is cohesive

Force can be replaced by the three componentsFx, Fy (tangent) Fz (normal)

Quotient of force and area is constantIndication of intensity of force

Consider a finite but very small area

Normal & Shear Stress

Normal StressIntensity of force acting normal to A

Shear StressIntensity of force acting tangent to A

dA

dF

A

F zz

Az

0

lim

dA

dF

A

F xx

Azx

0

limdA

dF

A

F yy

Azy

0

lim

General State of Stress

Set of stress components depend on orientation of cube

Basic Relationships of Elasticity Theory

Concentrated

Distributed on Surface

Distributed in Volume

Equilibrium

Equilibrium

Write Equations of Equilibrium

Fx=0

Fy=0

Fz=0

EquilibriumEquilibrium - X

X

dAdzz

dA

dAdzz

dA

dAdxx

dA

xzxz

xz

xyxy

xy

xx

x

EquilibriumEquilibrium

Write Equations of Equilibrium

Fx=0

Fy=0

Fz=0

Boundary Conditions

Prescribed Displacements

Boundary Conditions

Equilibrium at Surface

Deformation

Intensity of Internal Loads is specified using the concept of

Normal and Shear STRESS

Forces applied on bodies tend to change the body’s

SHAPE and SIZE

Body Deforms

Deformation

• Deformation of body is not uniform throughout volume

• To study deformational changes in a uniform manner consider very short line segments within the body (almost straight)

Deformation is described by changes in length of short line segments and the

changes in angles between them

Strain

Deformation is specified using the concept of

Normal and Shear STRAIN

Normal Strain - Definition

Normal Strain: Elongation or Contraction of a line segment per unit of length

s

ss

'

avgε

s

ssAB

'

n along limε

ss ε1'

Normal Strain - Units

Dimensionless Quantity: Ratio of Length Units

Common Practice

SI

m/mm/m (micrometer/meter)

US

in/in

Experimental Work: Percent

0.001 m/m = 0.1%

=480x10-6: 480x10-6 in/in 480 m/m 480 (micros)

Shear Strain – DefinitionShear Strain: Change in angle that occurs between two line

segments that were originally perpendicular to one another

(rad) 'lim2

talongACnalongAB

nt

negative 2

'

positive 2

'

nt

nt

Cartesian Strain Components

zz

yy

xx

ε1

ε1

ε1

zx

yz

xy

2

2

2

Normal Strains: Change VolumeShear Strains: Change Size

Small Strain Analysis

Most engineering design involves application for which only small deformations are allowed

DO NOT CONFUSE Small Deformations with Small Deflections

Small Deformations => <<1

Small Strain Analysis: First order approximations are made about size

Strain-Displacement Relations

AssumptionSmall Deformations

For each face of the cube

Stress-Strain (Constitutive)Relations

Isotropic Material: E,

Generalized Hooke’s Law

Stress-Strain (Constitutive)Relations

Note that:

Equations (a) can be solved for

Stress-Strain (Constitutive)Relations

Or in matrix form

Stress-Strain (Constitutive)Relations

xy

xz

yz

z

y

x

xy

xz

yz

z

y

x

Stress-Strain: Material Matrix

Material Matrix

Special Cases

One Dimensional: v=0No Poisson Effect

= E

Reduces to:

Special CasesTwo Dimensional – Plane StressThin Planar Bodies subjected to in plane loading

Special CasesTwo Dimensional – Plane Strain

Long Bodies Uniform Cross Section subjected to transverse loading

Special Cases

For other situations such as inostropy obtain the appropriate material matrix

Two Dimensional – Plane Stress Orthotropic Material

Dm=

Strain EnergyDuring material deformation energy is stored (strain energy)

yxAF zL

V

zyx

zFU

2

1

2

1

2

1

Strain Energy Density

2

1

V

Uu

e.g. Normal Stress

Strain Energy

In the general state of stress for conservative systems

dVUV

T εσ2

1

Principle of Virtual Work

PFdxWPx

2

1

0

Load Applied Gradually Due to another Force

'PWP

Apply Real LoadsApply Virtual Load

PVW Concept

Real

Deformns

udLP '

Internal

Virtual

Forces

Principle of Virtual Work

A body is in equilibrium if the internal virtual work equals the external virtual work for every

kinematically admissible displacement field

Principle of Virtual Work

V

zx

yz

xy

z

y

x

zxyzxyzyx

V

Tie

dV

dVUW

εσ