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FE Review – Mechanics of

Materials

FE Review Mechanics of Materials 2

Resources

You can get the sample reference book:

www.ncees.org

– main site

http://www.ncees.org/exams/study_ma

terials/fe_handbook

Multimedia learning material web site:

http://web.umr.edu/~mecmovie/index.

html

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FE Review Mechanics of Materials 3

Normal Stress (normal to surface)

Shear Stress (along surface)

First Concept – Stress

FE Review Mechanics of Materials 4

Normal Strain – length change

Mechanical

Thermal

Shear Strain – angle change

Second Concept – Strain

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FE Review Mechanics of Materials 5

Material Properties

Hooke’s Law

Normal (1D)

Normal (3D)

Shear

FE Review Mechanics of Materials 6

Material Properties

Poisson’s ratio

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FE Review Mechanics of Materials 7

Axial Loading

Stress

Deformation

FF

P

L

A

E

δ

=

∑

x

P

A

σ

=

F

σ

x

FE Review Mechanics of Materials 8

Torsional Loading

Stress

Deformation

TL

J

G

θ

=

∑

θ

J

ρ

τ

=

TT

max

Tc

J

τ

=

ρ

τ

τ

max

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FE Review Mechanics of Materials 9

Bending Stress

Stress

Find centroid of cross-section

Calculate I about the Neutral Axis

r

x

M

y

I

σ

=−

max

r

M

c

I

σ

=

M

M

σ

x

FE Review Mechanics of Materials 10

Transverse Shear Equation

ave

V

A

τ

=

Average over entire cross-section

ave

VQ

Ib

τ

=

Average over line

V = internal shear force

b = thickness

I = 2

nd

moment of area

Q = 1

st

moment of area of partial

section

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FE Review Mechanics of Materials 11

Partial 1

st

Moment of Area (Q)

FE Review Mechanics of Materials 12

Max. Shear Stresses on Specific

Cross-Sectional Shapes

Rectangular Cross-Section

max

3

2

V

A

τ

=

τ

Circular Cross-Section

max

4

3

V

A

τ

=

τ

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FE Review Mechanics of Materials 13

Max. Shear Stresses on Specific

Cross-Sectional Shapes

Wide-Flange Beam

max

web

V

A

τ

≈

τ

FE Review Mechanics of Materials 14

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FE Review Mechanics of Materials 21

V & M Diagrams

dV

w

dx

=

V

M

dM

V

dx

=

FE Review Mechanics of Materials 22

Six Rules for Drawing V & M

Diagrams

1.w = dV/dx

The value

of the distributed load

at any point in the beam is equal to the

slope

of the shear force

curve.

2.V = dM/dx

The value

of the shear force

at any point in the beam is equal to the slope

of

the bending moment

curve.

3.The shear force curve

is continuous unless there is a point force

on the

beam. The curve then “jumps” by the magnitude of the point force (+ for

upward force).

4.The bending moment curve

is continuous unless there is a point moment

on

the beam. The curve then “jumps” by the magnitude of the point moment

(+ for CW moment).

5.The shear force

will be zero at each end of the beam unless a point force is

applied at the end.

6.The bending moment

will be zero at each end of the beam unless a point

moment is applied at the end.

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FE Review Mechanics of Materials 23

Deflection Equation

2

2

d y

M

EI

dx

=

y = deflection of midplane

M = internal bending moment

E = elastic modulus

I = 2

nd

moment of area with

respect to neutral axis

To solve bending deflection problems (find y):

1.Write the moment equation(s) M(x)

2.Integrate it twice

3.Apply boundary conditions

4.Apply matching conditions (if applicable)

FE Review Mechanics of Materials 24

Method of Superposition

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FE Review Mechanics of Materials 25

Stress Transformation

Plane Stress Transformation Equations:

cos2 sin2

2 2

x y x y

n xy

σ

σ σ σ

σ

θ τ θ

+

−

= + +

sin2 cos2

2

x y

xy

nt

σ

σ

τ

θ τ θ

⎛ ⎞

⎜ ⎟

⎝ ⎠

−

=− +

τ

xy

σ

x

σ

y

FE Review Mechanics of Materials 26

Stress Transformation

Principal Stresses:

2

2

1,2

2 2

x

y

x y x y

p p

σ σ σ σ

σ

τ

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎝ ⎠

+ −

=

+ +

( )

tan 2

2

xy

p

x

y

τ

θ

σ

σ

=

−

⎛ ⎞

⎜ ⎟

⎝ ⎠

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FE Review Mechanics of Materials 27

Stress Transformation

Max Shear Stress:

1 2

max

2

p p

σ

σ

τ

−

=

1

max

2

p

σ

τ

=

2

max

2

p

σ

τ

=

FE Review Mechanics of Materials 28

Stress Transformation

Mohr’s Circle

σ

τ

C

(

)

,

x

xy

σ

τ−

(

)

,

y xy

σ

τ

R

τ

xy

σ

x

σ

y

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FE Review Mechanics of Materials 29

Combined Loading

We have derived stress equations for four

different loading types:

x

P

A

σ

=

max

V

k

A

τ

=

FE Review Mechanics of Materials 30

x

M

c

I

σ = −

x

M

c

I

σ = +

Tc

J

τ

=

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FE Review Mechanics of Materials 31

Method for Solving Combined

Loading Problems

1.Find internal forces and moments at

cross-section of concern.

2.Find stress caused by each individual

force and moment at the point in

question.

3.Add them up.

FE Review Mechanics of Materials 32

Thin-Walled Pressure Vessels

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FE Review Mechanics of Materials 33

Column Buckling

FE Review Mechanics of Materials 34

σ

Y

σ

Y

−σ

Y

−σ

Y

Failure occurs when:

1

p

Y

σ

σ>

where

σ

p1

is the largest principal stress.

if σ

p1

and σ

p2

have the same sign

1 2

p

p Y

σ

σ σ

−

>

if σ

p1

and σ

p2

have different signs

σ

p1

σ

p2

Maximum Shear Stress Theory

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FE Review Mechanics of Materials 35

σ

Y

σ

Y

−σ

Y

−σ

Y

Failure occurs when:

2 2 2

1 1 2 2

p

p p p Y

σ

σ σ σ σ

−

+ >

σ

p1

σ

p2

Maximum Distortion Energy

Theory

This theory assumes that failure

occurs when the distortion

energy

of the material is

greater than that which causes

yielding in a tension test.

FE Review Mechanics of Materials 36

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