Mom Notes

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FE Review Mechanics ofMaterials

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_materials/fe_handbook

Multimedia learning material web site:http://web.umr.edu/~mecmovie/index.html

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Normal Stress (normal to surface)

Shear Stress (along surface)

First Concept Stress


Normal Strain length change

Mechanical

Thermal

Shear Strain angle change

Second Concept Strain

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Material Properties Hookes Law

Normal (1D)

Normal (3D)

Shear


Material Properties

Poissons ratio

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Axial Loading

Stress

Deformation

FF

PLAE

=

x PA

= F x


Torsional Loading

Stress

Deformation TL

JG

=

TJ=

TT

maxTcJ

=

max

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Bending Stress

Stress

Find centroid of cross-section

Calculate I about the Neutral Axis

rx

M yI

= max rM cI

=

MM

x


Transverse Shear Equation

ave V

A = Average over entire cross-section

ave VQIb =Average over line

V = internal shear force

b = thickness

I = 2nd moment of area

Q = 1st moment of area of partial section

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Partial 1st Moment of Area (Q)


Max. Shear Stresses on SpecificCross-Sectional Shapes

Rectangular Cross-Section

max 3

2VA

=

Circular Cross-Section

max 4

3VA

=

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Max. Shear Stresses on Specific

Cross-Sectional ShapesWide-Flange Beam

maxweb

VA


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V & M Diagrams

dVw

dx=V

M dMVdx

=


Six Rules for Drawing V & MDiagrams

1. w = dV/dx

The value of the distributed load at any point in the beam is equal to theslope 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 ofthe bending moment curve.

3. The shear force curve is continuous unless there is a point force on thebeam. The curve then jumps by the magnitude of the point force (+ forupward force).

4. The bending moment curve is continuous unless there is a point moment onthe 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 isapplied at the end.

6. The bending moment will be zero at each end of the beam unless a pointmoment is applied at the end.

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Deflection Equation2

2d y M

EIdx=

y = deflection of midplane

M = internal bending moment

E = elastic modulus

I = 2nd moment of area withrespect to neutral axis

To solve bending deflection problems (find y):

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

2. Integrate it twice3. Apply boundary conditions

4. Apply matching conditions (if applicable)


Method of Superposition

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Stress TransformationPlane Stress Transformation Equations:

cos2 sin22 2

x y x yn xy

+ = + +

sin2 cos22

x yxynt

= +xy

x

y


Stress Transformation

Principal Stresses:

2

21, 2 2 2 xy

x y x yp p

+ = + +

( )tan 2

2

xy

px y

=

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Stress TransformationMax Shear Stress:

1 2max

2p p = 1max

2p = 2max 2

p

=


Stress Transformation

Mohrs Circle

C

( ),x xy

( ),y xy

R

xy

x

y

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Combined LoadingWe have derived stress equations for fourdifferent loading types:

x

P

A =

max

Vk

A =


x

Mc

I =

xMc

I = +

Tc

J=

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Method for Solving Combined

Loading Problems1. Find internal forces and moments at

cross-section of concern.

2. Find stress caused by each individualforce and moment at the point inquestion.

3. Add them up.


Thin-Walled Pressure Vessels

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Column Buckling


Y

Y

Y

Y

Failure occurs when:

1p Y

>

where p1 is the largest principal stress.

if p1

and p2

have the same sign

1 2p p Y > if p1 and p2 have different signs

p1

p2

Maximum Shear Stress Theory

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Y

Y

Y

Y

Failure occurs when:2 2 2

1 1 2 2p p p p Y + >

p1

p2

Maximum Distortion Energy

TheoryThis theory assumes that failureoccurs when the distortionenergy of the material isgreater than that which causesyielding in a tension test.


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