Review of Mechanics of Materials

Stresses on Prismatic Bars

Review of Mechanics of Materials

Cylindrical Thin-Walled Pressure Vessels

Circumferential (Hoop) Stress

Meridional Stress

p

cσ

L

p mσ

Force pressing on section = p2rL

Internal force developed to oppose = σc(2tL)

tpr

c =σ

Force expanding section = pπr2

Internal force developed to oppose = σm(2πrt)

tpr

m 2=σ

Review of Mechanics of Materials

Spherical Thin-Walled Pressure Vessels

Force pressing on the section = Force expanding section = pπr2

Internal force developed to oppose = σm(2πrt)

tpr

mc 2=σ=σ

Resultant

Resultant stress

Review of Mechanics of Materials

Airbus Fuselage Section

The cross section of an airbus 380 aircraft is almost a cylindrical thin wall structure. Note the 3 deck structure.

Review of Mechanics of Materials

Collapsed Cylindrical Tank

Review of Mechanics of Materials

First Moment of AreaThe first moment of area about any axis is given by the summation of the first moments of all the elemental areas.

xx

y

dA

∫ ∫== ydAdQQ xx ∫ ∫== xdAdQQ yy

Review of Mechanics of Materials

Centroid• The centroid of an area is simply the point at which the area might be considered to be concentrated. • The centroid is used in connection with geometry • The center of gravity is used in connection with physical bodies

AQ

A

xdAx y== ∫

AQ

A

ydAy x== ∫

Review of Mechanics of MaterialsCentroids of Common Shapes

Review of Mechanics of Materials

Finding the centroid from a part

Incorrect positions of centroidscan cause serious prevent vibration problems in rotating partsOne method to determine the centroid of machined aviation parts is to record the image and use first moment calculationsAs the calculations are very involved, it is necessary to use computers for the task

Review of Mechanics of Materials

Second Moment of AreaThe second moment of an area is the sum of a number of terms each consisting of an area multiplied by a distance squared.

xx

y

dA

∫=A

x dAyI 2 ∫=A

y dAxI 2 ∫ +=∂=A

yxo IIArJ 2

Review of Mechanics of Materials

The radius of gyration represents the distance from axis to point where a concentrated area can be placed and have the same second moment of area with respect to the given area.

Radius of Gyration

xx

y

dA

I x A AkyA

y= =∫ 2 2∂I y A AkxA

x= =∫ 2 2∂

Review of Mechanics of Materials

Second Moment of Common Shapes

Review of Mechanics of Materials

Computer aided design

Computer aided design (CAD) is now always used to design aircraft components.Most CAD software have features to calculate the centroid and second moment of area about any axis.

Review of Mechanics of Materials

Parallel Axis TheoremWhen the second moment of an area with respect to an axis is known, the second moment with respect to a parallel axis can be obtained by the parallel-axis theorem.

x’x”

y”A

y’

x

y

AyII xx2

' )"(+= AxII yy2

' )"(+= AdJJ oo2

' )(+=

Review of Mechanics of Materials

Product Moment of AreaThe product moment of area for the elemental area which is located at point x, y is given by

x’x”

y”A

y’

x

y

∫= Axy xydAI

The parallel-axis theorem may be applied to determine the product moment of area at some other set of x-y axis

""'' yAxII xyyx +=

Review of Mechanics of Materials

Second Moment of Area About An Inclined Axis (1)

x

y

x

y

dA

y’

x’

Establish the x,y axis for the area and determine xI , yI and xyI

Mark off the coordinates of points X( xyx II , ) and Y( xyy II −, ).

Join XY. XY cuts the horizontal axis at O

With OX or OY as radius, draw the circle

Review of Mechanics of Materials

Second Moment of Area About An Inclined Axis (2)

x

y

x

y

dA

y’

x’

• The radius of the circle is Ixy

• The direction of rotation of the circle is the same as that of the element

• Rotation of the circle is twice of that in the element