Review of Mechanics of Materials Stresses on Prismatic Bars Review of Mechanics of Materials Cylindrical Thin-Walled Pressure Vessels

Circumferential (Hoop) Stress

Force pressing on section = p2rL

Internal force developed to oppose = σc(2tL) L pr p σc = t σc Meridional Stress

Force expanding section = pπr2

Internal force developed to oppose = σm(2πrt) p σm pr σ = m 2t 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)

pr σ = σ = c m 2t Resultan Resultan t t 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 Area

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

Q = dQ = ydA Q = dQ = xdA x ∫ x ∫ y ∫ y ∫

dA

y

x x 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

xdA Q x = ∫ = y A A

ydA Q y = ∫ = x A A Review of Mechanics of Materials Centroids of Common Shapes Review of Mechanics of Materials Finding the centroid from a part

Incorrect positions of centroids can cause serious prevent vibration problems in rotating parts One method to determine the centroid of machined aviation parts is to record the image and use first moment calculations As the calculations are very involved, it is necessary to use computers for the task Review of Mechanics of Materials Second Moment of Area

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

I = y 2 dA I = x 2 dA J = r 2∂A = I + I x ∫ y ∫ o ∫ x y A A A

dA

y

x x Review of Mechanics of Materials Radius of Gyration

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.

22 22 IyAAkx ==∫ ∂ x IxAAky ==∫ ∂ y A A

dA

y

x x 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 Theorem

When 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.

2 2 2 I x' = I x + (y") A I y' = I y + (x") A Jo' = Jo + (d) A

y’ y

x

y” A

x’ x” Review of Mechanics of Materials Product Moment of Area

The product moment of area for the elemental area which is located at point x, y is given by I xy = xydA ∫A The parallel-axis theorem may be applied to determine the product moment of area at some other set of x-y axis

I x'y' = I xy + Ax" y"

y’ y

x

y” A

x’ x” Review of Mechanics of Materials Second Moment of Area About An Inclined Axis (1)

Establish the x,y axis for the area and determine I x , I y and I xy

Mark off the coordinates of points X( I x , I xy ) and Y( I y ,−I xy ). Join XY. XY cuts the horizontal axis at O With OX or OY as radius, draw the circle

y’ y

dA

x’ y

x x Review of Mechanics of Materials Second Moment of Area About An Inclined Axis (2)

• 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

y’ y

dA

x’ y

x x