Clarkson University ES222, Strength of Materials Final Exam Formula Sheet

Axial Loading

Normal Stress: PA

= Splice joint: aveFA

= Single shear: aveFA

=

Double shear: 2aveFA

= Bearing stress: bPtd

=

2cos , sin coso o

P PA A

= =

Factor of Safety = F.S. = ultimate loadallowable load

Stress and Strain Axial Loading

Normal strain: L

= Normal stress: E = Shear stress: G =

Elongation: PLAE

= Rods in series: i ii i i

PLAE

=

Thermal elongation: ( )T T L = Thermal strain: ( )T T = Poissons ratio: lateral strain

axial strain =

Generalized Hookes Law: yx zx E E E

=

yx zy E E E

= +

yx zz E E E

= +

, ,xy yz xzxy yz xzG G G = = =

Units: k = 103 M = 106 G = 109 Pa = N/m2 psi = lb/in2 ksi = 103 lb/in2

Coordinates of the Centroid: i iiii

x Ax

A=

i iiii

y Ay

A=

Parallel Axis Theorem: 2'x xI I Ad= + , where d is the distance from the xaxis to the xaxis

3112z

I bh=

3112y

I hb=

Torsion:

L = max L

c = TJ

= maxTcJ

= G =

TLJG

= solid rod: 412J c= hollow rod: ( )4 412 o iJ c c=

Rods in Series: i ii i i

T LJ G

= Pure Bending:

xMyI

= maxMc MI S

= =

xy

= y z x = = E = 1 M

EI=

= radius of curvature General Eccentric Loading:

yzx

z y

M zP M yA I I

= +

z yM d P= !! !

y zM d P= !! !

Shear and Bending Moment Diagrams

T

y

z

b

h

x

y

MM

x

y

zC

PP

dy

dz

d

c

x

D C x

dV w V V wdxdx

= = = (area under load curve between C and D)

d

c

x

D C x

dM V M M Vdxdx

= = = +(area under shear curve between C and D) Shear Stress in Beams

aveVQIt

= VQqI

= = shear per unit length Q Ay=

Stress Transformation

Principal stresses: ( )2

2

max,min 2 2x y x y

xy

+ = +

Principal planes: 2

tan 2 xypx y

=

Planes of maximum in-plane shear stress: tan 22x y

sxy

=

Maximum in-plane shear stress: ( )2

2

max 2x y

xy R

= + =

Corresponding normal stress: '2

x yave

+= =

Thin Walled Pressure Vessels

Cylindrical: Hoop stress = 1prt

= Longitudinal stress = 2 2prt

=

Maximum shear stress (out of plane) = max 2 2prt

= =

Spherical: Principal stresses = 1 2 2prt

= =

Maximum shear stress (out of plane) = 2max 2 4prt

= =

Deflections of Beams ( ) 2

2

1 M x d yEI dx

= = slope = ( ) ( ) 1M xdyx dx Cdx EI = = + deflection = ( ) ( ) 2y x x dx C= + = elastic curve Columns

2

2cre

EIPL

=

For x > a, replace x with (L-x) and interchange a with b.

Axial LoadingStress and Strain Axial LoadingShear and Bending Moment DiagramsShear Stress in BeamsStress TransformationThin Walled Pressure VesselsDeflections of BeamsColumns