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### Transcript of Strength of Materials Formula Sheet

Clarkson University ES222, Strength of Materials Final Exam Formula Sheet Axial Loading Normal Stress: =P A

Splice joint: ave =

F A F 2A

Double shear: ave =

F A P Bearing stress: b = td

Single shear: ave =

=

P P cos 2 , = sin cos Ao Ao

Factor of Safety = F.S. =

L PL Elongation: = AE

Normal stress: = E Rods in series: = i

Shear stress: = GPLi i Ai Ei

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

Thermal strain: T = ( T )

Generalized Hookes Law:

x =

x y zE E +

y = z = xy =Units: k = 103 M = 106

xE

y zE E +

E

xE

yE

zE , xz =

xyG

, yz =

yzG

xzG psi = lb/in2 ksi = 103 lb/in2

G = 109

Pa = N/m2y=

Coordinates of the Centroid: x =

xA Ai i i i2

i

yA Ai i i i

i

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

y

1 3 bh 12 z 1 I y = hb3 12Iz =b

h

Torsion:L T = J TL = JG

=

max = max

c L Tc = J

=G

solid rod: J = 1 c 4 24 hollow rod: J = 1 ( co ci4 ) 2

T

Rods in Series: = i

Ti Li J i Gi

y x

Pure Bending:

x =

x =

My I y

max =

Mc M = I S

M = E1

M= M EIy

= radius of curvatureGeneral Eccentric Loading:P M y M z x = z + y A Iz Iy

y = z = x

dz dy P PC

! ! ! Mz = dy P

! ! ! M y = dz P

z

x

Shear and Bending Moment Diagrams

xd dV = w VD VC = wdx = (area under load curve between C and D) xc dx

dM =V dx

M D M C = Vdx = +(area under shear curve between C and D)xc

xd

Shear Stress in Beams

ave =

VQ It

q=

VQ = shear per unit length I

Q = Ay

Stress Transformation

Principal stresses: Principal planes:

max,min =tan 2 p =

x + y2 2 xy x y

2 y x + ( xy ) 2 2

Planes of maximum in-plane shear stress:

tan 2 s =

x y 2 xy2

Maximum in-plane shear stress: Corresponding normal stress:

2 y max = x + ( xy ) = R 2 + y ' = ave = x 2

Thin Walled Pressure Vessels Cylindrical: Hoop stress = 1 =pr t

Longitudinal stress = 2 =pr 2t

pr 2t

Maximum shear stress (out of plane) = max = 2 = Spherical: Principal stresses = 1 = 2 =pr 2t

Maximum shear stress (out of plane) = max =

22

=

pr 4t

Deflections of Beams 1 M ( x) d 2 y = = 2 EI dx

slope = ( x ) =

M ( x) dy = dx + C1 dx EI

deflection = y ( x ) = ( x ) dx + C2 = elastic curve

Columns Pcr =

2 EIL2 e

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