Appendix B – Analysis formulae (1/8)

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APPENDIX B – ANALYSIS FORMULAE

B.1 Elastic bending formulae

Bending about a principle axis:

κσ EI

My

== ; curvature-change 0

11RR

−=κ

In general, bending moment is section modulus Z times maximum bending stress.Longitudinal shear force S on material of area As , due to transverse shear force F on thebeam.

∫ ==SA

S

IyAF

dAyIF

S per unit length of beam.

B.2 Elastic torsion formulae

Round shafts: φτG

JT

r== where φ is the angle of twist per unit length

and ∫= dArJ 2 is the polar moment of area.

Circular area, radius R: 2

4RJ

π=

Thin circular tube, radius R thickness t: tRJ 32π=

Thin walled tube of arbitrary cross-section:

tA

T

e2=τ : φ

∫=

tdsA

GT e24

where Ae is the enclosed area to mid thickness, t isthe wall thickness. and s is the distance round thepermiter.

Appendix B – Analysis formulae (2/8)

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B.3 Taut wires, cables or chains

Uniformly loaded cables with horizontal chordss = span lengthf = cable sagn = f/s = sag ratioL = length of cable curve∆Ls = cable elongation due to axial stress∆Lt = cable elongation due to temperature change, tA = area of cableE = modulus of elasticity of cableε = thermal coefficient of linear expansiont = temperature change in °Fp = load per unit length

Uniformly loaded cables Triangular loading on cablesWith inclined chords with horizontal chords

B.4 Vibration

Typically for most structures Where: f is in cycles per second y is the static deflection in mm

Simply supportedMass concentrated in centre

Simply SupportedMass and stiffness distributed

CantileverMass concentrated at end

CantileverMass and stiffness distributed

( )

+ε≅ε=∆

+≅∆

+−+=

+=

=

−=

2

2

42

2

2

2

2

3

81

3

161

5

32

3

81

161

8

4

ntstLL.f

nAE

HsL.e

...nnsL.d

nHmaxT.c

f

psH.b

xsxs

fy.a

t

s

( )

2.

secsec3

81.

secsec3

161.

secsec3

81.

41.

8/.

4.

4

2

4

2

4

2

2

max

2

22

ps

s

HhVg

ntsLf

n

AE

HsLe

nsLd

ns

hHTc

fpsHb

xsxs

fya

s

t

s

+=

+≅∆

+≅∆

+≅

++=

=

−=

θθ

ε

θθ

θθ

+≅∆

+≅∆

+−+=

+=

=

−=

2

2

42

2max

2

3

3

5

181.

5

361.

...185

181.

361.

24/.

81.

ntsLf

nAE

HsLe

nnsLd

nTc

fpsHb

s

xfya

t

s

ε

yf

18=

yf

8.15=

yf

18=

yf

8.15=

yf

7.19=

Appendix B – Analysis formulae (3/8)

THIS DOCUMENT IS COPYRIGHT AND IS PUBLISHED FOR DISTRIBUTIONONLY WITHIN THE OVE ARUP PARTNERSHIP. IT IS NOT INTENDED FORAND SHOULD NOT BE RELIED UPON BY ANY THIRD PARTY.Ver 3.0 / Aug 98

B.5 Design formulae for beams - cantilever

Appendix B – Analysis formulae (4/8)

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B.6 Design formulae for beams - fixed both ends

Appendix B – Analysis formulae (5/8)

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B.7 Design formulae for beams - simply supported

Appendix B – Analysis formulae (6/8)

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[B.7 Design formulae for beams - simply supported (cont..)]

Appendix B – Analysis formulae (7/8)

THIS DOCUMENT IS COPYRIGHT AND IS PUBLISHED FOR DISTRIBUTIONONLY WITHIN THE OVE ARUP PARTNERSHIP. IT IS NOT INTENDED FORAND SHOULD NOT BE RELIED UPON BY ANY THIRD PARTY.Ver 3.0 / Aug 98

B.8 Design formulae for beams - propped cantilever

Appendix B – Analysis formulae (8/8)

THIS DOCUMENT IS COPYRIGHT AND IS PUBLISHED FOR DISTRIBUTIONONLY WITHIN THE OVE ARUP PARTNERSHIP. IT IS NOT INTENDED FORAND SHOULD NOT BE RELIED UPON BY ANY THIRD PARTY.Ver 3.0 / Aug 98

[B.8 Design formulae for beams - propped cantilever (cont..)]