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Appendix A – Mathematical formulae (1/5)
APPENDIX A – MATHEMATICAL FORMULAE Version Version 3.1. Jan 99. A.5. corrected equation equation for trapezium inertia .
A.1 Trigonometric functions sin x
eix
=
− e−ix
cos x
2i
sin ( A ± B ) = sin A cos B
± cos A sin B
=
e ix
+ e−ix
2 cos( A ± B ) = cos A cos B
1 m tan A tan B
sin A + sin B cos A + cos B
= 2 sin
A + B
= 2 cos
cos
2 A + B 2
A − B
cos
2 A − B 2
1
sin A sin B
= [cos( A − B) − cos( A + B)]
sin A cos B
= [sin( A + B ) + sin( A − B )]
2 1
2
1
sin 2 x
= [1 − cos 2 x]
sin 3 x
= [3 sin x − sin 3 x]
2 1
sin A − sin B
= 2 cos
cos A − cos B cos A cos B
cosh x
=
cosh ix =
2 cos x
= i sin x cosh 2 x − sinh 2 x = 1 cosh ( x ± y ) = cosh x cosh y ± sinh x sinh y sinh ( x ± y ) = sinh x cosh y ± cosh x sinh y cosh ( x ± iy ) = cosh x cos y ± i sinh x sin y sinh ( x ± iy ) = sinh x cos y ± i cosh x sin y sinh ix
2
1 2
1
= [1 + cos 2 x]
cos3 x
= [3 cos x + cos 3 x]
2 1 4
e x
− e x
2 cos ix = cosh x
sin ix = i sinh x
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Ver 3.1 / Jan 99
2 A + B
cos 2 x
sinh x =
sin
A − B
sin
2 A − B 2
= [cos( A + B )+ cos( A − B )]
A.2 Hyperbolic Hyperbolic functions
+ e− x
A + B
= − 2 sin
4
x
sin A sin B
tan A ± tan B
tan( A ± B ) =
e
m
Appendix A – Mathematical formulae (2/5)
A.3 Standard indefinite integral Integrand
Integral
Integrand
Integral
-cos x
sinh
cosh x
cos
sin
cosh
sinh
tan
-ln (cos
tanh
ln (cosh
)
cosec
ln(tan x / 2)
cosech
ln (tanh
/ 2)
sec
ln (tan x + sec )
sech x
2 tan-1(ex)
cot
ln (sin
coth
ln (sinh x )
sec2
tan
sech 2
tanh x
sec
tanh
sech
-sech
-cosec
coth
cosech
-cosech
sin x
tan
sec
cot
cosec x
1
sin
− x 2
a2
−a
2
x
+ a2
x − cos −1 a
(
+ a2 )
(
− a2 )
or ln x + x 2
cosh −1
1 2
or
x or a 1 − x tan 1 a a
1 x
x a
sinh −1
+ a2
2
)
x a
1 x 2
−1
)
ln x + x 2
A.4 Standard substitutions for integration If the integrand is a function of : Substitute:
(a − x ) (a + x ) ( x − a ) 2
2
or
a2
− x 2
x = a sin θ
or
x = a cos θ
2
2
or
a2
+ x 2
x = a tan θ
or
x = a sinh θ
2
2
or
x 2
− a2
x = a sec θ
or
x = a cosh θ
or of the form:
{(a x + b)
p x + q
{(a x + b)
p x 2
}−
1
p x + q = u 2
+ q x + r }
−1
a x + b =
1 u
or a rational function of: sin x
t = tan
and / or cos x
whence
sin x =
2t 1 + t
2
cos x =
1 − t
2
1 + t
2
x 2
dx =
2 dt 1 + t
2
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Ver 3.1 / Jan 99
Appendix A – Mathematical formulae (3/5)
A.5 Geometric properties of plane sections section
Position of centroid
Area
Moments of inertia I xx
e l g n a i r t
A =
bh
e x
2
=
h
I yy
3
I aa
= bh 3 / 36 = hb 3 / 48
Z xx
= bh 3 / 4 = bh 3 / 12
apex = bh 2 / 24
I bb
e l g n a t c e R
A = bd
e l g n a t c e R
A = bd
e l g n a t c e R
e r a u q S
e x
= bd 3 / 12 I yy = db 3 / 12 I xx
h
=
2
I bb
e x
A = bd
bd
=
e x
I xx
+ d 2
b2
+ d cosθ
b sin θ
2
=
e x
A = s 2
s
=
ev
= bd
=
d n o m a i D
n o g a x e H
d (a + b) e x1
2
=
= I yy = s 4 /12 I bb = s 4 / 3 4 I vv = s / 12
2
3(a + b )
2
I xx
2 s
d (2a + b )
( + d )
6 b2
= bh 2 / 24
Z yy
Z xx Z yy
Z xx
=
= bd 2 / 6 = db 2 / 6
b 2 d 2 6 b2
+ 4ab + b2 ) I xx = 36(a + b) 3 2 2 d (a + a b + ab + b I =
Z xx Z vv
= Z yy = =
2
yy
48
Z xx
s
bd
e x =
2
A = 0.866d
2
e x
d 2
= 0.866 s =
d 2
I xx I yy
I xx
= =
bd
48
=
48
= I yy = I vv = 0.0601d 4
6
3
I xx d − e x
(two values) Z yy
=
2 I yy
Z xx
=
Z yy
=
3
db
s 3
6 2
3
A =
+ d 2
Z xx
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Ver 3.1 / Jan 99
b 3d 3
d (a
A =
/3
=
3
m u i z e p a r T
3
base = bh 2 / 12
= b2 sin 2 θ + bd (b 2 sin 2 θ + d 2 cos 2 θ ) bd d 2 cos2 θ 12 6(b sinθ + d cosθ )
I xx
=
Section Moduli
b 2
bd
24 db 2 24
= 0.1203d 3 Z yy = Z vv = 0.1042d 3 Z xx
Appendix A – Mathematical formulae (4/5)
Geometric properties of plane sections (cont.) Section n o g a t c O
A = 0.8284d
2
n s 2 cot θ
A = n r 2 tanθ A =
e l c r i C
e l c r i C i m e S
t n e m g e S
r o t c e S
2
n R sin 2θ 2
= 0.0547d
depending on
=
the axis and
=
value of n
= r =
e
A = 0.7854d 2
= I 2 A(6 R 2 − s 2 )
2
e
48
d 4
π
Z =
64
I = 0.7854r
π
d 3
32
Z = 0.7854r 3
2
= 3 base = 0.2587 r 3 crown = 0.1907 r 3 Z yy = 0.3927 r Z xx
A = 1.5708r 2
A = 2
r
2
π θ ° − sinθ 180°
θ
°
360°
π
r 2
= 0.424r
e x
e0 e x
=
A =
π
2
r 4
c
A = 0.2146r 2
3
12 A
e x
= =
θ
2 c r 3 a r 2 c 3 A
= 0.424r ev = 0.6r eu = 0.707r e x
= 0.777 r ev = 1.098r eu = 0.707r e = 0 .316r eb = 0.391r σ
π θ ° − sin 2θ 16 90° 4 3 20r (1 − cosθ ) − π θ ° − 180° sin θ 4 π θ ° − r I yy = 30° 48 8 sin θ sin 2 θ + I xx
= e0 − r cos
e x
= 0.1098r 4 I yy = 0.3927r 4 I xx
2
I xx
=
r 4
= I o −
360°
°
θ π
sin 2
θ
Z xx base = I xx / e I crown Z vv
=
I xx
=
b − e I
2 I yy c
Z xx
4r 4
2 9
4 π θ° = r − sin θ 8 180° 4 r π θ ° + sin θ I 0 = 8 180°
I yy
centre
=
crown
=
Z yy
=
I xx e x I xx r − e x
2 I yy c
= I yy = 0.0549r 4 I bb = 0.1963r 4 I uu = 0.0714r 4 I vv = 0.0384r 4
Minimum Values
= I yy = 0.0076r 4 I uu = 0.012r 4 I vv = 0.0031r 4
MinimumValues
I xx
I xx
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Ver 3.1 / Jan 99
I
Z =
24
A(12r 2 + s 2 )
I =
d
= Z yy
= 0.1095d 3 Z vv = 0.1011d 3
4
I 1
Section Moduli Z xx
= I yy = I vv
I xx
= r or R
e x
t n e m e l p m o C
d
A = π r 2
A =
t n a r d a u Q
Moments of inertia
2 ev = 0.541d
e
4
=
e x
= 0.4142d
s
A =
n o g y l o P
Position of centroid
Area
= Z yy = 0.0953r 3 Z uu = 0.1009 r 3 Z vv = 0.064r 3 Z xx
= Z yy = 0.0097 r 3 Z uu = 0.017r 3 Z vv = 0.0079r 3 Z xx
m 2 m 2 2 / 2 2 m / N / m m m f m m 5 / g / N / 9 N k N k N 8 6 7 2 8 3 . 4 0 0 8 8 . 7 4 . 0 8 8 . . . 7 0 5 0 9 4 4 1 1 = = = = = =
2 2 n 2 2 2 2 t f i
t n / m t / i f f f / f / / / f f f f n n g b o t o l k b l b l t 1 1 1 1 1 1 2 t f 2
n / f i f n / o n f t n / m / i o f b / f l f g b 4 t l b 2 5 l k 8 9 8 3 7 0 2 4 0 9 4 . 0 2 0 6 5 0 4 1 . 2 . 0 . 0 . 0 . 1 0 0 0 0 0 = = = = = =
g n i m 2 m 2 m m d m / m / f / m / m a o / g N / L N N k N k N s s p e r 1 1 1 1 1 1 i r t t S S 2
2
2
m . f g m m m k N m N k N 2 5 0 N 1 7 1 3 6 3 0 1 1 5 5 8 . 0 . 1 . 3 . 2 . 9 0 0 1 0 = = = = =
t f t m t / f f / f / f / f f g b b n o k l l t 1 1 1 1
n n i n i n i . m i . f . f . f . n f f g b b o k b l l l t 1 1 1 1 1
t f / f t f n t m / f / f o f b / t f g l b 9 k 0 l 5 2 2 3 0 0 7 5 . 3 1 . 6 . 8 0 . 0 0 6 0 = = = =
2 t f 2 2 t f / 2
m m / m / m f N / / g k N N k 6 k 7 8 4 9 0 8 1 6 . 8 . 4 . 0 . 2 9 1 0 3 = = = =
m m m m / f / / t / g N N n N k k k e 1 1 1 1 m o M
m m 0 0 2 6 1 4 =
3
n i 1
4
3
4
2
n i / f b l 1
2
n i / f b l 0 0 . 5 y 4 t 1 i c = i t s a l E 2 f m o m . m s / f m g m m N l u m N k N N k u N 1 1 1 1 1 d o 1 M t i n n i n f . m . f i . f . f . b n f f g b l o k b l l 6 t 2 0 1 7 1 0 8 5 . 5 3 1 . 6 8 . 7 . 9 . 0 8 8 0 3 = = = = =
2
m m m 8 4 0 4 4 4 . 0 1 5 3 . 9 . 2 0 0 = = =
2 2 2 m
m m 0 9 1 2 6 . 2 3 5 9 0 4 . 8 . 6 0 0 = = =
m / e m n / g n o k t 2 9 0 . 2 6 3 . 1 1 = =
3
3 3
3 m m m 2 m 6 0 3 4 9 8 2 3 0 6 6 . 7 . 1 0 0 = = =
N N N k 7 8 4 0 4 6 8 . 4 . 9 . 9 4 9 = = =
3
d n t i f y 1 1 1
d y t / f / n b o l t 1 1
3
3 3 t d
3
n b o l t 1 1
n o t b l 2 5 4 0 8 2 . 9 . 2 0 = =
d n t i f 1 1 1
n i d 7 t 3 f 9 1 4 3 8 9 0 . 2 . 0 . 0 3 1 = = =
2 2
2
d n t i f y 1 1 1
2
n 2 i t d 5 2 5 f y 1 6 6 0 7 . 9 0 . 0 1 . 0 1 1 = = =
3 3
3
f f f n g o k b l t 1 1 1
y f / / b l n o 2 t 4 4 2 2 6 5 0 . 7 . 0 0 = =
n i 3 1 3 d 6 t f y 0 0 2 8 0 3 . 0 0 . 5 3 . 0 3 1 = = =
f n f f o g b l t k 8 4 2 4 0 0 2 0 1 . 2 . 1 . 0 0 0 = = =
3
e e 3 2 n h t m m m m 2 2 3 3 n s g o g a u m s k l t m m m m m m m m e n a e 1 1 1 r 1 1 1 o 1 1 1 1 1 M L A V
y t i s n e D
m / e m n / g n o k t 1 1
3
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Ver 3.1 / Jan 99
e c r o F
n i 1
n n i i 6 6 0 0 1 1 x x 1 a 3 0 0 . e r 4 1 A . 6 2 f = o = s t u n l e u d m o o 3 M m M 4 m n m d m o n i t 1 o 1 c c e e S S
