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ALGEBRA 1

Composition Analysis: Ax + By = Cz
WORK PROBLEMS

LOGARITHM

x = logb N → N =bx
Properties

log(xy) = log x + log
x
y

y log

Rate of doing work = 1/ time
Rate x time = 1 (for a complete job)
Combined rate = sum of individual
rates Man-hours (is always assumed
constant)
(Wor ker s1)(time1) (Wor ker s2 )(time2 )
=
quantity
...
work1
quantity
...
work2

= log x − log y

ALGEBRA 2

log xn = nlog x
logb x =

UNIFORM MOTION PROBLEMS
log
x logb

S =Vt

loga a =1
REMAINDER AND FACTOR THEOREMS

Traveling with the wind or downstream:

Given:

Vtotal = V1 +V2

f (x)
(x − r)

Traveling against the wind or upstream:

Vtotal = V1 −V2

Remainder Theorem: Remainder = f(r)
Factor Theorem: Remainder = zero
QUADRATIC EQUATIONS
Ax2 + Bx +C = 0
− B ± B 2 − 4AC
Root = 2A
Sum of the roots = - B/A
Products of roots = C/A
MIXTURE PROBLEMS
Quantity Analysis: A + B = C

DIGIT AND NUMBER PROBLEMS

100h +10t +u →
where:

2-digit number

h = hundred’s digit
t = ten’s
digit u =
unit’s digit

CLOCK PROBLEMS

an = a m r n−m

where:
x = distance traveled by the
minute hand in minutes x/12 = distance
traveled by the hour
hand in
minutes

PROGRESSION PROBLEMS

nth term

r=

a 2 a3
=
a1 a2

S=

a ( r n −1)
1
→ r >1
r −1

Sum of ALL
terms, r >1

a1 (1 −r n )
S=
→ r <1
1 −r

Sum of ALL
terms, r < 1

S=

a1 = first term an = nth term

a1
1 −r

ratio

→ r <1 & n = ∞

Sum of ALL
terms,
r<1,n=∞

am = any term before an d =
common difference
= sum of all “n” terms

S

HARMONIC PROGRESSION (HP)
-

ARITHMETIC PROGRESSION (AP)
difference of any 2 no
...


COIN PROBLEMS

d = a − a = a − a ,
...


ALGEBRA 3
Fundamental Principle:
“If one event can occur in m different ways, and
after it has occurred in any one of these ways, a
second event can occur in n different ways, and
then the number of ways the two events can occur
in succession is mn different ways”

PERMUTATION

Properties of a binomial expansion: (x +
y)n

Permutation of n objects taken r at a time nPr
1
...
The powers of x decreases by 1 in the
successive terms while the powers of y
increases by 1 in the successive terms
...
The sum of the powers in each term is always
equal to “n”

= n!

4
...
objects
are alike

P=

is equal to “n+1”

of the terms having a coefficient of 1
...


Permutation of n objects arrange in a circle

P = (n−1)!

r th term = nCr-1 (x)n-r+1 (y)r-1
term involving yr in the expansion (x +
y)n

COMBINATION

y r term = nCr (x)n-r (y)r

Combination of n objects taken r at a time

sum of coefficients of (x + y)n

nCr =

Sum = (coeff
...
of y) n

−nr!)!r!

sum of coefficients of (x + k)n

Sum = (coeff
...
The sets are drawn as
circles
...


PLANE
TRIGONOMETRY

FULL OR PERIGON

Measurement
θ = 0°
0° < θ < 90°
θ = 90°
90° < θ < 180°
θ =180°
180° < θ < 360°
θ = 360°

Pentagram – golden triangle (isosceles)

36 °

72° 72 °

TRIGONOMETRIC IDENTITIES

sin 2 A+ cos2 A = 1 1+

a

= =

cot2 A = csc2 A
1+ tan2 A = sec 2 A sin(A± B) = sin
AcosB ± cos Asin B cos(A± B) =

COSINE LAW

cosAcosB sin Asin B tan(A± B) =
tan A± tanB
1tan

b
c
sin A
sinB
sinC

a2 = b2 + c2 – 2 b c cos A b2 = a2 + c2
– 2 a c cos B c2 = a2 + b2 – 2 a b cos

AtanB

cot(A± B) =

C

cot AcotB 1 cot A± cotB
sin 2A = 2sin AcosB cos2A
= cos2 A−sin 2 A

AREAS OF TRIANGLES AND
QUADRILATERALS

tan2A =

TRIANGLES

2tan
2
A 1− tan A
cot2 A−1

1
...
Given two sides and included angle

Area = absinq

3
...
Quadrilateral circumscribing in a circle

Area = s(s − a)(s −b)(s − c)
Area = rs

Area = abcd

s=a+b+c
2
4
...
Triangle circumscribing a circle

Area=rs
6
...
Given diagonals and included angle

1

Area =

d1d2 sinq

2
2
...
Cyclic quadrilateral – is a quadrilateral inscribed in a
circle

Area = (s − a)(s −b)(s −c)(s − d)
s = a +b + c + d
2
(ab +cd)(ac +bd)(ad +bc)
r=
4(Area)

d1d 2= ac+bd →Ptolemy’s Theorem

q = (n − 2)(180°)

SIMILAR TRIANGLES

A1
C

A
2

2

B

n

2

2

H

Value of each exterior angle
A2

a

b

c

h

SOLID GEOMETRY

a =180°−q =

360°
n

Sum of exterior angles:
POLYGONS
3 sides – Triangle
4 sides –
Quadrilateral/Tetragon/Quadrangle
5 sides – Pentagon
6 sides – Hexagon
7 sides – Heptagon/Septagon
8 sides – Octagon
9 sides – Nonagon/Enneagon
10 sides – Decagon
11 sides – Undecagon
12 sides – Dodecagon
15 sides – Quidecagon/ Pentadecagon
16 sides – Hexadecagon
20 sides – Icosagon
1000 sides – Chillagon
Let: n = number of sides
θ = interior angle
α = exterior angle
Sum of interior angles:

S = n α = 360°

Number of diagonal lines (N):

N=

n

(n − 3)

2
Area of a regular polygon inscribed in a circle of
radius r

Area = 1 nr2 sin
360
2
n
Area of a regular polygon circumscribing a
circle of radius r

Area = nr2 tan

n

S = n θ = (n – 2) 180°
Value of each interior angle

180

Area of a regular polygon having each side
measuring x unit length

Area = 1 nx2 cot

180

4

1

A=

n

d1d2sinq

2

PLANE GEOMETRIC FIGURES
RHOMBUS

CIRCLES

pd 2 2 A =
=pr

1

A=

4

d1d2 = ah

2

Circumference =pd = 2pr

A = a2 sina

Sector of a Circle
SOLIDS WITH PLANE SURFACE

A = rs = r2q

Lateral Area = (No
...
The bounding
planes are referred to as the faces and the intersections of
the faces are called the edges
...


PRISM

V = Bh
A(lateral) = PL
A(surface) = A(lateral) + 2B
where: P = perimeter of the base L
= slant height
B = base area

Truncated Prism

V=B
number∑heightsof heights

PYRAMID

V

Bh

A(lateral) = ∑ Afaces
A(surface) =A(lateral) +B
Frustum of a Pyramid

V=

h

(A1 + A2 + A1A2 )

3
A1 = area of the lower base
A2 = area of the upper base

PRISMATOID

h
V = 6(A1 + A2 +4Am)
Where: x = length of one edge
Am = area of the middle section

SOLIDS WITH CURVED SURFACES

REGULAR POLYHEDRON

CYLINDER

a solid bounded by planes whose faces are congruent
regular polygons
...

B
...

D
...


Tetrahedron
Hexahedron (Cube)
Octahedron
Dodecahedron
Icosahedron

A(lateral) = PkL = 2 π r h
A(surface) = A(lateral) + 2B
Pk = perimeter of right section
K = area of the right section
B = base area
L= slant height
CONE

V

=Bh

A(lateral) =prL

V = 3 (3r − h)
FRUSTUM OF A CONE

V = h (A1 + A2 + A1 A2
3

p
V=

+h2 )

2

+3b2 + h2 )

(3a

p

h

V=

(3a
6

SPHERE

=pr3

V

2

6

A(lateral) =p(R + r)L
SPHERES AND ITS FAMILIES

h

SPHERICAL WEDGE
is that portion of a sphere bounded by a lune and the planes
of the half circles of the lune
...


2

A

V

=

V = 1 A(zone)r
3

° 90
SPHERICAL ZONE
is that portion of a spherical surface between two
parallel planes
...


A(surface) = A(zone) + A(lateralofcone)
SPHERICAL PYRAMID
is that portion of a sphere bounded by a spherical
3
polygon and the planes of its sides
...


ph 2

E = [(n-2)180°]
E = Sum of the angles E =
Spherical excess
n = Number of sides of the given spherical polygon

SOLIDS BY REVOLUTIONS

V12

AA12

2

3

V

TORUS (DOUGHNUT)
a solid formed by rotating a circle about an axis not
passing the circle
...

It is a special ellipsoid with

d = (x2 − x1)2 + (y2 − y1)2

c=a

V = pa2b

Slope of a line

PROLATE SPHEROID

m = tanq = yx22 −− xy11

a solid formed by rotating an ellipse about its major axis
...


x = x1rr12 ++rx2 r1

y=

y1rr12 ++ry22r1
V = pr2h

2

SIMILAR SOLIDS

V1
V2

A2

3

H
h

A1

r
H

h

Location of a midpoint
R

3

L

l

2

R
r

x1 +2 x2

3

l

2

L

2

x=
STRAIGHT LINES

y = y1 +
2

y2

d = Ax1 + By2 +1 B+2C

General Equation Ax + By + C

=0

±A

Point-slope form
Note: The denominator is given the sign of B

y – y1 = m(x – x1)
Two-point form

y − y1 = yx22 −−

Distance between two parallel lines d =

C1 −C2

xy11 (x − x1)

A2 + B2
Slope relations between parallel lines: m1 =
m2

Slope and y-intercept form

y = mx + b
Intercept form

x

Slope relations between perpendicular lines:
m1m2 = –1

y
+

a

=1
b

Slope of the line, Ax + By + C = 0 m

=−

Line 1 → Ax + By + C1 = 0 Line
2 → Ax + By + C2 = 0

Line 1 → Ax + By + C1 = 0
Line 2 → Bx – Ay + C2 = 0
PLANE AREAS BY COORDINATES

= 1 x1,x2,x3,
...
yn, y1

Angle between two lines
−1

m−1mm12
tan

1m+2
q=

Note: Angle θ is measured in a counterclockwise
direction
...


Distance of point (x1,y1) from the line
Ax + By + C = 0;

Note: The points must be arranged in a counter clockwise
order
...


SPACE COORDINATE SYSTEM
Length of radius vector r:

r = x2 + y2 + z2
Distance between two points P1(x1,y1,z1)
and
P2(x2,y2,z2)

Parabola

B2 - 4AC = 0

Ellipse

B2 - 4AC < 0, A ≠ C

A≠C
same sign
Sign of A
opp
...


ANALYTIC
GEOMETRY 2

Standard Equation:

CONIC SECTIONS
a two-dimensional curve produced by slicing a plane
through a three-dimensional right circular conical surface

(x – h)2 + (y – k)2 = r2
General Equation:

Ways of determining a Conic Section
1
...

3
...


x2 + y2 + Dx + Ey + F = 0

By Cutting Plane
Eccentricity
By Discrimination
By Equation

Center at (h,k):

General Equation of a Conic Section:
2

D

h =−

E

; k =−

2A

2

2A

Ax + Cy + Dx + Ey + F = 0 **
Radius of the circle:
Circle
Parabola
Ellipse
Hyperbola

Circle

Cutting plane

Eccentricity

Parallel to base

e→0

Parallel to element

e = 1
...
0

PARABOLA

Parallel to axis

e > 1
...


Discriminant

Equation**

B2 - 4AC < 0, A = C

A=C

r 2 = h2 + k 2 −

F or r = 1 D2 +E2 −4F
A

2

STANDARD EQUATIONS:
Opening to the right:
where: a = distance from focus to vertex
= distance from directrix to vertex

(y – k)2 = 4a(x – h)
Opening to the left:

AXIS HORIZONTAL:

Cy2 + Dx + Ey + F = 0
Coordinates of vertex (h,k):

(y – k)2 = –4a(x – h)
Opening upward:

(x – h) 2 = 4a(y – k)

k =−

Opening downward:

2EC

substitute k to solve for h
Length of Latus Rectum:

(x – h) 2 = –4a(y – k)
Latus Rectum (LR)
a chord drawn to the axis of symmetry of the curve
...


AXIS VERTICAL:

e=1

for a parabola

2

Ax + Dx + Ey + F = 0
Coordinates of vertex (h,k): h

=−

ELLIPSE
a locus of a moving point which moves so that the sum of its
distances from two fixed points called the foci is constant and
is equal to the length of its major axis
...


Coordinates of the center:

D

h=−
E

;k =−

2A

2C

If A > C, then: a2 = A; b2 = C If A < C,
then: a2 = C; b2 = A

d = distance from center to directrix a
= distance from center to vertex c =
distance from center to focus

STANDARD EQUATIONS
Transverse axis is horizontal

KEY FORMULAS FOR ELLIPSE
Length of major axis: 2a
Length of minor axis: 2b
Distance of focus to center:

c= a2 −b2

(x−h)2
2−

(y−k)2
b2
=1 a

Transverse axis is vertical:

(y − k)2

(x − h)2
− = 1 a2 b2

GENERAL EQUATION

Ax2 – Cy2 + Dx + Ey + F = 0
Length of latus rectum:

2b2

Coordinates of the center:

D
2 A; k =−

h =−
E
2 C

e = c = aa
d
POLAR COORDINATES SYSTEM

If C is negative, then: a2 = C, b2 = A
If A is negative, then: a2 = A, b2 = C
Equation of Asymptote:

x = r cos θ

y=r
sin θ

(y – k) = m(x – h)
Transverse

axis

is

horizontal:

m=± ba

Transverse

axis

vertical:

is

m=± a b

r = x2 + y2
tanq = x

y

KEY FORMULAS FOR HYPERBOLA
Length of transverse axis: 2a
Length of conjugate axis: 2b
Distance of focus to center:

c = a2 +b2

Important propositions
1
...


Length of latus rectum:

2b2
LR =
a
Eccentricity:

SPHERICAL
TRIGONOMETRY

2
...

3
...


a+b>c
4
...


0° < a + b + c < 360°
5
...


180° < A + B + C < 540°
6
...


A + B < 180° + C
SOLUTION TO RIGHT TRIANGLES

3
...


QUADRANTAL TRIANGLE
is a spherical triangle having a side equal to 90°
...


sinb
sinc
=
sin A sin B
sinC

Law of Cosines for sides:

cosa = cosbcosc + sinbsinccos A cosb
= cosacosc + sinasinccosB
cosc = cosacosb + sinasinbcosC
Law of Cosines for angles:

Napier’s Rules

cos A = −cos BcosC + sin Bsin C cosa cos
B = −cos AcosC + sin Asin C cosb
cosC = −cos Acos B + sin Asin Bcosc

1
...


Co-op
AREA OF SPHERICAL TRIANGLE
2
...


Tan-ad

p R 2E
A=

180°

Important Rules:
1
...

2
...


R = radius of the sphere
E = spherical excess in degrees,

E = A + B + C – 180°

TERRESTRIAL SPHERE
Radius of the Earth = 3959 statute miles
Prime meridian (Longitude = 0°)
Equator (Latitude = 0°)
Latitude = 0° to 90°
Longitude = 0° to +180° (eastward)
= 0° to –180° (westward)

d
dv
(uv) = u
dx
dv

v
−u
d u = dx 2 dx
dx v
v
d (u n ) = nun−1
du dx dx du d dx u
=
dx
2u

1 min
...
x→a g(x) x→a g'(x) x→a g"(x)
Shortcuts
Input equation in the calculator
TIP 1: if x → 1, substitute x = 0
...
)
Y’
MAX
0

Y”
(-) dec

Concavity
down

MIN

0

(+) inc

up

INFLECTION

-

No change

-

du

u 1−u 2 dx
d −1u) =
(csch

dx

−1

du
du
−1 u)

LIMITS
Indeterminate Forms

−1

du

u 1+u 2 dx

HIGHER DERIVATIVES nth
derivative of xn

d nn (xn) =
n! dx

∫[ f (u) + g(u)]du = ∫ f (u)du +∫ g(u)du
un

n

+

∫u du =

nth derivative of xe n

∫

1

n +1 +C
...


RADIUS OF CURVATURE

[1+ ( y')2]
y"
R=

INTEGRAL
CALCULUS 1
∫du = u +C
∫adu = au +C

∫secu tanudu =secu +C
∫cscucotudu =− cscu +C
∫ tanudu = ln secu +C
∫cotudu = ln sin u +C
∫secudu = ln secu +

tanu+C

∫cscudu = ln cscu −

cotu+C

∫
du
1
−1 u
tan
=
+ du
2
∫a + u a
a
=
2

sin−1 u +C a2 −u 2 a
C

∫ du = 1 sec

∫

u +C u u 2

−1

2

=

1

−1

u

+C
...
u

CENTROID OF PLANE AREAS (VARIGNON’S
THEOREM)

Using a Vertical Strip:
x2

x= b= b

A• x = ∫dA• x

y
=h

x1

x2

A• y = ∫dA• 2y

=h
LENGTH OF ARC

x
1

Using a Horizontal Strip:
y2

A• x = ∫dA• 2x
y1

x2

2

S=∫ 1

dy

y2

A• y

=

dx
x1

∫dA• y

y2

2

dx

y1

S=

∫
1

CENTROIDS

y

Half a Parabola

z2

x= b

dx

1

dy

dy

2

S=∫

2

dx
dz

dy

dz

dz

z1

y =h
Whole Parabola

y= h
Triangle

INTEGRAL
CALCULUS 2

V = A•2pr

TIP 1: Problems will usually be of this nature:
•
“Find the area bounded by”
• “Find the area revolved around
...


A = S •2pr
A =∫dS •2pr

W = k(x22 − x12)
k = spring constant x1 = initial
value of elongation
x2 = final value of elongation

Work done in pumping liquid out of the
container at its top

Work = (density)(volume)(distance) Force
= (density)(volume) = ρv
Specific Weight:

Second Proposition: If a plane area is revolved
about a coplanar axis not crossing the area, the volume
generated is equal to the product of the area and the
circumference of the circle described by the centroid of
the area
...
81 kN/m2 SI
γwater = 45 lbf/ft2 cgs

3
bh3
Ixo =

Density:

12
Triangle bh3

r =Volume

mass

ρwater = 1000 kg/m3 SI
ρwater = 62
...


Ix =
I y=
4
FLUID PRESSURE

Ix = Ixo = Ad2
Moment of Inertia for Common Geometric
Figures
Square bh3

Ix =

F = whA =ghA F
=∫whdA
F = force exerted by the fluid on one side of
the area h = distance of the c
...
to the surface
of liquid w = specific weight of the liquid (γ)
A = vertical plane area

Specific Weight:

EQUILIBRIUM OF COPLANAR FORCE
SYSTEM
Conditions to attain Equilibrium:

g= Volume

Weight

∑F

(x−axis)

γwater = 9
...


Cross or Vector product

Uneven elevation of supports

P×Q = P Q sinq i j
k
P×Q = Px

Py Pz

Qx

Qy Qz

H = wx12 = wx22
2d1

2d2

T1 =

(wx1)2 + H 2

2
8d 2

T2 =

(wx2)2 + H 2

S = L + 3L

Even elevation of supports

−

32d 4
5L3

L = span of cable d = sag of cable T
= tension of cable at support H =
tension at lowest point of cable w =
load per unit length of span
S = total length of cable

L
>10
d
wL2
H=
8d

T

2

wL

+H2

CATENARY
the load of the cable is distributed along the entire length

MECHANICS 2

RECTILINEAR MOTION
Uneven elevation of supports
Constant Velocity

T1 = wy1
T2 = wy2

H = wc

S = Vt
Constant Acceleration: Horizontal Motion y12 = S12 + c2

y22 = S22 + c2

c

x1 = cln

S1 + y1

S =V0t ± 1 2 c

S2 + y2

2 at

x2 = ln
Span = x

1

+ xc2

V =V0 ± at

of the cable
...
81
9
...
806

English (ft/s2)
32
...
16

ROTATION (PLANE MOTION)

V=
dt Relationships between linear & angular dV

parameters:

a =

V = rw
PROJECTILE MOTION

a = ra
V = linear velocity
ω = angular velocity (rad/s)

dt

a = linear acceleration
α = angular acceleration (rad/s2)
r = radius of the flywheel
Linear Symbol

Angular Symbol

S
V
A
t

θ
ω
α
t

Distance
Velocity
Acceleration
Time

x = (V0 cosq)t

Constant Velocity

θ = ωt

2

± y = (V0 sinq)t

−gt

Constant Acceleration ±

y = xtanq −

2V 2gxcos2 2 q

q = w0t ± 12at 2
0

Maximum Height and Horizontal Range

V02 sin2q max ht

w = w0 ±at
w2 = w02 ± 2aq

y=

=

2g

+ (sign) = body is speeding up

V02 sing 2q

– (sign) = body is slowing down D’ALEMBERT’S PRINCIPLE

x
Maximum Horizontal Range

“Static conditions maybe produced in a body possessing
acceleration by the addition of an imaginary force called
reverse effective force

(REF) whose magnitude is

Assume: Vo = fixed (and parallel but opposite in direction to the acceleration
...


F V2
tanq = = W gr

mV 2 WV 2
Fc = = r gr
V 2
ac =
r

f

=

1p gh
2

frequency

Fc = centrifugal force
V = velocity m =
mass W = weight
r = radius of track ac = centripetal
acceleration g = standard
gravitational acceleration

BOUYANCY
A body submerged in fluid is subjected by an
unbalanced force called buoyant force equal to
the weight of the displaced fluid

BANKING ON HI-WAY CURVES

Fb = W
Fb = γVd

Ideal Banking: The road is frictionless

V2
tanq=

Fb = buoyant force W = weight of body or
fluid γ = specific weight of fluid Vd = volume
displaced of fluid or volume of submerged
body

Specific Weight:

gr
Non-ideal Banking: With Friction on the road

V
tan(q +f ) =
gr

2

;

tanf = m

V = velocity r = radius of track g =
standard gravitational acceleration θ
= angle of banking of the road

g= Volume

Weight

γwater = 9
...


I = mass moment of inertia ω
= angular velocity

Work Done = ΔKE
Mass moment of inertia of rotational INERTIA
for common geometric figures:

Positive Work – Negative Work = ΔKE
Total Kinetic Energy = linear + rotation

Solid sphere: I
sphere: I

=

= mr

2

2 mr2 Thin-walled hollow
Solid disk: I

Solid Cylinder: I =

Sensible Heat is the heat needed to change the
2

temperature of the body without changing its phase
...
156 kJ/kg
50% Cwater
48% Cwater

Latent Heat is the heat needed by the body to change
its phase without changing its temperature
...
186 Joules
1 BTU = 252 calories
= 778 ft–lbf
LAW OF CONSERVATION OF HEAT ENERGY
When two masses of different temperatures are combined
together, the heat absorbed by the lower temperature
mass is equal to the heat given up by the higher
temperature mass
...


In thermodynamics, there are four laws of very
general validity
...

ZEROTH LAW OF THERMODYNAMICS
stating that thermodynamic equilibrium is an
equivalence relation
...

FIRST LAW OF THERMODYNAMICS
about the conservation of energy
The increase in the energy of a closed system is
equal to the amount of energy added to the
system by heating, minus the amount lost in the
form of work done by the system on its
surroundings
...


STRENGTH OF
MATERIALS
SIMPLE STRESS

Stress =

Force
Area

Axial Stress
the stress developed under the action of the force acting
axially (or passing the centroid) of the resisting area
...

This law is more clearly stated as: "the entropy
of a perfectly crystalline body at absolute zero
temperature is zero
...


s =

P
s

A
Pappliedl ║ Area
σs = shearing stress P =
applied force or load
A = resisting area (sheared area)

Bearing stress
the stress developed in the area of contact (projected area)
between two bodies
...
325 kPa

=
=
=
=
=
=
=

14
...
032 kgf/cm2
780 torr
1
...
92 in

d
e=

Thin-walled Pressure Vessels
A
...
Longitudinal stress (also for Spherical)
Yield Point – at his point there is an appreciable
elongation or yielding of the material without any

corresponding increase in load; ductile materials and
continuous deformation

b
...
Due to changes in temperature

d=

La (Tf −Ti )
Types of elastic deformation:
a
...
Biaxial and Triaxial Deformation

sae
s =Ye Young'sModulus of Elasticity s =
Ee Modulus of Elasticity
s s =Eses

ey

ez
m=−ex = −ex

Modulus inShear

s V =EVeV

BulkModulus of Elasticity

1
Ev

δ = elongation
α = coefficient of linear expansion of the body
L = original length
Tf = final temperature
Ti = initial temperature

compressibility

d=
δ = elongation P =
applied force or load
A = area
L = original length E =
modulus of elasticity σ =
stress
ε = strain

AEPL

μ = Poisson’s ratio
μ
= 0
...
3 for steel
= 0
...
20 for concrete
μmin = 0
μmax = 0
...


P = Tw

Solid shaft

Hollow shaft

t = 16T3
pd

Prpm = 2pTN

rps
rpm

Prpm = 2pTN
60

t = 16TD
p(D 4 −d 4 )

ft −lb
sec

Php = 2pTN

ft −lb
min

550

τ = torsional shearing stress T
= torque exerted by the shaft D
= outer diameter
d = inner diameter

Php = 2pTN
3300

Maximum twisting angle of the shaft’s fiber:

HELICAL SPRINGS

t = 16pdPR3

TL

T = torque
N=
revolutions/time

1+ 4dR

q=
JG

t = 16pdPR3

θ = angular deformation (radians)
T = torque
L = length of the shaft
G = modulus of rigidity
J = polar moment of inertia of the cross

0

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