MATEMÁTICAS

PROPIEDADES DE GEOMETRÍAS PLANAS

ESTRUCTURAS

PROPIEDADES DE LOS MATERIALES

UNIDADES FUNDAMENTALES Y DERIVADAS

962

Mathematical Constantsp � 3.14159 . . . e 5 2.71828 . . . 2p radians � 360 degrees

1 radian � �1p80� degrees � 57.2958° 1 degree � �

1p80� radians � 0.0174533 rad

Conversions: Multiply degrees by �1p80� to obtain radians

Multiply radians by �1p80� to obtain degrees

Exponents

AnAm � An�m �AA

m

n� � Am�n (Am)n � Amn A�m � �A1m�

(AB)n � AnBn ��AB

��n

� �AB

n

n� Am/n � �n Am� A0 � 1 (A ; 0)

Logarithms

log � common logarithm (logarithm to the base 10) 10x � y log y � x

ln � natural logarithm (logarithm to the base e) ex � y ln y � x

eln A � A 10log A � A ln eA � A log 10A � A

log AB � log A � log B log �AB

� � log A � log B log �A1

� � �log A

log An � n log A log 1 � ln 1 � 0 log 10 � 1 ln e � 1

ln A � (ln 10)(log A) � 2.30259 log A log A � (log e)(ln A) � 0.434294 ln A

Mathematical Formulas

C

APPENDIX C Mathematical Formulas 963

Trigonometric Functions

tan x � �csoins

xx

� cot x � �csoins

xx

� sec x � �co

1s x� csc x � �

sin1

x�

sin2 x � cos2 x � 1 tan2 x � 1 � sec2 x cot2 x � 1 � csc2 x

sin (�x) � �sin x cos (�x) � cos x tan (�x) � �tan x

sin (x / y) � sin x cos y / cos x sin y cos (x / y) � cos x cos y > sin x sin y

sin 2x � 2 sin x cos x cos 2x � cos2 x � sin2 x tan 2x � �1

2�

tatannx2 x

�

tan x � �1 �

sinco

2sx

2x� � �

1 �

sinco

2sx

2x�

sin2 x � �12

� (1 � cos 2x) cos2 x � �12

� (1 � cos 2x)

For any triangle with sides a, b, c and opposite angles A, B, C:

Law of sines �sin

aA

� � �sin

bB

� � �sin

cC

�

Law of cosines c2 � a2 � b2 � 2ab cos C

Quadratic Equation and Quadratic Formula

ax2 � bx � c � 0 x ���b / �

2ab2�� 4ac��

Infinite Series

�1 �

1x

� � 1 � x � x2 � x3 � . . . (�1 4 x 4 1)

�1 � x� � 1 � �2x

� � �x8

2

� � �1x6

3

� � . . . (�1 4 x 4 1)

��1

1

� x�� � 1 � �

2x

� � �38x2

� � �51x6

3

� � . . . (�1 4 x 4 1)

ex � 1 � x � �x2!

2

� � �x3!

3

� � . . . (�3 4 x 4 3)

sin x � x � �x3!

3

� � �x5!

5

� � �7x7

!� � . . . (�3 4 x 4 3)

cos x � 1 � �x2!

2

� � �x4

4

!� � �

6x6

!� � . . . (�3 4 x 4 3)

Note: If x is very small compared to 1, only the first few terms in the series areneeded.

B

C

A

a

c

b

964 APPENDIX C Mathematical Formulas

Derivatives

�ddx� (ax) � a �

ddx� (xn) � nxn�1 �

ddx� (au) � a�

ddux�

�ddx� (uv) � u �

ddvx� � v�

ddux� �

ddx� ��

uv

�� �

�ddx� (un) � nun�1 �

ddux� �

d

d

y

x� � �

d

d

u

y� �

ddux� �

ddux� � �

dx1/du�

�ddx� (sin u) � cos u �

ddux� �

ddx� (cos u) � �sin u �

ddux�

�ddx� (tan u) � sec2 u �

ddux� �

ddx� (cot u) � �csc2 u �

ddux�

�ddx� (sec u) � sec u tan u �

ddux� �

ddx� (csc u) � �csc u cot u �

ddux�

�ddx� (arctan u) � �

1 �

1u2� �

ddux� �

ddx� (log u) � �

loug e� �

ddux� �

ddx�(ln u) � �

1u

� �ddux�

�ddx� (au) � au ln a �

ddux� �

ddx� (eu) � eu �

ddux�

Indefinite IntegralsNote: A constant must be added to the result of every integration

�a dx � ax �u dv � uv � �v du (integration by parts)

�xn dx � �nx

�

n�1

1� (n ; �1) ��

dxx� � ln ?x? (x ; 0)

��dxxn� � �

1x1

�

�n

n� (n ; 1) �(a � bx)n dx � �

(a

b

�

(n

b

�

x)

1

n�

)

1

� (n ; �1)

��a �

dxbx

� � �1b

� ln (a � bx) ��(a �

dxbx)2� � ��

b(a �

1bx)

�

��(a �

dxbx)n� � � (n ; 1)

��a2 �

dxb2x2� � �

a1b� tan�1 �

bax� (x in radians) (a 0 0, b 0 0)

��a2 �

dxb2x2� � �

21ab� ln ��aa

�

�

bbxx

�� (x in radians) (a 0 0, b 0 0)

��a

x�

dxbx

� � �b12� [bx � a ln (a � bx)] ��

(a �

x dbxx)2� � �

b12���a �

abx

� � ln (a � bx)���

(a �

x dbxx)3� � ��

2ba2(

�

a �

2bbxx)2� ��

(a �

x dbxx)4� � ��

6b

a2(

�

a �

3b

b

x

x)3�

1���(n � 1)(b)(a � bx)n�1

v(du/dx) � u(dv/dx)���

v2

APPENDIX C Mathematical Formulas 965

��ax�

2dbxx

� � �21b3� [( a � bx)(�3a � bx) � 2a2 ln(a � bx)]

��(a

x�

2dbxx)2� � �

b13� ��bx(

a

2a

�

�

bx

bx)� � 2a ln (a � bx)�

��(a

x�

2dbxx)3� � �

b13� ��a2((3aa�

�

b

4

x

b

)

x2

)� � ln (a � bx)�

��(a

x�

2dbxx)4� � ��

a2 �

3b

33(

a

a

bx

�

�

bx

3

)

b3

2x2

�

�sin ax dx � ��cos

aax� �cos ax dx � �

sinaax�

�tan ax dx � �1a

� ln (sec ax) �cot ax dx � �1a

� ln (sin ax)

�sec ax dx � �1a

� ln (sec ax � tan ax) �csc ax dx � �1a

� ln (csc ax � cot ax)

�sin2 ax dx � �2x

� � �sin

42aax

� �cos2 ax dx � �2x

� � �sin

42aax

� (x in radians)

�x sin ax dx � �sin

a2ax� � �

x coas ax� (x in radians)

�x cos ax dx � �co

as2ax� � �

x sian ax� (x in radians)

�eax dx � �ea

ax

� �xeax dx � �ea

a

2

x

� (ax � 1) �ln ax dx � x(ln ax �1)

��1 �

dsxin ax� � ��

1a

� tan ��p4

� � �a2x�� ��a � bx� dx � �

32b�(a � bx)3/2

��a2 � b�2x 2� dx � �2x

� �a2 � b�2x 2� � �2ab

2

� ln ��bax� � �1 � �

ba

2� x2

2

���� � �

1b

� ln ��bax� � �1 � �

ba

2�x2

2

�����a2 � b�2x2� dx � �

2x

� �a2 � b�2x2� � �2ab

2

� sin�1 �bax�

Definite Integrals

�b

af(x) dx � ��a

bf(x) dx �b

af (x) dx � �c

af (x) dx � �b

cf (x)dx

dx����a2 � b2x2

966

Notation: A � areax�, y� � distances to centroid CIx, Iy � moments of inertia with respect to the x and y axes,respectivelyIxy � product of inertia with respect to the x and y axesIP � Ix � Iy � polar moment of inertia with respect to the origin ofthe x and y axesIBB � moment of inertia with respect to axis B-B

1 Rectangle (Origin of axes at centroid)

A � bh x� � �b2

� y� � �h2

�

Ix � �b1h2

3

� Iy � �h1b2

3

� Ixy � 0 IP � �b1h2� (h2 � b2)

2 Rectangle (Origin of axes at corner)

Ix � �b3h3

� Iy � �h3b3

� Ixy � �b2

4h2

� IP � �b3h� (h2 + b2)

IBB � �6(b

b2

3

�

h3

h2)�

3 Triangle (Origin of axes at centroid)

A � �b2h� x� � �

b �

3c

� y� � �h3

�

Ix � �b3h6

3

� Iy � �b3h6� (b2 � bc � c2)

Ixy � �b7h2

2

� (b � 2c) IP � �b3h6� (h2 � b2 � bc � c2)

yc

h

b

C x

x

y

y

x

h

b

O

B

B

y

x

xhy

b

C

Properties of Plane Areas

D

APPENDIX D Properties of Plane Areas 967

4 Triangle (Origin of axes at vertex)

Ix � �b1h2

3

� Iy � �b1h2� (3b2 � 3bc � c2)

Ixy � �b2h4

2

� (3b � 2c) IBB � �b4h3

�

5 Isosceles triangle (Origin of axes at centroid)

A � �b2h� x� � �

b2

� y� � �h3

�

Ix � �b3h6

3

� Iy � �h4b8

3

� Ixy � 0

IP � �1b4h4

� (4h2 � 3b2) IBB � �b1h2

3

�

(Note: For an equilateral triangle, h � �3� b/2.)

6 Right triangle (Origin of axes at centroid)

A � �b2h� x� � �

b3

� y� � �h3

�

Ix � �b3h6

3

� Iy � �h3b6

3

� Ixy � � �b7

2h2

2

�

IP � �b3h6� (h2 � b2) IBB � �

b1h2

3

�

7 Right triangle (Origin of axes at vertex)

Ix � �b1h2

3

� Iy � �h1b2

3

� Ixy � �b2

2h4

2

�

IP � �b1h2� (h2 � b2) IBB � �

b4h3

�

8 Trapezoid (Origin of axes at centroid)

A � �h(a

2� b)� y� � �

h

3

(

(

2

a

a

�

�

b

b

)

)�

Ix ��h3(a2

36�

(a4�

abb�

)b2)

� IBB � �h3(3

1

a

2

� b)�

y

yxh

b

a

B B

C

y

x

h

b

B B

O

y

y x

xh

b

BC

B

B B

C

y

x

b

hy

x

y c

h

bO

B B

x

968 APPENDIX D Properties of Plane Areas

9 Circle (Origin of axes at center)

A � pr2 � �p

4d2

� Ix � Iy � �p

4r4

� � �p6d4

4

�

Ixy � 0 IP � �p

2r4

� � �p3d2

4

� IBB � �5p

4r 4

� � �5p

64d 4

�

10 Semicircle (Origin of axes at centroid)

A � �p

2r2

� y� � �34pr�

Ix � �(9p 2

7�

2p64)r4

� 0.1098r4 Iy � �p

8r4

� Ixy � 0 IBB � �p

8r 4

�

11 Quarter circle (Origin of axes at center of circle)

A � �p

4r 2

� x� � y� � �34pr�

Ix � Iy � �p1r6

4

� Ixy � �r8

4

� IBB � �(9p2

14�

4p64)r4

� 0.05488r4

12 Quarter-circular spandrel (Origin of axes at point of tangency)

A � �1 � �p4

��r2 x� � �3(4

2�

rp)

� 0.7766r y� � �(1

3

0

(4

�

�

3

p

p

)

)r� 0.2234r

Ix � �1 � �51p6��r4 0.01825r4 Iy � IBB � ��

13

� � �1p6��r4 0.1370r4

13 Circular sector (Origin of axes at center of circle)

a � angle in radians (a , p/2)

A � ar2 x� � r sin a y� � �2r

3sian a�

Ix � �r4

4

� (a � sin a cos a) Iy � �r4

4

� (a � sin a cos a) Ixy � 0 IP � �a2r 4

�

a a

C

O

y

r

x

x

y

x

y

x

r

x

OC

B B

y

y

yx

r

B BC

O

x

y

y xr C

B B

y

xr

d = 2r

C

B B

APPENDIX D Properties of Plane Areas 969

14 Circular segment (Origin of axes at center of circle)

a � angle in radians (a , p/2)

A � r2(a � sin a cos a) y� � �23r� ��a � s

siinn3

aacos a

��Ix � �

r4

4

� (a � sin a cos a � 2 sin3 a cos a) Ixy � 0

Iy � �1r2

4

�(3a � 3 sin a cos a � 2 sin3 a cos a)

15 Circle with core removed (Origin of axes at center of circle)

a � angle in radians (a , p/2)

a � arccos �ar

� b � �r2 � a�2� A � 2r2�a � �arb2��

Ix � �r6

4

� �3a � �3ra2b

� � �2a

r 4b3

�� Iy � �r2

4

� �a � �arb2� � �

2ar 4

b3

�� Ixy � 0

16 Ellipse (Origin of axes at centroid)

A � pab Ix � �pa

4b3

� Iy � �p b

4a3

�

Ixy � 0 IP � �p

4ab� (b2 � a2)

Circumference p [1.5( a � b) � �a�b� ] ( a/3 , b , a)

4.17b2/a � 4a (0 , b , a/3)

17 Parabolic semisegment (Origin of axes at corner)

y � f (x) � h�1 � �bx2

2��A � �

23bh� x� � �

38b� y� � �

25h�

Ix � �1160b5h3

� Iy � �21h5b3

� Ixy � �b1

2h2

2

�

y

x

b

a a

bC

a

a

y

x

a

2a

br

b

C

a a

C

O

y

y

x

r

y = f (x)

y

x

y

x

C

O b

h

Vertex

970 APPENDIX D Properties of Plane Areas

18 Parabolic spandrel (Origin of axes at vertex)

y � f (x) � �hbx2

2

�

A � �b3h� x� � �

34b� y� � �

31h0�

Ix � �b2h1

3

� Iy � �h5b3

� Ixy � �b1

2h2

2

�

19 Semisegment of nth degree (Origin of axes at corner)

y � f (x) � h�1 � �xbn

n

�� (n 0 0)

A � bh��n �

n1

�� x� � �b2((nn

�

�

12))

� y� � �2n

h�

n1

�

Ix � Iy � �3(

hnb�

3n3)

� Ixy ��4(n �

b2

1h)

2

(nn

2

� 2)�

20 Spandrel of nth degree (Origin of axes at point of tangency)

y � f (x) � �hbxn

n

� (n 0 0)

A � �n

b�

h1

� x� � �b(

n

n

�

�

2

1)� y� � �

2

h

(

(

2

n

n

�

�

1

1

)

)�

Ix � �3(3

bnh�

3

1)� Iy � �

nh�

b3

3� Ixy � �

4(nb2

�

h2

1)�

21 Sine wave (Origin of axes at centroid)

A � �4pbh� y� � �

p8h�

Ix � ��98p� � �

1p6��bh3 0.08659bh3 Iy � ��

p4

� � �p32

3��hb3 0.2412hb3

Ixy � 0 IBB � �89bph3

�

22 Thin circular ring (Origin of axes at center) Approximate formulas for case when t is small

A � 2prt � pdt Ix � Iy � pr 3t � �p

8d3t�

Ixy � 0 IP � 2pr3t � �pd

4

3t�

y

x

t

Cr

d = 2r

2bh3n3

���(n � 1)(2n � 1)(3n � 1)

y

y

h

bx

x

CO

Vertex

y = f (x)

C

O

yy = f (x)

y

x

x

b

h

y

x

x h

bO

C y

y = f (x)

y

yh

b b

xB B

C

APPENDIX D Properties of Plane Areas 971

23 Thin circular arc (Origin of axes at center of circle)Approximate formulas for case when t is small

b � angle in radians (Note: For a semicircular arc, b � p/2.)

A � 2brt y� � �r si

b

n b�

Ix � r3t(b � sin b cos b) Iy � r3t(b � sin b cos b)

Ixy � 0 IBB � r3t��2b �

2

sin2b� � �

1 � c

b

os2b��

24 Thin rectangle (Origin of axes at centroid)Approximate formulas for case when t is small

A � bt

Ix � �t1b2

3

� sin2 b Iy � �t1b2

3

� cos2 b IBB � �tb3

3

� sin2 b

25 Regular polygon with n sides (Origin of axes at centroid)

C � centroid (at center of polygon)

n � number of sides (n 5 3) b � length of a side

b � central angle for a side a � interior angle (or vertex angle)

b � �36

n0°� a � ��n �

n2

��180° a � b � 180°

R1 � radius of circumscribed circle (line CA) R2 � radius of inscribed circle (line CB)

R1 � �b2

� csc �b

2� R2 � �

b2

� cot �b

2� A � �

n4b2

� cot �b

2�

Ic � moment of inertia about any axis through C (the centroid C is a principal point andevery axis through C is a principal axis)

Ic � �1n9b2

4

��cot �b2

���3cot2 �b2

� � 1� IP � 2Ic

y

x

B B

C

b

t

b

y

y

x

B BC

b b

t

r

O

b

b

a

C

BA

R1R2

984

Deflections and Slopesof Beams

G

TABLE G-1 DEFLECTIONS AND SLOPES OF CANTILEVER BEAMS

v � deflection in the y direction (positive upward)v� � dv/dx � slope of the deflection curvedB � �v(L) � deflection at end B of the beam (positive downward)uB � �v�(L) � angle of rotation at end B of the beam (positive clockwise)EI � constant

1 v � ��2

q

4

x

E

2

I� (6L2 � 4Lx � x2) v� � ��

6

q

E

x

I�(3L2 � 3Lx � x2)

dB � �8

q

E

L4

I� uB � �

6

q

E

L3

I�

2 v � ��2

q

4

x

E

2

I� (6a2 � 4ax � x2) (0 , x , a)

v� � ��6

q

E

x

I�(3a2 � 3ax � x2) (0 , x , a)

v � ��2

q

4

a

E

3

I�(4x � a) v� � ��

6

q

E

a3

I� (a , x , L)

At x � a: v � ��8

q

E

a4

I� v� � ��

6

q

E

a3

I�

dB � �2

q

4

a

E

3

I�(4L � a) uB � �

6

q

E

a3

I�

q

a b

q

y

xA B

uB

dB

L

APPENDIX G Deflections and Slopes of Beams 985

3 v � ��1

q

2

b

E

x2

I�(3L � 3a � 2x) (0 , x , a)

v� � ��q

2

b

E

x

I�(L � a � x) (0 , x , a)

v � ��24

q

EI�(x4 � 4Lx3 � 6L2x2 � 4a3x � a4) (a , x , L)

v� � ��6

q

EI�(x3 � 3Lx2 � 3L2x � a3) (a , x , L)

At x � a: v � ��1

q

2

a

E

2b

I�(3L � a) v� � ��

q

2

a

E

b

I

L�

dB � �24

q

EI�(3L4 � 4a3L � a4) uB � �

6

q

EI�(L3 � a3)

4 v � ��6PExI

2

�(3L � x) v� � ��2PExI

�(2L � x)

dB � �P3E

LI

3

� uB � �P2E

LI

2

�

5 v � ��6PExI

2

�(3a � x) v� � ��2PExI

�(2a � x) (0 , x , a)

v � ��6PEaI

2

�(3x � a) v� � ��2PEaI

2

� (a , x , L)

At x � a: v � ��3PEaI

3

� v� � ��2PEaI

2

�

dB � �6PEaI

2

�(3L � a) uB � �2PEaI

2

�

6 v � ��M

2E0x

I

2

� v� � ��M

E0

I

x�

dB � �M

2E0L

I

2

� uB � �M

E0

I

L�

(Continued)

M0

P

a b

P

q

a b

986 APPENDIX G Deflections and Slopes of Beams

7 v � ��M

2E0x

I

2

� v� � ��M

E0

I

x� (0 , x , a)

v � ��M

2E0a

I�(2x � a) v� � ��

M

E0

I

a� (a , x , L)

At x � a: v � ��M

2E0a

I

2

� v� � ��M

E0

I

a�

dB � �M

2E0a

I�(2L � a) uB � �

M

E0

I

a�

8 v � ��12

q

00

L

x2

EI�(10L3 � 10L2x � 5Lx2 � x3)

v� � ��24

q

L0x

EI�(4L3 � 6L2x � 4Lx2 � x3)

dB � �3

q

00L

E

4

I� uB � �

2

q

40L

E

3

I�

9 v � ��12

q

00

L

x2

EI�(20L3 � 10L2x � x3)

v� � ��24

q

L0x

EI�(8L3 � 6L2x � x3)

dB � �1

1

1

2

q

00

E

L

I

4

� uB � �q

80

E

L

I

3

�

10 v � ��3

q

p04

L

EI� �48L3 cos �

p

2L

x� � 48L3 � 3p3Lx2 � p3x3�

v� � ��p

q30

E

L

I��2p 2Lx � p2x2 � 8L2 sin �

p

2L

x��

dB � �3

2

p

q04

L

E

4

I� (p 3 � 24) uB � �

p

q03

L

E

3

I� (p2 � 8)

q = q0 cos 9x—2L

q0

q0

q0

M0

a b

APPENDIX G Deflections and Slopes of Beams 987

TABLE G-2 DEFLECTIONS AND SLOPES OF SIMPLE BEAMS

v � deflection in the y direction (positive upward)v� � dv/dx � slope of the deflection curvedC � �v(L/2) � deflection at midpoint C of the beam (positive downward)x1 � distance from support A to point of maximum deflectiondmax � �vmax � maximum deflection (positive downward)uA � �v�(0) � angle of rotation at left-hand end of the beam

(positive clockwise)uB � v�(L) � angle of rotation at right-hand end of the beam

EI � constant (positive counterclockwise)

1 v � ��24

qx

EI�(L3 � 2Lx2 � x3)

v� � ��24

q

EI�(L3 � 6Lx2 � 4x3)

dC � dmax� �3

5

8

q

4

L

E

4

I� uA � uB � �

2

q

4

L

E

3

I�

2 v � ��38

q

4

x

EI�(9L3 � 24Lx2 � 16x3) �0 , x , �

L2

��v� � ��

38

q

4EI�(9L3 � 72Lx2 � 64x3) �0 , x , �

L2

��v � ��

38

q

4

L

EI�(8x3 � 24Lx2 � 17L2x � L3) ��

L2

� , x , L�v� � ��

38

q

4

L

EI�(24x2 � 48Lx � 17L2) ��

L2

� , x , L�dC � �

7

5

6

q

8

L

E

4

I� uA � �

1

3

2

q

8

L

E

3

I� uB � �

3

7

8

q

4

L

E

3

I�

3 v � ��24

q

L

x

EI�(a4 � 4a3L � 4a2L2 � 2a2x2 � 4aLx2 � Lx3) (0 , x , a)

v� � ��24

q

LEI�(a4 � 4a3L � 4a2L2 � 6a2x2 � 12aLx2 � 4Lx3) (0 , x , a)

v � ��24

q

L

a

E

2

I�(�a2L � 4L2x � a2x � 6Lx2 � 2x3) (a , x , L)

v� � ��24

q

L

a

E

2

I�(4L2 � a2 � 12Lx � 6x2) (a , x , L)

uA � �24

q

L

a

E

2

I�(2L � a)2 uB � �

24

q

L

a

E

2

I�(2L2 � a2)

(Continued)

q

a

q

L—2

L—2

q

y

x

uA uBA B

L

L—2

L—2

P

988 APPENDIX G Deflections and Slopes of Beams

4 v � ��4P8E

xI

�(3L2 � 4x2) v� � ��16

PEI�(L2 � 4x2) �0 , x , �

L2

��dC � dmax � �

4P8LE

3

I� uA � uB � �

1P6LE

2

I�

5 v � ��6PLbExI

�(L2 � b2 � x2) v� � ��6PL

bEI�(L2 � b2 � 3x2) (0 , x , a)

uA � �Pab

6

(

L

L

E

�

I

b)� uB � �

Pab

6

(

L

L

E

�

I

a)�

If a 5 b, dC ��Pb(3L

48

2

E�

I4b2)

� If a , b, dC ��Pa(3L

4

2

8E�

I4a2)

�

If a 5 b, x1 � ��L2 �

3

b� 2

�� and dmax �

6 v � ��6PExI

�(3aL � 3a2 � x2) v� � ��2PEI�(aL � a2 � x2) (0 , x , a)

v � ��6PEaI

�(3Lx � 3x2 � a2) v� � ��2PEaI

�(L � 2x) (a , x , L � a)

dC � dmax � �2P4Ea

I�(3L2 � 4a2) uA � uB ��

Pa(

2

L

E

�

I

a)�

7 v � ��6

M

L0

E

x

I� (2L2 � 3Lx � x2) v� � ��

6

M

LE0

I� (2L2 � 6Lx � 3x2)

dC � �M

160

E

L

I

2

� uA � �M

3E0L

I� uB � �

M

6E0L

I�

x1 � L�1 � ��33��� and dmax � �

9

M

�0

3�L

E

2

I�

8 v � ��2

M

4L0

E

x

I�(L2 � 4x2) v� � ��

24

M

L0

EI�(L2 � 12x2) �0 , x , �

L2

��dC � 0 uA � �

2

M

40

E

L

I� uB � ��

2

M

40

E

L

I�L

—2

L—2

M0

M0

P P

a a

Pb(L2 � b2)3/2

��9�3� LEI

P

a b

APPENDIX G Deflections and Slopes of Beams 989

9 v � ��6

M

L0

E

x

I�(6aL � 3a2 � 2L2 � x2) (0 , x , a)

v� � ��6

M

LE0

I�(6aL � 3a2 � 2L2 � 3x2) (0 , x , a)

At x � a: v � ��M

3L0a

E

b

I�(2a � L) v� � ��

3

M

LE0

I�(3aL � 3a2 � L2)

uA � �6

M

LE0

I�(6aL � 3a2 � 2L2) uB � �

6

M

LE0

I�(3a2 � L2)

10 v � ��M

2E0x

I�(L � x) v� � ��

2

M

E0

I�(L � 2x)

dC � dmax � �M

80

E

L

I

2

� uA � uB � �M

2E0L

I�

11 v � ��36

q

00

L

x

EI�(7L4 � 10L2x2 � 3x4)

v� � ��360

q

L0

EI�(7L4 � 30L2x2 � 15x4)

dC � �7

5

6

q

80L

E

4

I� uA � �

3

7

6

q

00L

E

3

I� uB � �

4

q

50L

E

3

I�

x1 � 0.5193L dmax � 0.00652�q

E0L

I

4

�

12 v � ��96

q

00

L

x

EI�(5L2 � 4x2)2 �0 , x , �

L2

��v� � ��

192

q

L0

EI� (5L2 � 4x2)(L2 � 4x2) �0 , x , �

L2

��dC � dmax � �

1

q

20

0

L

E

4

I� uA � uB � �

1

5

9

q

20 L

E

3

I�

13 v � ��p

q04

L

E

4

I� sin �

pLx� v� � ��

p

q03

L

E

3

I� cos �

pLx�

dC � dmax � �p

q04

L

E

4

I� uA � uB � �

p

q03

L

E

3

I�

q0

L—2

L—2

q0

a b

M0

M0M0

q = q0 sin9x—L

990

Propertiesof Materials

H

Notes:1. Properties of materials vary greatly depending upon manufacturing

processes, chemical composition, internal defects, temperature, previous load-ing history, age, dimensions of test specimens, and other factors. The tabulatedvalues are typical but should never be used for specific engineering or designpurposes. Manufacturers and materials suppliers should be consulted for infor-mation about a particular product.

2. Except when compression or bending is indicated, the modulus of elas-ticity E, yield stress sY, and ultimate stress sU are for materials in tension.

APPENDIX H Properties of Materials 991

TABLE H-1 WEIGHTS AND MASS DENSITIES

MaterialWeight density g Mass density r

lb/ft3 kN/m3 slugs/ft3 kg/m3

Aluminum alloys 160–180 26–28 5.2–5.4 2,600–2,8002014-T6, 7075-T6 175 28 5.4 2,8006061-T6 170 26 5.2 2,700

Brass 520–540 82–85 16–17 8,400–8,600

Bronze 510–550 80–86 16–17 8,200–8,800

Cast iron 435–460 68–72 13–14 7,000–7,400

ConcretePlain 145 23 4.5 2,300Reinforced 150 24 4.7 2,400Lightweight 70–115 11–18 2.2–3.6 1,100–1,800

Copper 556 87 17 8,900

Glass 150–180 24–28 4.7–5.4 2,400–2,800

Magnesium alloys 110–114 17–18 3.4–3.5 1,760–1,830

Monel (67% Ni, 30% Cu) 550 87 17 8,800

Nickel 550 87 17 8,800

PlasticsNylon 55–70 8.6–11 1.7–2.2 880–1,100Polyethylene 60–90 9.4–14 1.9–2.8 960–1,400

RockGranite, marble, quartz 165–180 26–28 5.1–5.6 2,600–2,900Limestone, sandstone 125–180 20–28 3.9–5.6 2,000–2,900

Rubber 60–80 9–13 1.9–2.5 960–1,300

Sand, soil, gravel 75–135 12–21 2.3–4.2 1,200–2,200

Steel 490 77.0 15.2 7,850

Titanium 280 44 8.7 4,500

Tungsten 1,200 190 37 1,900

Water, fresh 62.4 9.81 1.94 1,000sea 63.8 10.0 1.98 1,020

Wood (air dry)Douglas fir 30–35 4.7–5.5 0.9–1.1 480–560Oak 40–45 6.3–7.1 1.2–1.4 640–720Southern pine 35–40 5.5–6.3 1.1–1.2 560–640

992 APPENDIX H Properties of Materials

TABLE H-2 MODULI OF ELASTICITY AND POISSON’S RATIOS

MaterialModulus of elasticity E Shear modulus of elasticity G

Poisson’sksi GPa ksi GPa ratio n

Aluminum alloys 10,000–11,400 70–79 3,800–4,300 26–30 0.332014-T6 10,600 73 4,000 28 0.336061-T6 10,000 70 3,800 26 0.337075-T6 10,400 72 3,900 27 0.33

Brass 14,000–16,000 96–110 5,200–6,000 36–41 0.34

Bronze 14,000–17,000 96–120 5,200–6,300 36–44 0.34

Cast iron 12,000–25,000 83–170 4,600–10,000 32–69 0.2–0.3

Concrete (compression) 2,500–4,500 17–31 0.1–0.2

Copper and copper alloys 16,000–18,000 110–120 5,800–6,800 40–47 0.33–0.36

Glass 7,000–12,000 48–83 2,700–5,100 19–35 0.17–0.27

Magnesium alloys 6,000–6,500 41–45 2,200–2,400 15–17 0.35

Monel (67% Ni, 30% Cu) 25,000 170 9,500 66 0.32

Nickel 30,000 210 11,400 80 0.31

PlasticsNylon 300–500 2.1–3.4 0.4Polyethylene 100–200 0.7–1.4 0.4

Rock (compression)Granite, marble, quartz 6,000–14,000 40–100 0.2–0.3Limestone, sandstone 3,000–10,000 20–70 0.2–0.3

Rubber 0.1–0.6 0.0007–0.004 0.03–0.2 0.0002–0.001 0.45–0.50

Steel 28,000–30,000 190–210 10,800–11,800 75–80 0.27–0.30

Titanium alloys 15,000–17,000 100–120 5,600–6,400 39–44 0.33

Tungsten 50,000–55,000 340–380 21,000–23,000 140–160 0.2

Wood (bending)Douglas fir 1,600–1,900 11–13Oak 1,600–1,800 11–12Southern pine 1,600–2,000 11–14

APPENDIX H Properties of Materials 993

TABLE H-3 MECHANICAL PROPERTIES

MaterialYield stress sY Ultimate stress sU

Percentelongation(2 in. gage

ksi MPa ksi MPa length)

Aluminum alloys 5–70 35–500 15–80 100–550 1–452014-T6 60 410 70 480 136061-T6 40 270 45 310 177075-T6 70 480 80 550 11

Brass 10–80 70–550 30–90 200–620 4–60

Bronze 12–100 82–690 30–120 200–830 5–60

Cast iron (tension) 17–42 120–290 10–70 69–480 0–1

Cast iron (compression) 50–200 340–1,400

Concrete (compression) 1.5–10 10–70

Copper and copper alloys 8–110 55–760 33–120 230–830 4–50

Glass 5–150 30–1,000 0Plate glass 10 70Glass fibers 1,000–3,000 7,000–20,000

Magnesium alloys 12–40 80–280 20–50 140–340 2–20

Monel (67% Ni, 30% Cu) 25–160 170–1,100 65–170 450–1,200 2–50

Nickel 15–90 100–620 45–110 310–760 2–50

PlasticsNylon 6–12 40–80 20–100Polyethylene 1–4 7–28 15–300

Rock (compression)Granite, marble, quartz 8–40 50–280Limestone, sandstone 3–30 20–200

Rubber 0.2–1.0 1–7 1–3 7–20 100–800

SteelHigh-strength 50–150 340–1,000 80–180 550–1,200 5–25Machine 50–100 340–700 80–125 550–860 5–25Spring 60–240 400–1,600 100–270 700–1,900 3–15Stainless 40–100 280–700 60–150 400–1,000 5–40Tool 75 520 130 900 8

Steel, structural 30–100 200–700 50–120 340–830 10–40ASTM-A36 36 250 60 400 30ASTM-A572 50 340 70 500 20ASTM-A514 100 700 120 830 15

(Continued)

994 APPENDIX H Properties of Materials

TABLE H-3 MECHANICAL PROPERTIES (Continued)

MaterialYield stress sY Ultimate stress sU

Percentelongation(2 in. gage

ksi MPa ksi MPa length)

Steel wire 40–150 280–1,000 80–200 550–1,400 5–40

Titanium alloys 110–150 760–1,000 130–170 900–1,200 10

Tungsten 200–600 1,400–4,000 0–4

Wood (bending)Douglas fir 5–8 30–50 8–12 50–80Oak 6–9 40–60 8–14 50–100Southern pine 6–9 40–60 8–14 50–100

Wood (compression parallel to grain)Douglas fir 4–8 30–50 6–10 40–70Oak 4–6 30–40 5–8 30–50Southern pine 4–8 30–50 6–10 40–70

TABLE H-4 COEFFICIENTS OF THERMAL EXPANSION

Coefficient ofMaterial thermal expansion a

10�6/°F 10�6/°C

Aluminum alloys 13 23

Brass 10.6–11.8 19.1–21.2

Bronze 9.9–11.6 18–21

Cast iron 5.5–6.6 9.9–12

Concrete 4–8 7–14

Copper and copper alloys 9.2–9.8 16.6–17.6

Glass 3–6 5–11

Magnesium alloys 14.5–16.0 26.1–28.8

Monel (67% Ni, 30% Cu) 7.7 14

Nickel 7.2 13

Coefficient ofMaterial thermal expansion a

10�6/°F 10�6/°C

PlasticsNylon 40–80 70–140Polyethylene 80–160 140–290

Rock 3–5 5–9

Rubber 70–110 130–200

Steel 5.5–9.9 10–18High-strength 8.0 14Stainless 9.6 17Structural 6.5 12

Titanium alloys 4.5–6.0 8.1–11

Tungsten 2.4 4.3

CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS

Times conversion factorU.S. Customary unit

Accurate PracticalEquals SI unit

Acceleration (linear)foot per second squared ft/s2 0.3048* 0.305 meter per second squared m/s2

inch per second squared in./s2 0.0254* 0.0254 meter per second squared m/s2

Areasquare foot ft2 0.09290304* 0.0929 square meter m2

square inch in.2 645.16* 645 square millimeter mm2

Density (mass)slug per cubic foot slug/ft3 515.379 515 kilogram per cubic meter kg/m3

Density (weight) pound per cubic foot lb/ft3 157.087 157 newton per cubic meter N/m3

pound per cubic inch lb/in.3 271.447 271 kilonewton per cubicmeter kN/m3

Energy; workfoot-pound ft-lb 1.35582 1.36 joule (N�m) Jinch-pound in.-lb 0.112985 0.113 joule Jkilowatt-hour kWh 3.6* 3.6 megajoule MJBritish thermal unit Btu 1055.06 1055 joule J

Forcepound lb 4.44822 4.45 newton (kg�m/s2) Nkip (1000 pounds) k 4.44822 4.45 kilonewton kN

Force per unit lengthpound per foot lb/ft 14.5939 14.6 newton per meter N/mpound per inch lb/in. 175.127 175 newton per meter N/mkip per foot k/ft 14.5939 14.6 kilonewton per meter kN/mkip per inch k/in. 175.127 175 kilonewton per meter kN/m

Lengthfoot ft 0.3048* 0.305 meter minch in. 25.4* 25.4 millimeter mmmile mi 1.609344* 1.61 kilometer km

Massslug lb-s2/ft 14.5939 14.6 kilogram kg

Moment of a force; torquepound-foot lb-ft 1.35582 1.36 newton meter N·mpound-inch lb-in. 0.112985 0.113 newton meter N·mkip-foot k-ft 1.35582 1.36 kilonewton meter kN·mkip-inch k-in. 0.112985 0.113 kilonewton meter kN·m

Temperature Conversion Formulas T(°C) � �59

�[T(°F) � 32] � T(K) � 273.15

T(K) � �59

�[T(°F) � 32] � 273.15 � T(°C) � 273.15

T(°F) � �95

�T(°C) � 32 � �95

�T(K) � 459.67

CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS (Continued)

Times conversion factorU.S. Customary unit

Accurate PracticalEquals SI unit

Moment of inertia (area)inch to fourth power in.4 416,231 416,000 millimeter to fourth

power mm4

inch to fourth power in.4 0.416231 � 10�6 0.416 � 10�6 meter to fourth power m4

Moment of inertia (mass)slug foot squared slug-ft2 1.35582 1.36 kilogram meter squared kg·m2

Powerfoot-pound per second ft-lb/s 1.35582 1.36 watt (J/s or N·m/s) Wfoot-pound per minute ft-lb/min 0.0225970 0.0226 watt Whorsepower (550 ft-lb/s) hp 745.701 746 watt W

Pressure; stresspound per square foot psf 47.8803 47.9 pascal (N/m2) Papound per square inch psi 6894.76 6890 pascal Pakip per square foot ksf 47.8803 47.9 kilopascal kPakip per square inch ksi 6.89476 6.89 megapascal MPa

Section modulusinch to third power in.3 16,387.1 16,400 millimeter to third power mm3

inch to third power in.3 16.3871 � 10�6 16.4 � 10�6 meter to third power m3

Velocity (linear)foot per second ft/s 0.3048* 0.305 meter per second m/sinch per second in./s 0.0254* 0.0254 meter per second m/smile per hour mph 0.44704* 0.447 meter per second m/smile per hour mph 1.609344* 1.61 kilometer per hour km/h

Volumecubic foot ft3 0.0283168 0.0283 cubic meter m3

cubic inch in.3 16.3871 � 10�6 16.4 � 10�6 cubic meter m3

cubic inch in.3 16.3871 16.4 cubic centimeter (cc) cm3

gallon (231 in.3) gal. 3.78541 3.79 liter Lgallon (231 in.3) gal. 0.00378541 0.00379 cubic meter m3

*An asterisk denotes an exact conversion factorNote: To convert from SI units to USCS units, divide by the conversion factor

PRINCIPAL UNITS USED IN MECHANICS

International System (SI) U.S. Customary System (USCS)Quantity

Unit Symbol Formula Unit Symbol Formula

Acceleration (angular) radian per second squared rad/s2 radian per second squared rad/s2

Acceleration (linear) meter per second squared m/s2 foot per second squared ft/s2

Area square meter m2 square foot ft2

Density (mass) kilogram per cubic meter kg/m3 slug per cubic foot slug/ft3

(Specific mass)

Density (weight) newton per cubic meter N/m3 pound per cubic foot pcf lb/ft3

(Specific weight)

Energy; work joule J N�m foot-pound ft-lb

Force newton N kg�m/s2 pound lb (base unit)

Force per unit length newton per meter N/m pound per foot lb/ft(Intensity of force)

Frequency hertz Hz s�1 hertz Hz s�1

Length meter m (base unit) foot ft (base unit)

Mass kilogram kg (base unit) slug lb-s2/ft

Moment of a force; torque newton meter N�m pound-foot lb-ft

Moment of inertia (area) meter to fourth power m4 inch to fourth power in.4

Moment of inertia (mass) kilogram meter squared kg�m2 slug foot squared slug-ft2

Power watt W J/s foot-pound per second ft-lb/s(N�m/s)

Pressure pascal Pa N/m2 pound per square foot psf lb/ft2

Section modulus meter to third power m3 inch to third power in.3

Stress pascal Pa N/m2 pound per square inch psi lb/in.2

Time second s (base unit) second s (base unit)

Velocity (angular) radian per second rad/s radian per second rad/s

Velocity (linear) meter per second m/s foot per second fps ft/s

Volume (liquids) liter L 10�3 m3 gallon gal. 231 in.3

Volume (solids) cubic meter m3 cubic foot cf ft3

SI PREFIXES

Prefix Symbol Multiplication factor

tera T 1012 � 1 000 000 000 000.giga G 109 � 1 000 000 000.mega M 106 � 1 000 000.kilo k 103 � 1 000.hecto h 102 � 100.deka da 101 � 10.deci d 10�1 � 0.1centi c 10�2 � 0.01milli m 10�3 � 0.001micro � 10�6 � 0.000 001nano n 10�9 � 0.000 000 001pico p 10�12 � 0.000 000 000 001

Note: The use of the prefixes hecto, deka, deci, and centi is not recommended in SI.

SELECTED PHYSICAL PROPERTIES

Property SI USCS

Water (fresh)weight density 9.81 kN/m3 62.4 lb/ft3

mass density 1000 kg/m3 1.94 slugs/ft3

Sea water weight density 10.0 kN/m3 63.8 lb/ft3

mass density 1020 kg/m3 1.98 slugs/ft3

Aluminum (structural alloys)weight density 28 kN/m3 175 lb/ft3

mass density 2800 kg/m3 5.4 slugs/ft3

Steelweight density 77.0 kN/m3 490 lb/ft3

mass density 7850 kg/m3 15.2 slugs/ft3

Reinforced concreteweight density 24 kN/m3 150 lb/ft3

mass density 2400 kg/m3 4.7 slugs/ft3

Atmospheric pressure (sea level)Recommended value 101 kPa 14.7 psiStandard international value 101.325 kPa 14.6959 psi

Acceleration of gravity(sea level, approx. 45° latitude)

Recommended value 9.81 m/s2 32.2 ft/s2

Standard international value 9.80665 m/s2 32.1740 ft/s2

Unidades fUndamentales

MAGNITUD UNIDAD SÍMBOLO

longitud metro m

masa kilogramo kg

tiempo segundo s

intensidad de corriente eléctrica amperio A

temperatura termodinámica kelvin K

cantidad de sustancia mol mol

intensidad luminosa candela cd

Unidades sUPlementaRias (Estas unidades se pueden utilizar como fundamentales)

MAGNITUD UNIDAD SÍMBOLO

ángulo plano radián rad

ángulo sólido estereorradián sr

Unidades deRiVadas mÁs fReCUentes

MAGNITUD UNIDAD EXPRESIÓN EN

UNIDADES FUNDAMENTALES

U OTRAS UNIDADES

NOMBRE SÍMBOLO

ESPACIO Y TIEMPO

Superficie, área metro cuadrado m2 m2

volumen metro cúbico m3 m3

velocidad angular radián por segundo rad/s s-1.rad

velocidad metro por segundo m/s m.s-1

aceleración metro por segundo por segundo m/s2 m.s -2

frecuencia hercio Hz s-1

frecuencia de rotación por segundo s-1 s-1

MAGNITUD UNIDAD EXPRESIÓN EN

UNIDADES FUNDAMENTALES

U OTRAS UNIDADES

NOMBRE SÍMBOLO

MECÁNICA

densidad kilogramo por metro cúbico kg/m3 m-3.kg

caudal másico kilogramo por segundo kg/s kg.s-1

caudal volúmico metro cúbico por segundo m3/s m3.s-1

cantidad de movimiento kilogramo-metro por segundo kg.m/s m.kg.s-1

momento cinético kilogramo-metro cuadrado por segundo kg.m2/s m2kg.s-1

momento de inercia kilogramo-metro cuadrado kg.m2 m2.kg

fuerza newton N m.kg.s-2

momento de una fuerza newton-metro N.m m2.kg.s-2

presión, tensión pascal Pa N/m2

viscosidad dinámica pascal-segundo Pa.s m-1.kg.s-1

viscosidad cinemática metro cuadrado por segundo m2/s m2.s-1

tensión superficial newton por metro N/m kg.s-2

energía, trabajo, cantidad de calor julio J m2.kg.s-2

potencia, flujo energético vatio W m2.kg.s-3

TERMODINÁMICA

coeficiente de dilatación lineal por kelvin K-1 K-1

conductividad térmica vatio por metro-kelvin W/(m.K) m.kg.s-3.K-1

entropía específica julio por kilogramo-kelvin J/(kg.K) m2.s-2 K-1

entropía julio por kelvin J/K m2.kg.s-2.K-1

energía interna, entalpía, energía libre, entalpía libre julio J m2.kg.s-2

ÓPTICA

flujo luminoso lumen lm cd.sr

luminancia candela por metro cuadrado cd/m2 m-2.cd

exitancia luminosa lumen por metro cuadrado lm/m2 m-2.cd.sr

iluminancia lux lx m-2.cd.sr

exposición luminosa lux-segundo lx.s m-2.s.cd.sr

eficacia luminosa lumen por vatio lm/W m-2.kg-1.s3.cd.sr

MAGNITUD UNIDAD EXPRESIÓN EN

UNIDADES FUNDAMENTALES

U OTRAS UNIDADES

NOMBRE SÍMBOLO

ELECTRICIDAD-MAGNETISMO

cantidad de electricidad, carga eléctrica culombio C s.A

intensidad de campo eléctrico voltio por metro V/m m.kg.s-3.A-1

tensión eléctrica, diferencia depotencial, fuerza electromotriz voltio V m2.kg.s-3.A-1

capacidad faradio F m-2.kg-1.s4.A2

intensidad de campo magnético amperio por metro A/m m-1.A

inducción magnética tesla T kg.s-2.A-1

flujo de inducción magnética weber Wb m2.kg.s-2.A-1

Inductancia, permeancia henrio H m2.kg.s-2.A-2

reluctancia por henrio H-1 m-2.kg-1.s2.A2

resistencia, impedancia, reactancia ohmio Ω m2.kg.s-3.A-2

conductancia, admitancia, susceptancia siemens S m-2.kg-1.s3.A2

resistividad ohmio-metro Ω.m m3.kg.s-3.A-2

conductividad siemens por metro S/m m-3.kg-1.s3.A2

RADIACIONES IONIZANTES

actividad (radiaciones ionizantes) becquerel Bq s-1

dosis absorbida, energía impartida, kerma gray Gy J/kg

dosis equivalente sievert Sv J/kg

QUÍMICA FÍSICA y FÍSICA MOLECULAR

masa molar kilogramo por mol kg/mol kg.mol-1

volumen molar metro cúbico por mol m3/mol m3.mol-1

concentración kilogramo por metro cúbico kg/m3 m-3.kg

molaridad mol por metro cúbico mol/m3 m-3.mol

molalidad mol por kilogramo mol/kg kg-1.mol

energía molar julio por mol J/mol m2.kg.s-2.mol-1

mÚltiPlOs Y sUBmÚltiPlOs

FACTOR PREFIJO SÍMBOLO

1018 exa E

1015 peta P

1012 tera T

109 giga G

106 mega M

103 kilo k

102 hecto h

10 deca da

FACTOR PREFIJO SÍMBOLO

10-1 deci d

10-2 centi c

10-3 mili m

10-6 micro µ

10-9 nano n

10-12 pico p

10-15 femto f

10-18 atto a

ReGlas PaRa el emPleO de lOs mÚltiPlOs Y sUBmÚltlPlOsSe recomienda que el prefijo esté ligado a la unidad del numerador.

Los errores de cálculo se pueden evitar más fácilmente si todas las magnitudes están expresadas en unidades SI con los prefijos reemplazados por potencias de 10.

La elección de un múltiplo o submúltiplo de una unidad SI está regida, ante todo, por la comodidad que resulte de su empleo.

El múltiplo conviene elegirlo de forma que su valor numérico esté comprendido entre 0.1 y 1.000.

No se deben utilizar dos prefijos consecutivos; por ejemplo, se escribe nanómetro (nm), en lugar de milimicrómetro (m µ m).

ReGlas de esCRitURa de lOs sÍmBOlOs de UnidadesLos símbolos de unidades se emplean únicamente a continuación del valor numérico de la magnitud considerada con relación a la unidad elegida. Son invariables, respecto al plural, se escriben sin pun-to final y se deja un espacio entre el valor numérico y el símbolo.

Se escriben en caracteres rectos, con letras minúsculas, excepto cuando el nombre de la unidad se deriva de un nombre propio, en cuyo caso, la primera letra se escribe con mayúscula.

No debe figurar más de una raya de fracción sobre la misma línea de escritura para representar una unidad compuesta, a menos que se utilicen paréntesis para evitar cualquier ambigüedad.

NOTA - Todas las unidades derivadas pueden ser expresadas en función de las unidades fundamentales y/o suplementarias.