apendices c y d

8/18/2019 Apendices C y D

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; s;mply Supported Beam,~I 1 . . . . .

. rL ) '~ •----~-;m·.:··l· : :f-x

2 ~ 2

2

v pl ) f~ x

. . . . 0 · - 0 -

I: . · --~----·c ~v

3 Mf-- ., ·· .,4-- - xE L I

vI w

4 ii/ }:, -~--~~-:~~ J,- xb_ ~IEvrnax b_ :___j2 2

v

I w

5 k r1 r r r r 1 {~ x.... ·- . .oI:;1· · : ~ ·:i , .I

v

Iwo

~ x

IE L

6

Slope

PL2ii = - 2 = - 16El

0 _ _ _b_ ~ 2_ _ 2~)

l - 6LEl

02 = + Pa L2 - a~6L EJ

ML

0i = - 3£1

MLIJi = +6EI

wOii = - 02 = - 24El

o1

= _ wa 224LEJ (2 l - a )2

wa 202 = +

(2l2 - a2)24Le/

01 = - 7WoL3360El

2 = + woL345£1

Deflection

PL3

vrn"" = - 48El

Pa 2b2v

3LEl

at x = a

ML2vnrn•. = - 9JJE1

w x = L(-~]

Sw l 4

Vmox = - 384 £/

wa 3

v = -24LEJ

4l2 - 7aL + 3a2 )

ill: x = a

vffill)( = - 0.1)0 652 woL4l

at x = 0.5 193l

Elastic Curve

Px (3L2 _ 4x 2 )v = - 48£1

forO : : S x : : S ~2

Pbx l2 _ b2 _ x 2 )v = 6LE1

forO s x s a

v - - M x (2 L2 - 3 Lx + x2 )- 6LEI

v - wx- - 24£1 L1 - lx 2 + x3 )

v = ~ L x 3 - 4a L x2 + 2a2x224LEJ

+ 4a 2L2 _ 4a 3L + a4)for O S x S a

v - - wa 2 (2x 3- 6Lx 2 + a2x- 24LEJ + 4 L2x - a 2 L }

for a s x s L

v = - 3; ~ / (7L4 - 10l2 x 2 + 3x4 }


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I

ca n til v r eams

Beam

7 ' ' '---------- -----~ · · ·[

8 d ··· · ~ , __f. ':_ J :_ ---- l - - - , ;2 2

v

9 I ( ( ... · · ·••• · · ·• ·· · ··•••·•• --~

10

11

...............OrnaxI E - - - - - - . - - - - -__ . r

v

l wqi }).JJ.U l : _ _ l l \· - ... _ ____ ~ max

£ L

ym x

vy--w o

-- I:---= f v :

Slope

PL 2max = - 2.EI

PL 2

rnax = - 8E/

MLontl.


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his page is intentionally left blank


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verage Properties ofSelected Materials

Mechanical properties of metalic engineering materials v ry significantly as a result ofmechanical working heat treatment chemical content and various other factors. The

values presented in Tab le D la and D lb should be considered represent tive v lues thatare intended for educational purposes only. Commercial design applications should bebased on appropriate values for specific materials and specific usages rather th n the averagevalues given here.


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Table D .1a Average Properties o f Selected Materials U.S. Customary Units)

Shear Percent Coeffi cientSpecific Yield Ultimate Modulus modulus elongation of thermalweight strength strength of elasticity (1,000 Poisson's over :Z.in. expansion

Materials Ohlft3) (ksi )• > (ksi)a (1,000 ksi ) ks i) ratio gage length ( lo- l°F)Aluminum Allo ys

Alloy 20 14-T4 (A9201 4) 175 42 62 10. 6 4 0.33 20 12.8Alloy 2014- T6 (A9 201 4) 175 60 70 10.6 4 0.33 13 12.8Alloy 606 1-T6 (A9606 1) 170 40 45 10 3 .8 0 .33 17 13.1

BrassRed Brass C23000 550 18 16.7 6.4 0.307 45 10.4Red Brass C83600 550 17 37 12 4. 5 0 .33 30 10.0

BronzeBroRZ£ C86 100 490 48 95 15.2 6 .5 0.34 20 12.2Bronze C95400 TQ 50 465 4 5 90 16 6 0.316 8 9.0

Cast IronGray, ASTM A48 Gra de 20 450 20 12.2 5 0.22


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Tab le D.1b Average Properties o e lected Materia ls I U nits

Percentelongation Coefficient

Specific Yi eld IBtimate Modulus Shear over 50- of thermalweight strength strength of elasticity modulus Poisson's mm gage expansio n

Materials (kN/m 3) (MPa)8h MPa)• GPa) GPa) ratio length (10~/0C)

Aluminum AlloysAlloy 2014-T4 (A92014) 27 290 427 73 28 0.33 20 23.0Alloy 2014-T6 (A92014) 27 414 483 73 28 0.33 13 23.0Alloy 6061 -T6 (A96061) 27 276 310 69 26 0.33 17 23.6

BrassRed Brass C23000 86 124 303 115 44 0.307 45 18.7Red Brass C83600 86 117 255 83 31 0.33 30 18.0

BronzeBronze C86100 77 331 655 105 45 0.34 20 22.0Bronze C95400 TQ50 73 310 621 110 41 0.316 8 16.2

Cast IronGray, ASTM A48 Grade 20 71 138 84 34 0.22


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Table D 2 Typical Properties of Selected ood Construction Materials

I Allowable Stresses ITension Compression Compression Iparallel Horizontal perpendicular parallel to odulus of

Bending to grain shear to grain grain elasticity

Type and grade psi I MPa psi I MPa psi I MPa psi I MPa psi I MPa ksi I GPaFramin L be 2 to 4 thi k b 2 d dum r ID ID c y ID an w1 er I

Douglas Fir-LarchSelect Structural 1,450 10.0 1,000 6.9 95 0.66 625 4.3 1,700 11.7 1,900 13.1

No 2 875 6.0 575 4.0 95 0.66 625 4.3 1,300 9.0 l @ 11.0

Hem -Fir

Select Structural 1,400 9.7 900 6.275

0.52 405 2.8 1.50010.3 l @

11.0No 2 850 5.9 500 3.4 75 0.52 405 2.8 1,250 8.6 1 300 9.0

Spruce -Pine -Fir South)Select Structural 1,300 9.0 575 4.0 70 0.48 335 2.3 1,200 8.3 1,300 9.0

No 2 750 5.2 325 2.2 70 0.48 335 2.3 975 6.7 1,100 7.6

Western CedarsSelect Structural 1,000 6.9 600 4.1 75 0.52 425 2.9 1,000 6.9 1,100 7.6

No 2 700 4.8 425 2.9 75 0.52 425 2.9 650 4.5 1,000 6.9

I ams ID an c er, w1 more an 10 grea r c ess Ie 5 d thi k dth th 2 te than thi kn

Douglas Fir -LarchSelect Structural 1,600 11.0 950 6.6 85 0 59 625 4.3 1,100 7.6 1,600 11.0

No 2 875 6.0 425 2.9 85 0.59 625 4.3 600 4.1 1 300 9.0

Hem-FirSelect Structural 1,250 8.6 72 5 5.0 70 0.48 405 2.8 925 6.4 1,300 9.0

No 2 675 4.7 325 2.2 70 0.48 405 2.8 475 3.3 1,100 7.6

Spruce -Pine-Fir South)Select Stru ctural 1,050 7.2 625 4.3 6.5 0.45 335 2.3 675 4.7 1,200 8.3

No 2 575 4.0 300 2.1 6 0.45 335 2.3 350 2.4 1,000 6.9

Western CedarsSelect Structural 1,150 7.9 700 4.8 70 0.48 425 2.9 875 6.0 1,000 6.9

No 2 625 4.3 325 2.2 70 0.48 425 2.9 475 3.3 800 5.5

s : ID y l dthan arger, w1 no mo re th 2 ·n m. grea e th thi knan c ess IDouglas Fir-Larch

Select Stru ctural 1,500 10.3 1,000 6 9 85 0 59 625 4.3 1,150 7.9 1,600 11.0No . 2 700 4.8 475 3.3 85 0.59 625 4.3 475 3.3 1,300 9.0

Hem-FirSelec t Structural 1,200 8. 3 800 5 5 7 0 0.48 405 2.8 975 6.7 1 300 9.0

No 2 525 3.6 350 2.4 70 0.48 405 2.8 375 2.6 1,100 7.6

Spruce-Pine-Fir South)Sele c t Stru ctural 1,000 6.9 675 4.7 65 0.45 335 2.3 700 4.8 1.200 8.3

No 2 350 2.4 225 1.6 65 0.45 335 2.3 225 1.6 1,000 6.9

Wes tern CedarsSelect Structural 1,100 7.6 720 5.0 70 0.48 425 2.9 925 6.4 1,000 6.9

No 2 500 3.4 350 2.4 70 0.48 425 2.9 375 2.6 800 5.5


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nswers to Odd umbered

Problems

Chapter1.1 = 172 .8 kN

1.3 d = 0 .691 in., di 1.545 in.

1.5 UJ = 5. 9 ksi C), Uz = 7.92 ksi T),U3 = 3 .68 ksi C)

1.7 d 1 = 19.96 mm, d 2 = 16.13 mm

1.9 U B = 5 .97 ksi C), U c = 7.51 ksi T),U B c = 7 .93 ksi C)

1.11 u B = 41.6 MPa T), u c = 87.3 MPa T),u B c

=62

.6MPa

C)1.13 max = 13 .5 kips

1.15 a) u = 4,380 psi T)b) u = 1,730 psi T)

1.17 = 14.92 kN

1.19 dmin = 16.77 mm

1.21 Pmin = 125 .3 kips

1.23 a 324 mm

1.25 5 in . X 5 in. plate at A 6 in . X 6 in. plate at B

1.27 ub = 23,900 psi

1.29 a) Urod = 10,190 psib) r 0011 = 3,260 psic) ub = 8,530 psi

1.31 a) Tpin = 11.58 MPab) ub = 18.19 MPa

1.33 a) dmin = 14.42 mmb) dmin = 16.33 mmc) dmin = 6 .6 mm

1.35 dmin = 2.21 in.

1.37 Un = 3.43 ksi, Tnr = 5 .94 ksi1.39 Pmax = 479 kN

1.41 tmin = 15.32 mm

1.43 tmin = 0.900 in .

1.45 a) = 187.5 kNb) u = 16.00 MPa

c) Umax = 25.0 MP a, Tmax = 12.50 MPa


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Chapter

2.1 8 = 0.1170 in., Bt = 825 µ £

2.3 (a) e = 1,147 µ.e(b) e = 2,260 µ.e(c) e = 35.6 µ.e

2.5 B2 = 3,040 JJ.,E

a)s=YL2.76E

(b) - fLeavg - 6E

(c) e = -yLmax E

2.9 } = 41 2,000 µ.rad, T = 627 kPa

2.11 1 Q = 2,300 µ.rad

2.13 (a) YR = -0.210 rad(b) 1 s = 0.210 rad

2. 15 8 = -5 2 .0mm

2.17 llD = 0.1999 mm, lld = 0.1762 mm,llL = 2.54mm

2.19 35.1°C

2.21 Vpointer= 0.0241 in. j

2.23 llT = 175 .9°C; T = 25°C + 175.9°C = 201°C

Chapter3

3.1 (a) E = 10,360 ksi(b) II 0.321(c) u PL = 43.0 ksi

3.3 a)E = 2.17 GPa(b) II 0.370(c) llthick:n ess = . 083 3 mm

3.5 (a) II = 0.306(b) E = 117 .3 GP a

3.7 P = 35.3 kips (C)

3.9 (a) perman ent set = 0.0035 mm/mm(b) bar length unloaded= 351.225 mm(c) uPL = 444 MP a

3.11 G = 64.7 psi

3.13 8 = l.336mm

3.15 (a) u 1 = 87.5 MPa(b) P = 20.4 kN(c) vc = 13 .82 mm, ,

3.17 (a) E = 30,000 ksi(b) UP = 60 ksi(c) u u = 159 ksi(d) U y 80 ksi(e) Ufracture= 135 ksi(f) true Ufracture = 270 ksi

3.19 (a) E = 138,400 MPa(b) uP = 234 MPa(c) u u = 394 MPa

(d) 0.05 % offset u y = 239 MPa(e) 0.20% offset u y = 259 MPaf) Ufracture= 350 MPa

(g) true Ufracture = 457 MPa

3.21 (a) E = 11,180 ksi(b) UP = 33.6 ksi(c) u u = 70.4 ksi(d) 0.05% offset u y = 44.4 ksi(e) 0.20% offset u y = 54.5 ksi(f) Ufracture = 70.4 ksi(g) true Ufracture = 87.9 ksi

3.23 (a) P = 5.74 kips(b) Bz = 1,833 µ £

3.25 (a) P = 18 .19 kN (b) T c 42 .7 MP a

Chapter4

4.1 (a) bar stress u = 362.5 MP a,U y 550 MP a, FSyield = 1.517

(b) U u = 1,100 MP a, FSu1timate= 3.03

4.3 bar 1) red brass FS = 1.494,bar (2) aluminum FS = 1.303

4.5 P allow = 7 9.3 kips,bar (1): FS = 1.825, bar (2): FS = 1.600

4.7 (a) FS = 4.46(b) FS = 2.60(c) FS = 3.90

4.9 (a) P max = 15 .03 kN(b) = 13.46 mm

4.11 P max = 5.65 kips


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4.13 (a) d 1 = 1.335 in. 5 .37 P = 21.1 kN l..(b) d 8 = 0.771 in.

5 .39 (a) F 1 = 40.0 kN, F 2 = 25.0 kN,C) dA = 1.033 in .F3 = 10.00kN

4.15 P max = 11.28 kips (b) v 8 = 3.40 mm J..4.17 (a) Amin = 6.01 in. 2 5.41 (a) u 1 = 12.25 ksi (T), u 2 = 8.33 ksi (T)

(b) Amin = 5.78 in .2 (h) Vv = o.m .B in . J..

4.19 (a) dmin = 1.578 in. 5.43 Pmax = 30.9 kN(b) drrun = 1.358 in.

5.45 (a) u 1 = 26.0 ksi (T), u 2 = 8.66 ksi (C)

Chapter(b) FS 1 = 2.38, FS 2 = 8.66(c) u 8 = 0.1458 in . J..

5.1 dmin = 21.9 mm 5 .47 UJ = 22 .3 ksi (T), U = 12.74 ksi (C)

5.3 P = 118.7 kN5.49 (a) F 1 = 45.2 kN

5.5 uA = 3.18 mm--+ (i.e., moves toward C (b) ubolt = 119 .0 M Pa(c) e 00 t = 0

5 .7 (a) P = 80.6 kN5.51 (a) 6 T = - 94 .0°F(b) u 8 = 0.497 mm J..

(b) dmin = 0.405 in.5 .9 (a) 8 = - 0 .0581 in. 5.53 (a) u = 561 psi (C)(b) U v = - 0 .0946 in.

(c) Umax = 29.0 ksi (C)(b) e = 3,125 µe

5.11 8 = 0.000393 in. J..5.55 (a) 108.6°F

(b) u 1 = 20.9 ksi (C), u 2 = 31.4 ksi (C)5.13 8 = 90 .6 X 10-6 in .J, (c) e 1 = 283 µe (C), e 2 = 703 µe (C)

5.15 (a) u 1 = 158.7 MPa (C), u 2 = 26.0 MPa (T)(d) 6.width = 0.00913 in. (increase)

(b) VA= 1.190 mm j.. 5 .57 6 T= -5 0 .1°C(c) P = 72.3 kN

5 .59 u 1 = 15 .24 MPa (T), u 2 = 19 .51 MPa (C)5.17 (a) P max = 15.05 kips (b) B t = -693 µe, B = -693 µe

(b) vv = 0.248 in. J..5.61 (a) U t = 1.550 ksi (T), U2 = 9.66 ksi (T)

(c) dmin = 0 .730 in.(b) vv = 0.1767 in. I

5.19 Pmax = 77 .3 kN 5.63 (a) u 1 = 74.5 MPa (T), u 2 = 9.97 MPa (C)5.21 (a) dmin = 1.309 in. (b) vv = 0.454 mm J..

5.23 (a) u 1 = 75.0 MPa (C), u 2 = 4.50 MPa (C) 5.65 (a) u 1 = 35.0 MPa (C), u 2 = 70.0 MPa (C)(b) u 8 = 0.450 mm J.. (b) VA= 0.365 mm t

5 .25 (a) u 1 = 74.8 MPa (T), u 2 = 6.58 MPa (C) 5 .67 (a) aluminum: u = 124 .0 MPa (C),(b) u 8 = 1.122 mm J.. cast iron: u 2 = 53.1 MPa (C),

5 .27 (a) u 1 = 7.84 ksi (T), u 2 = 17 .10 ksi (C) bronze: u 3 = 186.0 MPa (C)(b) U = 0.1254 in.--+ (b) force on supports= 148 .8 kN (C)

5 .29 dmin = 26.4 mm (c)u 8 = 0.01257 mm--+, uc = 0.1600 mm --+

5.31 (a) P max = 130.6 kips 5 .69 Pa11ow = 103.3 kN

(b) U = 0 .0720 in. I 5 .71 p a Jow = 51.1 kN

5 .33 L i = 28 .8 in. 5 .73 min= 9mm

5.35 (a) u 1 = 25.8 MP a (T), u 2 = 11.69 M Pa (T) 5.75 (a) dmax = 37 mm(b) VA= 0.554 mm I (b) rmin = 5 mm

[ i i i

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