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Unformatted text preview: Prob. 2.13 A block of linearly elastic material (E, v) is
compressed between two rigid, perfectly smooth surfaces
by an applied stress a, = —00, as depicted in Fig. P2.137.
The only other nonzero stress is the stress 0, induced by
the restraining surfaces at y = 0 and y = b. (a) Determine
the value of the restraining stress 0,. (b) Determine Aa, the
change in the x dimension of the block. (c) Determine the
changeAt in the thickness I in the z direction. U Sou/Hon: (a) remmining s+mas
From 50,. 1.355)
a: é, (<11, — 90;) =0 since valmined (Y a VG’, = '\?r0 Ty: "' .90", (b) Change in lea/39%
From Eq. Z. 38 a,
6x= _'_(0”x  V0,) = AOL E T
mug (01—002,) = 0L0} (12%)
E E
No+c¥ since, 0094! , 179% l, Mentor/e,
m is megaHm. (c) Change In mickncss
From "Ear. 2.38 o, 62: "_’\_7_ (O'X*U:I) “At— ‘Prob. 2.1340. A block of linearly elastic material (E, v) is z
placed under hydrostatic pressure: v. a a, = ti. = ‘p; I
‘17,, = r"  1,. = 0, as shown in Fig. P2.1310i (a) Determine
an expression for the extensional strain e, (  a, III e‘). (1:) Determine an expression for the dilatation, ey. (c) The bulk
modulus, k... of a material is deﬁned as the ratio of the hydrostatic pressure, p, to the magnitude of the volume
change per unit volume, Icy], that is, . .L
"‘ I»: Determine an expression for the bulk modulus of this
block of linearly elastic material. Express your answer in (Stresses onhiddcn faces n0t Shown) terms of and‘y. P113.”
Soil/how NW ex= _E[o’, wwwm] = (4;)(1194) e,< = (:EﬂﬂzvI) I b) an a+u+fon V: “be v“: (M éxa)(b+6~,b)(C*€LC)
=V(l+€x)(He\/)(+€z) “
‘3'V (“'EXf €v= V*V = (Hews—I ev: (1) Wk Modulus
Since 03‘:ng 0'; =  1a is unﬁt:va compressive, ex (= e,= e2)
is n69 a+Ive .ThCVOPOYG, (w \‘s m3aﬁv050 ‘4‘): i. = “.21”— lévl 6" WA column in a twostory building is fabricated
from square structural steel tubing having a modulus of elas
ticity E = 210 GPa. The cross~sectional dimensions of the
two segments are shown in Fig. P3.34b. Axial loads 1" = 200 kN and P, = 300 kN are applied to the column at levels —T
A and B, as shown in Fig. Pas4a. (:1) Determine the axial '1 = 3 m 150 m...
stress al in segment AB of the column and the axial stress Q or; in segment BC of the column. (b) Determine the amount
8 by which the column is shortened. arc/@5294 .' M} Iﬁc/a/J/ﬂrrcr.
EEu/'/'{r1‘am ,' :0 I. 400w ~F=o F,= 4001M '
I Mal/u 300"" 6 = {Oahu
530/:
. D '
(ff/aria: F6 1’ F; r F, P334 zao W ’300 éﬁl— F; ‘0 g — 2004M = ———————" = ‘4 , / I ﬂ/ [/I‘YOMM)L//HMAY'] ¢ 0 y é 5 —Jaoéu =—— . —————————; =.. ,. Ml?
l4»  [(ZOUM)‘(/76~~)] {$405 a @ [/4/ {AM/6,161] 9/ 60/401» . 57¢»: 64/ ﬁ/C(/ a/cﬂ'rme [ﬁéﬁﬂb/ I :1 = 2)" a5. _ (—4¢,mmn¢)/jm) 63= 5/ 2mm =0,6z:em
,2 a 021, = {rwoammm) “Mi/rm
fl 2/05“ a fee maﬂ’z I / rm 4,5214 ,' /=—/4/f’zl = A450“ Prob. 3.39. A rigid beam AB of total length 3 m is supported
by vertical rods at its ends, and it supports a downward load
at C of P = 60 kN, as shown in Fig. P3.39. The diameters
of the hanger rods are d1 = 25 mm and d, = 20 mm, and
both are made of steel (E = 210 GPa). Neglect the weight
of beam AB. (a) At what distance x from A must the load
be placed such that u. = ml? (1:) How much is this displace
ment of the beam? (c) What are the corresponding axial
stresses 0‘ and a: in the two hanger rods? fly/“5‘09: 7
M) [044/ A/ﬂ(¢i.0/l K {26/
l/sz/ a6 = ﬂ”. fiat54' [’12va
Wij, =0 : {a  P/d—x) =0
(I) mam/g =0; 52  a :0 __ X
*fffy‘o! EIE'P‘O L—_
57!»; én/ 24/4,  c/c KmaAwv o/aﬁawa.’ (I. =74/‘T = ,5 m pcﬂrme/Tan (aorﬂar'vrfl twang/6;, w Z ﬂl— I
. ,_ ’ l )
6 Fl [1.4. ) ’ 4,19
(0""44 {Ianxl/f’)! 6 zj ) d,  PX =0 "{4}
4
(owl/u (h) hue/[4)! _ ____
x ’ /f//1 9/ ) =
[14’ {ZOMM)”
m /5) ﬂ/‘J‘ﬂ/tczmzn/ gr) :uﬁ' lx:
ﬂow {/6} €= 1% = W = 57.3325 AM
on, 4/3 — c; — ’{Z‘ = 6235’07mm 44. = #4” QJWM ’71. fr. Ii.
[[7 ﬁer/Ef. :
F,= MAM—5 = 2’0. wag/«U ¢e ,
a”: : [Z,776f/”/’(/ Zara/’2 = 4554¢ﬂ7ﬂt ’ 27'4" Prob. 3.47. A steel pipe is ﬁlled with concrete, and the
resulting column is subjected to a compressive load P  80
hips. The pipe has an outer diameter of 12.75 in. and an
inside diameter of 12.00 in. The elastic moduli of the steel
and concrete are: E, ' 30 x 10’ ksi and E‘ a 3.6 x 10’ ksi.
(a) Determine the stress in the steel and the stress in the concrete due to this loading. (b) If the initial length of the
column is L = 12 ft, how much does the column shorten when the load is applied? (Ignore radial expansion of the
concrete and steel due to Poisson’s ratio effect.) Soluﬁon *
(a) norm! S‘h’css swam dJSvI'Yl’DU‘HOYl ' 6 (>0 = e = Consmm
s+ress w‘w'hm'on‘ rs = E56 r6 sEce
rcsuiWhvcey repiace “3,930pr 95199 .___._—_———— Pg 0} Ag 315260200 in)"  _4
d5 Ail: 02 ‘7} => Po l’s Ps=<Fs As
' =E_6ESZ\1.15I'n)"— 02.00 [109‘]
EﬁDj Column swh'on 7 +4“ zrzlzozP—Pc—Ps =0
9; AI; [mHEcEs) “115‘ an
e: 4?
1m 7.1 (go 63) H139 5;]
Y =>—Fvom S+re$$ dis‘rviwn’om
Lx P° “'c ‘ APE/1r [121(26— 557+ males]
WPS = _ 0.34:02 ksé
= — P
“3 4 ES/Trtnz‘cec—ESMuHS‘Es’J
= — 2.84195 ksc 4 a.
96 ‘ es "‘ e ‘ 5L ‘ 1V[21(E¢’Es)*2.151551 = 0.0I5HI in. a: 4,354 no" in. New: column shelflays. Prob. 3.54. A steel pipe (E, = 200 Oh) surrounds a solid
aluminumalloy rod (& = 70 GPa), and together they are
subjected to a compressive force of 200 kN acting on rigid
end caps. Determine the shortening of this bimetallic com pression member, and determine the normal stresses in the.
pipe and in the rod. “Va/wan;
.0557th {Ker éh/hj (0") dm/ 575532? (f, max/d; €= F =
I jf’ / //z) 1/664? / at;
4=ﬂ€
1;
7:1: ALL: P=2oom ' L=0.5m P3.54 and P3.8‘l3 F; *—
9 P
/0.rm)
//—[{0,0f0m) —/0, Mom) ‘J/Zoox/o ’5’,
5,:4/f/10“')f’7
0, ﬂ». [/aosnm )2 {701/0‘7») 5 A5560 / /0”)’:7 §€0mrfry a O/cﬂ/mdﬁlﬂ’} [ém C} = fl = —J— (ﬁfél/ffnl’ij‘) 6’)
(In) 11;”: f?) éno//f).
{Fm/Le w /=/—;/5 6") (ml/o: (I) 4476/6”).
ihfﬂ — —  ,— 5, w
7—) ’ —/D —7 6 ﬂlfz — fr—ﬁ—P = —g¢,53’w
 = a uwmm
03—6, =—/,€'&,//?me~ If I
F 455471”
I: I ' =— I ~
r 7’— /7[/aoromj‘—/aa¢om)‘ i7 7/ $~4Z7ﬂﬂ4él
J = 5— ’ M =—/é.7¢7/m $5/a77mﬂ/d
L ’91, 7 Mair/MP ...
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 Spring '06
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