a comparison of the efficiencies of a two-hinged and a three-hinged ... ·...
TRANSCRIPT
A COMPARISON OF THE EFFICIENCIESOF A TWO-HINGED AND A THREE-
HINGED STEEE ARCH
BY
ROBERT HERMAN RIESCHEi^RANK FAIKWELL BEEBV
THESIS
FOR TH K
DEGREE OP BACHELOR OF SCIENCE
IN
CIVIL ENGINEERING
IN THK
COLLEaE OF ENGINEERING
ITNITERSITY OF ILLINOIS
PRESENTED. JUNE. 1909
UNIVERSITY OF ILLINOIS
June 1, i9o9
THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION BY
ZRAM FAIRTOL BESBY and ROBERT HERlvIAN RIESCHE
ENTITLED A OQMPARISON OF THE EFFIOIENCIBS OF A TWO-HINGED AND A
THREE-HINGED STEEL ARCH
IS APPROVED BY ME AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE
DEGREE OF^ - -BaclielDr jc^^^ Engineering
(2<i>o^ aX^^Instructor in Charge
APPROVED:
HEAD OF DEPARTMENT OF Clvil Engineering
145157
UlUC
Digitized by the Internet Archive
in 2013
http://archive.org/details/comparisonofeffiOOries
_1-
The object of thia thesis ia to determine the relative
weights of the main .nembera of a steel roof-tru33 arch, firit
when acting as a three-hinged arch, and second when acting as a
two-hinged arch, the same loading being used in both cases.
The arch used i^ that of the Baltimore Armory (Eng.
Rec. Vol, 49 pp. 604), which was designed as a three-hinged arch.
This arch ha3 a span of 190 feet 3-3/4 inches, center to center
of hinges, and a rise of 86 feet 6 inches above the finished floor
line, the floor line being 1 foot 6 inches above the center of the
end pins. The tipper chord panel points are on the circumference
of a circle whose radius is IH feet 10 inches, the center being
17 feet 4 inches below the floor line. The lower chord panel pointa
are on the circumference of a circle whose radius is 91 feet 10
inchea, the center being 5 feet 4 inches below the floor line.
The arches are spaced 25 feet 10 inchea, center to center. The
upper chord is divided into nineteen equal panela, each panel being
12 feet 8-15/16 inches long. The roof is designed for a wind
load of 30 pounds per square foot of its vertical projection, and
for a snow load varying from 2 to 3n pounds per square foot of
horizontal projection of the upper six panels on each side of the
center. The specifications used in thi^ design were those given
by Ketchuni in his "Steel Mill Buildings", and are as follows:
Tensile 'tresses
Shapes, main niembera , net sections I6,ooo pounds
per square inch,
^^^^ 16,000 TDOunds per square inch.
-2-
Compre33ive Stre3ae3,
Shapes S = 16,000 - 70 l/r pounda per square inch.
Least allowable l/r = 125,
No metal leas than 1/4 inch thick shall be used.
All stresses in both the two and the three-hinged arch
are obtained graphically, and the stress and space diagrams <are
shown in Plate 1. Prom inspection, it was assured that a com-
bination of the dead and wind load stresses would give a maxim^jm
deflection at the free end when the two-hinged arch was designed
as a single truss. Therefore, the stresses resulting from this
combination were used in both designs.
The design of the three-hinged arch wa^ a simple theoret-
ical design. The air.i was to get the smallest section that would
carry the given loads, little attention being paid to the practical
considerations which might, in some cases, have required the use of
larger members. (See Tables 1 and 2).
The design of the truss, treated as a two-hinged arch,
was made according to the theory given by Ketchun in his "Steel
Mill Buildings." The stresses usei are given in Table 3. The
formulae used areA = PUL/e,^'= P'UL/e, and H =, where
^ is the horizontal deformation due to a horizontal thrust
H is the horizontal thrust required to keep the free end of
the arch, when designed as a simple truss, from moving;
H' is a horizontal thrust, arbitrarily chosen, (in this case
it is 20,000 pounds);
P is the unit stress in each member due to the loads;
~3-
P* 1-3 the unit atreag in each member due to the horizontal
thruat H'
;
U ia the unit 3treaa in a inenber due to a unit horizontal
thruat applied at the free end of the aimple truaa;
L ia the length of the member in inches
;
E ia the modulus of elasticity of atructural ateel ('2BOOOOno),
The method of procedure in thia caae ia aa followa:-
The members were fir^t dealgned and the deflectiona
calculated (aee Table 4), aaauming the arch to act aa a aimple
truaa free to move at one end. Then, uaing the aectiona choaen in
Table 4, the deflectiona, cauaed by a horizontal thruat of 2n,000
pounda applied at the free end of the truaa, were calculated
(aee Table 5). From the formula H = it waa found that the
true horizontal thruat was 53,000 pounda, or 2,65 x 2n,noo pounda.
The atreaaea due to a horizontal thruat of 53,000 pounds were then
combined with those cauaed by the given loada. The members were
then redesigned u^ing thia combination of ^treaaea, and the defor-
mation computed. Prom the formula H = H'^ it waa aeen that these
muat be an additional horizontal thrust of 13,300 pounds, making
a total of 66,400 pounda or 3,315 x 20,000 pounds. In this manner,
after four trials, the true horizontal thruat of 66,600 pounds waa
determined and the final design made faee Table 7), The total
weight of the main members waa then determined (aee Table 8).
Conclua ion.
The weight of the main membera of this arch aa here
designed, treated both aa a two-hinged and a three-hinged arch, is
considerably below that of the arch as designed and erected at
Baltimore. It is alao lean than that found by Mr, E. B, Jordan
-4-
in hi3 thesia of 1908 where he designed for a combination of dead
arid wind load 3 tresses.
When designed for dead and snow load stresses, the vveight
of the main nember of the truss, treated as a two hinged arch, is
27,200 poimda, and treated as a tliree-hinged arch, 25,400 pounds.
This gives an advantage of 1,800 pounds or, 7.1^, in favor of the
three-hinged arch.
r
rnDLEl THREE. hINQEO nRCH 5TPE35E3.
I\/IF' DeadLoad 3nowLoQd \NindLoad 01.-^3. L. D.L -^Wind MHX.5trc53 3trc5S 3tress
/7V -i-30/00 -1-3600 i-/63200 -h//a900 +163200/7V' /f If -^23800 -t/33900 '1
B-/ -hS7<000 -f- ^7O0 +233O0 -h 6/700 +82300 +6/700/' /f - 2300 II +32300
/-^ -65300 -28000 - 3000 -9/300 - 66300 - 9/300fi -/4200 II -773O0 ll
//-^ -hSSOCO -h38000 -h ^iOO H24/P00 -^-90400 +/24000n -h/9600 II +703600 /
1
2~d -i-3S/00 f34400 -7900^ •^69300 +76/00 +693002-J' ii -f22000 II +37/00 1
1
3-5 t/SMO -20000 -i-32/00 - 2000 +30/00 -20003-3' /I -/70W II + /000J-f -'/9300 -/sooo -f/0/00 '34S00 -9400 -34300
/< II - 9300 1
1
-E9300 1
1
H-^ -t7O000 -h/O3JO0 -9200 -h//3300 +60300 +/7330Of7^' -^36000 1
1
+/O6O00 $1
^'5 -^26200 -f20000 -7SOOO -h46200 + //200 +46200i'/3400 1
1
+39600 II
3-S - 8760 -3^000 h^2200 -42700 +33300 -427003'S // '26000 II '347O03-6 - 7200 - sooo -f- 69O0 -/2200 -300 ~722GO5-6' ll -37(^0 1
1
- /O900/7-€ -f9^6300 -i-79^00 -20200 -f/69900 +7O600 ^/69800/7-6' II -h-^6200 ll +/370006-7 -i- 8000 -h3300 - 7300 -h/3300 -t200 +/33006'-7' /I -h^/OO +/Z/00 1
1
3-7 -//600 -36200 i-46000 -47600 +34i-00 -47SOO3-7' // II -27300 II -39400 It
7-8 -h 300O -hS/OO 2300 -h70/00 + 7SOO +/0/007-8' /I ll II II II
/7-a 'h96000 -hSdOOO -26^00 t/790/?0 +7S600 +/79dOO/7-S' it II +743000 1
1
6'3 -+/8600 -29730 -hZ3/O0 ~2S200 +23/008-9' It /I -h/6300 ll +20800 ll
3-9 ~/6SOO -^3300 -h59^00 -60600 +42200 - 6O600
TFiBLEl 1
IVIL lYl.
DeadLoad 5nowLdad WindLoad D.L ^d.L. D.L^W.L JVtf^X^
stress Stress Stress 5fr€5S D.L.-^5.L3-S' -76300 ~-f3aoo -34600 ~ eoeoo -3/4-30 -S06009-/€ -tSSOO - 3S00 -t/9000 -H 3OOO -i-27S30 + 3C009-/0' ii ~6/00 ii + 2400/7-/0 ^98000 -fS^OO -4-7000 +/92eoo +3/00O -h/92600/7-/0' It -^39730 1
1
+J37700 II
/O-// - Z300 H-//200 -230OO +-8700 '2S3'00 -+S730/O'-//' ti II -13500 1 • -76300 4 *
3~// -2Z000 ~3/000 -f74<^00 - 73OOO +-32300 -73OOOB-//' ii II -43200 II -63200 II
//-/a -i-/4-000 -i'3300 -t/0300 +/7SOO +24300 i-z/aoo//-/^' " If - 2000 II +/2000 II
^'/^ -^93000 -t/0/300 -63200 +-/96SOO +3/aoo +-/96300II -i-eaaoo +/63SOO •
«
B'/J -22000 -33O00 -i-s^aoo -7SOOO -^62800 -7310003-/3' It -30000 It -72030 „
/2-/3 -sooo i-3000 -/3700 -6OOO -24730 -6330If -h/3000 -f/OOO
/3-/4 -t/ZSOO -t/osoo -t2300 +20300 +20000 +20300/3-/r +2000 +/9300 II
-f-87200 -t/C27O0 -74300 +/S9900 +2900 +/S9900f 1 #
»
+7seoo II +/62800 ti
3-/3^ -/a/oo +90400 -e/zoo t7/730 -677003-/3' -34600 -73300 n
/f-/3- -/2eoo - 6000 - 78OO -/9SOO -20600 -/88004 * -f6200 1' - 6eoo 'I
/3--/e -^/9000 'i'/7000 -4730 +-36000 +/4300 +36300/3-/e' +380O +24800 if
/7ye -i-7(^soa ^97000 '80000 t/73800 - J200 +/73830F-/6' /< -f-79aoo It +/3e600 II
B-/7 -//ooo -37200 -1-89300 -^8200 +7SOOO -492303-/7' It -3'4800 ii -63900 II
/ev7 -/<3000 -/ZOdO -f2O0O '3JOOO '74^300 -33003/evz' 4* 1' +300 ft '73-SOO 4 1
/7VS -f/900O -i-2/300 '/0800 -h40300 +-S230 f-4O300It + 9330 II +2630O
THBLE 1
\ AC h A DeadLoad 5nowLoad WindLoad D.L -^5.L /V13X.Ml M.
5tre55 Dtress 3trcs3 dtrc55 D.L^3L./7-/S i-62300 -^S2300 - 76^000 -f-/44SOO -h/3300 -+/44300
f
-f-Z9300 /I -+/420^00
B-/9 - 2200 -2070 -hSOSOO ~22900 -h73700 -229003-/9' -30S00 1
1
-32700 /f
/S-/9 -/7O00 -2^000 -h/0700 -4/000 -^6300 -^/OOO/6'/S' II -J6O0 /I -22600
-i/rooo i-23000 -/3700 -h40000 -+/300 -+>400OO
/9W It -t/3300 II -(-30300
-t4S300 -f-62O0O -easoo -I-//0300 -h20200 -h//0300
-h74300 II +/22600 .
3-2/ -i- 7000 -36CO -hSSSOO -f4000 -h73300 y-4^003-2/' -40SOO II -33JOO20-2/ -/6000 -26000 -t/SOOO -42^00 -^2000 -4^£^^020'2/' II -/2SOO 1
1
'23800 /f
2/-22 -h/4300 -t2/00O -/SOOO -+33300 ~3700 -+333002/'22' II +/eooo /I -^30300
/7'22 +34300 -+39600 -:32900 -+73900 -/S600 -h73900/7-22' -f-ejooo -h973003-2^ t/4300 'i-/O2O0 +^6800 -f24SOO -+6//00 i-24300B-23' // 1
1
-^^ooo /' ~9700 II
22-23 -/^2O0 -/9800 i~2SOOO -32000 -h/2300 -3200022''23' -2O300 1
'
-33OO0 II
23-24 -h/OOOO -*-/4-Soo -/azoo ^24600 - 8200 -+2460023-24' // -h/7SOO II +27600 II
fl^^4 -i-24000 -h22J00 -3/ZOO -+4^300 -7200 +4S300/7-24' '1 -^43000 -+2/000 //
3-2:3 -f-/S330 •f/SOOO -t2300O -+34300 -h4/300 -f-343003-23' /I /I - /J-^^ -t/ssoo It
24-15 -eroo -/OOOO -+23400 -/S700 -i-2/700 -767^024-23' -27SOO 1
1
-34200 //
23-2S -eooo - 2000 -/JO'OO - BOOO -/9SO0 - 300023-26' " -h 2600 1
1
-3400f/-26 -^2/200 -i-/7000 -eaoa +33200 +/4400 -t-36200/=/-26' f2S30a -h477003-26 -i-22000 -t33000 -f27SOO tS7000 -+^9300 -+37000
- -3500
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riemberD.L.3 tress
S.L.S fre33
W.L.Stress
H.T3treSs5
DLs5.L.S tress
D.L. + W.L.
3 tress
27OOO ~ ^600 + J3 SOO ^ III SOO + 7^SOO+ 27 000 V/ 600 + 15 SOO 90/00
H-2 + 7/ Soo f 21 200 ' 7 SOO + // doo + 93 000 + G-^ 300+ 7/ 3oo + 21 20O +9 OOO + / / 800 + 30 3oo+ /5'OOO ^ /d doo -30 200 + 30&00 + 33 800 - 20 000+/^ ooo ^ I880O + 300O + 30 300 -^23 000+ // 50(? t /09OO -62300 +00 2oo + 2G ^00 - S 1 300
/ JOO + M0OO + 6 OOO +00 200 ^ /7S0O+ 6 SO o + /O OOO - 73 300 -I- -09 OOO + /6 SOO - 6G800+ 6 soo -t- /O OOO -h ^ OOO + 09 OOO + /o soo
/f-/o -J/ OOO - S OOQ - / /0OO - 70 OOO - >40 000 ' /03 000~3J OOO - 3 OOO - OOO + 70 OOO - 3S 000
/f-/2 -30 OOO -3j2>20o -132 700 ^93 ^oo - //6200 -232 7000-/2' ^QO OOO -3e2O0 - /S300 +g3 0oo -0)S 300
/I-M ^/390OO - 72000 -/&9 OOO +//S200 -2/1 SOO -323 soo//-/4' '/3950O ~ 72 OOO - 3 / SOO + f/3200 - /7/ 000
~2o8 OOO '//830O -220300 ^ /^^ 700 - 226> 300 -023OOO0-/6' -203 OOO -//d300 - SI SOO -* /00 700 -2J9'Soo
//-/8 -279JO
O
- /72 OOO -2^^ 2oo + /7/ OOO - 0S/ SOO -323 7000-/S' -270300 - /72 OOO - 76&00 1-/7/ OOO -33 3000-2O -355 SOO -23MOO -2S9 2oo ^/99 700 -636 900 -6/0 700
-355500 -23I0OO 70(^ /o o + /99 700 - -46 / SOO0-22 '^28 700 -268 SOO '2629oo +222 700 - 777200 - 69/ 6000-22' -023 700 -2Sd SOO - /3d 3od ^222 700 -36 7SOO0-2^ "^3506 -330 OOO -2J3 060 239 300 - 3 /SOOO - 70 000
0-2^' -•0dSooo -330 OOO -/ 7-0 OOO ^239300 -6S9 0000-26 -3/7300 -33S OOO -237OOO "^209 000 -372 dOO - 7S0OOO/5z/j + 30 SOO + 27 OOO + -^O OOO ~ /Ssoo + /// SOO ^ /20SOO^-/' 30 soo + 27 OOO 12 OOO - /S soo +9G SOO3-3 f 73 200 + 23 OOO + 6>f OOO - 0O 000 ^9>6 200 •^/302003-3' + 7320O •^23 OOO + /o soo - 30 00 -1-33 700
3-d ^ 76 soo + 27 20O + 82 soo - 0/300 /03 700 +/390003-3' + 76 SOO +27 200 + /2 30 -0/300 + 33 3003-7 + 7S60O -^2S9O0 ^33300 - 00900 ^ /03 SOO -^/670SO3-7' + 70 600 +29900 ->-J3 0OO '00900 t92 000
T/1BLE3
MemberD.L.
Stressd.J-.
3 tressW. L.
v5 tressH.T
•3 tr^55
D.L + S.L.
3 1 re 55
D.L + W.L.
5 tress
3-S ^ 3S OOO +^33000 + //J J'OO -J3 Joo + //s 300 + 27OS0OB-S' coo + ^3S0O + 200 -33 300 1 M eoo3-// + / 3 (h ooo + 673 OO^Si + /S/ 300 -79300 + 203 300 +237 3003-//' -1- /J&ooo t- 67S'0o ^29 Sao - 79 800 +/63 300
B'/3 + 13^ ooo /OO 000 + /83'6oo '77^3 Soo +234000 + 369 Goo
B'/3' t/S^ ooo t- /oo aao + -73 700 -/03 300 +22 7 7003-/^ ^2^2 7/Po + /42400 +2/0200 '/2S 300 "-353/OO ^433 9003-J6' +7^2^00 ^ (h2200 ~/2d doo + 3049003-/7 + 3^5000 ~^JP/ 700 -^2 39S0O -/33 30O t-336 700 *384 3003-/7' ^3^JO60 ^/^/ 700 ^ S3 doo -/6 3 JOO + 430 3003-/3 ^2^3^^o ^30 73-00 ^ 233000 - /3 / 60 6 6 72900 + 622300B-/3' +2^ ^^00 + //3^oo - /3/ (boo + 480 3003-2/ ^2S72oo + ^39000 239(^00 - 203 800 + 73G20O " 693 6 003-2/' -^297200 + ^39000 + /4'4230 - 203300 + 333 2003-23 + 33^ 000 + 439-700 + 232 300 -219 700 ^323430 ^ 7422003-23' -1-33^ 000 + -73^-700 -> / 77000 - 2/9 700 + 666 4ao3-2d ^336 200 J/7200 + 236 700 -229000 -373400 + 7339003-23' 35(^20G ^S/7200 209^00 -F2^ooo + 723 6063-26 i-3^5 000 ^3/7300 + 209 000 -229000 ^372300 + 726 300/-2 - 7^0 - ^^Soo + J -400 - 3300 - 6O JOO -4^ 4002-3 - /2 300 - d)oo - 43 900 +26 000 - 70 300 - 364/003-^ ^ 3 300 ^ 30 -^23/00 - /2 300 + ysoo -26 600
- -73 00 - 3900^ -3/300 +-/ y 300 - 7 07^0 -36 300^ 3 800 * -4900 -^7/ 300 - 3 700 + 3 700 + /3 600
6-7 --^ doo - 3 6 00 -73 ^/?0 1 6-700 -^900 -77 yojo
7-6 + 38oo + ^Sooo -h / yo'<7 + doo + 3 6OO 3 300s-s - 33 -^oo -27 300 - 33 7^0 ^30 20 -30 roo - //2 7/^0
S'/O -^^0200 -f2/ 40O -^34 300 -76 407? + 6>7 6o>o +74 3-00
/^-// - 6O0 -3(^ 300 -3-4 300 ^32 /oo -704 400 -/2/ 90a//-/2 +^3 700 +2^ Soo ^ 24 300 -74 200 *63 900 + 63200/2-/3 - 7S Sao ~ ^7200 -49330 + 3-7400 -/26 000 -/23 300/3-/^ ^^2 odo +2V^oo ->- /4O0O - 6 700 +69 000 + .76) 000
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-20 soo ^ 3 300 ^- 3(^ 300 - YV^o-8 7^00 -6 8 000? - s 000 -1-23 000 - /33 000 - 93 000
^ 3 ^00 -30 oao + /2 600 - / So>o -3^ SO^o
- o (po - ^9 000 V 600 700 - /f^ 300 -37900-2SS0O - / 3 ^(PO • 3 7 000 -t-2(P 30a -^3900 - 63 ^500
2^-25 ~36 ^oo -29 J^O i-/2 6^00 t // - 6 3 30)6 - / 6^00E3-2^ -^3 JOO -22 ^00 - -^2 000 ^ 23 (P^o - 72 SOO -33 300/-2' - doo - 7^^^ - 6 Soo - 3 soo -6o> soo -J2 3002'-3' - /2 Sdo - ^^ao - / 7^0 ->-26 000 - /6 soo - 200
^ 3 J^o ^ -^300 ^ /90>O -/2 Soo + 7300 + 3'^00- ^300 ' 3900 - 2 Soo -h/7 SOO -/6^700 - 7^ 6 00* 3 8oo +^goo -i- 2 00 - 3 700 - 3 700 - 3 300- ^300 - ^3 3oo - 2 -^oo 6 -400 -9Soo - 6 70a
7'S' f- 3 Soo ^^3(^0 -* 2 000 + 300 ->-3 600 t S 3 ^
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- SS -33 800 -2 3 700 +3-^900 - //3 700^3S 7^ -+ J2 000 - (p 000 62 /oo -^^7 700
/6-/7' - 3d 6ao 300 - J / soo -^33 200 - /3^900 -/^^ /O^O
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TABLE 3
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TABLE 8
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UNIVERSrTY OF ILLINO«-URB*NA
3 0112 079093990