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BENDING IN BOTH AXIS

Beams Bending in Both Axis ( Unsymmetrical Bending ) When a beam is subjected to a normal load which causes bending in the x - axis and a tangential load which causes bending along the mirror axis, it is said that the member is subjected to an unsymmetrical bending.

The ollowing ex!ression can be written or the stress at any !oint in a beam subjected to an unsymmetrical bending.

". Bend Bendin ing g #tre #tress ss $ a. % lateral loads !asses through the centroid o the beam section.   b &' (x x * %x ) +*- '(y y * %y )+   b & 'x * #x+ *- ' y * #y +  b. % the lateral loads is a!!lied at the to! lange and does not !asses through the centroid o the  beam section.   b & 'x * #x+ *- ' y * (#y* ) +

 ote $ /nly one hal o the section modulus about the y-axis is considered eecti0e. eecti0e. . #hea #heari ring ng #tre #tress ss $  0 & ' 1x 2x * b lx + *- ' 1y 2y * b %y + 3. Using Using the the %nte %nterac ractio tion n 4x!re 4x!ressi ssion on $ a. '   bx * 5 bx +  '   by * 5 by + 6 ".7 or com!act laterally su!!orted sha!es $  b. '   bx * 7.885y +  '   by * 7.9:5y + 6 ".7 5or doubly symmetrical l and ; sha!e members with com!act langes continuously connected to the web and bent about their wea< axis, the allowable bending b ending stress is 7.9: 5y.  ote $ onsideration should be gi0en to the =uestion o lateral su!!ort or the com!ression lange which will indicate whether com!act or non - com!act sections.

Problem # 1

A W :7 x 33 beam carries a uniorm gra0ity load o > mm t & ." mm. #x & 39 x "73 mm3 tw & 8.8 mm. #y & 8?.9 x "73 mm3 ". eterm etermine ine the the bendi bending ng stres stresss o the section with res!ect to x - axis. . eterm etermine ine the the tota totall bendi bending ng stre stress ss o the section. 3. eterm etermine ine the the rati ratio o o the actu actual al to to the allowable bending stress using interaction 0alue. Solution :

5lexural stress o the beam about the x- axis. •

Wx & ( ?>77  :" ) cos 37C Wx & ?877.33  *m x & WD E * > x & ' ?877.33(:.?) + * > x & "8.98> x "78 .mm  x & x * #x x & "8.98> x "78 * :"" x "73 x & 3.>" @a

•

5lexural stress o beam about the y-

axis W" & ?>77 sin 37C W" & ?77  *m W & :" sin 37C W & :8  *m  y & ' "* (#y *)+ * '  * #y + " & WD E * > " & ' ?77(:.?) + * > " & >9?> .mm " & >9?> x "73 .mm  & WD E * >  & ' :8(:.?) + * >  & 33." .mm  & 33." x "73 .mm  y & ' "* (#y *)+ * '  * #y +  y & ' >9?> x "73 * ( "9? x "73 *  ) +  ' 33." x "73 * "9? x "73 +  y & "7:." @a Fatio o the actual to the allowable  bending stress using interaction e=uation. •

 b  *  t  & 73.9" * '(".:9)+ & >."7  b  *  t 6 "97 *

√ F y  & "7.9:

use 5 bx & 7.88 5y 5 by & 7.9: 5y (  x * 5 bx ) * (  y * 5 by ) 6*& ".7 3.>" * '7.88(:7)+  "7:." * '7.9:(:7)+ & 0.76

Problem # !

A 387 x " beam o A :>> steel ( 5y & 3?: @a ) su!!orts a su!er im!osed gra0ity load o ? (8) + * > x & "3 Hn.m (   bx * 5 bx )  (   by * 5 by ) 6*& ".7   bx & x * #x   bx &"3 x "78 * ":"7 x "73   bx & "9.>" 5 bx & 7.88 5y 5 bx & 7.88( 3?: ) 5 bx & 9.97 @a 5 by & 7.9: 5y 5 by & 7.9:( 3?: ) 5 by & :>.9: @a (   bx * 5 bx )  (   by * 5 by ) 6*& ".7 ( "9.>" * 9.97 )  (   by * :>.9: ) & ".7   by & ""3.:" @a y &   by #y y & ""3.:"(3:3) "73 y & ?7.79 x "78 .m " $ 0.07 %N.m Eateral concentrated that the beam could su!!ort i it will be acting on the lange at its mids!an. •

  by & ""3.:" y &   by #y * y & ""3.:"(3:3) * y & 7.73?: x "78 7 x "73 mm3 t & "8.3 mm. #y & 3:" x "73 mm3 d & 3"7 mm Assume that the to! lange is not laterally braced between end su!!orts. ". etermine the bending stress along the wea8 x "78 * ">7 x "73 bx & >:.>3 @a

 & bx  by

 & >:.>3  :.>3 ' $ 1(.66 "P+ Total ratio o actual bending to allowable bending stress using interaction e=uation. •

bx * 5bx  by * 5by 6*& ".7 allowable bending stress $ E b & 8777 mm Ec & 77 b *

√ Fy

Ec & 77 (:?) *

√ 250

Ec & 3"3 mm Eu & "3977 * ( 5y d ) * ( b  t ) Eu & "3977 * ( :7x3"7 ) * ( :?x"8.3 ) Eu & 9389 mm E b I E b E b 6 Eu Use 5bx & 7.87 5y 5by & 7.9: 5y ( bending in wea.? x "73 mm3 %y & 7."> x "78 mm?

". etermine the total actual bending stress o the !urlin assuming that the loads !asses through the ce ntroid o the !urlins. . etermine the total actual bending stress o the !urlins assuming the li0e load and rooing is acting on the lange o the !urlins. 3. etermine the ratio o the actual and allowable bending stress assuming that all the loads !asses through the centroid o the !urlins. Solution :

Total actual bending stress o the  !urlins assuming all the loads !asses through the centroid o the !urlins $ •

θ  & J θ= 26.6 °

tan

1ertical loads $ Fooing & 7.799 (".93) Fooing & 7."33 .? x "73) ' b $ 1&7.7* "P+ Total actual bending stress i the li0e load and rooing does not !ass through the centroid. •

Fooing load  li0e load & 7."33  ".883 Fooing load  li0e load & ".98 Hn*m

W" & ".?  ".98 cos 8.8 C  7."33 cos 8.8 C W" & .8: *m W & ".98 sin 8.8 C W & 7.>7? y & .9 7? (:) * 3 y & 7.8> .? x "73 ) ' b $ 17!.1! "P+ Fatio o actual bending stress to allowable bending stress i the !asses through the center. •

  bx & x * #x   bx & .: x "78 * ".: x "73   bx & "7".7 @a   by & y * #y   by & 7.89: x "78 * ">.? x "73   bx & 38.8> @a  b  *  t  & "77 * (:.:)

 & .7 6 "97 *

√  Fy

¿ 10.75 This is a com!act section $ 5 bx & 7.88 5y 5 bx & 7.88 (:7) 5 bx & "8: @a

5 by & 7.9: 5y 5 by & 7.9: (:7) 5 by & ">9.:7 @a   bx * 5 bx    by * 5 by 6*& ".7 ( "7".7 * "8: )  ( 38.8> * ">9.:7 ) & 7.>" 0.*1 , 1.0

Problem # !

A W :7 x 93 A 38 steel is used as a !urlin that is sim!ly su!!orted on a 8m. s!an between roo  is assumed to su!!ort a su!er im!osed dead load o 97 *m and a snow load o "8>7 *m. The slo!e o the roo truss or to! chord is " 0ertically and  horiKontally and the !urlins are s!aced 3m. on a center. 5y & :7 @a. Assume the !urlin com!ression lange has ull lateral su!!ort. @ro!erties o W :7 x 93 #x & >" x "73 mm3 #y & 378 x "73 mm3

". etermine the bending stress about the axis !er!endicular to the to! chord. . etermine the total bending stress o  the !urlin. 3. etermine the ratio o the actual to the allowable bending stress using interaction e=uation. Solution :

Bending stress about the axis  !er!endicular to the to! chord. •

θ  & J

θ  & 8.8 C

x & 3cos 8.8 C x & .8> m

W" & 97 (3) W" & "87 *m W & "8>7 (3) W & :7?7 *m

W3 & ( W"  W ) cos θ W3 & ( "87  :7?7 ) cos 8.8 C W3 & 8?3> *m W? & W sin 8.8 C W? & :7?7 sin8.8 C W? & :8.9 *m W: & "87 sin 8.8 C W: & 98. *m x & W3 E * > x & 8?3> (8) * > x & >.9 x "78 .mm

  bx & x * #x   bx & >.9 x "78 * >" x "73 ' b $ &!.(1 "P+ •

Total bending stress o the !urlin.

y" & W? E * > y" & :8.9 (8) * > y" & "7.":: x "78 .mm y & W: E * > y & 89. (8) * > y & ?.3: x "78 .mm

  by & ' y" * J #y +  'y * #y +   by & '"7.":: x "78 * J (378 x "73 +  '?.3: x "78 * 378 x "73 +   by & >7.: @a

Total bending stress $   b &   bx    by   b & 3.:"  >7.: ' b $ 11&.10 "P+ Fatio o the actual to the allowable  bending stress using interaction e=uation. •

'   bx * 5 bx +  '   by * 5 by + 6*& ".7 '   bx * 7.88 5y +  '   by * 7.9: 5y + 6*& ".7 ' 3.:" * 7.88 (:7) +  ' >7.:* 7.9: (:7) + & 0.6&

Problem # &

A W 8 x "8 section is to ser0e as a !urlin between roo trusses 9.m on centers. The roo is assumed to su!!ort a dead load o 87 *m o roo surace including its own weight and a li0e load o "777 *m o horiKontal roo surace !rojection. The slo!e o the roo truss is " 0ertical to  horiKontal and the !urlins are to be s!aced ".>m on centers. Use A 38 steel with 5y & ?> @a. Assume all loads !ass through the center o gra0ity o the section. #ag rods are to be !laced at the middle thirds between trusses. @ro!erties o W 8 x "8 A & 37:> mm d & ":.:" mm  b  & "7.38 mm t  & "7.mm

#x & "89 x "73 mm3 #y & 38 x "73 mm3 tw & 8.87 mm

". etermine the bending stress along the x G axis o the W section. . etermine the bending stress along the y G axis o the W section. 3. etermine the ratio o the actual to the allowable bending stress using intersection e=uation. Solution :

etermine the bending stress along the x G axis o the W section. •

x & Wx E * > x & >?.8 ( 9.) * > x & "3?" .m   bx & x * #x   bx & "3?" x "73* "89 x "73   bx & "":. >" @a

tan

θ=

1 2

θ  & 8.8 C

x & ".> cos 8.8 C x & ".8"( horiKontal roo surace !rojection )

ead load & 87 (".>) Ei0e load & "777(".8") Total 0ertical load

& "9> *m & "8"7 *m & 333> *m

Wx & 333> cos 8.8 C  $ !)*.6) N2m

Bending stress along y G axis o the W section. •

Wy & 333> sin 8.8 C Wx & "??.8 *m y & Wy E * > ( sag rods at middle thirds ) y & "??.8 ( 9.) * 7 y & >87. .m   by & y * #y   by & >87. x "73* 38 x "73 ' b $ !&.)1 "P+ Fatio o the actual to the allowable  bending stress using intersection e=uation. •

'   bx * 5 bx +  '   by * 5 by + 6*& ".7 hec< i com!acted or non-com!act section$  b *  t 6 "97 * √ Fy "7.38 * ("7.) 6 "97 * √ 248 ?.9 6 "7.> This is a com!act section $ 5 bx & 7.88 5y 5 bx & 7.88 (?>) 5 bx & "83.8> @a 5or bending along wea< axis $ 5 by & 7.9: 5y 5 by & 7.9: (?>) 5 by & ">8 @a Using interaction 0alue $ '   bx * 5 bx +  '   by * 5 by + 6*& ".7 ' "":.>" * "83.8> +  ' 3." * ">8 + 0.*&6 , 1.0

 TENSION WITH BENDING Usually bottom chords of trusses are subjected to tension ith bendin! but bendin! in tension members are not as serious as those in com"ression members due to the fact that tension tends to reduce lateral de#ections unli$e the com"ression members% they tend to cause more or bi!!er lateral de#ections&  The combined stress is due to a'ial stress hich causes tensile stresses and due to bendin! hich causes both tensile and com"ressi(e stresses& f ) T * + ,*- ./ * I

.embers subject to both a'ial tension and bendin! shall be "ro"ortioned at all "oints alon! their len!th to satisfy the folloin! e0uation&

Bending in one axis only. 1a * f t , fb' * 1b' 2 3&4 f a ) com"uted a'ial stress f a ) T*+ f b ) com"uted bendin! stress 1t ) alloable tensile stress 1t ) 4&54 1y 1b' ) alloable bendin! stress

Bending in both axis 1a * f t , fb' * 1b' , fb 6 * 1b 6 2 3&4 1t ) 4&54 1y 1b' ) 4&55 1y for com"act section 1by ) 4&78 1y 1b' ) 4&54 1y for non - com"act section

Problem # 1 + W 984 ' 54 beam ha(in! a sim"le s"an of 5m& carries a uniform load :W: $N*m throu!hout its s"an& The beam is also subjected to an a'ial tensile force of 345 $N& The com"ression #an!e is su""orted laterally at 0uarter "oints only& Use + ;5 steel&1y ) 44 3C* @ mm

τ 2=

V Q2 lb

Q2=60 ( 20 ) ( 80 ) 1 & max. 0ertical shear  A0erage shearing stress o web $

2

 Ave . τ = τ 2+ ( τ 1− τ 2) 3

#hear low $ % the shearing

τ  is multi!lied by the width b, we obtain a =uantity q, 7

0 & ' :77* ( h* tw ) + ' √  H 0 * 5y + when 0 I 7.>7

H 0 & ?  ' :.3? (a*h) when a*h 6 ".7  H 0 & :.3?  ' ?.7 * (a*h)  when a*h I ".7 5or a rolled section without stieners H0 & :.3?  ote $ h* tw I 87 intermediate stieners are re=uired. Where$ tw & thic1r38Rr: =$&52>r!8 o%+?8

Problem # !

A uniormly distributed load o 77 lb*t is carried on a sim!ly su!!orted beam s!an. % the crosssection is as shown in 5ig. @-:97, determine the maximum length o the beam i the shearing stress is limited to >7 !si. Assume the load acts o0er the entire length o the beam.

0&12%b0&12%b

Where$ 0&>7!si 1&"77E 2& >(:)(.:)O8(?)() 2 & in3 % & ""(>)("73)O""(8)(>3) % & "33in? B & >O8 B & in Thus, >7&"77E(:)"33() /$1!.6 't/$1!.6 't

+n9er

Problem # &

The cross-section o a beam is an isosceles triangle with 0ertex u!!ermost, o altitude h and base  b. % 1 is the 0ertical shear, show that the maximum shearing stress is 31 * bh located at the mid!oint o the altitude. 0&12*%b 0&12*%b