Chaper 2: Stress in beams
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1 Chaper : Stress n eams FLEURE Beams suject to enng wll fle COPRESSON TENSON On the lower surface the eam s stretche lengthwse. Ths sujects t to tensle stress. N.A. N.A. s the neutral as On the upper surface the eam s squashe lengthwse. Ths sujects t to compressve stress. Between the compresson zone an the tenson zone s the neutral as (N.A). Ths remans at ts orgnal length. n smple enng: the neutral as s ent nto a crcular arc where R s ts raus of curvature. s the stance from e neutral as to another pont n the secton of the eam The stress nuce enng, σ s gven σ E / R E s the moulus of elastct. The stress s hgher at those ponts locate further from the neutral as. We shoul focus on the mamum value of stress, hence we select ma as our value. Elastc moulus, also known as Young s moulus, s characterstc of a materal. ts value vares greatl. t has the same unts of pressure, that s Pascal (Pa). Eamples: ateral Ruer ABS plastc Bone Alumnum Glass Ttanum Allo Alumnum Bronza Caron Fer renforce plastc Stanless steel AS 0 amon Young s moulus (GPa) Note the unts Pa PASCAL N /m kpa KLOPASCAL 0 N/m Pa EGAPASCAL 0 N/m GPa GGAPASCAL 0 9 N/m QUESTON Calculate the smallest raus to whch a rectangular secton steel strp of epth can e ent f the mamum stress allowe s 00 Pa. ts elastc moulus s 0 GPa. amum stress wll occur at the surface / 0 - m Usng Hence σ E/R R E σ 9 ( 0 m)(0 0 N / m ). m 00 0 N / m KNS OF OENTS st OENT OF FORCE FORCE STANCE n OENT OF FORCE FORCE STANCE (Also calle moment of nerta) st OENT OF AREA AREA STANCE Ths s use to fn the centro of a comple shape. n OENT OF AREA AREA STANCE Relate wth a eam s alt to resst enng For moments of area, the stance s measure from the NEUTRAL AS.
2 SECON OENT OF AREA Conser eams wth an equal cross-sectonal area. OENTS OF AREA FOR FFERENT SHAPES: () RECTANGULAR BEAS A BEA A A A + A A BEA B Values an are greater n case A, hence: NEUTRAL AS PASSES THROUGH THE CENTRO eam A has a much greater value than eam B. NEUTRAL AS. Conser a ruler wth a cross-secton of 0. 0 Wth the eam n vertcal orentaton: () -BEA (0), - Wth the eam n horzontal orentaton: 0(). Wth the eam n the frst orentaton t s 00 tmes more ale to resst vertcal enng. s that a etter opton? for the rectangle enclosng the eam B B for the two mssng rectangles Hence the value for the -eam s B Note that an alternatve metho s to suate the values for the rectangles n the secton. However B / onl apples to the neutral as. An ajustment must e mae for other aes. PARALLEL AS THEORE: How to calculate the n moment of area respect a as parallel to the neutral as AREA A NEUTRAL AS OF PART NEUTRAL AS OF WHOLE + A.
3 () T-BEA EAPLE: 0 00 A 0 The neutral as of the composte passes through the centro. Steps: ) Fn the locaton of the centro ) Fn the secon moment of area mensons n NOTE: How to get the centro usng the st moment of area a A. f a. f ( ) a f ( ) a f(). f ( ) A a centro f (. A ) A f ( ) a For smple shapes we can use rectl A. A. () T-BEA (contnuaton) EAPLE: 00 0 A 0 Fn centro usng the st moment of area A A A A + A mensons n Note: we are takng reference at the ase of the secton Takng moments aout the ase A. A + A * *80 *80 + 8*80. Takng moments aout the left se A * * * + 8*. A + A 0 * So the centro s at pont (,.) s ths value what ou epect? Wh? () T-BEA (contnuaton) EAPLE: 0 00 A 0 + total mensons n,, Usng the st moment of area we foun out that the centro s at pont (,.) Now we can calculate the n moment of area respect As of nether of the sectons passes through the centro, we nee to use the parallel as theorem for each secton., total, ( + A, + A ) For the stance etween the N.A. of part an whole s Usng the parallel theorema A , 00(0) / A. Hence the contruton of to the n moment of area of the whole s,, + A
4 For A the stance etween the N.A. of part an whole s. 80. Usng the parallel theorema A + A () CYLNRCAL BEAS, (0) / Hence the contruton of A to the n moment of area s,, + A π Then, the total n moment of area of the whole secton s TOTAL ( + A ).7 0, ( ) 0 +, (5) TUBE BEAS Also calle hollow sectons Avantage: The stll have a hgh -value ut the weght less BEA TERNOLOGY Y FLANGE WEB B B FLLET Y FLANGE π π apples when the loang s vertcal an the - neutral as s horzontal. YY apples when the loang s vertcal an the Y-Y neutral as s horzontal. BENNG EQUATON From the fleure equaton: Notng that: E σ R σ E R QUESTON A m long eam wth a rectangular cross-secton, 00 we an eep, s smpl supporte at ts ens. t carres a loa of 0 kn at ts mpont. Calculate the mamum tensle an compresson stress. SOLUTON Step : Fn secon moment of area m
5 Step : Fn mamum value from the enng moment agram SUPPORT REACTONS R R 5kN 0kN m m S.F.. 5kN 5kN S.F.. +5kN Step : Fn the mamum stress means of the formula gven earler USNG ma σ ma ma 0 0 (75 0 ) ±.7 Pa B... 0kNm -5kN ma 0 kn.m Benng causes tensle stress (stretchng) on one surface an compressve stress (squashng) on the other. STRETCHE SQUASHE NEUTRAL AS (UNCHANGE) Benng stress changes wth epth from the largest postve value to zero to the largest negatve value. Benng falure often starts as a tear on the surface. (En of chapter ) 5
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