BNDING OF BA Compessed laye N Compession longation Un-stained laye Unchanged longated laye NA Neutal Axis Two Dimensional View A When a beam is loaded unde pue moment, it can be shown that the beam will bend in a cicula ac. If we assume that plane coss-sections will emain plane afte bending, then to fom the cicula ac, the top layes of the beam have to shoten in length (compessive stain) and the bottom layes have to elongate in length (tensile stain) to poduce the cuvatue. The compession amount will gadually diminish as we go down fom the top laye, eventually changing fom compession to tension, which will then gadually incease as we each the bottom laye. Thus, in this type of loading, the top laye will have imum compessive stain, the bottom laye will have imum tensile stain and thee will be a middle laye whee the length of the laye will emain unchanged and hence no nomal stain. This laye is known as Neutal Laye, and in D epesentation, it is known as Neutal Axis (NA). Because the beam is made of elastic mateial, compessive and tensile stains will also give ise to compessive and tensile stesses (stess and stain is popotional Hook s Law), espectively. oe the applied moment load moe is the cuvatue, which will poduce moe stains and thus moe stesses. Ou objective is to estimate the stess fom bending.
We can detemine the bending stain and stess fom the geomety of bending. Let us take a small coss section of width dx, at a distance x fom the left edge of the beam. Afte the beam is bent, let the section dx, subtends an angle dφ at the cente of cuvatue with a adius of cuvatue at NA. Then, dφdx dφ 1 o, dx Let us conside an abitay laye at a distance v fom the NA. If we daw a line BC paallel to AO 1, then the angle CBD dφ. The elongation of this laye CD v.dφ. The oiginal length of this laye was dx. Hence the stain vd φ v ε..(1) dx Within elastic limit stess σ ε v () whee elastic constant. Since is fixed fo a loading condition, and is also a constant, then the stess will be popotional to the distance v of any laye fom the neutal axis. If we know the adius of cuvatue due to bending, we can find the bending stess and bending stain using these fomulas.
We can now apply the static equilibium condition to one half of the beam and can detemine the elationship between the stess and the applied bending moment. Let da is an elemental aea at a distance v fom the neutal plane. The foce acting on the element da is the aea multiplied by the bending stess, i.e. σ da vda/. The total foce can be obtained by integation of all foces acting on the face. quating foces in X diection σ da As 0 vda vda 0 vda 0 0 : This can only happen if the NA passes though the CG of the coss section. Thus NA must pass though the CG of the beam coss section. F x
Now if we equate sum total of moments due to bending stess with the applied moment, then fom 0 : I omentof σ vkk(3) I At thefathest laye Thus, σ vσda I v da Inetia aboutna ckk(4) v da v fom NA, v c, & σ σ I v Using equation (3) we can find bending stess at any laye at a distance v fom the neutal axis. quation (4) is the fomula fo imum bending stess, which will occu at the futhest laye fom the NA, whee c v. da σ I
ONT OF INRTIA (I) I, fo ectangula and cicula sections about thei NA can be found using following fomulae: I fo I-sections, Box sections and channel sections can be found using following fomulae: NA passes though the Cente of gavity (CG) of the beam coss section. Fo ectangula o cicula cosssection of the beam, CG is at the geometic cente of the section. Fo a composite section, the location of the CG can be detemined by the following fomula, y A1 y1 + A y + A3 y3 +... A + A + A +... 1 3 Tansfe of axis fo oment of Inetia This fomula is used to find I of a T o othe sections, whose NA o located at the geometic symmetically. I 1 I 0 + Ay CG is not
ADDITIONAL BA QUATION Slope of the beam at x φ, dy dφ d y φ & dx dx dx dφ 1 d y dx dx d y dx I fundamental beamequation When thee ae applied foces and moments
TRANSVRS SHAR STRSS τ V Ib va a VQ Ib ax. tansvese shea stess, τ always occus at NA (because Q is at NA)
ax Tansvese shea stesses (t ): Solid ectangula coss-section Solid cicula coss-section Cicula coss section with thin wall * I coss-section 3 V A 4 V 3 A τ V A V τ A τ A aea of the coss-section b.d τ A aea of the coss-section A aea of the coss-section π ( do d i ) 4 A t.d t thickness of the web, and d total depth of the I-beam π 4 d When finding tansvese shea stess (τ) fo a composite section, the following fomula can be used: τ V va Ib a V Ib ( v A + v A ) 1 a1 a...