006 The Graw-Hill Copanies, n. ll rights reserved. Fourth E CHTER ure ECHNCS OF TERLS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit
ECHNCS OF TERLS ure Bending ure Bending Other Loading Tpes Setri eber in ure Bending Bending Deforations Strain Due to Bending Bea Setion roperties roperties of erian Standard Shapes Deforations in a Transverse Cross Setion Saple roble. Bending of ebers ade of Several aterials Eaple.0 Reinfored Conrete Beas Saple roble. Stress Conentrations lasti Deforations ebers ade of an Elastoplasti aterial Eaple.0 Reinfored Conrete Beas Saple roble. Stress Conentrations lasti Deforations ebers ade of an Elastoplasti aterial lasti Deforations of ebers With a Single lane of S... Residual Stresses Eaple.05,.06 Eentri ial Loading in a lane of Setr Eaple.07 Saple roble.8 Unsetri Bending Eaple.08 General Case of Eentri ial Loading 006 The Graw-Hill Copanies, n. ll rights reserved. -
ECHNCS OF TERLS ure Bending ure Bending: risati ebers subjeted to equal and opposite ouples ating in the sae longitudinal plane 006 The Graw-Hill Copanies, n. ll rights reserved. -
ECHNCS OF TERLS Other Loading Tpes Eentri Loading: ial loading whih does not pass through setion entroid produes internal fores equivalent to an aial fore and a ouple Transverse Loading: Conentrated or distributed transverse load produes internal fores equivalent to a shear fore and a ouple riniple of Superposition: The noral stress due to pure bending a be obined with the noral stress due to aial loading and shear stress due to shear loading to find the oplete state of stress. 006 The Graw-Hill Copanies, n. ll rights reserved. -
ECHNCS OF TERLS Setri eber in ure Bending nternal fores in an ross setion are equivalent to a ouple. The oent of the ouple is the setion bending oent. Fro statis, a ouple onsists of two equal and opposite fores. The su of the oponents of the fores in an diretion is zero. The oent is the sae about an ais perpendiular to the plane of the ouple and zero about an ais ontained in the plane. These requireents a be applied to the sus of the oponents and oents of the statiall indeterinate eleentar internal fores. F z z d d 0 d 0 006 The Graw-Hill Copanies, n. ll rights reserved. - 5
ECHNCS OF TERLS Bending Deforations Bea with a plane of setr in pure bending: eber reains setri bends uniforl to for a irular ar ross-setional plane passes through ar enter and reains planar length of top dereases and length of botto inreases a neutral surfae ust eist that is parallel to the upper and lower surfaes and for whih the length does not hange stresses and strains are negative (opressive) above the neutral plane and positive (tension) below it 006 The Graw-Hill Copanies, n. ll rights reserved. - 6
ECHNCS OF TERLS Strain Due to Bending Consider a bea segent of length L. fter deforation, the length of the neutral surfae reains L. t other setions, L L L L or ρ (strain varies linearl) 006 The Graw-Hill Copanies, n. ll rights reserved. - 7
ECHNCS OF TERLS Stress Due to Bending For a linearl elasti aterial, E E (stress varies linearl) For stati equilibriu, F 0 0 d d d First oent with respet to neutral plane is zero. Therefore, the neutral surfae ust pass through the setion entroid. For stati equilibriu, Substituting S d d d 006 The Graw-Hill Copanies, n. ll rights reserved. - 8
ECHNCS OF TERLS Bea Setion roperties The aiu noral stress due to bending, S S setion oent of setion odulus inertia bea setion with a larger setion odulus will have a lower aiu stress Consider a retangular bea ross setion, S h bh 6 bh Between two beas with the sae ross setional area, the bea with the greater depth will be ore effetive in resisting bending. Strutural steel beas are designed to have a large setion odulus. 6 h 006 The Graw-Hill Copanies, n. ll rights reserved. - 9
ECHNCS OF TERLS roperties of erian Standard Shapes 006 The Graw-Hill Copanies, n. ll rights reserved. - 0
ECHNCS OF TERLS Deforations in a Transverse Cross Setion Deforation due to bending oent is quantified b the urvature of the neutral surfae E E E lthough ross setional planes reain planar when subjeted to bending oents, in-plane deforations are nonzero, z Epansion above the neutral surfae and ontration below it ause an in-plane urvature, antilasti urvature 006 The Graw-Hill Copanies, n. ll rights reserved. -
ECHNCS OF TERLS Saple roble. SOLUTON: Based on the ross setion geoetr, alulate the loation of the setion entroid and oent of inertia. Y d ast-iron ahine part is ated upon b a kn- ouple. Knowing E = 65 Ga and negleting the effets of fillets, deterine (a) the aiu tensile and opressive stresses, (b) the radius of urvature. 006 The Graw-Hill Copanies, n. ll rights reserved. ppl the elasti fleural forula to find the aiu tensile and opressive stresses. Calulate the urvature E -
ECHNCS OF TERLS Saple roble. SOLUTON: Based on the ross setion geoetr, alulate the loation of the setion entroid and oent of inertia. 0 0 rea, 90 0 800 00 000, 50 0, 90 0 0 0 Y 0 000 8 868 90 0 d 0 800 868 bh 0-9 d 0 0 00 8 006 The Graw-Hill Copanies, n. ll rights reserved. -
ECHNCS OF TERLS Saple roble. ppl the elasti fleural forula to find the aiu tensile and opressive stresses. B B kn 868 9 868 0 kn 0.0 0 0.08 9 B 76.0 a. a Calulate the urvature E kn 65 Ga 868 0-9 - 0.95 7.7 0 006 The Graw-Hill Copanies, n. ll rights reserved. -
ECHNCS OF TERLS Eentri ial Loading in a lane of Setr Stress due to eentri loading found b superposing the unifor stress due to a entri load and linear stress distribution due a pure bending oent Eentri loading F d entri bending Validit requires stresses below proportional liit, deforations have negligible effet on geoetr, and stresses not evaluated near points of load appliation. 006 The Graw-Hill Copanies, n. ll rights reserved. - 5
ECHNCS OF TERLS Eaple.07 SOLUTON: Find the equivalent entri load and bending oent Superpose the unifor stress due to the entri load and the linear stress due to the bending oent. n open-link hain is obtained b bending low-arbon steel rods into the shape shown. For 60 lb load, deterine (a) aiu tensile and opressive stresses, (b) distane between setion entroid and neutral ais 006 The Graw-Hill Copanies, n. ll rights reserved. Evaluate the aiu tensile and opressive stresses at the inner and outer edges, respetivel, of the superposed stress distribution. Find the neutral ais b deterining the loation where the noral stress is zero. - 6
ECHNCS OF TERLS Eaple.07 Noral stress due to a entri load 0 0.96in 85psi 0.5in 60lb 0.96in Equivalent entri load and bending oent 60 lb d 0 lb 60 lb in 0.65in Noral stress due to bending oent.068 875 psi 0 0 lb.068 0.5 in in 0 0.5in in 006 The Graw-Hill Copanies, n. ll rights reserved. - 7
ECHNCS OF TERLS Eaple.07 aiu tensile and opressive stresses t 0 85 0 85 875 875 t 960 psi 7660 psi Neutral ais loation 0 0 0 0 0.00in 85psi.068 0 in 05lb in 006 The Graw-Hill Copanies, n. ll rights reserved. - 8
ECHNCS OF TERLS Saple roble.8 The largest allowable stresses for the ast iron link are 0 a in tension and 0 a in opression. Deterine the largest fore whih an be applied to the link. SOLUTON: Deterine equivalent entri load and bending oent. Superpose the stress due to a entri load and the stress due to bending. Fro Saple roble., 0 Y 0.08 868 0 9 Evaluate the ritial loads for the allowable tensile and opressive stresses. The largest allowable load is the sallest of the two ritial loads. 006 The Graw-Hill Copanies, n. ll rights reserved. - 9
ECHNCS OF TERLS Saple roble.8 Deterine equivalent entri and bending loads. d 0.08 0.00 entri load d 0.08 0.08 bending oent Superpose stresses due to entri and bending loads Evaluate ritial loads for allowable stresses. B B 77 559 0 a 0 0 0 a 0.08 868 0 0.08 0.0 868 0 79.6kN 77.0kN 0.0 9 9 77 559 The largest allowable load 77.0 kn 006 The Graw-Hill Copanies, n. ll rights reserved. - 0