00 The Graw-Hill Copanies, n. All rights reserved. Third E CHAPTER 4 Pure ECHANCS OF ATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit
ECHANCS OF ATERALS Pure Bending Pure Bending Other Loading Tpes Setri eber in Pure Bending Bending Deforations Strain Due to Bending Bea Setion Properties Properties of Aerian Standard Shapes Deforations in a Transverse Cross Setion Saple Proble 4. Bending of ebers ade of Several aterials Eaple 4.0 Reinfored Conrete Beas Saple Proble 4.4 Stress Conentrations Plasti Deforations ebers ade of an Elastoplasti aterial Eaple 4.0 Reinfored Conrete Beas Saple Proble 4.4 Stress Conentrations Plasti Deforations ebers ade of an Elastoplasti aterial Plasti Deforations of ebers With a Single Plane of S... Residual Stresses Eaple 4.05, 4.06 Eentri Aial Loading in a Plane of Setr Eaple 4.07 Saple Proble 4.8 Unsetri Bending Eaple 4.08 General Case of Eentri Aial Loading 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Pure Bending Pure Bending: Prisati ebers subjeted to equal and opposite ouples ating in the sae longitudinal plane 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Other Loading Tpes Eentri Loading: Aial 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 Priniple 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. 00 The Graw-Hill Copanies, n. All rights reserved. 4-4
ECHANCS OF ATERALS Setri eber in Pure Bending F da 0 da 0 da 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 ero. The oent is the sae about an ais perpendiular to the plane of the ouple and ero 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. 00 The Graw-Hill Copanies, n. All rights reserved. 4-5
ECHANCS OF ATERALS 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 00 The Graw-Hill Copanies, n. All rights reserved. 4-6
ECHANCS OF ATERALS Strain Due to Bending Consider a bea segent of length L. After deforation, the length of the neutral surfae reains L. At other setions, L L L L or ρ (strain varies linearl) 00 The Graw-Hill Copanies, n. All rights reserved. 4-7
ECHANCS OF ATERALS Stress Due to Bending For a linearl elasti aterial, E E (stressvaries linearl) For stati equilibriu, F 0 0 da da da First oent with respet to neutral plane is ero. Therefore, the neutral surfae ust pass through the setion entroid. For stati equilibriu, Substituting da S da da 00 The Graw-Hill Copanies, n. All rights reserved. 4-8
ECHANCS OF ATERALS Bea Setion Properties The aiu noral stress due to bending, S setion oent of inertia S setion odulus A 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 Ah 00 The Graw-Hill Copanies, n. All rights reserved. 4-9
ECHANCS OF ATERALS Properties of Aerian Standard Shapes 00 The Graw-Hill Copanies, n. All rights reserved. 4-0
ECHANCS OF ATERALS Deforations in a Transverse Cross Setion Deforation due to bending oent is quantified b the urvature of the neutral surfae E E E Although ross setional planes reain planar when subjeted to bending oents, in-plane deforations are nonero, Epansion above the neutral surfae and ontration below it ause an in-plane urvature, antilasti urvature 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Saple Proble 4. SOLUTON: Based on the ross setion geoetr, alulate the loation of the setion entroid and oent of inertia. A A Ad A ast-iron ahine part is ated upon b a kn- ouple. Knowing E = 65 GPa and negleting the effets of fillets, deterine (a) the aiu tensile and opressive stresses, (b) the radius of urvature. 00 The Graw-Hill Copanies, n. All rights reserved. Appl the elasti fleural forula to find the aiu tensile and opressive stresses. Calulate the urvature E 4 -
ECHANCS OF ATERALS Saple Proble 4. SOLUTON: Based on the ross setion geoetr, alulate the loation of the setion entroid and oent of inertia. Area, 090 800 400 00 A 000, 50 0 A, 900 40 A 40 A A 40 000 8 Ad bh Ad 900 800 040 008 8680 8680-9 4 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Saple Proble 4. Appl the elasti fleural forula to find the aiu tensile and opressive stresses. A B A kn 0.0 9 4 8680 kn 0.08 9 4 8680 B A B 76.0 Pa. Pa Calulate the urvature E kn 65 GPa 9 4 868 0 - - 0.950 47.7 00 The Graw-Hill Copanies, n. All rights reserved. 4-4
ECHANCS OF ATERALS Bending of ebers ade of Several aterials Consider a oposite bea fored fro two aterials with E and E. Noral strain varies linearl. Pieewise linear noral stress variation. E E E E Neutral ais does not pass through setion entroid of oposite setion. n Eleental fores on the setion are df E E da da df da da Define a transfored setion suh that ne E E df da nda n E 00 The Graw-Hill Copanies, n. All rights reserved. 4-5
ECHANCS OF ATERALS Eaple 4.0 SOLUTON: Transfor the bar to an equivalent ross setion ade entirel of brass Evaluate the ross setional properties of the transfored setion Bar is ade fro bonded piees of steel (E s = 90 6 psi) and brass (E b = 50 6 psi). Deterine the aiu stress in the steel and brass when a oent of 40 kip*in is applied. Calulate the aiu stress in the transfored setion. This is the orret aiu stress for the brass piees of the bar. Deterine the aiu stress in the steel portion of the bar b ultipling the aiu stress for the transfored setion b the ratio of the oduli of elastiit. 00 The Graw-Hill Copanies, n. All rights reserved. 4-6
ECHANCS OF ATERALS Eaple 4.0 SOLUTON: Transfor the bar to an equivalent ross setion ade entirel of brass. E n E b T s b 0.4 in.90.75in 0.4 in.5in Evaluate the transfored ross setional properties b h.5in. in 90 50 T 5.06in 4 6 6 psi.9 psi Calulate the aiu stresses 40 kip in.5 in 5.06in 4.85ksi b a b a n.9.85ksi.9 ksi s a s a.85ksi 00 The Graw-Hill Copanies, n. All rights reserved. 4-7
ECHANCS OF ATERALS Reinfored Conrete Beas Conrete beas subjeted to bending oents are reinfored b steel rods. The steel rods arr the entire tensile load below the neutral surfae. The upper part of the onrete bea arries the opressive load. n the transfored setion, the ross setional area of the steel, A s, is replaed b the equivalent area na s where n = E s /E. To deterine the loation of the neutral ais, b n A d 0 s b n A n A d s s The noral stress in the onrete and steel s n 0 00 The Graw-Hill Copanies, n. All rights reserved. 4-8
ECHANCS OF ATERALS Saple Proble 4.4 A onrete floor slab is reinfored with 5/8-in-diaeter steel rods. The odulus of elastiit is 906psi for steel and.606psi for onrete. With an applied bending oent of 40 kip*in for -ft width of the slab, deterine the aiu stress in the onrete and steel. SOLUTON: Transfor to a setion ade entirel of onrete. Evaluate geoetri properties of transfored setion. Calulate the aiu stresses in the onrete and steel. 00 The Graw-Hill Copanies, n. All rights reserved. 4-9
ECHANCS OF ATERALS Saple Proble 4.4 SOLUTON: Transfor to a setion ade entirel of onrete. E n E na s s 8.06 4 6 90.60 6 5 in 4.95in 8 psi 8.06 psi Evaluate the geoetri properties of the transfored setion. 4.954 0.450in 4 in.45in 4.95in.55in 44.4 in Calulate the aiu stresses. s n 40kip in.45in 4 44.4in 40kip in.55in 8.06 4 44.4in s.06ksi 8.5ksi 00 The Graw-Hill Copanies, n. All rights reserved. 4-0
ECHANCS OF ATERALS Stress Conentrations Stress onentrations a our: in the viinit of points where the loads are applied in the viinit of abrupt hanges in ross setion K 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Plasti Deforations For an eber subjeted to pure bending strain varies linearl aross the setion f the eber is ade of a linearl elasti aterial, the neutral ais passes through the setion entroid and For a aterial with a nonlinear stress-strain urve, the neutral ais loation is found b satisfing F da 0 For a eber with vertial and horiontal planes of setr and a aterial with the sae tensile and opressive stress-strain relationship, the neutral ais is loated at the setion entroid and the stressstrain relationship a be used to ap the strain distribution fro the stress distribution. da 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Plasti Deforations When the aiu stress is equal to the ultiate strength of the aterial, failure ours and the orresponding oent U is referred to as the ultiate bending oent. The odulus of rupture in bending, R B, is found fro an eperientall deterined value of U and a fititious linear stress distribution. R B U R B a be used to deterine U of an eber ade of the sae aterial and with the sae ross setional shape but different diensions. 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS ebers ade of an Elastoplasti aterial Retangular bea ade of an elastoplasti aterial aiu elastioent f the oent is inreased beond the aiu elasti oent, plasti ones develop around an elasti ore. elastiore half - thikness n the liit as the oent is inreased further, the elasti ore thikness goes to ero, orresponding to a full plasti deforation. k p p plasti oent shape fator (depends onl on rosssetion shape) 00 The Graw-Hill Copanies, n. All rights reserved. 4-4
ECHANCS OF ATERALS Plasti Deforations of ebers With a Single Plane of Setr Full plasti deforation of a bea with onl a vertial plane of setr. The neutral ais annot be assued to pass through the setion entroid. Resultants R and R of the eleentar opressive and tensile fores for a ouple. A A R R The neutral ais divides the setion into equal areas. The plasti oent for the eber, p A d 00 The Graw-Hill Copanies, n. All rights reserved. 4-5
ECHANCS OF ATERALS Residual Stresses Plasti ones develop in a eber ade of an elastoplasti aterial if the bending oent is large enough. Sine the linear relation between noral stress and strain applies at all points during the unloading phase, it a be handled b assuing the eber to be full elasti. Residual stresses are obtained b appling the priniple of superposition to obine the stresses due to loading with a oent (elastoplasti deforation) and unloading with a oent - (elasti deforation). The final value of stress at a point will not, in general, be ero. 00 The Graw-Hill Copanies, n. All rights reserved. 4-6
ECHANCS OF ATERALS Eaple 4.05, 4.06 A eber of unifor retangular ross setion is subjeted to a bending oent = 6.8 kn-. The eber is ade of an elastoplasti aterial with a ield strength of 40 Pa and a odulus of elastiit of 00 GPa. Deterine (a) the thikness of the elasti ore, (b) the radius of urvature of the neutral surfae. After the loading has been redued bak to ero, deterine () the distribution of residual stresses, (d) radius of urvature. 00 The Graw-Hill Copanies, n. All rights reserved. 4-7
ECHANCS OF ATERALS Eaple 4.05, 4.06 Thikness of elasti ore: 6.8kN 8.8kN 0.666 60 80 aiu elasti oent: b 00 6 8.8 kn 500 600 6 00 40Pa Radius of urvature: E.0 6 400 000 9 400.0 Pa Pa. 00 The Graw-Hill Copanies, n. All rights reserved. 4-8
ECHANCS OF ATERALS Eaple 4.05, 4.06 = 6.8 kn- 40 40Pa = -6.8 kn- 6.8kN 6 00 06.7Pa = 0 At the edge of E 77.50 the elasti ore, 6 6 5.50 Pa 9 000 Pa 400 77.50 6 5 00 The Graw-Hill Copanies, n. All rights reserved. 4-9
ECHANCS OF ATERALS Eentri Aial Loading in a Plane of Setr Eentri loading F P Pd 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 P A entri bending Validit requires stresses below proportional liit, deforations have negligible effet on geoetr, and stresses not evaluated near points of load appliation. 00 The Graw-Hill Copanies, n. All rights reserved. 4-0
ECHANCS OF ATERALS Eaple 4.07 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. An 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 00 The Graw-Hill Copanies, n. All 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 ero. 4 -
ECHANCS OF ATERALS Eaple 4.07 Noral stress due to a entri load A 0.5in 0 0.96in P A 85psi 60lb 0.96in Equivalent entri load and bending oent P 60lb Pd 04lbin 60lb0.6in Noral stress due to bending oent 4 4 0.5 4.0680 8475psi 4 in 4 04lbin0.5in.0680 in 4 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Eaple 4.07 aiu tensile and opressive stresses t 0 858475 0 858475 t 960psi 7660psi Neutral ais loation 0 0 P A P A 0 0 0.040in 85psi.0680 in 05lbin 4 00 The Graw-Hill Copanies, n. All rights reserved. 4 -
ECHANCS OF ATERALS Saple Proble 4.8 The largest allowable stresses for the ast iron link are 0 Pa in tension and 0 Pa in opression. Deterine the largest fore P whih an be applied to the link. SOLUTON: Deterine an equivalent entri load and bending oent. Superpose the stress due to a entri load and the stress due to bending. Fro Saple Proble.4, A 0 0.08 8680 9 4 Evaluate the ritial loads for the allowable tensile and opressive stresses. The largest allowable load is the sallest of the two ritial loads. 00 The Graw-Hill Copanies, n. All rights reserved. 4-4
ECHANCS OF ATERALS Saple Proble 4.8 Deterine an equivalent entri and bending loads. d 0.08 0.00 0.08 P entriload Pd 0.08P bending oent Superpose stresses due to entri and bending loads P 0.08P0.0 Evaluate ritial loads for allowable stresses. A B A B P A P A 77P 559P A A 0 P 0 0Pa 0Pa 8680 0.08P0.0 8680 P 79.6kN P 79.6kN 9 9 77P 559P The largest allowable load P 77.0 kn 00 The Graw-Hill Copanies, n. All rights reserved. 4-5
ECHANCS OF ATERALS Unsetri Bending Analsis of pure bending has been liited to ebers subjeted to bending ouples ating in a plane of setr. ebers reain setri and bend in the plane of setr. The neutral ais of the ross setion oinides with the ais of the ouple Will now onsider situations in whih the bending ouples do not at in a plane of setr. Cannot assue that the eber will bend in the plane of the ouples. n general, the neutral ais of the setion will not oinide with the ais of the ouple. 00 The Graw-Hill Copanies, n. All rights reserved. 4-6
ECHANCS OF ATERALS Unsetri Bending 0 F da or 0 da da neutral ais passes through entroid Wish to deterine the ons under whih the neutral ais of a ross setion of arbitrar shape oinides with the ais of the ouple as shown. or da σ oent defines stress distribution of inertia The resultant fore and oent fro the distribution of eleentar fores in the setion ust satisf F 0 applied ouple 0 da da or 0 da produt of inertia ouple vetor ust be direted along a prinipal entroidal ais 00 The Graw-Hill Copanies, n. All rights reserved. 4-7
ECHANCS OF ATERALS Unsetri Bending Superposition is applied to deterine stresses in the ost general ase of unsetri bending. Resolve the ouple vetor into oponents along the priniple entroidal aes. os sin Superpose the oponent stress distributions Along the neutral ais, 0 tan tan os sin 00 The Graw-Hill Copanies, n. All rights reserved. 4-8
ECHANCS OF ATERALS Eaple 4.08 SOLUTON: Resolve the ouple vetor into oponents along the priniple entroidal aes and alulate the orresponding aiu stresses. os sin A 600 lb-in ouple is applied to a retangular wooden bea in a plane foring an angle of 0 deg. with the vertial. Deterine (a) the aiu stress in the bea, (b) the angle that the neutral ais fors with the horiontal plane. 00 The Graw-Hill Copanies, n. All rights reserved. Cobine the stresses fro the oponent stress distributions. Deterine the angle of the neutral ais. tan tan 4-9
ECHANCS OF ATERALS Eaple 4.08 Resolve the ouple vetor into oponents and alulate the orresponding aiu stresses. 600lbin 600lbin.5in.5in.5in.5in os0 86lbin sin0 800lbin The largest tensile stressdue to 86lbin.75in 5.59in The largest tensile stressdue to 800lbin0.75in 0.9844in 5.59in 0.9844in 4 4 4 4 oursalong 45.6psi oursalong 609.5psi The largest tensile stress due to the obined loading ours at A. a 45.6 609.5 AB AD a 06 psi 00 The Graw-Hill Copanies, n. All rights reserved. 4-40
ECHANCS OF ATERALS Eaple 4.08 Deterine the angle of the neutral ais. tan.4 o 7.4 tan 5.59in 0.9844in 4 4 tan 0 00 The Graw-Hill Copanies, n. All rights reserved. 4-4
ECHANCS OF ATERALS General Case of Eentri Aial Loading Consider a straight eber subjet to equal and opposite eentri fores. The eentri fore is equivalent to the sste of a entri fore and two ouples. P entri fore Pa Pb B the priniple of superposition, the obined stress distribution is P A f the neutral ais lies on the setion, it a be found fro P A 00 The Graw-Hill Copanies, n. All rights reserved. 4-4