Strength of Materials

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1 Strength of Materials Session Pure Bending 04 Leture note : Praudianto, M.Eng. g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà

2 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà Pure Bending: Prisati ebers subjeted to equal and opposite ouples ating in the sae longitudinal plane

g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà Eentri Loading: Aial loading whih does not pass through setion entroid produes internal fores equivalent to an aial fore and a ouple Transverse

3 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà 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.

The su of the oponents of the fores in an diretion is zero.

4 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà 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 M 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 M M z da 0 z da 0 da M

5 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà 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

6 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) g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà

7 For a linearl elasti aterial, E E (stress varies linearl) For stati equilibriu, F 0 0 da da da First oent with respet to neutral plane is zero. Therefore, the neutral surfae ust pass through the setion entroid. g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà For stati equilibriu, M M M Substituting M da M S da da

g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà The aiu noral stress due to bending, M M S setion oent of inertia S setion odulus A bea setion with a larger setion odulus will have a lower aiu

8 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà The aiu noral stress due to bending, M M 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 6 Ah 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.

9 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà Deforation due to bending oent M is quantified b the urvature of the neutral surfae E M E E M Although 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

10 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. g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà SOLUTON: Based on the ross setion geoetr, alulate the loation of the setion entroid and oent of inertia. Y A A Ad Appl the elasti fleural forula to find the aiu tensile and opressive stresses. M Calulate the urvature M E

11 SOLUTON: Based on the ross setion geoetr, alulate the loation of the setion entroid and oent of inertia. Area, A 000, 50 0 A, A 40 A A g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà Y Ad bh Ad

12 Appl the elasti fleural forula to find the aiu tensile and opressive stresses. A B M M A M kn kn B A B 76.0 MPa. MPa Calulate the urvature M E kn 65 GPa g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà

13 BENDNG OF A MEMBER MADE FROM SEVERAL MATERAL g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà

M n g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà Consider a oposite bea fored fro two aterials with E and E. Noral strain varies linearl. Pieewise linear noral stress variation.

14 M n g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà 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. Eleental fores on the setion are df E da da df da Define a transfored setion suh that ne E E df da n da n E da E

15 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. SOLUTON: g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà Transfor the bar to an equivalent ross setion ade entirel of brass Evaluate the ross setional properties of the transfored setion 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.

9 0.75 in 0.4 in.5 in Evaluate the transfored ross setional properties b h.5 in. in T 5.

16 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà SOLUTON: Transfor the bar to an equivalent ross setion ade entirel of brass. E n E b T s b 0.4 in in 0.4 in.5 in Evaluate the transfored ross setional properties b h.5 in. in T 5.06 in psi.9 psi Calulate the aiu stresses M 40 kip in.5 in 5.06 in 4.85 ksi b a b a n.9.85 ksi.9 ksi s a s a.85 ksi

The upper part of the onrete bea arries the opressive load.

17 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà 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 M s n 0

18 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. g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà SOLUTON: Transfor to a setion ade entirel of onrete. Evaluate geoetri properties of transfored setion. Calulate the aiu stresses in the onrete and steel.

06 psi Evaluate the geoetri properties of the transfored setion. 4.954 0.450in 4 in.45in 4.

19 g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà SOLUTON: Transfor to a setion ade entirel of onrete. E n E na s s in 4.95in 8 psi 8.06 psi Evaluate the geoetri properties of the transfored setion in 4 in.45in 4.95in.55in 44.4 in Calulate the aiu stresses. s M M n 40kip in.45in in 40kip in.55in in s.06ksi 8.5ksi

20 That s for now THANK YOU g{ V ä Ä tçw ÄtÇÇ Çz XÇz ÇÜ Çz Xwâvtà ÉÇ WÑtÜàÅÇà

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 00 The Graw-Hill Copanies, n. All rights reserved. Third E CHAPTER Pure ECHANCS OF ATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit