LECTURE+19+COE-3001-A

# Force x mc 0 m 0570026263 90052522

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Unformatted text preview: re rela\on (note that we have not assumed any speciﬁc cross sec\onal proﬁle and so 2/22/13 M. Mello/Georgia Tech Aerospace 25 Moment- Curvature Rela\onship (cont.) 1 M = = •  Moment- curvature equa\on: ⇢ EI (5- 12) Bending moment resul\ng from sta\c equilibrium 1 M ( x) ( x ) = = ⇢( x ) EI “actually func\ons of x” (I = Izz assumed) Beam curvature Flexural rigidity •  Recall other “rigidity” factors that we have encountered ²་  Axial rigidity: EA ²་  Torsional rigidity: GIp •  Sign conven\on for bending moment is perfectly consistent with curvature sign conven\on ²་  Posi\ve bending moment posi\ve curvature ²་  Nega\ve bending moment nega\ve curvature 2/22/13 M. Mello/Georgia Tech Aerospace 26 And ﬁnally… stress in terms of the bending moment ²་  Now, we can subs\tute for curvature (κ) in terms of stress (σ) in the moment- curvature equa\on. ²་  i.e., recall x = = E y “Bending stress” or “Flexural stress” “Flexure formula”: 1 M = ⇢ EI Ey = M EI Bending moment depth within the beam = My (5- 13) I Moment of iner\a Tr ²་  Note how strikingly analogous this equa\on is to the torsion rela\on: ⌧max = Ip 2/22/13 M. Mello/Georgia Tech Aerospace 27 Consistency of sign conven\ons between normal stress and bending moment Rela\onships between the signs of bending moments and direc\ons of normal stresses: = (A) Posi\ve bending moment 2/22/13 My (5- 13) I (B) Nega\ve bending moment M. Mello/Georgia Tech Aerospace 28 Maximum Stresses at a Cross Sec\on ²་  Maximum tensile and compressive bending stresses ac\ng at any given cross sec\on occur along the top and bo=om edges of the beam along points furthest from the neutral axis (c1,c2 in the ﬁgure below) M c1 M M c2 M ²་  Maximum normal stress from ﬂexure formula: 1 = = = = (5- 14) 2 I c1 ²་  where: S1 = and S2 = I S1 I S2 I are called secGon moduli. Units of secGon moduli: [Length]3 c2 ²་  Convenient quanGty used in beam design lumps beams cross secGonal properGes into a single term ²་  SecGon moduli listed in tables and handbooks as a property of the beam. 2/22/13 M. Mello/Georgia Tech Aerospace 29 Doubly Symmetric Beams ²་  Doubly symmetric cross sec\onal proﬁles: Cross sec\on symmetric with respect to z axis as well as y...
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## This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Tech.

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