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Course: ARCH 331, Fall 2008
School: Texas A&M
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Word Count: 691

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STRUCTURES: FOUNDATIONS FORM, BEHAVIOR, AND DESIGN ARCH 331 Beam Bending Galileo relationship between stress and depth2 DR. ANNE NICHOLS SPRING 2009 ten lecture can see top squishing bottom stretching beams: bending and shear stress Beam Stresses 1 Lecture 10 Foundations Structures ARCH 331 F2008abn what are the stress across the section? Beam Stresses 2 Lecture 10 Foundations Structures ARCH 331...

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STRUCTURES: FOUNDATIONS FORM, BEHAVIOR, AND DESIGN ARCH 331 Beam Bending Galileo relationship between stress and depth2 DR. ANNE NICHOLS SPRING 2009 ten lecture can see top squishing bottom stretching beams: bending and shear stress Beam Stresses 1 Lecture 10 Foundations Structures ARCH 331 F2008abn what are the stress across the section? Beam Stresses 2 Lecture 10 Foundations Structures ARCH 331 F2008abn Pure Bending bending only no shear axial normal stresses from bending can be found in homogeneous materials plane of symmetry follow Hookes law Beam Stresses 3 Lecture 10 Foundations Structures ARCH 331 Bending Moments sign convention: + V M size of maximum internal moment will govern our design of the section y x F2008abn Beam Stresses 4 Lecture 10 Foundations Structures ARCH 331 F2008abn 1 Normal Stresses geometric fit plane sections remain plane stress varies linearly Neutral Axis stresses vary linearly zero stress occurs at the centroid neutral axis is line of centroids (n.a.) Beam Stresses 5 Lecture 10 Foundations Structures ARCH 331 F2008abn Beam Stresses 6 Lecture 10 Foundations Structures ARCH 331 F2008abn Derivation of Stress from Strain pure bending = arc shape R Derivation of Stress zero stress at n.a. L = R f = E = L y Loutside = ( R + y ) c = L = Loutside L (R + y ) R y = = L R R Foundations Structures ARCH 331 F2008abn Beam Stresses 8 Lecture 10 Ey R Ec f max = R y f = f max c R c L y Beam Stresses 7 Lecture 10 Foundations Structures ARCH 331 F2008abn 2 Bending Moment resultant moment from stresses = bending moment! Bending Stress Relations 1 M = R EI curvature My fb = I general bending stress I S= c section modulus M = fyA = f f yf max yA = max y 2 A = max I = f max S c c c Foundations Structures ARCH 331 F2008abn M fb = S maximum bending stress Beam Stresses 10 Lecture 10 Foundations Structures ARCH 331 S required M Fb F2008abn required section modulus for design Beam Stresses 9 Lecture 10 Transverse Loading and Shear Bending vs. Shear in Design bending stresses dominate perpendicular loading internal shear along with bending moment shear stresses exist horizontally with shear no shear stresses with pure bending F2008abn Beam Stresses 12 Lecture 10 Foundations Structures ARCH 331 F2008abn Beam 11 Stresses Lecture 10 Foundations Structures ARCH 331 3 Shear Stresses horizontal & vertical Shear Stresses horizontal & vertical Beam Stresses 13 Lecture 10 Foundations Structures ARCH 331 F2008abn Beam Stresses 14 Lecture 10 Foundations Structures ARCH 331 F2008abn Beam Stresses horizontal with bending Equilibrium horizontal force V needed Vlongitudinal = VT Q x I Q is a moment area Beam Stresses 15 Lecture 10 Foundations Structures ARCH 331 F2008abn Beam Stresses 16 Lecture 10 Foundations Structures ARCH 331 F2008abn 4 Moment of Area Q is a moment area with respect to the n.a. of area above or below the horizontal Qmax at y=0 (neutral axis) q is shear flow: Shearing Stresses V V fv = = A b x VQ f v ave = Ib f v ave 0 on the top/bottom = b min may not be with Q max with h/4 b, fv-max 1.008 fv-ave q= Beam Stresses 17 Lecture 10 Vlongitudinal x = VT Q I F2008abn Foundations Structures ARCH 331 Beam Stresses 18 Lecture 10 Foundations Structures ARCH 331 F2008abn Rectangular Sections bh 3 I= 12 Q = Ay = bh 2 Steel Beam Webs W and S sections 8 b varies d tweb fv = VQ 3V = Ib 2 A fv-max occurs at n.a. stress in flange negligible presume constant f v max stress in web F2008abn Beam Stresses 20 Lecture 10 Foundations Structures ARCH 331 = 3V V 2 A Aweb F2008abn Beam Stresses 19 Lecture 10 Foundations St...

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