ten top squishing bottom stretching beams: what are the stress - PowerPoint PPT Presentation

A RCHITECTURAL S TRUCTURES : Beam Bending F ORM, B EHAVIOR, AND D ESIGN ARCH 331 • Galileo D R. A NNE N ICHOLS – relationship between S PRING 2019 stress and depth 2 lecture • can see ten – top squishing – bottom stretching beams: • what are the stress across the section? bending and shear stress Beam Stresses 1 Architectural Structures F2009abn Beam Stresses 2 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 Bending Moments Pure Bending • sign convention: • bending only V • no shear + - M • axial normal stresses from bending can be found in • size of maximum internal moment will – homogeneous materials govern our design of the section – plane of symmetry y – follow Hooke ’ s law x Beam Stresses 4 Foundations Structures F2008abn Beam Stresses 3 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 1

Normal Stresses Neutral Axis • geometric fit • stresses vary linearly – plane sections remain plane • zero stress occurs at – stress varies linearly the centroid • neutral axis is line of centroids (n.a.) Beam Stresses 5 Foundations Structures F2008abn Beam Stresses 6 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 Derivation of Stress from Strain Derivation of Stress • pure bending = • zero stress at n.a. arc shape   R Ey R    f E L   R R    Ec y L outside ( R y ) c c L L y  f max ½  ½  ½  ½  R   y       f  L L R y R y f      outside max c  L L R R Beam Stresses 7 Foundations Structures F2008abn Beam Stresses 8 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 2

Bending Moment Bending Stress Relations • resultant moment My I 1 M f b  S  R  from stresses = I c EI bending moment! general bending stress section modulus curvature    M fy A M M  f b  S required F S b yf f f         2 max max max y A y A I f S maximum bending stress required section max c c c modulus for design Beam Stresses 9 Foundations Structures F2008abn Beam Stresses 10 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 Transverse Loading and Shear Bending vs. Shear in Design • bending stresses dominate • shear stresses exist • perpendicular loading horizontally with shear • internal shear • along with bending moment • no shear stresses with pure bending Beam Stresses 11 Foundations Structures F2008abn Beam Stresses 12 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 3

Shear Stresses Shear Stresses • horizontal & vertical • horizontal & vertical Beam Stresses 13 Foundations Structures F2008abn Beam Stresses 14 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 Equilibrium Beam Stresses • horizontal • horizontal with bending force V needed V Q   T V x longitudinal I • Q is a moment area Beam Stresses 16 Foundations Structures F2008abn Beam Stresses 15 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 4

Moment of Area Shearing Stresses • Q is a moment area with respect to the n.a. V V   f v of area above or below the horizontal    A b x VQ  f • Q max at y=0  v ave Ib (neutral axis) • f  = 0 on the top/bottom v ave • q is shear flow: • b min may not be with Q max V V Q   longitudinal T q • with h/4  b, f v-max  1.008 f v-ave  x I Beam Stresses 17 Foundations Structures F2008abn Beam Stresses 18 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 Rectangular Sections Steel Beam Webs • W and S sections 3 bh 2   bh I  Q A y – b varies 8 12 VQ 3 V   d f v Ib 2 A t web • f v-max occurs at n.a. – stress in flange negligible 3 V V – presume constant   f  v max stress in web A A 2 web Beam Stresses 19 Foundations Structures F2008abn Beam Stresses 20 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 5

Shear Flow Shear Flow Quantity • loads applied in plane of symmetry • sketch from Q • cut made perpendicular VQ q  VQ q  I I f v f v f v f v f v f v Beam Stresses 21 Foundations Structures F2008abn Beam Stresses 22 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 Connectors Resisting Shear Vertical Connectors • plates with • isolate an area with vertical interfaces p – nails p y p VQ 8” p – rivets 2”   connected area 4.43” nF p 4” p x connector y a – bolts 2” p I 12” • splices V VQ  longitudinal p I VQ   connected area nF p connector I Beam Stresses 23 Foundations Structures F2008abn Beam Stresses 24 Foundations Structures F2008abn Lecture 10 ARCH 331 Lecture 10 ARCH 331 6

Unsymmetrical Shear or Section • member can bend and twist – not symmetric – shear not in that plane • shear center – moments balance Beam Stresses 25 Foundations Structures F2008abn Lecture 10 ARCH 331 7