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E LEMENTS OF A RCHITECTURAL S TRUCTURES : Mechanics of Materials F ORM, B EHAVIOR, AND D ESIGN ARCH 614 • MECHANICS • MATERIALS D R. A NNE N ICHOLS S PRING 2019 lecture four mechanics www.carttalk.com of materials Mechanics of Materials 1 Elements of Architectural Structures S2009abn Lecture 4 ARCH 614 Mechanics of Materials 7 Elements of Architectural Structures S2004abn Lecture 14 ARCH 614 Mechanics of Materials Knowledge Required • material properties • external loads and their effect on deformable bodies • member cross sections • use it to answer question if structure • ability of a material to resist breaking meets requirements of • structural elements that resist excessive – stability and equilibrium – deflection – strength and stiffness – deformation • other principle building requirements • economy, functionality and aesthetics Mechanics of Materials 9 Elements of Architectural Structures S2004abn Mechanics of Materials 8 Elements of Architectural Structures S2004abn Lecture 14 ARCH 614 Lecture 14 ARCH 614 1

Stress Problem Solving • stress is a term for the intensity of a 1. STATICS: force, like a pressure equilibrium of external forces, • internal or applied internal forces, stresses 2. GEOMETRY: • force per unit area cross section properties, deformations and conditions of geometric fit, strains P   stress f 3. MATERIAL PROPERTIES: A stress-strain relationship for each material obtained from testing Mechanics of Materials 11 Elements of Architectural Structures S2004abn Mechanics of Materials 10 Elements of Architectural Structures S2004abn Lecture 14 ARCH 614 Lecture 14 ARCH 614 Design Design (cont) • we ’ d like • materials have a critical stress value  f F where they could break or yield actual allowable – ultimate stress • stress distribution may – yield stress vary: average acceptance – compressive stress vs. failure • uniform distribution – fatigue strength exists IF the member is – (creep & temperature) loaded axially (concentric) Mechanics of Materials 13 Elements of Architectural Structures S2004abn Mechanics of Materials 12 Elements of Architectural Structures S2004abn Lecture 14 ARCH 614 Lecture 14 ARCH 614 2

Scale Effect Normal Stress • model scale • normal stress is normal to the cross section – material weights by volume, small section areas – stressed area is • structural scale perpendicular to the load – much more material weight, bigger section areas P  • scale for strength is not f t or c   A proportional:   3 L   L 2 L Mechanics of Materials 9 Elements of Architectural Structures S2009abn Mechanics of Materials 16 Elements of Architectural Structures S2004abn Lecture 4 ARCH 614 Lecture 14 ARCH 614 Shear Stress Bearing Stress • • stress parallel to a surface stress on a surface by contact in compression P P P P     f v f p     A td A td   ave Mechanics of Materials 17 Elements of Architectural Structures S2004abn Mechanics of Materials 18 Elements of Architectural Structures S2004abn Lecture 14 ARCH 614 Lecture 14 ARCH 614 3

Bending Stress Torsional Stress • • normal stress caused by bending shear stress caused by twisting  Mc M T    f b f v     I S J   Mechanics of Materials 19 Elements of Architectural Structures S2004abn Mechanics of Materials 20 Elements of Architectural Structures S2004abn Lecture 14 ARCH 614 Lecture 14 ARCH 614 Structures and Shear Bolts • • connected members in tension cause what structural elements see shear? shear stress – beams – bolts connections – splices – slabs – footings • connected members in V – walls  compression cause • wind bearing stress • seismic loads Mechanics of Materials 16 Elements of Architectural Structures S2011abn Mechanics of Materials 21 Elements of Architectural Structures S2004abn Lecture 4 ARCH 614 Lecture 14 ARCH 614 4

Double Shear Single Shear • • seen when 3 members are connected seen when 2 members are connected • two areas F= P P P 2 2    f v P P  2 2 A A d   f 4  v 2 A d 4 Mechanics of Materials 18 Elements of Architectural Structures S2011abn Mechanics of Materials 17 Elements of Architectural Structures S2011abn Lecture 4 ARCH 614 Lecture 4 ARCH 614 Bolt Bearing Stress Strain • materials deform • compression & contact • • axially loaded materials change projected area F= length • bending materials deflect  • STRAIN: P P L      f strain – change in length p A td L (S) projected over length Mechanics of Materials 15 Elements of Architectural Structures S2004abn Mechanics of Materials 19 Elements of Architectural Structures S2011abn Lecture 14 ARCH 614 Lecture 4 ARCH 614 5

Shearing Strain Shearing Strain • deformations • deformations  s  with shear with torsion L • parallelogram • twist • change in angles • change in angle of line    • stress: • stress:            • strain: • strain: s L tan L – unitless (radians) – unitless (radians) Stress & Strain 10 Elements of Architectural Structures S2004abn Stress & Strain 11 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 Load and Deformation Material Behavior • for stress, need P & A • every material has its own response • for strain, need  & L – 10,000 psi – L = 10 in – how? – Douglas fir vs. – TEST with load and steel? measure – plot P/A vs.  Stress & Strain 13 Elements of Architectural Structures S2005abn Stress & Strain 12 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 6

Behavior Types Behavior Types • ductile - “necking” • brittle • true stress P f  A • engineering stress • semi-brittle – (simplified) P f  A o Stress & Strain 14 Elements of Architectural Structures S2004abn Stress & Strain 15 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 Hooke’s Law Stress to Strain • important to us in f -  diagrams: • straight line has constant slope – straight section • Hooke’s Law f – LINEAR-ELASTIC E  E   f – recovers shape 1 (no permanent  • E deformation) – Modulus of elasticity – Young’s modulus – units just like stress Stress & Strain 16 Elements of Architectural Structures S2004abn Stress & Strain 17 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 7

Isotropy & Anisotropy Stiffness • ISOTROPIC • ability to resist strain – materials with E same at f u any direction of loading • steels – ex. steel – same E – different • ANISOTROPIC yield points – materials with different E – different at any direction of loading ultimate strength – ex. wood is orthotropic Stress & Strain 19 Elements of Architectural Structures S2004abn Stress & Strain 18 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 Elastic, Plastic, Fatigue Plastic Behavior • ductile • elastic springs back • plastic has permanent deformation at yield stress • fatigue caused by reversed loading cycles Stress & Strain 21 Elements of Architectural Structures S2004abn Stress & Strain 20 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 8

Poisson’s Ratio Lateral Strain • or “ what happens to the cross section • constant relationship between with axial stress ” longitudinal strain and lateral strain f x    x  lateral strain        y z E   axial strain    x x f f 0 f x      y z • strain in lateral direction y z E – negative    y z – equal for isometric materials    • sign! 0 0 . 5 Stress & Strain 15 Elements of Architectural Structures S2006abn Stress & Strain 23 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Lecture 15 ARCH 614 Calculating Strain Orthotropic Materials • from Hooke’s law • non-isometric  E   f • directional values of E and  • substitute  P • ex:   E – plywood A L – laminates PL   • get  – polymer (  L) AE composites Stress & Strain 24 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 Stress & Strain 25 Elements of Architectural Structures S2004abn Lecture 15 ARCH 614 9