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A RCHITECTURAL S TRUCTURES : F ORM, B EHAVIOR, AND D ESIGN A RCH 331 D R. A NNE N ICHOLS S PRING 2018 lecture six mechanics www.carttalk.com of materials Mechanics of Materials 1 Architectural Structures S2018abn Lecture 6 ARCH 331

Mechanics of Materials • MECHANICS • MATERIALS Mechanics of Materials 2 Architectural Structures S2018abn Lecture 6 ARCH 331

Mechanics of Materials • external loads and their effect on deformable bodies • use it to answer question if structure meets requirements of – stability and equilibrium – strength and stiffness • other principle building requirements • economy, functionality and aesthetics Mechanics of Materials 3 Architectural Structures S2018abn Lecture 6 ARCH 331

Knowledge Required • material properties • member cross sections • ability of a material to resist breaking • structural elements that resist excessive – deflection – deformation Mechanics of Materials 4 Architectural Structures S2018abn Lecture 6 ARCH 331

Problem Solving 1. STATICS: equilibrium of external forces, internal forces, stresses 2. GEOMETRY: cross section properties, deformations and conditions of geometric fit, strains 3. MATERIAL PROPERTIES: stress-strain relationship for each material obtained from testing Mechanics of Materials 5 Architectural Structures S2018abn Lecture 6 ARCH 331

Stress • stress is a term for the intensity of a force, like a pressure • internal or applied • force per unit area P   stress f A Mechanics of Materials 6 Architectural Structures S2018abn Lecture 6 ARCH 331

Design • materials have a critical stress value where they could break or yield – ultimate stress – yield stress acceptance – compressive stress vs. failure – fatigue strength – (creep & temperature) Mechanics of Materials 7 Architectural Structures S2018abn Lecture 6 ARCH 331

Design (cont) • we’d like  f F actual allowable • stress distribution may vary: average • uniform distribution exists IF the member is loaded axially (concentric) Mechanics of Materials 8 Architectural Structures S2018abn Lecture 6 ARCH 331

Scale Effect • model scale – material weights by volume, small section areas • structural scale – much more material weight, bigger section areas • scale for strength is not proportional:  3 L   L 2 L Mechanics of Materials 9 Architectural Structures S2018abn Lecture 6 ARCH 331

Normal Stress (direct) • normal stress is normal to the cross section – stressed area is perpendicular to the load P  f t or c   A  Mechanics of Materials 10 Architectural Structures S2018abn Lecture 6 ARCH 331

Shear Stress • stress parallel to a surface P P   f v   A td  ave Mechanics of Materials 11 Architectural Structures S2018abn Lecture 6 ARCH 331

Bearing Stress • stress on a surface by contact in compression P P   f p   A td  Mechanics of Materials 12 Architectural Structures S2018abn Lecture 6 ARCH 331

Bending Stress • normal stress caused by bending Mc M   f b   I S  Mechanics of Materials 13 Architectural Structures S2018abn Lecture 6 ARCH 331

Torsional Stress • shear stress caused by twisting  T  f v   J  Mechanics of Materials 14 Architectural Structures S2018abn Lecture 6 ARCH 331

Structures and Shear • what structural elements see shear? – beams – bolts connections – splices – slabs – footings V – walls  • wind • seismic loads Mechanics of Materials 15 Architectural Structures S2018abn Lecture 6 ARCH 331

Bolts • connected members in tension cause shear stress • connected members in compression cause bearing stress Mechanics of Materials 16 Architectural Structures S2018abn Lecture 6 ARCH 331

Single Shear • seen when 2 members are connected P P   f  v 2 A d 4 Mechanics of Materials 17 Architectural Structures S2018abn Lecture 6 ARCH 331

Double Shear • seen when 3 members are connected • two areas F= P P P 2 2    f  v 2 2 A A d 4 Mechanics of Materials 18 Architectural Structures S2018abn Lecture 6 ARCH 331

Bolt Bearing Stress • compression & contact • projected area F= P P   f p A td projected Mechanics of Materials 19 Architectural Structures S2018abn Lecture 6 ARCH 331

Strain • materials deform • axially loaded materials change length • bending materials deflect • STRAIN:  L    strain – change in length L over length + UNITLESS Mechanics of Materials 20 Architectural Structures S2018abn Lecture 6 ARCH 331

Shearing Strain • deformations  s  with shear L • parallelogram • change in angles  • stress:         • strain: s tan L – unitless (radians) Mechanics of Materials 21 Architectural Structures S2018abn Lecture 6 ARCH 331

Shearing Strain • deformations with torsion • twist • change in angle of line   • stress:    • strain: L – unitless (radians) Mechanics of Materials 22 Architectural Structures S2018abn Lecture 6 ARCH 331

Load and Deformation • for stress, need P & A • for strain, need  & L – how? – TEST with load and measure – plot P/A vs.  Mechanics of Materials 23 Architectural Structures S2018abn Lecture 6 ARCH 331

Material Behavior • every material has its own response – 10,000 psi – L = 10 in – Douglas Fir vs. steel? Mechanics of Materials 24 Architectural Structures S2018abn Lecture 6 ARCH 331

Behavior Types • ductile - “necking” • true stress P f  A • engineering stress – (simplified) P f  A o Mechanics of Materials 25 Architectural Structures S2018abn Lecture 6 ARCH 331

Behavior Types • brittle • semi-brittle Mechanics of Materials 26 Architectural Structures S2018abn Lecture 6 ARCH 331

Stress to Strain • important to us in -  diagrams: f – straight section – LINEAR-ELASTIC – recovers shape (no permanent deformation) Mechanics of Materials 27 Architectural Structures S2018abn Lecture 6 ARCH 331

Hooke’s Law • straight line has constant slope • Hooke’s Law f  E   E f 1  • E – Modulus of elasticity – Young’s modulus – units just like stress Mechanics of Materials 28 Architectural Structures S2018abn Lecture 6 ARCH 331

Stiffness • ability to resist strain f u • steels – same E – different yield points – different ultimate strength Mechanics of Materials 29 Architectural Structures S2018abn Lecture 6 ARCH 331

Isotropy & Anisotropy • ISOTROPIC – materials with E same at any direction of loading – ex. steel • ANISOTROPIC – materials with different E at any direction of loading – ex. wood is orthotropic Mechanics of Materials 30 Architectural Structures S2018abn Lecture 6 ARCH 331

Elastic, Plastic, Fatigue • elastic springs back • plastic has permanent deformation • fatigue caused by reversed loading cycles Mechanics of Materials 31 Architectural Structures S2018abn Lecture 6 ARCH 331

Plastic Behavior • ductile at yield stress Mechanics of Materials 32 Architectural Structures S2018abn Lecture 6 ARCH 331

Lateral Strain • or “what happens to the cross section with axial stress” f x  x  E   0 f f y z • strain in lateral direction – negative    y z – equal for isometric materials Mechanics of Materials 33 Architectural Structures S2018abn Lecture 6 ARCH 331

Poisson’s Ratio • constant relationship between longitudinal strain and lateral strain   lateral strain        y z   axial strain  x x f x      y z E    • sign! 0 0 . 5 Mechanics of Materials 34 Architectural Structures S2018abn Lecture 6 ARCH 331

Calculating Strain • from Hooke’s law  E   f • substitute  P   E A L PL   • get  AE Mechanics of Materials 35 Architectural Structures S2018abn Lecture 6 ARCH 331

Orthotropic Materials • non-isometric • directional values of E and  • ex: – plywood – laminates – polymer composites Mechanics of Materials 36 Architectural Structures S2018abn Lecture 6 ARCH 331

Stress Concentrations • why we use f ave • increase in stress at changes in geometry – sharp notches – holes – corners – Mechanics of Materials 37 Architectural Structures S2018abn Lecture 6 ARCH 331

Maximum Stresses • if we need to know where max f and f v F happen: P        f 0 cos 1 max A o         45 cos sin 0 . 5 P f   max f  v max 2 2 A o Mechanics of Materials 38 Architectural Structures S2018abn Lecture 6 ARCH 331

Maximum Stresses Mechanics of Materials 39 Architectural Structures S2018abn Lecture 6 ARCH 331