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Structural Analysis (Statics & Mechanics) APPLIED ARCHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS Structural Requirements ARCH 631 • serviceability DR. ANNE NICHOLS FALL 2013 – strength – deflections lecture two • efficiency – economy of materials • construction structural analysis • cost (statics & mechanics) • other www.pbs.org/wgbh/buildingbig/ Analysis 1 Applied Architectural Structures F2009abn Analysis 2 Architectural Structures III F2009abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Structure Requirements Structure Requirements • strength & • stability & equilibrium stiffness – safety – stability of – stresses components not greater – minimum than deflection and strength vibration – adequate – adequate foundation foundation Analysis 3 Architectural Structures III F2008abn Analysis 4 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 1 Structure Requirements Relation to Architecture • economy and “The geometry and arrangement of the construction load-bearing members, the use of – minimum material materials, and the crafting of joints all represent opportunities for buildings to – standard sized express themselves. The best members buildings are not designed by – simple connections architects who after resolving the and details formal and spatial issues, simply ask – maintenance the structural engineer to make sure it – fabrication/ erection doesn’t fall down.” - Onouy & Kane Analysis 5 Architectural Structures III F2008abn Analysis 6 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Structural Loads - STATIC Structural Loads – STATIC & DYNAMIC • dead load • wind loads – static, fixed, includes – dynamic, wind pressures material weights, fixed treated as lateral static equipment loads on walls, pressure or suction • live load – pressure determined – transient and moving from wind velocity, q loads (including h – dynamic effects include occupants) motion from buffeting or FW Cd qh A • snow load “vortex shedding” Analysis 8 Architectural Structures III F2008abn Analysis 7 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 2 Structural Loads - DYNAMIC Dynamic Response • earthquake loads – seismic, movement of ground (3D) – building mass responds – static models often used, V is static shear • impact loads ZICW – rapid, energy loads V RW Analysis 9 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 10 Architectural Structures III F2008abn Lecture 2 ARCH 631 Dynamic Response Statics & Mechanics Review • period of vibration or • how loads affect our structures frequency – statics: things don’t move – wave • forces – sway/time period • supports & connections • equilibrium • damping – mechanics: things can change shape – reduction in sway • stress & strain • resonance • deflections – amplification of sway • buckling Analysis 11 Architectural Structures III F2008abn Analysis 12 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 3 Structural Math Physical Math • quantify environmental loads • physics takes observable phenomena – how big is it? and relates the measurement with rules: • evaluate geometry and angles mathematical relationships – where is it? • need – what is the scale? – reference frame – what is the size in a particular direction? – measure of length, mass, time, direction, • quantify what happens in the structure velocity, acceleration, work, heat, – how big are the internal forces? electricity, light – how big should the beam be? – calculations & geometry Analysis 13 Architectural Structures III F2008abn Analysis 14 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Units Vectors • measures • scalars – any quantity – US customary & SI • vectors - quantities with direction – like displacements Units US SI – summation results in Length in, ft, mi mm, cm, m the “straight line path” Volume gallon liter from start to end Mass lb mass g, kg – normal vector is perpendicular to y Force lb force N, kN something Temperature F C z x Analysis 15 Architectural Structures III F2008abn Analysis 16 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 4 Forces & Reactions Force Components • Newton’s 3rd law: • convenient to resolve into 2 vectors – for every force of action • at right angles there is an equal and • in a “nice” coordinate system F opposite reaction along y F the same line • is between F and F from F y x x x F • external forces act on bodies F Fx Fx F cos Fy – can cause moments F F F sin y • internal forces are y Fx 2 2 Fx – in bodies F Fx Fy F – between bodies (connections) tan y Analysis 18 F Architectural Structures III F2008abn Analysis 17 Architectural Structures III F2008abn Lecture 2 x ARCH 631 Lecture 2 ARCH 631 Load Types Load Tracing • weight (F = ma) • tributary load W tA – think of water flow • concentrated – “concentrates” load of area into center • distributed – uniform load w tributary width – linear area width w x W 2w • friction w x W w 2 2 w w – F=N 0 x x x W W Analysis 19 Architectural Structures III F2008abnW/2 W/2Analysis 20 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 x/2 x/2 2x/3 x/3 x/2 x/6 x/3 5 Moments Equilibrium • defined by magnitude and direction • analytically • units: Nm, kft Rx Fx 0 • direction: Ry Fy 0 + ccw (right hand rule) F M M 0 - cw C • value found from F A B • free body diagrams and distance d M F d • d also called “lever” or “moment” arm Analysis 21 Architectural Structures III F2008abn Analysis 22 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Free Body Diagram Supports and Connections • FBD (sketch) • tool to see all forces on a body or a point including – external forces – weights – force reactions – external moments – moment reactions – internal forces Analysis 23 Architectural Structures III F2008abn Analysis 24 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 6 Supports and Connections Centroid • “average” x & y of an area • for a volume of constant thickness – W tA where is weight/volume – center of gravity = centroid of area xA x A yA y A Analysis 26 Architectural Structures III F2008abn Analysis 25 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Moments of Inertia Internal and Pin Forces • 2nd moment area – 3 equations per three-force body – math concept – two-force body forces in line – area x (distance)2 – 2 reactions per pin + support forces • need for behavior of – beams B – columns G C E 2 F I x Icx Ad y A D Ax Ay D Analysis 27 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 28 Architectural Structures III F2008abn Lecture 2 ARCH 631 7 Internal Beam V & M (+P) Deflected Shape • maximums needed for design • Mmax at V = 0 V + L • positive bending moment - – tension in bottom, compression in top • negative bending moment +V M – tension in top, compression in bottom + L +M L/2 • zero bending moment – inflection point Analysis 29 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 30 Architectural Structures III F2008abn Lecture 2 ARCH 631 Stress Stress Types • stress is a term for the intensity of a • normal stress is normal force, like a pressure to the cross section • internal or applied P f • force per unit area t or c A • shear stress parallel P to a surface stress A P P f v A td Analysis 31 Architectural Structures III F2008abn Analysis 32 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 8 Stress Types Bolt Stresses • bearing stress on a • single shear surface by contact in P P f compression P P v d 2 f p A 4 A td • double shear • torsional stress by P P P shear from twisting f 2 2 v d 2 T 2A A 4 fv P P J • bearing f p Aprojected td Analysis 33 Architectural Structures III F2008abn Analysis 34 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Bending Stresses Connectors Resisting Shear • tension and compressive stress caused • plates with p y p by bending – nails p 8” 2” Mc M – rivets 4.43” x 4” f ya b – bolts 2” I S 12” • shear stress from bending • splices Vlongitudinal VQ VQ fvave p I Ib VQ nF connected area p connector I Analysis 35 Architectural Structures III F2008abn Analysis 36 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 9 Strain Problem Solving • materials deform 1. STATICS: • axially loaded materials change equilibrium of external forces, length internal forces, stresses • bending materials deflect 2. GEOMETRY: cross section properties, deformations and conditions of geometric fit, strains 3. MATERIAL PROPERTIES: • STRAIN: L stress-strain relationship for each material – change in length over length strain L obtained from testing Analysis 37 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 38 Architectural Structures III F2008abn Lecture 2 ARCH 631 Stress to Strain Behavior Types • important to us in f - diagrams: • brittle – straight section – LINEAR-ELASTIC f E – recovers shape (no permanent • semi-brittle deformation) PL AE Analysis 39 Architectural Structures III F2008abn Lecture 2 ARCH 631 Analysis 40 Architectural Structures III F2008abn Lecture 2 ARCH 631 10 Plastic Behavior Maximum Stresses • ductile • if we need to know where max f and f v F happen: at yield stress P 0 cos 1 fmax Ao 45 cos sin 0.5 P fmax fvmax 2Ao 2 Analysis 42 Architectural Structures III F2008abn Analysis 41 Architectural Structures III F2008abn Lecture 2 ARCH 631 Lecture 2 ARCH 631 Thermal Deformation Beam Deflections • -
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