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A RCHITECTURAL S TRUCTURES : Structural Math F ORM, B EHAVIOR, AND D ESIGN ARCH 331 quantify environmental loads D R. A NNE N ICHOLS how big is it? S PRING 2019 evaluate geometry and angles lecture where is it? three what is

  1. A RCHITECTURAL S TRUCTURES : Structural Math F ORM, B EHAVIOR, AND D ESIGN ARCH 331 • quantify environmental loads D R. A NNE N ICHOLS – how big is it? S PRING 2019 • evaluate geometry and angles lecture – where is it? three – what is the scale? – what is the size in a particular direction? • quantify what happens in the structure forces and – how big are the internal forces? – how big should the beam be? moments Forces & Moments 1 Architectural Structures F2009abn Lecture 3 ARCH 331 Structural Planning 32 Foundations Structures F2008abn Lecture 3 ARCH 331 Physics for Structures Structural Math • measures • physics takes observable phenomena and relates the measurement with rules: – US customary & SI mathematical relationships • need Units US SI Length in, ft, mi mm, cm, m – reference frame Volume gallon liter – measure of length, mass, time, direction, Mass lb mass g, kg velocity, acceleration, work, heat, Force lb force N, kN electricity, light Temperature F C – calculations & geometry Structural Planning 34 Foundations Structures F2008abn Structural Planning 33 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 1

  2. Physics for Structures Language • scalars – any quantity • symbols for operations: +,-, /, x • symbols for relationships: (), =, <, > • vectors - quantities with direction • algorithms   – like displacements 2 5 2 2 1     – cancellation    – summation results in 5 6 6 2 3 3 – factors the “ straight line path ” x 1 6  – signs from start to end 3 – ratios and proportions y 10 3  – normal vector is perpendicular to – power of a number 1000 something – conversions, ex. 1X = 10 Y 10 Y 1 X  or 1 – operations on both sides of equality z 1 X 10 Y x Structural Planning 35 Foundations Structures F2008abn Structural Planning 36 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 On-line Practice Geometry • eCampus / Study Aids • angles – right = 90º – acute < 90º – obtuse > 90º –  = 180º • triangles B b  h  – area 2 A C – hypotenuse   2 2 2 AB AC BC – total of angles = 180º Structural Planning 37 Architectural Structures F2008abn Structural Planning 38 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 2

  3. Geometry Geometry • lines and relation to angles – intersection of a line with    parallel lines results in identical  – parallel lines can ’ t intersect   angles   – perpendicular lines cross at 90º – two lines intersect in the same – intersection of two lines is a point way, the angles are identical – opposite angles are equal when    two lines cross     Structural Planning 39 Foundations Structures F2008abn Structural Planning 40 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 Geometry Geometry – sides of two angles are parallel and – sides of two angles bisect a right angle (90 ), the angles are complimentary intersect opposite way, the angles are supplementary - the sum is 180°      90      – right angle bisects a straight line, – two angles that sum to 90° are said to be remaining angles complimentary are complimentary        90   Structural Planning 41 Foundations Structures F2008abn Forces & Moments 12 Foundations Structures F2009abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 3

  4. Geometry Trigonometry – similar triangles have proportional sides • for right triangles B A AB AC BC   opposite side AB     sin sin AD AE DE C hypotenuse CB  B E C A  A adjacent side AC     D  cos cos hypotenuse CB A C   AB AC BC   opposite side AB            tan tan A B A C B C  adjacent side AC   C B SOHCAHTOA B  Structural Planning 43 Foundations Structures F2008abn Structural Planning 44 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 Trigonometry Trigonometry • cartesian coordinate system • for angles starting at positive x – sin is y side – origin at 0,0 Y Y 6 – cos is x side 6 – coordinates 5 5 4 4 in (x,y) pairs Quadrant II Quadrant I 3 3 2 2 – x & y have sin<0 for 180-360° 1 1 cos<0 for 90-270° 0 X 0 signs X -6 -5 -4 -3 -2 -1 -1 0 1 2 3 4 5 6 -1 tan<0 for 90-180° -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -2 -2 tan<0 for 270-360° -3 -3 Quadrant III Quadrant IV -4 -4 -5 -5 -6 -6 Structural Planning 46 Foundations Structures F2008abn Structural Planning 45 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 4

  5. Trigonometry Algebra • equations (something = something) • for all triangles  C • constants – sides A, B & C are opposite B angles  ,  &   – real numbers or shown with a, b, c...  • unknown terms, variables A – LAW of SINES – names like R, F, x, y    sin sin sin   • linear equations A B C – unknown terms have no exponents – LAW of COSINES • simultaneous equations     2 2 2 A B C 2 BC cos – variable set satisfies all equations Structural Planning 47 Foundations Structures F2008abn Structural Planning 48 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 Algebra Algebra • solving one equation • solving one equations – only works with one variable – only works with one variable      – ex: – ex: 2 x 1 0 2 x 1 4 x 5     2 1 1 0 1 x • add to both sides • subtract from both sides 2       x 1 2 x 1 2 x 4 x 5 2 x  x • divide both sides • subtract from both sides 2 1        1 5 2 x 5 5 2 2      • get x by itself on a side  • divide both sides 6 3 2 2 x x 1   2   2 2 2 • get x by itself on a side   x 3 Structural Planning 49 Foundations Structures F2008abn Structural Planning 50 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 3 ARCH 331 5

  6. Algebra Forces • solving two equation • statics – only works with two variables – physics of forces and reactions on bodies  y  – ex: 2 x 3 8 and systems  y  – equilibrium (bodies at rest) • look for term similarity 12 x 3 6 • can we add or subtract to eliminate one term? • forces – something that exerts on an object:      2 x 3 y 12 x 3 y 8 6 • add • motion 14  x 14 • tension 14 x 14 • get x by itself on a side    x 1 • compression 14 14 Structural Planning 51 Foundations Structures F2008abn Point Equilibrium 2 Foundations Structures F2008abn Lecture 3 ARCH 331 Lecture 4 ARCH 331 Force Force Characteristics • applied at a point • “ action of one body on another that affects the state of motion or rest of the • magnitude body ” – Imperial units: lb, k (kips) • Newton ’ s 3 rd law: – SI units: N (newtons), kN – for every force of action • direction there is an equal and opposite reaction along the same line (tail) (tip) http://www.physics.umd.edu Point Equilibrium 3 Foundations Structures F2008abn Point Equilibrium 4 Foundations Structures F2008abn Lecture 4 ARCH 331 Lecture 4 ARCH 331 6

  7. Forces on Rigid Bodies Transmissibility • for statics, the bodies are ideally rigid • the force stays on the same line of action • can translate • truck can ’ t tell the difference and rotate • internal forces are translate rotate – in bodies = – between bodies (connections) • external forces act on bodies • only valid for EXTERNAL forces Point Equilibrium 5 Foundations Structures F2008abn Point Equilibrium 6 Foundations Structures F2008abn Lecture 4 ARCH 331 Lecture 4 ARCH 331 Force System Types Force System Types • collinear • coplanar Point Equilibrium 7 Foundations Structures F2008abn Point Equilibrium 8 Foundations Structures F2008abn Lecture 4 ARCH 331 Lecture 4 ARCH 331 7

  8. Force System Types Adding Vectors • space • graphically – parallelogram law R • diagonal F • long for 3 or more vectors P – tip-to-tail • more convenient F R with lots of vectors P Point Equilibrium 9 Foundations Structures F2008abn Point Equilibrium 10 Foundations Structures F2008abn Lecture 4 ARCH 331 Lecture 4 ARCH 331 Force Components Trigonometry • convenient to resolve into 2 vectors • F x is negative   • at right angles – 90 to 270 Y 6 • F y is negative • in a “ nice ” coordinate system F 5 y 4   Quadrant II Quadrant I •  is between F x and F from F x – 180 to 360 3 F y 2  • tan is positive 1 x F x   F cos F x F F 0 X -1 F y – quads I & III -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 F y   -2 F sin F y -3 • tan is negative Quadrant III Quadrant IV -4   F x 2 2 F x F F F -5 x y – quads II & IV -6 F   y tan F Point Equilibrium 11 Foundations Structures F2008abn x Point Equilibrium 12 Foundations Structures F2008abn Lecture 4 ARCH 331 Lecture 4 ARCH 331 8

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