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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 three forces and moments Forces & Moments 1 Architectural Structures S2018abn Lecture 3 ARCH 331 Structural Math quantify

  1. A RCHITECTURAL S TRUCTURES : F ORM, B EHAVIOR, AND D ESIGN A RCH 331 D R. A NNE N ICHOLS S PRING 2018 lecture three forces and moments Forces & Moments 1 Architectural Structures S2018abn Lecture 3 ARCH 331

  2. Structural Math • quantify environmental loads – how big is it? • evaluate geometry and angles – where is it? – what is the scale? – what is the size in a particular direction? • quantify what happens in the structure – how big are the internal forces? – how big should the beam be? Systems & Planning 2 Architectural Structures S2018abn Lecture 3 ARCH 331

  3. Structural Math • physics takes observable phenomena and relates the measurement with rules: mathematical relationships • need – reference frame – measure of length, mass, time, direction, velocity, acceleration, work, heat, electricity, light – calculations & geometry Forces & Moments 3 Architectural Structures S2018abn Lecture 3 ARCH 331

  4. Physics for Structures • measures – US customary & SI Units US SI Length in, ft, mi mm, cm, m Volume gallon liter Mass lb mass g, kg Force lb force N, kN Temperature F C Forces & Moments 4 Architectural Structures S2018abn Lecture 3 ARCH 331

  5. Physics for Structures • scalars – any quantity • vectors - quantities with direction – like displacements – summation results in the “straight line path” from start to end y – normal vector is perpendicular to something z x Forces & Moments 5 Architectural Structures S2018abn Lecture 3 ARCH 331

  6. Language • symbols for operations: +,-, /, x • symbols for relationships: (), =, <, > • algorithms   2 5 2 2 1     – cancellation    5 6 6 2 3 3 – factors x 1 6  – signs 3 – ratios and proportions 3  – power of a number 10 1000 – conversions, ex. 1X = 10 Y 10 Y 1 X  or 1 – operations on both sides of equality 1 10 X Y Forces & Moments 6 Architectural Structures S2018abn Lecture 3 ARCH 331

  7. On-line Practice • eCampus / Study Aids Forces & Moments 7 Architectural Structures S2018abn Lecture 3 ARCH 331

  8. Geometry • angles – right = 90º – acute < 90º – obtuse > 90º –  = 180º • triangles B b  h  – area 2 A C – hypotenuse   2 2 2 A B A C BC – total of angles = 180º Forces & Moments 8 Architectural Structures S2018abn Lecture 3 ARCH 331

  9. Geometry • lines and relation to angles – parallel lines can’t intersect – perpendicular lines cross at 90º – intersection of two lines is a point – opposite angles are equal when   two lines cross   Forces & Moments 9 Architectural Structures S2018abn Lecture 3 ARCH 331

  10. Geometry – intersection of a line with    parallel lines results in identical    angles   – two lines intersect in the same way, the angles are identical    Forces & Moments 10 Architectural Structures S2018abn Lecture 3 ARCH 331

  11. Geometry – sides of two angles are parallel and intersect opposite way, the angles are supplementary - the sum is 180 °    – two angles that sum to 90 ° are said to be complimentary       90  Forces & Moments 11 Architectural Structures S2018abn Lecture 3 ARCH 331

  12. Geometry – sides of two angles bisect a right angle (90 ° ), the angles are complimentary      90   – right angle bisects a straight line, remaining angles are complimentary   Forces & Moments 12 Architectural Structures S2018abn Lecture 3 ARCH 331

  13. Geometry – similar triangles have proportional sides A AB AC BC   AD AE DE C B E A  D  A  C  AB AC BC          A B A C B C    C B B  Forces & Moments 13 Architectural Structures S2018abn Lecture 3 ARCH 331

  14. Trigonometry • for right triangles B opposite side AB     sin sin  hypotenuse CB C A adjacent side AC     cos cos hypotenuse CB opposite side AB     tan tan adjacent side AC SOHCAHTOA Forces & Moments 14 Architectural Structures S2018abn Lecture 3 ARCH 331

  15. Trigonometry • cartesian coordinate system – origin at 0,0 Y 6 – coordinates 5 4 in (x,y) pairs Quadrant II Quadrant I 3 – x & y have 2 1 0 signs X -1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -2 -3 Quadrant III Quadrant IV -4 -5 -6 Forces & Moments 15 Architectural Structures S2018abn Lecture 3 ARCH 331

  16. Trigonometry • for angles starting at positive x – sin is y side Y 6 – cos is x side 5 4 3 2 sin<0 for 180-360 ° 1 cos<0 for 90-270 ° 0 X -6 -5 -4 -3 -2 -1 -1 0 1 2 3 4 5 6 tan<0 for 90-180 ° -2 tan<0 for 270-360 ° -3 -4 -5 -6 Forces & Moments 16 Architectural Structures S2018abn Lecture 3 ARCH 331

  17. Trigonometry • for all triangles  C – sides A, B & C are opposite B angles  ,  &    A – LAW of SINES    sin sin sin   A B C – LAW of COSINES     2 2 2 A B C 2 BC co s Forces & Moments 17 Architectural Structures S2018abn Lecture 3 ARCH 331

  18. Algebra • equations (something = something) • constants – real numbers or shown with a, b, c... • unknown terms, variables – names like R, F, x, y • linear equations – unknown terms have no exponents • simultaneous equations – variable set satisfies all equations Forces & Moments 18 Architectural Structures S2018abn Lecture 3 ARCH 331

  19. Algebra • solving one equation – only works with one variable   – ex: 2 x 1 0     2 x 1 1 0 1 • add to both sides  2 x 1  x • divide both sides 2 1   2 2 • get x by itself on a side  x 1 2 Forces & Moments 19 Architectural Structures S2018abn Lecture 3 ARCH 331

  20. Algebra • solving one equations – only works with one variable    – ex: 2 x 1 4 x 5 • subtract from both sides      2 x 1 2 x 4 x 5 2 x • subtract from both sides      1 5 2 x 5 5      • divide both sides 6 3 2 2 x     2 2 2 • get x by itself on a side   3 x Forces & Moments 20 Architectural Structures S2018abn Lecture 3 ARCH 331

  21. Algebra • solving two equation – only works with two variables   – ex: 2 x 3 y 8   • look for term similarity 12 x 3 y 6 • can we add or subtract to eliminate one term?      2 x 3 y 1 2 x 3 y 8 6 • add  14 14 x 14 x 14 • get x by itself on a side    x 1 14 14 Forces & Moments 21 Architectural Structures S2018abn Lecture 3 ARCH 331

  22. Forces • statics – physics of forces and reactions on bodies and systems – equilibrium (bodies at rest) • forces – something that exerts on an object: • motion • tension • compression Forces & Moments 22 Architectural Structures S2018abn Lecture 3 ARCH 331

  23. Force • “action of one body on another that affects the state of motion or rest of the body” • Newton’s 3 rd law: – for every force of action there is an equal and opposite reaction along the same line http://www.physics.umd.edu Forces & Moments 23 Architectural Structures S2018abn Lecture 3 ARCH 331

  24. Force Characteristics • applied at a point • magnitude – Imperial units: lb, k (kips) – SI units: N (newtons), kN • direction (tail) (tip) Forces & Moments 24 Architectural Structures S2018abn Lecture 3 ARCH 331

  25. Forces on Rigid Bodies • for statics, the bodies are ideally rigid • can translate and rotate • internal forces are translate rotate – in bodies – between bodies (connections) • external forces act on bodies Forces & Moments 25 Architectural Structures S2018abn Lecture 3 ARCH 331

  26. Transmissibility • the force stays on the same line of action • truck can’t tell the difference = • only valid for EXTERNAL forces Forces & Moments 26 Architectural Structures S2018abn Lecture 3 ARCH 331

  27. Force System Types • collinear Forces & Moments 27 Architectural Structures S2018abn Lecture 3 ARCH 331

  28. Force System Types • coplanar Forces & Moments 28 Architectural Structures S2018abn Lecture 3 ARCH 331

  29. Force System Types • space Forces & Moments 29 Architectural Structures S2018abn Lecture 3 ARCH 331

  30. Adding Vectors • 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 Forces & Moments 30 Architectural Structures S2018abn Lecture 3 ARCH 331

  31. Force Components • convenient to resolve into 2 vectors • at right angles • in a “nice” coordinate system F y •  is between F x and F from F x F y  x F x   F F cos F x F F y F y   F sin F y   F x 2 2 F x F F F x y F   y tan F Forces & Moments 31 Architectural Structures S2018abn x Lecture 3 ARCH 331

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