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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

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SLIDE 1

S2018abn

three

forces and moments

Forces & Moments 1 Lecture 3 Architectural Structures ARCH 331

lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN ARCH 331

  • DR. ANNE NICHOLS

SPRING 2018

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SLIDE 2

S2018abn Systems & Planning 2 Lecture 3 Architectural Structures ARCH 331

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?

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SLIDE 3

S2018abn Forces & Moments 3 Lecture 3 Architectural Structures ARCH 331

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

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SLIDE 4

S2018abn Forces & Moments 4 Lecture 3 Architectural Structures ARCH 331

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

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SLIDE 5

S2018abn Forces & Moments 5 Lecture 3 Architectural Structures ARCH 331

Physics for Structures

  • scalars – any quantity
  • vectors - quantities with direction

– like displacements – summation results in the “straight line path” from start to end – normal vector is perpendicular to something

y x z

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SLIDE 6

S2018abn Forces & Moments 6 Lecture 3 Architectural Structures ARCH 331

Language

  • symbols for operations: +,-, /, x
  • symbols for relationships: (), =, <, >
  • algorithms

– cancellation – factors – signs – ratios and proportions – power of a number – conversions, ex. 1X = 10 Y – operations on both sides of equality

3 1 3 2 2 6 2 6 5 5 2         

3 1 6  x

1000 10

3 

1 10 1 1 10  Y X

  • r

X Y

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SLIDE 7

S2018abn Forces & Moments 7 Lecture 3 Architectural Structures ARCH 331

On-line Practice

  • eCampus / Study Aids
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SLIDE 8

S2018abn Forces & Moments 8 Lecture 3 Architectural Structures ARCH 331

Geometry

  • angles

– right = 90º – acute < 90º – obtuse > 90º –  = 180º

  • triangles

– area – hypotenuse – total of angles = 180º

2 h b 

2 2 2

BC A C A B  

A B C

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SLIDE 9

S2018abn Forces & Moments 9 Lecture 3 Architectural Structures ARCH 331

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

   

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SLIDE 10

S2018abn Forces & Moments 10 Lecture 3 Architectural Structures ARCH 331

Geometry

– intersection of a line with parallel lines results in identical angles – two lines intersect in the same way, the angles are identical

          

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SLIDE 11

S2018abn Forces & Moments 11 Lecture 3 Architectural Structures ARCH 331

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  

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SLIDE 12

S2018abn Forces & Moments 12 Lecture 3 Architectural Structures ARCH 331

– sides of two angles bisect a right angle (90°), the angles are complimentary – right angle bisects a straight line, remaining angles are complimentary

Geometry

 

   90  

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SLIDE 13

S2018abn Architectural Structures ARCH 331

– similar triangles have proportional sides

Geometry

B C E A A B C A C B

DE BC AE AC AD AB   C B BC C A AC B A AB        

D      

Forces & Moments 13 Lecture 3

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SLIDE 14

S2018abn Architectural Structures ARCH 331

Trigonometry

  • for right triangles

C

B A

CB AB hypotenuse side

  • pposite

    sin sin CB AC hypotenuse side adjacent     cos cos AC AB side adjacent side

  • pposite

    tan tan

SOHCAHTOA

Forces & Moments 14 Lecture 3

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SLIDE 15

S2018abn Architectural Structures ARCH 331

Trigonometry

  • cartesian coordinate system

– origin at 0,0 – coordinates in (x,y) pairs – x & y have signs

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6

  • 6 -5 -4 -3 -2 -1 0

1 2 3 4 5 6

X Y

Quadrant I Quadrant II Quadrant III Quadrant IV

Forces & Moments 15 Lecture 3

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SLIDE 16

S2018abn Architectural Structures ARCH 331

Trigonometry

  • for angles starting at positive x

– sin is y side – cos is x side

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6 X Y

sin<0 for 180-360° cos<0 for 90-270° tan<0 for 90-180° tan<0 for 270-360°

Forces & Moments 16 Lecture 3

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SLIDE 17

S2018abn Architectural Structures ARCH 331

Trigonometry

  • for all triangles

– sides A, B & C are opposite angles ,  &  – LAW of SINES – LAW of COSINES

C B A    sin sin sin  

 co s 2

2 2 2

BC C B A   

A

C B

 

Forces & Moments 17 Lecture 3

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SLIDE 18

S2018abn Architectural Structures ARCH 331

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 Lecture 3

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SLIDE 19

S2018abn Architectural Structures ARCH 331

Algebra

  • solving one equation

– only works with one variable – ex:

  • add to both sides
  • divide both sides
  • get x by itself on a side

1 2   x

1 1 1 2     x

2 1

 x

2 1 2 2    x

1 2  x

Forces & Moments 19 Lecture 3

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SLIDE 20

S2018abn Architectural Structures ARCH 331

Algebra

  • solving one equations

– only works with one variable – ex:

  • subtract from both sides
  • subtract from both sides
  • divide both sides
  • get x by itself on a side

5 4 1 2    x x x x x x 2 5 4 2 1 2      5 5 2 5 1      x

2 2 2 2 3 2 6          x

3   x

Forces & Moments 20 Lecture 3

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SLIDE 21

S2018abn Architectural Structures ARCH 331

Algebra

  • solving two equation

– only works with two variables – ex:

  • look for term similarity
  • can we add or subtract to eliminate one term?
  • add
  • get x by itself on a side

8 3 2   y x 6 3 12   y x

6 8 3 1 2 3 2      y x y x

14 14  x

1 14 14 14 14    x x

Forces & Moments 21 Lecture 3

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SLIDE 22

S2018abn

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

Architectural Structures ARCH 331 Forces & Moments 22 Lecture 3

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SLIDE 23

S2018abn

Force

  • “action of one body on another that

affects the state of motion or rest of the body”

  • Newton’s 3rd law:

– for every force of action there is an equal and

  • pposite reaction along

the same line

Architectural Structures ARCH 331 Forces & Moments 23 Lecture 3 http://www.physics.umd.edu

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SLIDE 24

S2018abn

Force Characteristics

  • applied at a point
  • magnitude

– Imperial units: lb, k (kips) – SI units: N (newtons), kN

  • direction

Architectural Structures ARCH 331

(tail) (tip)

Forces & Moments 24 Lecture 3

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SLIDE 25

S2018abn

Forces on Rigid Bodies

  • for statics, the bodies are ideally rigid
  • can translate

and rotate

  • internal forces are

– in bodies – between bodies (connections)

  • external forces act on bodies

Architectural Structures ARCH 331

translate rotate

Forces & Moments 25 Lecture 3

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SLIDE 26

S2018abn

Transmissibility

  • the force stays on the same line of

action

  • truck can’t tell the difference
  • only valid for EXTERNAL forces

Architectural Structures ARCH 331

=

Forces & Moments 26 Lecture 3

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S2018abn

Force System Types

  • collinear

Architectural Structures ARCH 331 Forces & Moments 27 Lecture 3

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S2018abn

Force System Types

  • coplanar

Architectural Structures ARCH 331 Forces & Moments 28 Lecture 3

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SLIDE 29

S2018abn

Force System Types

  • space

Architectural Structures ARCH 331 Forces & Moments 29 Lecture 3

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S2018abn

Adding Vectors

  • graphically

– parallelogram law

  • diagonal
  • long for 3 or more vectors

– tip-to-tail

  • more convenient

with lots of vectors

Architectural Structures ARCH 331

F P R F P R

Forces & Moments 30 Lecture 3

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SLIDE 31

S2018abn

Force Components

  • convenient to resolve into 2 vectors
  • at right angles
  • in a “nice” coordinate system
  •  is between Fx and F from Fx

Architectural Structures ARCH 331

Fy Fx  F x y Fy Fx F Fy Fx F

 cos F F x   sin F F y 

2 2 y x

F F F  

x y

F F   tan

Forces & Moments 31 Lecture 3

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SLIDE 32

S2018abn

Trigonometry

  • Fx is negative

– 90 to 270

  • Fy is negative

– 180 to 360

  • tan is positive

– quads I & III

  • tan is negative

– quads II & IV

Architectural Structures ARCH 331

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5 6

  • 6 -5 -4 -3 -2 -1 0

1 2 3 4 5 6

X Y

Quadrant I Quadrant II Quadrant III Quadrant IV

   

Forces & Moments 32 Lecture 3

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SLIDE 33

S2018abn

Component Addition

  • find all x components
  • find all y components
  • find sum of x components, Rx (resultant)
  • find sum of y components, Ry

Architectural Structures ARCH 331

2 2 y x

R R R  

x y

R R   tan

Fy Fx R Py Px  Rx Ry

Forces & Moments 33 Lecture 3

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SLIDE 34

S2018abn

Alternative Trig for Components

  • doesn’t relate angle to axis direction
  •  is “small” angle between F and

EITHER Fx or Fy

  • no sign out of calculator!
  • have to choose RIGHT

trig function, resulting direction (sign) and component axis

Architectural Structures ARCH 331

+/-? Fy +/-? Fx  F x y

Forces & Moments 34 Lecture 3

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SLIDE 35

S2018abn

Friction

  • resistance to movement
  • contact surfaces determine 
  • proportion of normal force ()

– opposite to slide direction – static > kinetic

Architectural Structures ARCH 331

N μ F 

Forces & Moments 35 Lecture 3

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SLIDE 36

S2018abn

Cables

  • simple
  • uses

– suspension bridges – roof structures – transmission lines – guy wires, etc.

  • have same tension all along
  • can’t stand compression

Architectural Structures ARCH 331 Forces & Moments 36 Lecture 3 http:// nisee.berkeley.edu/godden

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SLIDE 37

S2018abn

Cables Structures

  • use high-strength steel
  • need

– towers – anchors

  • don’t want movement

Architectural Structures ARCH 331 Forces & Moments 37 Lecture 3 http:// nisee.berkeley.edu/godden

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SLIDE 38

S2018abn

Cable Structures

Architectural Structures ARCH 331 Forces & Moments 38 Lecture 3

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SLIDE 39

S2018abn

Cable Loads

  • straight line

between forces

  • with one force

– concurrent – symmetric

Architectural Structures ARCH 331 Forces & Moments 39 Lecture 3

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SLIDE 40

S2018abn

Cable Loads

  • shape directly

related to the distributed load

Architectural Structures ARCH 331 Forces & Moments 40 Lecture 3

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SLIDE 41

S2018abn

Cable-Stayed Structures

  • diagonal cables support horizontal

spans

  • typically symmetrical
  • Patcenter,

Rogers 1986

Architectural Structures ARCH 331 Forces & Moments 41 Lecture 3 www.columbia.edu

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SLIDE 42

S2018abn

Patcenter, Rogers 1986

  • column free space
  • roof suspended
  • solid steel ties
  • steel frame supports masts

Architectural Structures ARCH 331 Forces & Moments 42 Lecture 3

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S2018abn

Patcenter, Rogers 1986

  • dashes – cables pulling

Architectural Structures ARCH 331 Forces & Moments 43 Lecture 3

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SLIDE 44

S2018abn Architectural Structures ARCH 331

Moments

  • forces have the tendency to make a

body rotate about an axis

– same translation but different rotation

Forces & Moments 44 Lecture 3 http://www.physics.umd.edu

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SLIDE 45

S2018abn Architectural Structures ARCH 331

Moments

Forces & Moments 45 Lecture 3

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SLIDE 46

S2018abn Architectural Structures ARCH 331

Moments

  • a force acting at a different point causes

a different moment:

Forces & Moments 46 Lecture 3

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SLIDE 47

S2018abn Architectural Structures ARCH 331

Moments

  • defined by magnitude and direction
  • units: Nm, kft
  • direction:

+ ccw (right hand rule)

  • cw
  • value found from F

and  distance

  • d also called “lever” or “moment” arm

F A C B

d F M  

d

Forces & Moments 47 Lecture 3

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SLIDE 48

S2018abn Architectural Structures ARCH 331

Moments

  • with same F:

2 1

d F M d F M

A A

    

(bigger)

Forces & Moments 48 Lecture 3

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SLIDE 49

S2018abn Architectural Structures ARCH 331

Moments

  • additive with sign convention
  • can still move the force

along the line of action

=

+

MA = Fd MB = Fd d F A B d MA = Fd MB = Fd d F A B d

Forces & Moments 49 Lecture 3

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SLIDE 50

S2018abn Architectural Structures ARCH 331

Moments

  • Varignon’s Theorem

– resolve a force into components at a point and finding perpendicular distances – calculate sum of moments – equivalent to original moment

  • makes life easier!

– geometry – when component runs through point, d=0

Forces & Moments 50 Lecture 3

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SLIDE 51

S2018abn Architectural Structures ARCH 331

Moments of a Force

  • moments of a force

– introduced in Physics as “Torque Acting on a Particle” – and used to satisfy rotational equilibrium

Forces & Moments 51 Lecture 3

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SLIDE 52

S2018abn Architectural Structures ARCH 331

Physics and Moments of a Force

  • my Physics book:

Forces & Moments 52 Lecture 3

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SLIDE 53

S2018abn Architectural Structures ARCH 331

  • 2 forces

– same size – opposite direction – distance d apart – cw or ccw – not dependant on point of application

Moment Couples

d F A d1 d2 F

Forces & Moments 53 Lecture 3

2 1

d F d F M    

d F M  

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SLIDE 54

S2018abn Architectural Structures ARCH 331

Moment Couples

  • equivalent couples

– same magnitude and direction – F & d may be different

100 mm 300 N 300 N 150 mm 200 N 200 N 250 mm 120 N 120 N

Forces & Moments 54 Lecture 3

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SLIDE 55

S2018abn Architectural Structures ARCH 331

Moment Couples

  • added just like moments caused by one

force

  • can replace two couples with a single

couple

+ = 100 mm 300 N 300 N 150 mm 200 N 200 N 250 mm 240 N 240 N

Forces & Moments 55 Lecture 3

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SLIDE 56

S2018abn Architectural Structures ARCH 331

Moment Couples

  • moment couples in structures

Forces & Moments 56 Lecture 3

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SLIDE 57

S2018abn Architectural Structures ARCH 331

Equivalent Force Systems

  • two forces at a point is equivalent to the

resultant at a point

  • resultant is equivalent to two

components at a point

  • resultant of equal & opposite forces at a

point is zero

  • put equal & opposite forces at a point

(sum to 0)

  • transmission of a force along action line

Forces & Moments 57 Lecture 3

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SLIDE 58

S2018abn Architectural Structures ARCH 331

  • single force causing a moment can be

replaced by the same force at a different point by providing the moment that force caused

  • moments are shown as arched arrows

Force-Moment Systems

  • F

F F d A A F A A

Forces & Moments 58 Lecture 3

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SLIDE 59

S2018abn Architectural Structures ARCH 331

Force-Moment Systems

  • a force-moment pair can be replaced by

a force at another point causing the

  • riginal moment

F d

  • F

F A A F A A F M=Fd A A

Forces & Moments 59 Lecture 3

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SLIDE 60

S2018abn Architectural Structures ARCH 331

Parallel Force Systems

  • forces are in the same direction
  • can find resultant force
  • need to find location for equivalent

moments

R=A+B C D x A B a b

a A b B

x B A   ) (

C D

Forces & Moments 60 Lecture 3