a structures orm, ehavior, and design rch 331 r. a...
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S2015abn
three
forces and
momentsForces & Moments 1Lecture 3
Architectural StructuresARCH 331
lecture
ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 331
DR. ANNE NICHOLS
SPRING 2015
S2015abnSystems & Planning 2Lecture 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?
S2015abnForces & Moments 3Lecture 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
S2015abnForces & Moments 4Lecture 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
S2015abnForces & Moments 5Lecture 3
Architectural Structures ARCH 331
Physics for Structures
• scalars – any quantity
• vectors - quantities with direction
– like displacements
– summation results inthe “straight line path” from start to end
– normal vector is perpendicular to something
y
xz
S2015abnForces & Moments 6Lecture 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
32
2
6
2
6
5
5
2=
×/
/==
/×
/
3
1
6=
x
1000103 =
110
1
1
10=
Y
Xor
X
Y
S2015abnForces & Moments 7Lecture 3
Architectural Structures ARCH 331
On-line Practice
• eCampus / Study Aids
S2015abnForces & Moments 8Lecture 3
Architectural Structures ARCH 331
Geometry
• angles
– right = 90º
– acute < 90º
– obtuse > 90º
– π = 180º
• triangles
– area
– hypotenuse
– total of angles = 180º
2
hb×=
222BCACAB =+
A
B
C
S2015abnForces & Moments 9Lecture 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 α
βα
β
S2015abnForces & Moments 10Lecture 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
α
βα
β
α
βα
β
α α
α
S2015abnForces & Moments 11Lecture 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
α β
βγ
α
°=+ 90γβ
S2015abnForces & Moments 12Lecture 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
αγ
γ
°=+ 90γα
α
S2015abnArchitectural Structures ARCH 331
– similar triangles have proportional sides
Geometry
B
C
E
A
A
BC
A′
C′
B′
DE
BC
AE
AC
AD
AB==
CB
BC
CA
AC
BA
AB
′′=
′′=
′′
D
α
α
ββγ
γ
Forces & Moments 13Lecture 3
S2015abnArchitectural Structures ARCH 331
Trigonometry
• for right triangles
Cα
B
A
CB
AB
hypotenuse
sideopposite=== αsinsin
CB
AC
hypotenuse
sideadjacent=== αcoscos
AC
AB
sideadjacent
sideopposite=== αtantan
SOHCAHTOAForces & Moments 14Lecture 3
S2015abnArchitectural Structures ARCH 331
Trigonometry
• cartesian coordinate system
– origin at 0,0
– coordinatesin (x,y) pairs
– x & y havesigns
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
X
Y
Quadrant IQuadrant II
Quadrant III Quadrant IV
Forces & Moments 15Lecture 3
S2015abnArchitectural Structures ARCH 331
Trigonometry
• for angles starting at positive x
– sin is y side
– cos is x side
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 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 16Lecture 3
S2015abnArchitectural Structures ARCH 331
Trigonometry
• for all triangles
– sides A, B & C are opposite
angles α, β & γ
– LAW of SINES
– LAW of COSINES
CBA
γβα sinsinsin==
αcos2222BCCBA −+=
A
α C
B
γβ
Forces & Moments 17Lecture 3
S2015abnArchitectural 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 18Lecture 3
S2015abnArchitectural 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
012 =−x10112 +=+−x
21=x2
1
2
2=
/
/ x
12 =x
Forces & Moments 19Lecture 3
S2015abnArchitectural 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
5412 +=− xx
xxxx 254212 −+=−−
55251 −+=−− x
2
2
2
23
2
6
/
/=
/
/⋅−=
− x
3−=xForces & Moments 20Lecture 3
S2015abnArchitectural 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
832 =+ yx
6312 =− yx
6831232 +=−++ yxyx
1414 =x
114
14
14
14=== x
x
Forces & Moments 21Lecture 3
S2015abn
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 StructuresARCH 331
Forces & Moments 22Lecture 3
S2015abn
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 opposite reaction alongthe same line
Architectural StructuresARCH 331
Forces & Moments 23Lecture 3
http://www.physics.umd.edu
S2015abn
Force Characteristics
• applied at a point
• magnitude
– Imperial units: lb, k (kips)
– SI units: N (newtons), kN
• direction
Architectural StructuresARCH 331
(tail) (tip)
Forces & Moments 24Lecture 3
S2015abn
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 StructuresARCH 331
translate rotate
Forces & Moments 25Lecture 3
S2015abn
Transmissibility
• the force stays on the same line of
action
• truck can’t tell the difference
• only valid for EXTERNAL forces
Architectural StructuresARCH 331
=
Forces & Moments 26Lecture 3
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Force System Types
• collinear
Architectural StructuresARCH 331
Forces & Moments 27Lecture 3
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Force System Types
• coplanar
Architectural StructuresARCH 331
Forces & Moments 28Lecture 3
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Force System Types
• space
Architectural StructuresARCH 331
Forces & Moments 29Lecture 3
S2015abn
Adding Vectors
• graphically
– parallelogram law
• diagonal
• long for 3 or more vectors
– tip-to-tail
• more convenientwith lots of vectors
Architectural StructuresARCH 331
F
P
R
F
P
R
Forces & Moments 30Lecture 3
S2015abn
Force Components
• convenient to resolve into 2 vectors
• at right angles
• in a “nice” coordinate system
• θ is between Fx and F from Fx
Architectural StructuresARCH 331
Fy
Fx
θ
F
x
y
Fy
Fx
F
Fy
Fx
FθcosFFx =
θsinFFy =22
yx FFF +=
x
y
F
F=θtan
Forces & Moments 31Lecture 3
S2015abn
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 StructuresARCH 331
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
X
Y
Quadrant IQuadrant II
Quadrant III Quadrant IV
° °
° °
Forces & Moments 32Lecture 3
S2015abn
Component Addition
• find all x components
• find all y components
• find sum of x components, Rx (resultant)
• find sum of y components, Ry
Architectural StructuresARCH 331
22
yx RRR +=
x
y
R
R=θtan
Fy
Fx
R
Py
Pxθθθθ
Rx
Ry
Forces & Moments 33Lecture 3
S2015abn
Alternative Trig for Components
• doesn’t relate angle to axis direction
• φ 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 StructuresARCH 331
+/-? Fy
+/-? Fx
φ
F
x
y
Forces & Moments 34Lecture 3
S2015abn
Friction
• resistance to movement
• contact surfaces determine µ
• proportion of normal force (⊥)
– opposite to slide direction
– static > kinetic
Architectural StructuresARCH 331
NµF =
Forces & Moments 35Lecture 3
S2015abn
Cables
• simple
• uses
– suspension bridges
– roof structures
– transmission lines
– guy wires, etc.
• have same tension all along
• can’t stand compression
Architectural StructuresARCH 331
Forces & Moments 36Lecture 3
http:// nisee.berkeley.edu/godden
S2015abn
Cables Structures
• use high-strength steel
• need
– towers
– anchors
• don’t want movement
Architectural StructuresARCH 331
Forces & Moments 37Lecture 3
http:// nisee.berkeley.edu/godden
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Cable Structures
Architectural StructuresARCH 331
Forces & Moments 38Lecture 3
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Cable Loads
• straight line
between forces
• with one force
– concurrent
– symmetric
Architectural StructuresARCH 331
Forces & Moments 39Lecture 3
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Cable Loads
• shape directly
related to the
distributed load
Architectural StructuresARCH 331
Forces & Moments 40Lecture 3
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Cable-Stayed Structures
• diagonal cables support horizontal
spans
• typically symmetrical
• Patcenter,
Rogers 1986
Architectural StructuresARCH 331
Forces & Moments 41Lecture 3
www.columbia.edu
S2015abn
Patcenter, Rogers 1986
• column free space
• roof suspended
• solid steel ties
• steel frame supports masts
Architectural StructuresARCH 331
Forces & Moments 42Lecture 3
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Patcenter, Rogers 1986
• dashes – cables pulling
Architectural StructuresARCH 331
Forces & Moments 43Lecture 3
S2015abnArchitectural StructuresARCH 331
Moments
• forces have the tendency to make a body rotate about an axis
– same translation but different rotation
Forces & Moments 44Lecture 3
http://www.physics.umd.edu
S2015abnArchitectural StructuresARCH 331
Moments
Forces & Moments 45Lecture 3
S2015abnArchitectural StructuresARCH 331
Moments
• a force acting at a different point causes
a different moment:
≠
Forces & Moments 46Lecture 3
S2015abnArchitectural StructuresARCH 331
Moments
• defined by magnitude and direction
• units: N⋅m, k⋅ft
• direction: + ccw (right hand rule)
- cw
• value found from Fand ⊥ distance
• d also called “lever” or “moment” arm
F
A
C
B
dFM ⋅=
d
Forces & Moments 47Lecture 3
S2015abnArchitectural StructuresARCH 331
Moments
• with same F:
21 dFMdFM AA ⋅=<⋅=(bigger)
Forces & Moments 48Lecture 3
S2015abnArchitectural StructuresARCH 331
Moments
• additive with sign convention
• can still move the force
along the line of action
=
+MA = F⋅dMB = F⋅d′
d
F
A
Bd′
MA = F⋅d
MB = F⋅d′
d
F
A
B
d′
Forces & Moments 49Lecture 3
S2015abnArchitectural StructuresARCH 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 50Lecture 3
S2015abnArchitectural StructuresARCH 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 51Lecture 3
S2015abnArchitectural StructuresARCH 331
Physics and Moments of a Force
• my Physics book:
Forces & Moments 52Lecture 3
S2015abnArchitectural StructuresARCH 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 53Lecture 3
21 dFdFM ⋅−⋅=
dFM ⋅=
S2015abnArchitectural StructuresARCH 331
Moment Couples
• equivalent couples
– same magnitude and direction
– F & d may be different
100 mm
300 N
300 N
150 mm200 N
200 N
250 mm
120 N120 N
Forces & Moments 54Lecture 3
S2015abnArchitectural StructuresARCH 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 mm200 N
200 N
250 mm
240 N240 N
Forces & Moments 55Lecture 3
S2015abnArchitectural StructuresARCH 331
Moment Couples
• moment couples in structures
Forces & Moments 56Lecture 3
S2015abnArchitectural StructuresARCH 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 57Lecture 3
S2015abnArchitectural StructuresARCH 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
FF
d
A
A′F
A
A′
Forces & Moments 58Lecture 3
S2015abnArchitectural StructuresARCH 331
Force-Moment Systems
• a force-moment pair can be replaced by
a force at another point causing the
original moment
F
d
-F
F
A
A′F
A
A′
F M=F⋅d
A
A′
Forces & Moments 59Lecture 3
S2015abnArchitectural StructuresARCH 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 Dx
AB
ab
aA ⋅
bB ⋅
xBA ⋅+ )(
C D
Forces & Moments 60Lecture 3