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S2009abnMechanics of Materials 1
Lecture 4
Elements of Architectural Structures
ARCH 614
ELEMENTS OF ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 614
DR. ANNE NICHOLS
SPRING 2014
four
mechanics
of materials
lecture
www.carttalk.com
Mechanics of Materials 7
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Elements of Architectural Structures
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S2004abn
Mechanics of Materials
• MECHANICS • MATERIALS
Mechanics of Materials 8
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Mechanics of Materials
• external loads and their
effect on deformable bodies
• use it to answer question if structure
meets requirements of
– stability and equilibrium
– strength and stiffness
• other principle building requirements• economy, functionality and aesthetics
Mechanics of Materials 9
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Knowledge Required
• material properties
• member cross sections
• ability of a material to resist breaking
• structural elements that resist excessive
– deflection
– deformation
2
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Problem Solving
1. STATICS:
equilibrium of external forces,
internal forces, stresses
2. GEOMETRY:
cross section properties, deformations and
conditions of geometric fit, strains
3. MATERIAL PROPERTIES:
stress-strain relationship for each material
obtained from testing
Mechanics of Materials 11
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Stress
• stress is a term for the intensity of a
force, like a pressure
• internal or applied
• force per unit area
A
Pfstress
Mechanics of Materials 12
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• materials have a critical stress value
where they could break or yield
– ultimate stress
– yield stress
– compressive stress
– fatigue strength
– (creep & temperature)
Design
acceptance
vs. failure
Mechanics of Materials 13
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allowableactual Ff • we’d like
• stress distribution may
vary: average
• uniform distribution
exists IF the member is
loaded axially
(concentric)
Design (cont)
3
Mechanics of Materials 9
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Scale Effect
• model scale
– material weights by volume,
small section areas
• structural scale
– much more material weight,
bigger section areas
• scale for strength is not
proportional:L
L
L
2
3
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Normal Stress
• normal stress is normal
to the cross section
– stressed area is
perpendicular to the
load
A
Pf cort
Mechanics of Materials 17
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• stress parallel to a surface
Shear Stress
td
P
A
Pfv
ave
Mechanics of Materials 18
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Elements of Architectural Structures
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• stress on a surface by
contact in compression
Bearing Stress
td
P
A
Pf p
4
Mechanics of Materials 19
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• normal stress caused by bending
Bending Stress
S
M
I
Mcfb
Mechanics of Materials 20
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• shear stress caused by twisting
Torsional Stress
J
Tfv
Mechanics of Materials 21
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• what structural elements see shear?
– beams
– bolts
– splices
– slabs
– footings
– walls
• wind
• seismic loads
Structures and Shear
connections
V
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• connected members in tension cause
shear stress
• connected members in
compression cause
bearing stress
Bolts
Mechanics of Materials 16
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• seen when 2 members are connected
Single Shear
Mechanics of Materials 17
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2dv
P
A
Pf
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Double Shear
Mechanics of Materials 18
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F=
• seen when 3 members are connected
• two areas
4d
v 2
2P
A
2P
A2
Pf
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td
P
A
Pf
projected
p
• compression & contact
• projected area
Bolt Bearing Stress
Mechanics of Materials 19
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F=
Mechanics of Materials 15
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Strain
• materials deform
• axially loaded materials change
length
• bending materials deflect
• STRAIN:
– change in length
over length L
Lstrain
(S)
6
Stress & Strain 10
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Shearing Strain
• deformations
with shear
• parallelogram
• change in angles
• stress:
• strain:
– unitless (radians)
s
L
tanL
s
Stress & Strain 11
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Elements of Architectural Structures
ARCH 614
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Shearing Strain
• deformations
with torsion
• twist
• change in angle of line
• stress:
• strain:
– unitless (radians)
L
Stress & Strain 12
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Load and Deformation
• for stress, need P & A
• for strain, need & L
– how?
– TEST with load and
measure
– plot P/A vs.
Stress & Strain 13
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Material Behavior
• every material has its own response
– 10,000 psi
– L = 10 in
– Douglas fir vs.
steel?
7
Stress & Strain 14
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Behavior Types
• ductile - “necking”
• true stress
• engineering stress
– (simplified)
A
Pf
oA
Pf
Stress & Strain 15
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Behavior Types
• brittle
• semi-brittle
Stress & Strain 16
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Stress to Strain
• important to us in f- diagrams:
– straight section
– LINEAR-ELASTIC
– recovers shape
(no permanent
deformation)
Stress & Strain 17
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Hooke’s Law
• straight line has constant slope
• Hooke’s Law
• E
– Modulus of elasticity
– Young’s modulus
– units just like stress
f
E
1
Ef
8
Stress & Strain 18
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Stiffness
• ability to resist strain
• steels
– same E
– different
yield points
– different
ultimate strength
uf
Stress & Strain 19
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Isotropy & Anisotropy
• ISOTROPIC
– materials with E same at
any direction of loading
– ex. steel
• ANISOTROPIC
– materials with different E
at any direction of loading
– ex. wood is orthotropic
Stress & Strain 20
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Elastic, Plastic, Fatigue
• elastic springs back
• plastic has permanent
deformation
• fatigue caused by
reversed loading
cycles
Stress & Strain 21
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Plastic Behavior
• ductile
at yield stress
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Stress & Strain 15
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Elements of Architectural Structures
ARCH 614
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Lateral Strain
• or “what happens to the cross section
with axial stress”
• strain in lateral direction
– negative
– equal for isometric materialszy
E
f xx
0 zy ff
Stress & Strain 23
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Elements of Architectural Structures
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Poisson’s Ratio
• constant relationship between
longitudinal strain and lateral strain
• sign!
x
z
x
y
strainaxial
strainlateral
E
f xzy
5.00
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Elements of Architectural Structures
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Calculating Strain
• from Hooke’s law
• substitute
• get
Ef
LE
A
P
AE
PL
(L)
Stress & Strain 25
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Elements of Architectural Structures
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Orthotropic Materials
• non-isometric
• directional values of
E and
• ex:
– plywood
– laminates
– polymer
composites
10
Stress & Strain 26
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• why we use fave
• increase in stress atchanges in geometry
– sharp notches
– holes
– corners
–
Stress Concentrations
Stress & Strain 27
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22
max
o
maxv
f
A
Pf
• if we need to knowwhere max f and fv
happen:
Maximum Stresses
F
o
maxA
Pf 10 cos
5045 .sincos
Stress & Strain 28
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Elements of Architectural Structures
ARCH 614
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Maximum Stresses
Torsion & Temp 18
Lecture 16
Elements of Architectural Structures
ARCH 614
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Deformation Relationships
• physical movement
– axially (same or zero)
– rotations from axial changes
• relates to P
steel 20 kN
aluminum
AE
PL
11
Torsion & Temp 19
Lecture 16
Elements of Architectural Structures
ARCH 614
S2004abn
Deformations from Temperature
• atomic chemistry reacts
to changes in energy
• solid materials• can contract with decrease in temperature
• can expand with increase in temperature
• linear change can
be measured per
degree
Torsion & Temp 20
Lecture 16
Elements of Architectural Structures
ARCH 614
S2004abn
Thermal Deformation
• - the rate of strain per degree
• UNITS : ,
• length change:
• thermal strain:
– no stress when movement allowed
LTT
TT
F C
Torsion & Temp 21
Lecture 16
Elements of Architectural Structures
ARCH 614
S2004abn
Coefficients of Thermal Expansion
12.8 x 10-6Aluminum
10.4 x 10-6Brass
10.1 x 10-6Bronze
9.3 x 10-6Copper
6.7 x 10-6Wrought Iron
6.5 x 10-6Steel
5.9 x 10-6Cast Iron
5.5 x 10-6Concrete
4.4 x 10-6Glass
3.0 x 10-6Wood
Coefficients () [in./in./F]Material
Torsion & Temp 22
Lecture 16
Elements of Architectural Structures
ARCH 614
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ELA
Pf
Stresses and Thermal Strains
• if thermal movement is restrained
stresses are induced
1. bar pushes on supports
2. support pushes back
3. reaction causes internal
stress
12
Torsion & Temp 23
Lecture 16
Elements of Architectural Structures
ARCH 614
S2004abn
Superposition Method
– can remove a support to
make it look determinant
– replace the support with a
reaction
– enforce the geometry
constraint
Torsion & Temp 24
Lecture 16
Elements of Architectural Structures
ARCH 614
S2004abn
Superposition Method
– total length change restrained to
zero0 TP
0 LTAE
PL
ETA
Pf
LTT AE
PLp
constraint:
sub:
Mechanics of Materials 43
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Elements of Architectural Structures
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S2007abn
Dynamics
• kinematics
– time, velocity, acceleration
– linear motion
– angular rotation
• kinetics
– forces causing motion
W = m·g
– work
– conservation of energy
2
21)0()( attvts
Mechanics of Materials 44
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Dynamic Response
13
Mechanics of Materials 45
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Dynamic Response
• period of vibration or
frequency
– wave
– sway/time period
• damping
– reduction in sway
• resonance
– amplification of sway
Mechanics of Materials 46
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Frequency and Period
• natural period of vibration
– avoid resonance
– hard to predict seismic period
– affected by soil
– short period
• high stiffness
– long period
• low stiffness
“To ring the bell, the sexton must pull
on the downswing of the bell in time
with the natural frequency of the bell.”
Stress & Strain 29
Lecture 15
Elements of Architectural Structures
ARCH 614
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Design of Members
• beyond allowable stress...
• materials aren’t uniform 100% of the
time
– ultimate strength or capacity to failure may
be different and some strengths hard to
test for
• RISK & UNCERTAINTY
A
Pf uu
Stress & Strain 30
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Factor of Safety
• accommodate uncertainty with a safety
factor:
• with linear relation between load and
stress:
SF
loadultimateloadallowable
.
stressallowable
stressultimate
loadallowable
loadultimateSF .
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Stress & Strain 31
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Load and Resistance Factor Design
• loads on structures are
– not constant
– can be more influential on failure
– happen more or less often
– UNCERTAINTY
- resistance factor
- load factor for (D)ead & (L)ive load
nLLDDu RRRR