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Strain 1Lecture 16
Architectural Structures IENDS 231
S2008abn
ARCHITECTURAL STRUCTURES I:STATICS AND STRENGTH OF MATERIALS ENDS 231
DR. ANNE NICHOLS
SPRING 2008
sixteen
elasticity& strain
lecture
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Strain 2Lecture 16
Architectural Structures IENDS 231
S2008abn
Deformations • materials deform• axially loaded materials change
length• normal stress is load per unit
area• STRAIN:
– change in length over length– UNITLESS
Lδε =
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Strain 3Lecture 16
Architectural Structures IENDS 231
S2008abn
Shearing Strain• deformations
with shear• parallelogram• change in angles• stress:• strain:
– unitless (radians)
sδ
L
γ
φ
τφφδγ ≅== tan
Ls
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Strain 4Lecture 16
Architectural Structures IENDS 231
S2008abn
Shearing Strain• deformations
with torsion• twist • change in angle of line• stress:• strain:
– unitless (radians)
γτ
Lρφγ =
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Strain 5Lecture 16
Architectural Structures IENDS 231
S2008abn
Load and Deformation• for stress, need P & A• for strain, need δ & L
– how?– TEST with load and
measure– plot P/A vs. ε
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Strain 6Lecture 16
Architectural Structures IENDS 231
S2008abn
Material Behavior• every material has its own response
– 10,000 psi– L = 10 in– Douglas Fir vs.
steel?
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Strain 7Lecture 16
Architectural Structures IENDS 231
S2008abn
Behavior Types• ductile - “necking”• true stress
• engineering stress– (simplified)
APf =
oAPf =
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Strain 8Lecture 16
Architectural Structures IENDS 231
S2008abn
Behavior Types• brittle
• semi-brittle
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Strain 9Lecture 16
Architectural Structures IENDS 231
S2008abn
Stress to Strain• important to us in -ε diagrams:
– straight section– LINEAR-ELASTIC– recovers shape
(no permanentdeformation)
f
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Strain 10Lecture 16
Architectural Structures IENDS 231
S2008abn
Hooke’s Law• straight line has constant slope• Hooke’s Law
• E – Modulus of elasticity– Young’s modulus– units just like stress
f
ε
E
1 ε⋅= Ef
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Strain 11Lecture 16
Architectural Structures IENDS 231
S2008abn
Stiffness• ability to resist strain
• steels– same E– different
yield points– different
ultimate strength
uf
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Strain 12Lecture 16
Architectural Structures IENDS 231
S2008abn
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
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Strain 13Lecture 16
Architectural Structures IENDS 231
S2008abn
Elastic, Plastic, Fatigue• elastic springs back• plastic has permanent
deformation• fatigue caused by
reversed loading cycles
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Strain 14Lecture 16
Architectural Structures IENDS 231
S2008abn
Plastic Behavior• ductile
at yield stress
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Strain 15Lecture 16
Architectural Structures IENDS 231
S2008abn
Lateral Strain• or “what happens to the cross section
with axial stress”
• strain in lateral direction – negative– equal for isometric materials
Efx
x =ε
0== zy ff
zy εε =
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Strain 16Lecture 16
Architectural Structures IENDS 231
S2008abn
Poisson’s Ratio• constant relationship between
longitudinal strain and lateral strain
• sign!
x
z
x
y
strainaxialstrainlateral
εε
εε
μ −=−=−=
Efx
zyμεε −==
5.00 << μ
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Strain 17Lecture 16
Architectural Structures IENDS 231
S2008abn
Calculating Strain• from Hooke’s law
• substitute
• get ⇒
ε⋅= Ef
LE
AP δ
⋅=
AEPL
=δ
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Strain 18Lecture 16
Architectural Structures IENDS 231
S2008abn
Orthotropic Materials• non-isometric• directional values of
E and μ• ex:
– plywood– laminates– polymer
composites
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Strain 19Lecture 16
Architectural Structures IENDS 231
S2008abn
• why we use fave
• increase in stress atchanges in geometry– sharp notches– holes– corners
–
Stress Concentrations
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Strain 20Lecture 16
Architectural Structures IENDS 231
S2008abn
22max
maxf
APf
ov ==−
• if we need to knowwhere max f and fvhappen:
Maximum Stresses
F
oAPf =max1cos0 =→°= θθ
5.0sincos45 ==→°= θθθ
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Strain 21Lecture 16
Architectural Structures IENDS 231
S2008abn
Maximum Stresses
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Strain 22Lecture 16
Architectural Structures IENDS 231
S2008abn
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
APf u
u =
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Strain 23Lecture 16
Architectural Structures IENDS 231
S2008abn
Factor of Safety• accommodate uncertainty with a safety
factor:
• with linear relation between load and stress:
SFloadultimateloadallowable
.=
stressallowablestressultimate
loadallowableloadultimateSF ==.
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Strain 24Lecture 16
Architectural Structures IENDS 231
S2008abn
Load and Resistance Factor Design• loads on structures are
– not constant– can be more influential on failure– happen more or less often– UNCERTAINTY
φ - resistance factorγ - load factor for (D)ead & (L)ive load
nLLDDu RRRR φγγ ≤+=