fifteen lecture - faculty.arch.tamu.edu

Strain 1Lecture 15

Architectural Structures IENDS 231

Su2006abn

ARCHITECTURAL STRUCTURES I:STATICS AND STRENGTH OF MATERIALS ENDS 231

DR. ANNE NICHOLS

SUMMER 2006

fifteen

elasticity& strain

lecture

Strain 2Lecture 16


S2006abn

Deformations • materials deform• axially loaded materials change

length• normal stress is load per unit

area• STRAIN:

– change in length over length– UNITLESS

Lδε =

Strain 3Lecture 16


S2006abn

Shearing Strain• deformations

with shear• parallelogram• change in angles• stress:• strain:

– unitless (radians)

sδ

L

γ

φ

τφφδγ ≅== tan

Ls

Strain 4Lecture 16


S2006abn

Shearing Strain• deformations

with torsion• twist • change in angle of line• stress:• strain:

– unitless (radians)

γτ

Lρφγ =

Strain 5Lecture 16


S2006abn

Load and Deformation• for stress, need P & A• for strain, need δ & L

– how?– TEST with load and

measure– plot P/A vs. ε

Strain 6Lecture 16


S2006abn

Material Behavior• every material has its own response

– 10,000 psi– L = 10 in– Douglas Fir vs.

steel?

Strain 7Lecture 16


S2006abn

Behavior Types• ductile - “necking”• true stress

• engineering stress– (simplified)

APf =

oAPf =

Strain 8Lecture 16


S2006abn

Behavior Types• brittle

• semi-brittle

Strain 9Lecture 16


S2006abn

Stress to Strain• important to us in -ε diagrams:

– straight section– LINEAR-ELASTIC– recovers shape

(no permanentdeformation)

f

Strain 10Lecture 16


S2006abn

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

Strain 11Lecture 16


S2006abn

Stiffness• ability to resist strain

• steels– same E– different

yield points– different

ultimate strength

uf

Strain 12Lecture 16


S2006abn

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

Strain 13Lecture 16


S2006abn

Elastic, Plastic, Fatigue• elastic springs back• plastic has permanent

deformation• fatigue caused by

reversed loading cycles

Strain 14Lecture 16


S2006abn

Plastic Behavior• ductile

at yield stress

Strain 15Lecture 16


S2006abn

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 =

Strain 16Lecture 16


S2006abn

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

Strain 17Lecture 16


S2006abn

Calculating Strain• from Hooke’s law

• substitute

• get ⇒

ε⋅= Ef

LE

AP δ

⋅=

AEPL

=δ

Strain 18Lecture 16


S2006abn

Orthotropic Materials• non-isometric• directional values of

E and μ• ex:

– plywood– laminates– polymer

composites

Strain 19Lecture 16


S2006abn

• why we use fave

• increase in stress atchanges in geometry– sharp notches– holes– corners

–

Stress Concentrations

Strain 20Lecture 16


S2006abn

22max

maxf

APf

ov ==−

• if we need to knowwhere max f and fvhappen:

Maximum Stresses

F

oAPf =max1cos0 =→°= θθ

5.0sincos45 ==→°= θθθ

Strain 21Lecture 16


S2006abn

Maximum Stresses

Strain 22Lecture 16


S2006abn

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 =

Strain 23Lecture 16


S2006abn

Factor of Safety• accommodate uncertainty with a safety

factor:

• with linear relation between load and stress:

SFloadultimateloadallowable

.=

stressallowablestressultimate

loadallowableloadultimateSF ==.

Strain 24Lecture 16


S2006abn

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 ≤+=

fifteen lecture - faculty.arch.tamu.edu

Documents

Transcript of fifteen lecture - faculty.arch.tamu.edu