Building MaterialsBuilding MaterialsBuilding MaterialsBuilding Materials

Lecture 3Lecture 3

MechanicalMechanical propertiespropertiesSt thSt th titiStrengthStrength propertiesproperties

Statistical strengthStatistical strength

from the single samples properties the property of the whole population can beproperty of the whole population can be estimated by thestatistical methodsstatistical methods

StatistikaStatistika

The only statistics you can trust are those you falsified yourself. y y

attributed to Winston Churchillattributed to Winston Churchill

Statistics - glossaryStatistics glossary random experiment - an random experiment - an

experiment whose outcome is not perfectly predictableperfectly predictable

population - the entire collection ofpopulation the entire collection of items that is the focus of concern

random sample - a sample whose members are chosen at random from a given population in such a way that the chance of obtaining any particular g y psample can be computed

Statistical evaluation of strenght

only part of the population is tested y p p prandom sample

f f from the results of random sample can be estimated a corresponding parameter p g pof the population

i l l i h l typical population has normal distribution (Gaussian function)( )

Normal distributionNormal distribution for the whole population

y

u

e

n

c

y

F

r

e

q

u

mean Measured value

Gaussian curveGaussian curve

the narrower and higher the curve is, the t ti ti ll h th d tmore statistically homogenous the data

HistogramHistogram

from testing of random sample the g pdistribution curve could not be made

the more numerous the random sample is, the closer to the curve the histogram is

Normal and other distributionNormal and other distribution

normal distribution

non symmetrical non-symmetrical

modemodemedian

meanmean

Statistical parametersStatistical parametersValues: 4 8 6 2 5 11Values: 4, 8, 6 2, 5,11

Mean x = 6 x = 6Deviations -2,+2, 0 -4,-1,+5Sum of deviations 0 0Sum of deviations 0 0Deviations square 4, 4, 0 16, 1, 25Sum of squares 8 42Sum of squares 8 42

2,67 14 Variance

1 63 3 74 Standard 1,63 3,74deviation

Statistical parametersStatistical parameters

mean

variancevariancestandard deviation

Standard deviation sStandard deviation s measure of variability or

diversity of a data sety low standard deviation indicates that the data pointsindicates that the data points tend to be very close to the meanmean high standard deviation indicates that the data points areindicates that the data points are spread out over a large range of valuesvalues

Normal distributionNormal distribution

symetrical

+s to -s : 68,26 % of area2 t 2 95 6 % f +2s to 2s : 95,6 % of area

+3s to 3s : 99,7 % of area

Guaranteed strengthGuaranteed strength

95 %

-1,645 s +1,645 s

Guaranteed strengthGuaranteed strength the value of the strength, for which can be

statistically guaranteed, that 95 % of whole production will have the same or higher value of the strength

95 %

5 %

guaranteed strength

Strength testingStrength testingaccording the loadingg g

compressive

t il tensile

bending bending

torsion

shear

Compressive strengthCompressive strength maximum resistance of a material

F

a u es s a ce o a a e ato axial compressive loading

FR maxA

Fmax

AR maxc [MPa]

F i f [N]

AFmax .... maximum force [N]A ...... compressed area

(cross-sectional) [mm] Fmax

Compressive strength - testingCompressive strength testing Test specimen for i t tcompressive test

Regular shape: cubes (concrete) cubes (concrete)

cylinders (concrete, )lightweight concretes)

beams (lightweight concretes)

prism halves (cement)p ( ) whole product (block,

brick)brick)

Test specimen for i t tcompressive test

irregular shapeg p auxiliary plates

compressed area Acompressed area Agiven by the area ofplatesplates

Compressive strenghtp gConcrete C 25/30

cylinder strength < cube strength h i fl f f i i b i l the influence of friction between material

surface and testing machine decreases ith th h i ht f th lwith the height of the sample

Compressive strength of building materials

8000 200 400 600

Tensile strengthTensile strengthF

A

Fmax

FR maxA

[MPa]A

Rt [MPa]

Fmax .... maximum force [ N]FA ...... area [mm] Fmax

Tensile strenght - testingTensile strenght - testing Test specimen for tensile testTest specimen for tensile test

round or flat, slim special shoulders for gripping in the machinespecial shoulders for gripping in the machine

Flexural strengthFlexural strength

tensile strength of materials with distinctively higher compressive strength y g p gthan tensile strength

fracture in the place of the maximum fracture in the place of the maximum bending moment

Flexural strength

M

Flexural strength

FmaxWMR maxy maxWy

Fmax Fmax

Mmax..maximum bending moment [N.mm]W ... section modulus [mm3]

Flexural strengthFlexural strength Bending moment M

according type of loading (three point test, four point test)

S Section modulus W according the shape and i f isize of cross section

Calculation of M and WCalculation of M and W

Splitting tensile strengthSplitting tensile strengthF fragile materials F

F2lD

F2R maxt F

l.D.D L

F

Brazilian testBrazilian test stones s o es

Deformation propertiesDeformation propertiesDeformation propertiesDeformation properties

Deformation propertiesd ib h b h i f h i l

Deformation properties describe the behavior of the materials

before the fracture

DeformationDeformation

irreversible irreversibleplastic deformation

reversible elastic deformation

Stress-strain diagramStress-strain diagram

graphical representation of the relationship between stress derivedrelationship between stress, derived from measuring the load applied on the sample, and strain, derived from measuring the deformation of the gsample (elongation, compression, or distortion)distortion)

Stress strain diagramStress strain diagram the relationship between stress ()the relationship between stress ()

and strain () (load F and deformation l )l )

[MP ] deformations [MPa]F [N]

deformations yield strength ultimate strength toughness

[ ] toughness Young modulus [-]

l [mm]

Stress-strain diagramStress-strain diagram

Load F[kN][ ]

l l0

lFdA

Work = force x distance (through which it acts)(through which it acts)

Deformation l [mm]

Stress-strain diagramStress-strain diagram

Stress [MPa][MPa]

AA

Strain [-,%]

Stress-strain curvesStress-strain curves Stress-strain curves of different materials

Stress-strain diagramstress

g

4

5

3

12

4 - ultimate strength5 - failure

3 - yield strength2 - elastic limit

4 ultimate strength

1 - proportionality limit2 elastic limit

strain

Stress strain curve without well-d fi d i ld i tdefined yield point

R1 mild steel2R0,2 2 cold-formed steel

R0,2 offset yield strength0,2 y g

A line is drawn parallel to the linear elastic 0,2 % portion of the curve and intersecting the x-

axis at some arbitrary value (0.1% or 0.2%)

Deformationstress

Deformationt t l

1total total

deformation

total1

el - elastic deformation

eldeformation

l tipl - plastic deformation

strainstrain (deformation)pl el

M d l f l ti itModulus of elasticity

Elastic behavior of materials describes ast c be a o o ate a s desc bes

Hooke's Law :

.E .E ... strain [unitless]... stress [MPa]E modulus of elasticity [MPa]E ... modulus of elasticity [MPa]

(Youngs modulus)

Elastic modulusElastic modulus

E el

E the mathematical description of athe mathematical description of a

material's tendency to be deformed elastically when a force is applied to itelastically when a force is applied to it

Hooke's law is valid only for elastic range of material

Elastic modulusElastic modulustension or compression Young's modulus

E el

shear modulus of rigidity or shear modulus

G

Graphical determination of Y ' M d l

Young's Modulus

loading1 loadingE

el

unloadingE

g

tgE

totalel

Young's modulus determination

statical

E el

E t [MP ]

el

. stress [MPa] .. strain [-][ ]

Young's modulus determination

Stress : according the type of loadingaccording the type of loading F

compression AcompressiontensionMWM

bending

Young's modulus determination

strain 01

lll

ll

l change of the length [mm]00 ll

l .... change of the length [mm]l1 .. length after elongation [mm]l0 original (initial) length [mm]l0 ...... original (initial) length [mm]

FF

l l0

Measuring of elongation lMeasuring of elongation l deformations l have to be measured

by special devices - strain gaugeby special devices - strain gauge

Strain gauge:Strain gauge:

mechanicalmechanical

electricalelectrical

opticaloptical

Mechanical strain gaugeMechanical strain gaugeFirmFirm point

l

e

n

g

t

h

)

)

g

a

u

g

e

Dial

O

r

i

g

i

n

a

l

)

(

O

Movable TestedMovable point

Tested material

Measuring of deformationsMeasuring of deformations

Electrical resistance gaugeElectrical resistance gauge

Rl 1RR

Kll 1

RKl

Optical strain gaugeOptical strain gauge

optical fibers laser

Dynamic Young's modulusDynamic Young s modulus lt i ultrasonic waves

2d .cE vdyn .cE

c sound velocity [m2 s-1]c... sound velocity [m2.s 1]v... bulk density [kg.m-3]

Measuring of dynamicYoung's modulus

Elastic modulus is influenced by:

temperature

Elastic modulus is influenced by:

temperature thermoplastics: with rising temperature

E significantly decreasesE significantly decreases

concrete : -20 C - +70 C - E constantunder -50 C - ca. about 20 % highergabove 300 C - ca. 50 % of initial valuei t i t i l moisture in porous materials

Young's modulus of some materials

[GPa]210

200

[GPa]

150

70100

25 2515 10

50

15 10 3 1.5 0.030

Material Young's modulus [GPa]

[GPa] Diamond 1050-1200 Steel 210Steel 210 Glass 50 -85 Aluminium and light alloys 65 -73Aluminium and light alloys 65 -73 Brass nad bronze 103-124 Concrete 15 60Concrete 15 - 60 Ceramic brick 8 - 12 Wood 7 18Wood 7 -18Glass laminate 10 - 30 Th t 4 13Thermosets 4 - 13 Thermoplastics solid 0,1 - 4 Th l ti f d 0 02 0 3Thermoplastics foamed 0,02 0,3Rubber 0,002 0,005

DuctilityDuctility percentage elongation after tensile testpercentage elongation after tensile test

0 LLL)(A u 00 LL

)(A

Reduction of areaReduction of area change of cross sectional area as a change of cross sectional area as a

percentage of the original cross-sectional area S i i l ti l

0

SSS)(Z u

S0..... original cross-sectional area before testing

S minimal cross sectional0S Su..... minimal cross-sectional area after failure

Brittleness and toughnessBrittleness and toughness

brittle material, subjected to stress, breaks without significant deformation g

tough material deforms plastically and absorbs energy before fractureabsorbs energy before fracture

ToughnessToughness the amount of energy per volume that a the amount of energy per volume that a

material can absorb before rupturing units: kJ/m3 units: kJ/m3

Test of toughnessTest of toughnessi t t h impact toughness Charpy, Izod test

notch toughness (ability to absorb energy in the presence of a flaw)

Charpy impact testCharpy impact test BrittlenessBrittlenesst d f t i l t f t f il tendency of a material to fracture or fail upon the application of a relatively small

t f f i t h kamount of force, impact, or shock opposite of toughnesspp g no numerical value

Rough criterion for brittle materials: compressive strength : tensile strengthcompressive strength : tensile strength

> 8 : 1

HardnessHardness

defines the materials resistance to penetrationp

depends on temperature and moisture

Methods:et ods scratch hardness

indentation hardness indentation hardness rebound hardness

Scratch h Mohs scale1. talc

Scratch h. Mohs scale

2. gypsum3 calcite3. calcite4. fluorite5 apatite5. apatite6. feldspar (orthoclase)7 t7. quartz8. topaz9. corundum10.diamond

used for minerals

Indentation h Vickers testIndentation h. Vickers test indenter: diamond point with a

136 point angle abbreviation VHN metals, hard materials

Indentation h Brinell testIndentation h. Brinell testi d t t l (t t ) indenter: steel (tungsten) ball (10 mm )

abbreviation: HBW (HBS) abbreviation: HBW, (HBS) metals, wood , hard

polymers

)(,

22

F21020HB polymers

)(. 22 dDDD

Indentation h Rockwell test diamond cone

Indentation h. Rockwell test diamond cone abb.: HR(A,B,C..G)

d th f i d t ti depth of indentation metals

Shore durometer spring+ steel rod abb.: SH polymers, elastomers, rubber

Conversion of Brinell hardness (HB) t Vi k R k ll(HB) to Vickers or Rockwell

hardness

Indentation h POLDI hammerIndentation h.POLDI hammer Comparison of the indentation size of testedComparison of the indentation size of tested

material and reference material with known hardness

calibrated bartested material

Rebound h SchmidtRebound h. Schmidt hammer

measures the rebound of a spring-l d d i ti i t thloaded mass impacting against the surface of the sample

Correlation between Schmidt rebound number and the compressive strength

th b d l b d t d t i the rebound value can be used to determine the compressive strength (by reference to the conversion chart)conversion chart)

D dDepends on: orientation of

th hthe hammer water content

F tiFatigue

f ti h t i l i bj t d t fatigue occurs when a material is subjected to repeated loading and unloading

cyclic stress causes the decrease of the strength

typical for metalsFatigue limit (strength) = the amplitude (orFatigue limit (strength) the amplitude (or

range) of cyclic stress that can be applied to the material without causing fatigue failurethe material without causing fatigue failure

Fatigue if the loads are above a certain threshold,

microscopic cracks will begin to form

Fatiguemicroscopic cracks will begin to form

after reaching critical size, and the structure willsuddenly (without warning) fracturesuddenly (without warning) fracture

the shape of the structure affect the fatigue life(square holes, sharp corners)(square holes, sharp corners)

the greater the applied stress range, the shorter the life

damage is cumulative, materials do not recover when rested

f. is influenced by a variety of factors (temperature, surface finish, microstructure, presence of oxidizing

i t h i l id l t t )or inert chemicals, residual stresses, etc.)

Endurance limit some materials (ferrous and titanium alloys)

h di ti t li it b l hi h thhave a distinct limit below which there appears to be no number of cycles that will

f ilcause failure some structural metals (aluminium, copper)

do not have a distinct limit and will eventually fail even from small stress amplitudes

Whler curves

Fatigue cracksFatigue cracks Fatigue testingFatigue testing

Infamous fatigue failuresg

B t M l Boston Molasses Disaster (Boston, 1919)

Alexander L Kielland Alexander L. Kiellandoil platform capsize (N 1980)

InterCity expres

(Norway, 1980)

InterCity expres(Germany, E h d 1998)Eschede,1998)

Dynamic strengthDynamic strength Tacoma narrows bridge (USA Washington Tacoma narrows bridge (USA,Washington,

1940)

1950