responses of structural materials mechanical ... courses/0_fundamentals of... · department of...
Post on 17-May-2018
214 Views
Preview:
TRANSCRIPT
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 1
Budapest University of Technology and Economics
Department of Mechanics Materials and Structures English courses General course 2018 Fundamentals of Structures BMEEPSTG201
Responses of structural materials Mechanical characteristics testing strength
evaluation
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 2
Introduction
Structural materials are used to construct the loadbearing structural part of building constructions
Loadbearing structures of buildings should safely support loads acting on
the structure (without damage rupture loss of stability falling over or down) and transmit loads to the subsoil under the building
Their most important characteristic is having adequate strength
Strength is the greatest force per unit area that the material can bear
without damage rupture
Tensile strength compression strength and shear strength can be
distinguished according to the way of actuation the force is acting on to the cross-section of the investigated member
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 3
Normal stress and normal strength
Normal stress σ (Nmm2) force per unit area acting perpendicular to the plane
under consideration It can be compression or tensile stress
Normal strength f (Nmm2) capacity force per unit area acting perpendicular
to the plane under consideration It can be compression or tensile strength
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 4
Shear stress and shear strength
Shear stress τ (Nmm2) force per unit area acting in the plane under
consideration Shear strength τR (Nmm
2) capacity force per unit area acting in the plane
under consideration
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 5
Stresses and deformations
Tension Compression
Compression stress provokes shortening
Tensile stress provokes elongation
Torsion
Shear stress ndash due to torsion for example ndash provokes distortion
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 6
Stress-deformation diagrams
visualize the better the mechanical behaviour of the structural materials
Strain is the specific deformation ε (mmm=o) = 119897
119871 or deformation
(that is elongation or shortening) per unit length
Stress-strain diagrams σ=f(ε)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 2
Introduction
Structural materials are used to construct the loadbearing structural part of building constructions
Loadbearing structures of buildings should safely support loads acting on
the structure (without damage rupture loss of stability falling over or down) and transmit loads to the subsoil under the building
Their most important characteristic is having adequate strength
Strength is the greatest force per unit area that the material can bear
without damage rupture
Tensile strength compression strength and shear strength can be
distinguished according to the way of actuation the force is acting on to the cross-section of the investigated member
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 3
Normal stress and normal strength
Normal stress σ (Nmm2) force per unit area acting perpendicular to the plane
under consideration It can be compression or tensile stress
Normal strength f (Nmm2) capacity force per unit area acting perpendicular
to the plane under consideration It can be compression or tensile strength
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 4
Shear stress and shear strength
Shear stress τ (Nmm2) force per unit area acting in the plane under
consideration Shear strength τR (Nmm
2) capacity force per unit area acting in the plane
under consideration
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 5
Stresses and deformations
Tension Compression
Compression stress provokes shortening
Tensile stress provokes elongation
Torsion
Shear stress ndash due to torsion for example ndash provokes distortion
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 6
Stress-deformation diagrams
visualize the better the mechanical behaviour of the structural materials
Strain is the specific deformation ε (mmm=o) = 119897
119871 or deformation
(that is elongation or shortening) per unit length
Stress-strain diagrams σ=f(ε)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 3
Normal stress and normal strength
Normal stress σ (Nmm2) force per unit area acting perpendicular to the plane
under consideration It can be compression or tensile stress
Normal strength f (Nmm2) capacity force per unit area acting perpendicular
to the plane under consideration It can be compression or tensile strength
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 4
Shear stress and shear strength
Shear stress τ (Nmm2) force per unit area acting in the plane under
consideration Shear strength τR (Nmm
2) capacity force per unit area acting in the plane
under consideration
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 5
Stresses and deformations
Tension Compression
Compression stress provokes shortening
Tensile stress provokes elongation
Torsion
Shear stress ndash due to torsion for example ndash provokes distortion
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 6
Stress-deformation diagrams
visualize the better the mechanical behaviour of the structural materials
Strain is the specific deformation ε (mmm=o) = 119897
119871 or deformation
(that is elongation or shortening) per unit length
Stress-strain diagrams σ=f(ε)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 4
Shear stress and shear strength
Shear stress τ (Nmm2) force per unit area acting in the plane under
consideration Shear strength τR (Nmm
2) capacity force per unit area acting in the plane
under consideration
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 5
Stresses and deformations
Tension Compression
Compression stress provokes shortening
Tensile stress provokes elongation
Torsion
Shear stress ndash due to torsion for example ndash provokes distortion
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 6
Stress-deformation diagrams
visualize the better the mechanical behaviour of the structural materials
Strain is the specific deformation ε (mmm=o) = 119897
119871 or deformation
(that is elongation or shortening) per unit length
Stress-strain diagrams σ=f(ε)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 5
Stresses and deformations
Tension Compression
Compression stress provokes shortening
Tensile stress provokes elongation
Torsion
Shear stress ndash due to torsion for example ndash provokes distortion
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 6
Stress-deformation diagrams
visualize the better the mechanical behaviour of the structural materials
Strain is the specific deformation ε (mmm=o) = 119897
119871 or deformation
(that is elongation or shortening) per unit length
Stress-strain diagrams σ=f(ε)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 6
Stress-deformation diagrams
visualize the better the mechanical behaviour of the structural materials
Strain is the specific deformation ε (mmm=o) = 119897
119871 or deformation
(that is elongation or shortening) per unit length
Stress-strain diagrams σ=f(ε)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 7
General requirements of structural materials
structural requirements non-structural requirements
strength space limitation
tensile strength (ft) thermal insulation
compression strength (fc) sound insulation shear strength (fτ) economy prize
deformability density specific weight (γ)
elastic elastic-plastic plastic brittle behaviour ultimate deformation
fire resistance
durability resistance to corrosion re-modelling possibilities
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 8
Mechanical characteristics of some structural materials
Materials Strength Deformability fd γ ratio
compression tension at σ=fd at rupture (kNm2)(kNm
3)=
Nmm2
mmm=o =m
Brick walls ceramic brick+lime mortar
02-1 0 08 4 120
Concrete 10-50 07-4 1 25-35 320
Timber (parallel to fibre direction)
15-30 15-30 2 4 1900
Steel mild 180-280 180-280 1-2 25 2500
high strength steel 1400-1800 1400 1800 7-9 15 17200
Reinforced concrete see concrete see concrete 1200
1kNcm2 = 10 Nmm
2
Max column length that the material can support its weight In compression continuous
lateral support against buckling
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 9
Numerical example
Gmanasymp08 kN= 800 N
Apalm 100x100 =10 000 mm2
soil 1010000
800
A
G
soil
man Nmm2=001kNcm
2
Compare this stress You can feel with material strengths indicated above
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 10
1 Uniaxial stress-strain relationships of structural materials
Definitions
Stress = σ = force per unit area = A
F (Nmm
2)
Stress at rupture= strength (compression or tensile strength according to
the direction of F)
Strain = ε = specific deformation =length
ncontractioor alongation=
l
l
Perfectly plastic behaviour the material deforms under compression or tension without stress increment
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 11
Characteristic mechanical behaviours in uniaxial stress state
elastic rigid-plastic elastic-plastic nonlinear
Hooks law describes the linear elastic mechanical behaviour
E
where E modulus of elasticity (Nmm2)
Definition of E the value of the stress to be applied to double (or ndash in case of compression ndash to halve) the length of the member (ε=1)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 12
Characteristic σ-ε relationships of some important structural materials
timber steel concrete Idealized stress-strain relationships
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 13
Test specimens used for uniaxial tests
timber steel concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 14
Static bending test of a timber specimen Tension test of steel specimen
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 15
Mechanical behaviour of the tested materials
Timber Strength perpendicular to grains is different (very low) Linear elastic
behaviour Strength is similar to compression strength of concrete
Steel
After linear elastic behaviour yield produces great plastic deformation The same strength in compression or tension The strength is approximately
10 times greater than that of timber
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 16
Concrete
Compression and tensile strength is very different After short linear
behaviour non-linear σ-ε curve can be observed Ultimate deformation is
much smaller than that of steel Brickwork
Negligible tensile strength The strength of brickwork is much smaller than
that of ceramic bricks Compression strength is at about by one order of magnitude smaller than that of concrete
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 17
2 Strength as probability variable
21 Statistical strength data no of occurrence relative occurrence
nom
A=1 An
fnom
strength f (Nmm2) strength
Strength distribution diagram Density function of strength
(histogram) distribution
The probability of not exceeding fnom= nn A
A
A
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 18
22 Statistical evaluation of test results (Statistical strength evaluation9
Characteristics of the strength distribution
Mean value fm=n
fi n number of test data (min 10 if
qualification is based on unknown scatter)
Scatter s=1n
)ff( 2mi
Threshold value (characteristic or nominal value)
fk= fnom = fm ndash ts
where t is the so called Student factor which for n=10 and 5 risk is t=179
(5 threshold value the probability of the occurrence of a strength value smaller than fk is 5)
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 19
Relation of characteristic and design value
relative occurrence
strength distribution curve
An
fd fk f =fm strength (Nmm2)
design (d) characteristic (k) and mean (m) strengths
Design value fd= m
kf
The safety factor γm for some important structural materials
115 for reinforcement (steel in reinforced concrete)
13 for timber 15 for concrete
The probability of fd not being exceeded is 01
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 20
Example test data evaluation of a timber test specimen subjected to axial compression in direction of the grains
Specimen data 32x32x5 cm timber No Cross-section Rupture load Rupture strength
A Nu fu=NuA fui- fm (fui- fm)2
mmxmm kN Nmm2 Nmm
2 N
2mm
4
1 32 x 32 350 3418 151 228
2 32 x 32 312 3047 -22 484
3 32 x 32 284 2773 -494 2440
4 32 x 32 375 3662 395 1560
5 32 x 32 348 3398 131 172
6 32 x 32 334 3262 -005 0031
7 32 x 32 372 3633 366 1340
8 32 x 32 294 2871 -396 1568
9 32 x 32 337 3291 024 0058
10 32 x 32 340 3320 053 0281
fm= nfuii =3267510=3267 Nmm2 2
mui )ff( 7829
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 21
Mean value of the rupture strength
673210
75326m
n
ff u Nmm
2
The scatter
s= 1n
)ff( 2mi
=
9
2978 295 Nmm
2
The characteristic compression strength
fk= fm- ts= 3267 ndash 179295middot = 2739 Nmm2
for n= 10 and 5 risc the Student factor t=179 Design compression strength
fd= m
kf
=
31
3927 2107 Nmm
2= 2107 kNcm
2
the safety factor of timber γm= 13
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 22
TESTING OF PREFABRICATED REINFORCED
CONCRETE BEAMS IN2005 IN THE LABORATORY
OF THE DEPARTMENT OF STRENGTH OF
MATERIALS AND STRUCTURES
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 23
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 24
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 25
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 26
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 27
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 28
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 29
FUNDAMENTALS OF STRUCTURES 2018 STRUCTURAL MATERIALS page 30
top related