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1
Prestressed Concrete
Prof. Ing. Jaroslav Procházka, CSc.
Department of Concrete and Masonry Structures
Part 1
(Introduction ; Materials )
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2
Literature:
Standards:
• EN 1992-1-1: Design of concrete structures -Part 1-1: General rules and rules for buildings
• EN 1992-1-2: Design of concrete structures -Part 1-2: Fire design
Monograph, textbook:• Navrátil,J.: Prestressed
concrete structures, CERM Brno, ISBN 80-7204-462-1, 2006
• Procházka, J., Štemberk, P.: Concrete Structures 1, ČVUT, ISBN 978-80-01-03607-5, 2009
• Procházka, J., Štemberk, P.: Design Procedures for Reinforced Concrete Structures, ČVUT, ISBN 978-80-01-04240-3, 2009
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3Plain and reinforced concrete beams
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Prestressed concrete beamprestressing in centre of gravity of cross section
t – topb – bottomz – distance to fiberI – moment of inertia
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Prestressed concrete beam prestressing with eccentricity ep
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Plain and prestressed concrete beams
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Prestressed concrete beam prestressing with eccentricity ep
Elastic section modulus:
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Prestressed concrete beam - prestressing with eccentricity ep
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Prestressed concrete beam - prestressing with eccentricity ep
�� =�
�+
� × × ��
� ⇒ 0 =
�
�+
� ×�� × ��
� ⟶ �� = −
��
� × ��
�, ℎ;�� = −
112
× � × ℎ�
� × ℎ × (−ℎ2)
="
#
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Prestressed concrete beam - prestressing with eccentricity ep
�$ =�
�+
� × × �$
� ⇒ 0 =
�
�+
� ×%$ × �$
�
⟶ �& = −��
� × �&
�, ℎ;�$ = −
112
× � × ℎ�
� × ℎ ×ℎ2
= −"
#
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Calculation of stress in bottom and top fibres
'� =�
��(−)
�� =�
� × �� −=
'�
�(−) �$ =
�
� × �$ +=
'$
�(+)
'$ =�
�$(+)
�$ =�
�+
� × × �$
�+
( × �$
�=
�
�+
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'$
+(
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=�
'$
×'$
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× �$ + +(
'$
�� =�
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× �� + +(
'�
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Assumptions for the calculation of stresses:
1. The strain is directly proportional to the distan ce from neutral axis2. The strain in bonded reinforcement is the same as that in the
surrounding concrete3. Concrete and reinforcing steel is assumed as idea ly elastic material4. The tension concrete is neglected when the tensio n stress reaches
the strength of concrete
h
b
As2
As1
1
2
2
strain stress in concrete
4
3
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h
b
As2
As1
strain stress in concrete
d2
d1 zs1
Non crack section
Fs2
Fcc
FctFs1
forces
zt
zb
idealized (transformed)cross section
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h
b
As2
As1
d2
d1
strainFs2
Fcc
Fs1
stress
xcr
Cracked section
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check crack !!
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Materials
• Concrete• Steel - reinforcing
- prestressing
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Concrete(EN 1992-1-1)
≥ C100/115
very brittle behaviour
≥ C30/37
prestressed concrete
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27R, N, S - depends on cement strength classes: R – CEM 42,5 R, CEM 52,5 N, CEM 52,5 R; N – CEM 32,5R, CEM 42,5N; S - CEM 32,5N
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Reinforcing steel (EN 10 080)Product forms:• Bar - rebar plain, ribbed,
∅ > 8 mm
• Wire – decoiled rods,
∅ ≤ 14 mm• Wire fabrics
Mark of steel
B reinforcing steel
fyk (f0,2k) in MPa
A, B, C ductility class
e.g. B 500 B
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Prestressing steel (EN 10 138)
Product forms:• Wires - prestressing
elements delivered in coil form, ∅ ≤ 10 mm
• Strands
• Bars
Mark: Y prestressing steelnominal tensile strengthe. g. Y 1860 (fpk)
Connection with concrete:
- tendons with bond
- tendons without bond
Location in structure:
- Internal tendons
- External tendons
- Combined tendons
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Wire
Standard strand
Drawn strand
Cable of 7 strands
Dividag bar (ribbed)
Macalloy bar
Types of tendons
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Wire: Ø
3,0; 3,5; 4,0; 4,5; 4,7; 5,0
6,0; 6,3; 6,5; 7,0; 7,5; 8,0;
9,0; 10,0
Mark: DIN
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Strand : e.g. Y 1770S7 15,2core Ø P 5,2 + 6 x Ø P 5,0
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Bar:
e.g. Ø 20, 32 etc.
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External tendons
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Stress strain diagram – prestressing steel
Prestressing tendons- Strength, denoting the
value of the 0,1proof stress (fp0,1k)
- Tensile strength to proof strength (fpk/fp0,1)k
- Elongation at maximum load (εuk)
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εud
σ
fpd/ Ep
fp 0,1k
fpk
fpd = fp 0,1k/γs
A
B
εεuk
fpk/γs
Idealised and design stress-strain diagrams for prestressing steel(absolute values are shown for tensile stress and strain)
A Idealised
B Design