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Design of Reinforced Concrete Structures 1 BEAMS, FLOORS, BRACING
BME Department of Mechanics, Materials and Structures
Dr András Draskóczy
Lecture 5:
DESIGN OF RC BEAMS AND FLOORS, BRACING
Design of Reinforced Concrete Structures 2 BEAMS, FLOORS, BRACING
CONTENT:
BEAMS
Examples
General characteristics
Approximate design of beam dimensions Constructional rules
R.c. cantilevers used as architectural motifs.
FLOORS
Characteristic span ranges of reinforced concrete structures
Advantages of different floor systems
Design of variable height and variable thickness RC floor systems
1 Monolithic solid slabs
2 Partially prefabricated floors
3 Cobiax (or bubble deck) floor slabs
4 Steel-concrete composite floors
Design of Reinforced Concrete Structures 3 BEAMS, FLOORS, BRACING
5 Post-tensioned floors (lecture of Máté Borbás Pannon Freyssinet Kft.)
BRACING
Ways of bracing Bracing systems of reinforced concrete load bearing structures.
Connection of tilting and bracing
Design of Reinforced Concrete Structures 4 BEAMS, FLOORS, BRACING
BEAMS
Examples
Design of Reinforced Concrete Structures 5 BEAMS, FLOORS, BRACING
Series of continuous arched diaphragm beams of Nervi, Orvieto airplane
hangars
Design of Reinforced Concrete Structures 6 BEAMS, FLOORS, BRACING
Monolithic rc cantilever of the Flaminio stadium, Rome by Nervi
Design of Reinforced Concrete Structures 7 BEAMS, FLOORS, BRACING
Variable section arched continuous rc beams of Calatrava,Lyon railway
station
Design of Reinforced Concrete Structures 8 BEAMS, FLOORS, BRACING
Variable section rc frame beam of Calatrava, Lyon railway station
Design of Reinforced Concrete Structures 9 BEAMS, FLOORS, BRACING
Metro line 4 Budapest, Fővám square, slurry wall supporting beams
Design of Reinforced Concrete Structures 10 BEAMS, FLOORS, BRACING
General characteristics of beams
DEFINITION, INTERNAL FORCES Beams are linear members generally supporting vertical loads, with their axis running
horizontally and mainly being subjected to moments and shear forces along their axis
(M+V), but axial force (in frame beams) can not be excluded either. Try to avoid torsion!
TYPES
-Geometry -Axis: straight, curved or eventu-
ally poligonal
-Cross-section: constant, variable, haunched near supports, flanged
(when cast integral with the floor slab)
The way of variation is mainly in connection with the variation
of moments, which can also be
exploited for rainwater canalisation -Static models:
Design of Reinforced Concrete Structures 11 BEAMS, FLOORS, BRACING
-single span and continuous with or without cantilever
-beams can also be part of frames, joining to other members by hinged or rigid joint.
Principal beams and secundary beams, joist floors
Design of Reinforced Concrete Structures 12 BEAMS, FLOORS, BRACING
Ribs
Column
Principal beam
Design of Reinforced Concrete Structures 13 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 14 BEAMS, FLOORS, BRACING
Approximate design of beam dimensions
1 Estimation of the width (b, bw) and height (h):
b, bw is generally determined by technological requirement: b= wall thickness or column width
h estimate by approximate value of the slenderness ratio: l/d≈l/(h-5 cm):
-heavily loaded simple supported beam: l/d≈10 to 12
-simple supported beam with low load intensity: l/d≈14 to 16 -continuous beam, flanged beam: l/d≈ 16 to 18
-prefabricated prestressed beam: l/d≈ 18 to 22
2 Rigidity check in SLS of deformations (limitation of deflections)
wmax l /250
ℓ
Design of Reinforced Concrete Structures 15 BEAMS, FLOORS, BRACING
Simplified check of the deflection by limiting the slenderness ratio ℓ/d
allowable)d/( d
K/
where: ℓ/K distance between 0-moment points,
K: tabulated, (ℓ/d)allowable tabulated (see below!), qp
Ed
p
p
2
1
ykEd
Rd
f
500
M
M
ykrequ,s
prov,s
f
500
A
A
pqp= gk + ψ2qk quasi perm.load
ψ2qk long term part of the variable load
Design of Reinforced Concrete Structures 16 BEAMS, FLOORS, BRACING
Basic values of the allowable slenderness ratio (l/d)allowable for rectangular sections
Concrete
strength
grade
b
pEd [kN/m2] (by beams b is the width of the beam in m, by slabs b=1,0 m)
300 250 200 150 100 50 25 20 15 10 5
≥C40/50 13 14 14 15 17 20 25 27 30 35 47
C35/45 13 14 14 15 16 19 24 26 29 34 45
C30/37 13 13 14 15 16 19 23 25 28 33 43
C25/30 13 14 14 16 18 22 24 27 31 41
C20/25 14 14 15 18 21 23 25 29 39
C16/20 14 15 17 21 22 24 28 37
――„beam” ―――――――→ ←――――――„slab” ―――――
For T-sections and flanged beams use another table of the design aids (DA)
In case of applying pre-camber of the extent l/500, we can add Δ(l/d)allow =4 to the
values of the previous table, whereas in case of a pre-camber by l/250, Δ(l/d)allow =8 can be added to the tabulated values
Design of Reinforced Concrete Structures 17 BEAMS, FLOORS, BRACING
Constructional rules of beams (for information only) 1. Reinforcement designed for bending
Minimum quantity of tensile reinforcement
As,min=ρminbtd
ρmin = max 0,26 fctm/fyk ; 0,0015 see table below
bt medium width of the tension zone
Minimum steel ratio min (%o)
fyk
concrete
C12 C16 C20 C25 C30 C35 C40 C45 C50
500 1,5 1,5 1,5 1,5 1,51 1,66 1,82 1,98 2,13
400 1,5 1,5 1,5 1,69 1,89 2,08 2,28 2,47 2,67
240 1,73 2,06 2,38 2,82 3,14 3,47 3,79 4,12 4,44
The allowable maximum quantity of the total steel cross-section:
As,max=0,04Ac , where Ac is the area of the total concrete cross-section.
At overlaps the double of this quantity is allowed.
Design of Reinforced Concrete Structures 18 BEAMS, FLOORS, BRACING
Partial restraint at beam ends:
Partially restrained ends of monolithic beams must be designed for the calculated restrain moment. The respected restrain moment can not be smaller than 15% of
maximum moment in the span. The section design for 15% of the span moment
should be made even if the beam was considered simply supported. The rule concerning the minimum reinforcement must be respected.
Anchorage of bent-up bars with straight end
The anchorage length in the tension zone must be at least 1,3lbd, in the compression zone at least 0,7 lbd , which should be measured from the intersection point with the
longitudinal reinforcement.
Design of Reinforced Concrete Structures 19 BEAMS, FLOORS, BRACING
Anchorage of the bottom longitudinal reinforcement above support
At least 1/3 of the span reinforcement must be continued beyond the theoretical support point.
At extreme support reinforcement must be anchored for the tensile force FEd given in
point 6.5.1. , and the anchorage should be measured from the internal face of the support.
At intermediate supports one of the solutions indicated below can be applied.
The mostly recommended solution is (d). The figures do not indicate the top reinforcement. Values of db= Dmin are given in section 8.2.
Design of Reinforced Concrete Structures 20 BEAMS, FLOORS, BRACING
2. Shear reinforcement
In case of designed shear reinforcement at least half of the shear force should be
equilibrated by links.
The shear steel ratio: w=Asw/(s . bw
. sin) , where α is the angle between axis
of the beam and axis of elements of the shear reinforcement
The minimum shear steel ratio: w,min=max
001,0;f
f08,0
yk
ck , which can be taken from
the table below.
Values of the minimum shear steel ratio: w,min (%o)
fyk
Concrete
C12/16
C16/20
C20/25
C25/30
C30/37
C35/45
C40/50
C45/55
C50/67
500 1,00 1,00 1,00 1,00 1,00 1,00 1,01 1,07 1,13
400 1,00 1,00 1,00 1,00 1,10 1,18 1,26 1,34 1,41
240 1,15 1,33 1,48 1,67 1,81 1,95 2,05 2,21 2,33
Design of Reinforced Concrete Structures 21 BEAMS, FLOORS, BRACING
If bw>h: w,min=
ww
yk
ckb/h0003,00007,0
001,0
b/h0003,00007,0
f
f08,0,
The greatest allowable spacing of elements of the shear reinforcement
Maximum spacing of links
in general sl,max=0,75d )cot1( mm300;b5,1min w
in case of designed compression steel sl 15 [13], [15]*
is the smallest diameter of the compression steel perpendicular to the beam axis sl,max=0,75d 1000 mm.
Maximum spacing of 45 bent-up bars sb,max=1,5d
Detailing of links
a) b) c) d) e) f) g)
* The Eurocode does not give a rule for beams, the given proposal is the same as the similar one to be applied by compression bars of
columns (see chapter 8.7).
.
Design of Reinforced Concrete Structures 22 BEAMS, FLOORS, BRACING
Open links can only be applied in flanged beams, if there is transverse reinforcement
in the slab. Links cages can also be bent using spot-welded meshes. Related constructional rules
see in point 8.2.
At junction of principal and secondary beam the links of the principal beam must go through, at column-beam junctions links of the column must go through.
3. Torsion reinforcement Links must be anchored with overlap.
Spacing of links can not be greater than a) 1/8 of the concrete perimeter
b) the smaller side length Distance between elements of the longitudinal reinforcement uniformly distributed
along the link perimeter must be smaller than 350 mm.
(In the cross-section indicated on the figure elements of flexural and torsion reinforcement can also be seen.)
Design of Reinforced Concrete Structures 23 BEAMS, FLOORS, BRACING
R.c. cantilevers used as architectural motifs
Design of Reinforced Concrete Structures 24 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 25 BEAMS, FLOORS, BRACING
Some design principles
(Beams)
5
h cc
(Deep beams, walls)
15 to
12h cc
c
(Slabs)
Smaller Fc better:
Greater compression zone better
Design of Reinforced Concrete Structures 26 BEAMS, FLOORS, BRACING
Pile foundation or heavy mass foundation may be necessary to anchor Ft!
Downloading by G better ( tF2
G )
Design of Reinforced Concrete Structures 27 BEAMS, FLOORS, BRACING
Greater restrain length better:
< restrain length >
Symmetric arrangement better:
Design of Reinforced Concrete Structures 28 BEAMS, FLOORS, BRACING
Storey-high cantilever (deep beam cantilever) better:
Design of Reinforced Concrete Structures 29 BEAMS, FLOORS, BRACING
FLOORS Characteristic span ranges of reinforced concrete structures Advantages of different floor systems
Design problems of variable height and variable thickness
RC floor systems 1 Monolithic solid slabs
2 Partially prefabricated floors
3 Cobiax (or bubble deck) floor slabs 4 Steel-concrete composite floors
5 Post-tensioned floors
Design of Reinforced Concrete Structures 30 BEAMS, FLOORS, BRACING
CHARACTERISTIC SPAN RANGES OF RC FLOOR STRUCTURAL SYSTEMS
Construction
type
one-way
two-way
Char.
slender
ness l/d
Approximate maximum span l(m) 5 7,5 10 12,5 15 17,5 20 25 30 35 >35
monolithic rc
solid slab
one-way 25-30
30-35
two-way
prefab.
prestressed
hollow core
floor panels
one-way 35-40
monolithic rc
beams
one-way 10-15
solid flat slab two-way 25-35
prestressed
solid slab
one-way 30-35
prestressed
solid flat slab
two-way 35-40
hollow core
(flat) slab
two-way 30-40
prestressed h.
core flat slab
two-way 35-40
Design of Reinforced Concrete Structures 31 BEAMS, FLOORS, BRACING
CHARACTERISTIC SPAN RANGES OF RC FLOOR STRUCTURAL SYSTEMS (cont.)
Construction
type
one-way,
two way
Char.
slender
ness l/d
Approximate maximum span (m) 5 7,5 10 12,5 15 17,5 20 25 30 35 >35
prestressed rc
beams (used
in building
construction)
one-way 18-22
prestressed rc
beams (used
in bridge
construction)
one-way 18-22
deep beams one-way 5
prefabricated
box-culvert
construction
with post-
tensioning
one-way
Design of Reinforced Concrete Structures 32 BEAMS, FLOORS, BRACING
ADVANTAGES OF DIFFERENT FLOOR SYSTEMS
monolithic or prefabricated floors?
Cheaper, mainly if manpower is cheaper More rapid construction Increases safety through better structural integrity
Better possibilities for individual design Higher level of weather independency
No problem in jointing members Partial pre-fabrication can integrate well all advantages!
one-way or two-way floors?
Mass production by application of hollow Load intensity and moment reduction
core pre-stressed floor panels Deflection reduction
→ structural height reduction
normal or pre-stressed floor structures?
Cheaper Deflection reduction
Better possibilities for individual design → structural height reduction
Design of Reinforced Concrete Structures 33 BEAMS, FLOORS, BRACING
DESIGN OF VARIABLE HEIGHT AND VARIABLE THICKNESS
Beams
Variable beam height for rainwater
canalization (cca 5% fall) from flat
roof of an industrial hall
Favourable effect: shear capacity increase a)
variation on side of the compression zone
tan,cEdEd NVV
b) variation on side of the tension zone ssEdEd NVV tan,
Design of Reinforced Concrete Structures 34 BEAMS, FLOORS, BRACING
Slabs
Favourable effects of the self weight reduction of cantilever slabs achieved by 50% reduction of the thickness at the free extremity
The 33% moment and deflection reduction can also be exploited by cca 10 to 13% reduction of the slab thickness itself.
The floor slab becoming thin looks lighter, having also a positive aesthetical effect.
Design of Reinforced Concrete Structures 35 BEAMS, FLOORS, BRACING
RC STRUCTURAL FLOOR SYSTEMS 1 MONOLITHIC SOLID SLABS
Design of Reinforced Concrete Structures 36 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 37 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 38 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 39 BEAMS, FLOORS, BRACING
2 PARTIALLY
PREFABRICATED
FLOORS
Omnia floors 5 to 6 cm thick prefab
formwork panels strength-
ened by lattice-like reinforcement, provisory
supported at 1,5 to 2 m axis
distances
Design of Reinforced Concrete Structures 40 BEAMS, FLOORS, BRACING
Stadium Debrecen
Design of Reinforced Concrete Structures 41 BEAMS, FLOORS, BRACING
Details of Omnia floors
Design of Reinforced Concrete Structures 42 BEAMS, FLOORS, BRACING
Slim-floor construction details
Hollow core rc floor panels supported by flanged steel beams to reduce the overall
structural height of the floor. In situ concreting constitutes the compression zone and impreves fire resistance. With use of bent-up bars F90 fire resistance can be achieved
Design of Reinforced Concrete Structures 43 BEAMS, FLOORS, BRACING
3 COBIAX (OR BUBBLE DECK) FLOOR SLABS
Design of Reinforced Concrete Structures 44 BEAMS, FLOORS, BRACING
Installation guide
Post-tensioning
Mainova · Frankfort, Germany
Design of Reinforced Concrete Structures 45 BEAMS, FLOORS, BRACING
SPAN RANGE – SPHERE DIAMETER – LOAD INTENSITY DIAGRAM
OF COBIAX FLOORS
Spans [m] with the same slab thickness 22 C
20
18
Loads [kN/m22] 16 B
14
12
10 A
8
6
4
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
d = deck thickness dia. = sphere diameter
A d 23.0 cm / dia. 18.0 cm
B d 40.0 cm / dia. 31.5 cm
C d 58.0 cm / dia. 45.0 cm
Design of Reinforced Concrete Structures 46 BEAMS, FLOORS, BRACING
Omnia products incorporate
the triangular Omnia lattice girder that is attached to a
lower layer of reinforcement
before wet concrete is poured to create the Omnia panel
Design of Reinforced Concrete Structures 47 BEAMS, FLOORS, BRACING
The wet-method The dry-method THREE ALTERNATIVES OF PARTIAL PREFABRICATION AND MOUNTING
The in situ-method
Design of Reinforced Concrete Structures 48 BEAMS, FLOORS, BRACING
Lifting of the ready-assambled floor panel by application of the dry construction
method .
Design of Reinforced Concrete Structures 49 BEAMS, FLOORS, BRACING
Mayor advantages of COBIAX (Bubble-deck) floors:
- up to 18 m span without beams
- biaxial load-bearing, reduced deflections
- up to 30% selfweight reduction - unification of the advantages of prefabrication and monolithic technology
STEEL-CONCRETE COMPOSITE FLOORS
Design of Reinforced Concrete Structures 50 BEAMS, FLOORS, BRACING
4 STEEL – CONCRETE COMPOSITE FLOORS
Flexural resistance of encased steel beam
Alternative solutions
Design of Reinforced Concrete Structures 51 BEAMS, FLOORS, BRACING
5 PRESETRESSED RC SLABS WITH BOUNDED AND UNBOUNDED TENDONS
50 cm thick transition slab with unbounded post-tensioned cables, Jerusalem
Design of Reinforced Concrete Structures 52 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 53 BEAMS, FLOORS, BRACING
The manual pre-stressing jack
Design of Reinforced Concrete Structures 54 BEAMS, FLOORS, BRACING
BRACING
Ways of bracing
Beside solid sheared-walls, bracing of buildings can be assured by use of
diagonals rigid frames frame filling walls
(Andrew-crosses, characteristic for
steel constructions)
Design of Reinforced Concrete Structures 55 BEAMS, FLOORS, BRACING
EXAMPLE: ORVIETO AIRPLANE HANGAR BY NERVI
Design of Reinforced Concrete Structures 56 BEAMS, FLOORS, BRACING
Bracing systems of reinforced concrete loadbearing structures
Design of Reinforced Concrete Structures 57 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 58 BEAMS, FLOORS, BRACING
Design of Reinforced Concrete Structures 59 BEAMS, FLOORS, BRACING
Connection of tilting and bracing
Design of Reinforced Concrete Structures 60 BEAMS, FLOORS, BRACING
END