INTRODUCTION
α RCC= 10 x 10-6 /°C
α STEEL= 12 x 10-6 /°C → Steel used because α is reasonable equal
γPCC = 24 kn/m³= 2400 kg/m³
γRCC = 25 kn/m³=2500kg/m³
μ = 0.1 - 0.3 → Design strength = 0.15, Serviceability Criteria = 0.2
Poisson ratio (μ) → ↑es with a richer mix
E↑es → More elasticity
Francois caignet → 1st to use iron reinforced concrete,developed RCC
Reinforcing steel gives ductility to concrete
IS 432 → Mild steel in RCC
Reinforcement is represented by two horizontal parallel lines.
Steel member t ≥ 6mm (exposed to weather)
Heavily reinforced sectⁿ Compaction factor = 0.85 - 0.92.
f = PL/bd² ← MOR tension test of concrete
When not specified → Steel = 0.6 - 1 % of RCC vol, Slabs = 0.7 - 1%, beams = 1-2%, column = 1-5%.
Properties of concrete can broadly be divided into two → Fresh state & Harden State
CRRI Charts → Concrete strength vs W/C ratio
HYSD are less ductile than mild steel but have more strength
Fe250(hot rolled) mild steel bar →IS 432 & member is designed for working stress.
Number of bars in any direction = (Perpendicular distance or centre to centre distance) + 1
Filler joist → Steel beam of light section
Spacing of main reinforcement → Controls cracking width
Mild steel Fe250 is more ductile → Hence preferred for EQ zones or where there are possibilities of vibration, impact, blast
Structural member is generally designed so that the material → stressed up to working stress
Quality of reinforcing steel is evaluated by → Yield strength, ductility
Strength of commonly used concrete, for constructing low rise residential building = 15000 psi
Live load to be considered for an inaccessible reinforced concrete roof = 75 kg/m²
RBC = reinforced Brick Concrete
Plastering t = 6mm underside of Rcc work
RBLL: reinforced brick lime concrete
RCC stair case max BM = wl²/8
Reinforced Band thickness or depth ≥ 75mm → Masonry building
RCC
Acid soluble Chloride content < 0.6 kg/cum → To avoid corrosion and decaying
Type of Rcc = 2 → Cast in situ & precast
RCC → Equally strong in taking tensile, compressive and shear stresses
min cement content in RCC = 300kg/m³
M40 → Highway (RCC)
Min grade of concrete → IS 456:1978 = M15, IS 456:2000 = M20
Cracks → Shrinkage → Flexure → Settlement → Corrosion
Tensile strength = 0 → Concrete doesn't take any tensile strength in RCC
Aggregate size
Max size of aggregate = 1/4 of minimum thickness of member
For RCC max size = 20mm & PCC = 25mm
Concrete cube size 100mm ≤ 20mm
Concrete cube size 150mm = 20 to 40mm
Cement concrete dam ≤ 40mm
Changing max size 20 mm to 40 mm → min cement content reduced by = 30 kg/cum
Impurities max permissible limit in water (IS 456 : 2000)
IS 456 gives details regarding water to be used in concrete
PH ≥ 6 (6 - 9) → Prevent Sulphate attack
Calcium chloride = 2% weight of cement
Organic solid ≤ 200 mg/ltr
Sugar = 500 ppm
Sulphate = 400 ppm
Chloride = 500(RCC), 2000(PCC)
Sodium & Potassium Carbonate & Bicarbonates ≤ 1000 ppm
Salt & Suspended particles ≤ 2000 ppm
inorganic matter/Sulphuric anhydride ≤ 3000 ppm
Dissolved salts ≤ 15000 ppm
Diff in 7 day CS prepared with impure & pure waters ≤ 10%, Diff in setting time ≤ ± 30 min
1 ppm = 1 mg/L
To prevent corrosion of steel Reinforcement pH value → Alkaline
Water required per 50 kg of cement
M5 → 60 kg
M7.5 → 45 kg
M10 → 34 kg
M15 → 32 kg
M20 → 30 kg
Grade of concrete
Ordinary Concrete (M10 - M20) = 03 → without carrying out preliminary tests
Standard (M25 - M60) = 08
High strength > M60 → design parameter not applicable
Minimum Grade → Reinforced concrete ≥ M20, Water tank and Post-tension ≥ M30, Pre-tension and Road Pavement ≥ M40
Design Method
Factor affecting concrete mix design → Cement grade, Agg shape and size, Workability of concrete, fck, Degree of quality control of concrete
Maximum cement content = 450 kg/m³
i. Nominal mix
Up to M20 only
M5 =1:5:10
M7.5 =1:4:8 → foundation and flooring
M10 = 1:3:6 → flooring
M15 = 1:2:4 → Foundation, PCC
M20 = 1:1.5:3 → Nominal mix, RCC Str (Columns, beams, slabs, cantilever chajja, porch, balcony)
M25 = 1:1:2
M25 → M is mix and fck = 25N/mm² → 150mm size @28days
ii. Design mix (IS 10262:1982)
Guidelines for mix design of non air entertained medium and high strength concrete
Alkali content of cement is not considered/use in design calculation of concrete mix design
Strength
TS = 10% CS
BS = 15% CS
SS = 20% CS
Fatigue ≈ 0 negligible
If the CS of concrete increases then TS also increases but at a decreasing rate
Compressive Strength
Strength of cube → Avg of 3 specimen, individual variation < (± 15)% of average otherwise test rejected
Cube is always tested on sides
Concrete cube = 150 x 150 x 150mm
Cylinder size = 150 x 300mm
Temp = 27 ± 3°C @ 90% humidity for 24±½hr
3days = 1/2 of 28 days strength
7days = 2/3 of 28days strength
3 months = 1.15 of 28 days strength
1 year = 1.2 of 28 days strength = 20-25 % more than 28 days strength
100mm cube > 150mm cube
Cube strength = 1.25 of Cylinder Strength (25%more) → Due to the difference in area of cross section
Cylinder = 0.8 of Cube strength
Core strength = 85% of Cube strength → Consider acceptable
Permissible CS = 0.60 Design CS → fac = 0.60fcd
Number of samples
Depends on the volume of concrete
1-5m³ = 1 Sample , 6-15 = 2, 16-30 = 3, 31-50 = 4 & >50 = 4+1 (for every 50m³ additional sample)
Tensile Strength test
Applying third point loading on a prism
Sample → Briquette (cylinder)
Bending Strength = 0.45√fck
Flexure(modulus of rupture) > Splitting > Direct tensile Strength
Modulus of rupture/direct tensile strength ≈ 2
Uniaxial test on mild steel bar → Lueders line will be inclined at 45 to the direction of tensile stress applied
Flexural tensile strength (fcr)/Modulus of Rupture
fcr = 0.7fck
Used to determine the load at which cracking starts in concrete (onset of cracking)
Modulus of rupture of concrete gives → Flexural tensile strength or TS of concrete under Bending
Modulus of Rupture → Specimen size = 150 x 150 x 700 mm
Cracking Moment = fcrI/y=fcrZ
Splitting tensile strength(fct)
Measured by testing cylinder (150, 300mm) under diametral compression
Split test or Brazilian test → fct = 2P/πDL
fct= 0.66fcr =0.462fck= (7-11)% of CS
Direct tensile strength
TS = k(CS)ⁿ = 0.50fcr=0.35fck
Characteristic strength(fck)
Not more than 5% of test result fail & for concrete it is measured at 28days
fck = fm - 1.65σ
Value of σ → M10 & M15 = 3.5, M20 & M25 = 4, > M25 = 5
σ ∝ mean strength
Calculation of σ → 30 Sample requires
Coeff of variation Cv = σ/μ
Partial safety factor (γ) for material strength → Collapse = 1.5 & Serviceability = 1
Twisted (TOR steel) = 50% more fy than mild
HYSD ↑es bond strength by 60%
Modulus of elasticity (Short term)
Based on initial tangent modulus
IS 456:2000 → E = 5000√fck
IS 456:1978 → E = 5700√fck
E = 5000√fck ± 20%
Relaxation
Loss of stress with time at constant strain in steel
Creep
Creeping → Constant load, Yielding → Not constant load
Creep in concrete → Time dependent component of strain (due to permanent dimension change), Deform under influence of mechanical stress
Steam curing under pressure reduces the effect of creep
Creep ↑es (small,low) → Relative humidity, size/ thickness ratio, aggregate content.
Creep ↑es (high,large) → Temp, w/c, cement content, loading at an early age
Terminal value of creep = 5 years
Creep Coeff (ϕ) = Ultimate creep strain/Elastic strain
ϕ → 7 days = 2.2, 28 days = 1.6, 1 Year = 1.1 → Varies logarithm with time
Ec = E/(1+ ϕ) = 5000√fck/(1+ ϕ) → Long term E
Shrinkage
Time dependent phenomenon ,reduce volume of C without impact of external force due to loss of capillary water
Shrinkage strain = 0.0003
Max axial or direct compression strain = 0.0020 → RCC column
Bending or Flexural strain = 0.0035
Due to shrinkage stresses → SSB having reinforcement only at the bottom tends to deflect downward
Unequal top and bottom reinforcement in RC section results into shrinkage deflection
Shrinkage deflection in case of rectangular beams and slabs can be eliminated by putting → Compression steel = Tensile steel
Type of shrinkage
Plastic s: very soon after curing
Carbonation s: reaction of CO2
Drying s: setting & hardening of cement due to capillary water loss
Autogenous: minor can be ignored.
Factor affecting Shrinkage of Concrete
S↑es → W/C ↑es
S↑es → With addition of accelerating admixture.
S↓es → Relative humidity ↑es (H = 100%, S =0)
S↓es → Agg size ↑es
S↓es → Time↑es → Shrinkage strain ↑es
S↓es → Strength of concrete ↑es
Different method of curing have different rate of shrinkage
Note
Tension steel → Shrinkage reduces tensile stress, Creep increases/produces tensile stress
Compression steel → Shrinkage and creep causes more stresses
Impact factor
Reinforced concrete structure → IF = 4.5/(6+L), L→ Span of bridge
IF ∝ 1/Span of bridge
Moment of Resistance (MOR)
Moment of couple by longitudinal Compression & Tension Force
By over reinforcement MOR can be ↑es max to 25%
RC braced frames/beams maximum Redistribution of moment = 30%
Equivalent Shear force & Moment
Ve = V + 1.6T/Be=Ve/bd
Me = M + (T/1.7)(1 + D/B)
Where V = SF, M = BM,T = Torque, D = overall depth, B = Width of section
Shafts → torque, Ties → tension, Strut → compression, Beams → BM & SF
Types of Reinforcement
Carbon fibre reinforced polymer composite → Repair of column
Fibre Reinforced polymeric(FRP) composite → Replacement of defective/corroded reinforcement
Micro concrete → Flowable, shrinkage free, high early strength concrete
High performance concrete → Heavy duty floors with congested reinforcement
Fibre Reinforced concrete
Composite material consisting of cement, mortar or concrete, discontinuous, discrete, uniformly dispersed suitable fibre
increases → Tensile strength, CS, FS, Toughness & durability of concrete
Controls → Plastic shrinkage Cracking, Dry shrinkage Cracking
Reduces → Bleeding of water, e , Vv, Vw
Asbestos cement fibres → Commercially successful fibres
Glass fibre RC → Cement + Polymers + Glass fibres, Used in ornamental str, fountain, domes
Steel fibre RC
Polypropylene fibre RC
Fibre = 2ndry reinforcement (FRC)
Rock Reinforcement
To stabilise Tunnels, surface, underground mines, and mine roadways intersections
Controlled concrete
for which preliminary tests are performed for designing the mix & it is used for all the seven types of grades of cement
Bacterial concrete
Self healing concrete for crack repair
Underwater concreting methods
Tremie pipe method, Direct placement with pump, Drop bottom bucket, grouting
DESIGN METHOD
WSM & LSM are suggested by IS 456
Ultimate load method (ULM)/Whitney's theory/Load factor method/Ultimate strength method
is more economical than elastic theory method
Optimum use of inherent strength of both steel & concrete is made
Use of Non linear region of stress-strain curves of steel & concrete.
Ultimate strain of concrete = 0.3%
Depth of stress block for a balanced section of a concrete beam = 0.537d
Max MOR for balanced section = σcybd²/3 → σcy = Cylinder CS of concrete
Load used in RCC design = Load factor x Working load
Limitations of ULM
No factor of safety for material stresses
Gives very thin sectⁿ, leads to excessive deformation & cracking thus makes structure unserviceable
Load factor
LF = Ultimate strength/Service load = Avg load /max load = Theoretical design strength/max load expected in service
For Live load = 2.2
For Dead load = 1.5
WORKING STRESS METHOD (WSM)
Elastic Method, Critical method, Modular ratio method
Based on linear-elastic theory → Deterministic approach
Assumes both steel & concrete are linear elastic & obey hooke's law
Stress based method
Stress in concrete, steel is within permissible limit
Working stress of mild steel is determined from the lower yield stress
Working stress < yield Stress
Drawbacks of WSM
Assumes concrete is elastic which is not true
Gives uneconomical section
FOS for stresses only & No FOS for loads
Factor of safety
Direct compression = 4, Bending compression = 3
Concrete/Flexural compression = 3
Steel = 1.82
Permissible stress = Ultimate stress/FOS
Ratio of permissible stress in direct compression and bending compression < 1
Design load = Characteristic load with FOS
Modular ratio
Modular ratio = Ratio of E of two materials
Short term → m = Es/Ec = Es/5000√fck=fsteel/fconcrete=Aconcret/Asteel
Long Term → m = 280/3σcbc → Partially takes into account the long-term effects such as creep → for tensile bars
m↑es → Due to creep
Reinforced brick → m = 40
Loss of stress due to elastic deformation of concrete depends upon modular ratio
Equivalent Area = Ac + m.Asc
fsteel=mfconcret
Neutral axis
BX²/2 =mAst(d-X)
Xc=280d / (3σst + 280) = kd → NA depends only on σst
Xc = mcd / (mc+t) = md/m+r
k= m/m+r=280/(3σst + 280) → Depends on only σ st
r = t/c, c =cbc, t =st
Eff depth (d) → Centroid of the area of tension in reinforcement and the maximum compressive fibre
NA of the reinforced beam passes through → Centroid of the transform section
MOR
M = Qbd²=(1/2)cjkbd2
Lever arm → J = d - X/3
MOR factor → Q = cjk/2
Q = 0.87 (M15 & fe250), = 0.91 (M20)
MOR = σ st Ast (d-X/3) → Under reinforced section
MOR = (1/2) σ cbc B X (d-X/3) → Over reinforced section
Economical percentage(%) of Steel (P)
P =Ast/bd = 50K²/m(1-K) = 50x² / md(d-x)
m = modular ratio, K = mc/(mc+t) = x/d
LIMIT STATE METHOD (LSM)
Probabilistic Approach, Strain based
Gives most economical sectⁿ
In LSM basis of analysis of structure → Linear elastic theory
Max principal strain theory predominant
Uses multiple safety factor format
Bearing stress at bends for LSM = 1.5 x WSM
Probability of failure = 0.0975 → Order of failure 10-2(0.01)
End of side covers for steel bar in RCC work = 4cm - 5 cm
LSM takes concrete to → Higher stress level than WSM
Failure criteria for beam and column → Based on max principle strain theory
Spacing of main reinforcement → Controls Cracking
Deflection is completed by using sort and long term values of young modulus
Assumption (LSM)
Plan sectⁿ before = After → Strain ∝ Distance from N.A.(y) → Strain distribution is Linear
TS of Concrete = 0
Max Axial strain concrete = 0.0020
Max Flexural or bending strain concrete = 0.0035
Max strain in concrete = 0.0035 - 0.75 x (strain at least compression side) → if no tension in section
Max strain in reinforcement(Steel) → ε > fy/1.15E + 0.002 = 0.87fy/E + 0.002
Characteristic
Strength → fck = fm - 1.65σ
Load → fck = fm + 1.65σ
Design load max of (Limit state of collapse)
= 1.5(DL + LL)
= (1.5 or 0.9)DL + 1.5EL/WL → 0.9 when Stability against overturning or stress reversal is critical
= 1.2(DL + LL + EL/WL)
DL is permanent & constant assumed as per IS:875 (part-1)
Wind and seismic loads are not considered simultaneously
Rain load isn't considered in design
Ordinary Building → Staircase load = DL + 0.5LL
Massey irregulatory shall be considered to exist → When the seismic weight of any story > 200% of that of its adjacent story
Note:- Limit state of serviceability → factors for loads = 1
Design strength (fd)
Compressive strength of concrete = 0.67fck = fck/1.5 → in actual structure
Design Compressive strength of concrete = 0.45fck = 0.67fck/1.5 → Permissible stress in Concrete
Design Tensile strength of steel = 0.87fy = fy/1.15 → Permissible stress in Steel
Allowable Tensile stress σ st = fy/1.8 = 0.55fy → σ st = fy/1.78 appx
Concrete Stress Block
CF = 0.36 fck b Xu → Act at 0.42 Xu = 3Xu/7 from top
Area of concrete stress block = 0.36 fck Xu
T = 0.87 fy Ast → Act at tensile reinforcement
Max strain at top fibre = 0.0035
Max strain upto point having Uniform stress = 0.002
Depth of uniform stress = 3/7 of Xu from top
Depth of parabolic = 4/7 of Xu from N.A
Stress vs Strain curve for concrete
Up to 0.002 strain → Parabolic
0.002 - 0.0035 strain → Straight
Mu lim
Mu = 0.36 fck b Xu (d - 0.42Xu) = 0.87 fy Ast (d - 0.42Xu)
C = F → 0.36fckbXu = 0.87fyAst
Ast/bd=0.36fckXu / 0.87fyd
Mu > Mulim → Either section dimensions need to be modified or higher grade of steel/concrete to be used
Xu lim
Xu ∝ Ast ∝ fck ∝ 1/fy ∝ Es
Xulim = k.d
k=Xulim/d = (0.0035)/(0.0055 + 0.87fy/Es)=700/(1100+0.87fy) → Depend only upon grade of steel
Xu = 0.87fyAst / 0.36fckb
t → Direct tensile strength, cc→ Direct compression, cbc→ Bending compression
Eff depth
Singly reinforcement beam → measured from compression edge to tensile reinforcement
d = D - Eff cover
Effective cover = Clear cover + ϕ/2
Beam Section
Ast ↑es → N.A. ↑es
N.A. shift upwards as load ↑es beyond Fy
Ductility/Design criterion → Under reinforced section is preferred
i. Under reinforcement section
Xu < Xulim, Ast < Ast lim, MOR < MOR balanced
Tensile strain in steel reaches yield value while maximum compressive strain in concrete < its ultimate crushing strain
Steel attains max stress earlier(σ st = fy )
Tensile or ductile failure or secondary compression failure → Failure of steel
ii. Over reinforcement
Xu > Xulim, Ast > Ast lim, MOR > MOR balanced
Concrete reaches a maximum strain of 0.0035 before Steel yields
Concrete attains max stress earlier (σ c = fck)
Compressive or brittle failure, primary compression failure or flexural collapse → Failure of concrete
iii. Balanced/economic/Critical reinforcement
Xu = Xulim, Ast = Ast lim
Both steel & Concrete attains max stress simultaneously
Gives → Smallest Concrete section and Max area of reinforcement(Ast)
Nominal/Clear cover
Minimum cover ≥ dia of bar(ϕ)
Concrete cover at end of Reinforcement bars > (25mm, 2ϕ)
Slab > 15mm or ϕ
Beam > 25mm or ϕ
Column > 40mm (generally) & 25mm(d <12mm) or ϕ
Rcc water tank > 40mm or ϕ
Footing > 50 mm or ϕ
Other reinforcement > 15 mm or ϕ
Durability criteria
Mild if main12mm→ Then cover reduced by 5mm
Severe and very severe if > M35 → Then cover reduced by 5mm
Surface width of cracks
Under mild exposure or in General < 0.3 mm
Structure exposed to Ground water or continuous moisture condition < 0.2 mm
Aggressive environment (Severe,very severe, extreme exposure) < 0.1 mm
Permissible deviation
Dimension of c/s Column and Beams = ±12mm, -6mm
Footing Dimension in plan = +50mm, -12mm
Minimum width for fire exposure
Beam → Fire exposure ≤ 2 hr = 200mm, 3hr = 240mm, 4hr = 280mm
Floor → for 2hr Fire exposure = 125mm
Column → for 2hr = 300mm
Expansion joint
Expansion joint → movement joints
i) RCC Structures → L > 45m
ii) Load bearing brick structure ---- 30m
iii) Boundary wall ---- 10m
iv) Overhanging members--- 6m
Note
For HYSD Fe500 → Permissible stress in direct tension and flexure tension = 0.55fy
SHEAR, BOND & ANCHORAGE
Shear design for a prestressed concrete is based on Elastic theory
Reinforcement provided in the compression zone which extends into the support also provides shear strength to the sectⁿ
When shear stress exceeds the permissible limit in a slab, then it is reduced by → Increasing the depth
Shear span → SF constant, Contraflexure → BM changes sign
Type → flexure, torsion, punching shear
Max shear stress in concrete = Shear force/(lever arm x width)
q = S.F./ L.A.xB (Rcc beam) or Bs = Q/(JD*S)
Shear stress is ↓esd → by ↑esing d
Torsion resistance capacity → Increases with the increase in stirrup and longitudinal steel
Torsion : both longitudinal & shear reinforcement
Shear stress distⁿ in RCC sectⁿ
Compression zone (above N.A) → Parabolic
Tensile zone (below N.A) → Rectangular or Constant
Zero at top of compression zone
Max shear stress → At the neutral axis of the section
Diagonal Tension
Caused in the tensile zone of the beam due to shear, at or near the supports → Vertical and horizontal shear stress
Prevent → by shear reinforcement or Diagonal tension reinforcement
Diagonal tension → increases below NA and Decreases above NA
Permissible diagonal tensile stress in reinforced brick work = 0.1 N/mm²
The chance of diagonal tension cracks in RCC member reduce → Axial compression and shear force acts
Form of Shear Reinforcement
Vertical bar, bent-up bar with stirrups, inclined bar
Shear failure without shear reinforcement = Plane inclined 30 degree to the horizontal → At sections of SSB, Cantilever
Shear force resist by → Concrete, Vertical stirrups and inclined bars, Reinforcement provided into tension zone which extend into support
Spacing of shear stirrups → Decreases towards support(min), Increase towards centre(max)
Minimum shear reinforcement in beam → To resist Principle tension, cracks due to shrinkage, sudden failure of beam, hold the main reinforcement
For Reversible shear → Combination of vertical and inclined stirrups
i. Vertical stirrups
Best for load reversal cases
Spacing = less of (0.75d, 300mm & 0.87fyAsv/0.4b)
ii. inclined stirrup
Spacing = less of (d, 300mm & .87fyAsv/0.4b)
Asv = Area of stirrup bar
iii Bent up bar with vertical stirrups
At support @45° → Resists SF & -ve BM
Main tensile reinforcement bend at appropriate location & always provide in combination with vertical stirrups
Bent Up bar Shear resistance contribution < 50% total shear
EQ resistant design both end of vertical stirrups on a beam should be bent → at 135 degree
Cranked bar
Bending of a bar near at support is 45° → To resist shear cracks
Crank/inclined length for 1 bent up (=d/sin) → 45° = 1.42d, 30° = 2d
Length covered by crank → (=d/tan) → 45° = 1d, 30° = 1.73d
Extra length required for 1 bent up (=d/sin - d/tan) → 45° = 0.42d, 30°= 0.27d
Total length of bar → 45° = L + 2 x 0.42d, 30° = L + 2 x 0.27d
Curtailment → at eff d or 12 x bar dia
Additional length
Straight bar = 0
Bent-up at one end = 0.42D - Cover
Double bent-up = 2 x 0.42D - Cover = 0.84D - Cover
Critical section for shear
RCC beam frame buildings → d from edge of support
Design For Shear
Min spacing is provided near support bcz SF is max at support
LSM → τc max ≈ 0.63√fck > τ → Based on Diagonal compression failure
τ > τc max → Dimension of beam needs to be changed, Design for shear stress
Permissible shear stress (τc) & shear strength of rcc beam → Depends on grade of concrete(fck), % Steel in tension(Ast) and Shear reinforcement provided
Max allowable shear stress (τc max ) → Depends only on Type/Grade of Concrete
Design SF (Vu)
Vu = 1.5 x V = 1.5 (wl/2) → for UDL
Max shear stress → max=1.5avg= 1.5 V/BD
Shear design for prestressed concrete beams → Based on elastic theory
Shear force resist by concrete only → Fc=bdc
Case 1 → τv > τc
Design for SF = (τv - τc)bd = Vu
Case 2 → 0.5τc < τv < τc
Provide min. shear reinforcement
Asv/b.Sv ≥ 0.4/0.87fy --> Asv = 40Sv/fyd
Case 3 → τv > τc max → Redesign
τv = V/bd
Design SF = V - w x d
High shear case → V > 0.6 Vs
Nominal shear stress = Vu/bd = 1.5V/bd
Bond strength
Bond strength b/w reinforcement and concrete → Affected by Steel properties, concrete properties and shrinkage of concrete
Pullout test → Bond b/w steel & concrete
Adhesion of the concrete to steel is not perfect within elastic limit → Concrete will be overstressed
Bond stresses → The longitudinal shearing stresses acting on the surface b/w steel and concrete
HYSD in place of mild steel → increases Bond strength but decreases ductility
Methods To Improve Bond Strength
Most economical method to ↑es τbd is use → More no of thinner/smaller bar
↑es grade of concrete, Use deformed bar, ↑es cover
Provide → bends, hooks, mechanical anchorage
Design bond stress (τbd)
LSM → τbd (MPa) = 1.0, 1.2, 1.4, 1.5, 1.7 & 1.9 for M15, M20, M25, M30, M35, M40 & above
WSM → τbd (MPa) = 0.6, 0.8, 0.9, 1.0, 1.1 & 1.2 for M15, M20, M25, M30, M35, M40 & above
HYSD(Deformed bar) → ↑es above value of τbd by 60%
Bars in Compression → ↑es above value of τbd by 25% for HYSD
The main reason for providing number of reinforcing bars at a support in a simply supported beam is to resist in that zone Bond stress
Development length
Ld = ϕσs/4τbd = 0.87fyϕ/4τbd
σs = 0.87fy = fy/1.15
Deformed or HYSD bar → Ld=0.87fyϕ/6.4τbd → τbd ↑es 60% for Deformed bar
HYSD bar in compression → Ld=0.87fyϕ/8τbd → τbd ↑es 25% more for Deformed bar in Compression
Ld for bundled bar is ↑es By → 2 bar in contact = 10%, 3 bar = 20%, 4bar in contact = 33%
Ld → HYSD < Mild steel, Compression < Tension
Deformed bars may be used without end anchorage → Development length required is satisfied
Load → P=bdLd
Embedment Length
Development length due to flexure
Ld ≤ M1/V + Lo
Lo = max (d, 12ϕ)
V = SF, M1 = MOR ... Stressed to 0.87fy
Ends of reinforcement confined by compression → Then M1↑es by 30% → Ld ≤ 1.3M1/V + Lo.
Bends & Hooks
Anchorage value of bend/hook = 4ϕ for each 45°turn → 90°hook = 8ϕ , 135° = 12ϕ, Std. or U Type or 180° hook = 16ϕ
Length of one Hook = 9ϕ
Total length of bar having hook at both end = L + 18ϕ = L + 18D
For compression → Anchorage length is not considered
Length of straight portion beyond end of hook ≥ 4ϕ & ≤16ϕ
Radii = kϕ (mild steel → k = 2 & HYSD → k = 4 )
Tensile bar must be anchored at support → Cantilever = Ld, SSB = Ld/3, Continuous = Ld/4
RCC roof straight bar length of hook = 9D
Lap Length
minimum length that must be provided if two bars are joined together such that forces can transfer safely.
Straight length of lap ≥ (15ϕ, 200mm)
Compression ≥ (Ld, 24ϕ)
Flexural tension ≥ (Ld, 30ϕ)
Direct tension ≥ (2Ld, 30ϕ)
ϕ = dia of smaller bar
two diff dia bars used lap length should be calculated on the basis of avg dia
Lap Splices → Not permitted for rebar if bar Dia > 32mm
Splicing → Done by Lapping of bars in a RCC beam
Splicing of flexure member Is taken at location → Where bending moment < 50% of the moment of resistance at that section → Not more than 50% of bars are spliced at any particular section
When reinforcement bars placed short of their required length need to be extended → use Splices
BEAM
Beam → Resists BM & SF
Curved beams → Designed for BM, Shear and Torsion
The pin of rocker bearing in a bridge → Design for Bearing, Shearing and Bending
Assumption for beam → d = 2B = Span/30
Spandrel Beam → Beam supporting load from the floor, slab, as well as from wall
Ring bean → TS concrete σ t = σ h / (bd + (m-1)xAst)
Max moment redistribution in beam = 30 %
Most economical type of RCC beam → Singly reinforced T-Beam
Material efficiency → T-beam > Rectangular beam > 2 way Slab
Cantilever porch and beam → Main reinforcement at Top surface
If beams are laterally unsupported → Lateral torsional buckling will occur
Deep beams
Acc to is 456 Deep beam → L/D < 2 (SSB), L/D < 2.5 (Continuous Beam)
Deep beams → Designed for Bending moment only & checked for Shear Deflection
Design takes in account → Lateral buckling, Temp stresses, Non-linear distribution of stresses
Continuous beam
Length of End span = 0.9 x intermediate
Top reinforcement at the support section of CB shall be extended → Span/3 from either face of the column
Moment and shear coefficients used → Spans do not differ more than 15% of longest span
Shear Force Coefficient at the support next to the end support and inner side of mid span = (0.55Wd + 0.60WL)Lc
Max BM next to end support = WDL2/10 + WLL2/9 → Three span cb
Bending moment coeff for CB
Singly Reinforcement Beam
Compression by Concrete , Tension by steel
Limiting reinforcement index ∝ fck/fy
Depth ratio at the limit state of collapse → Parabolic/rectangular portion block = 4/3
Effective length (Leff)
SSB = min of (Lo + d) or (Lo + w1/2 + w2/2) or Centre to centre distance b/w supports.
Cantilever = Lo + d/2
Continuous Beam or slab = Same as ssb if w < Lo/12, Otherwise min of (Lo + d) or (Lo + w1/2 + w2/2)
Deflection(δ)
δ ≤ Span/250 → Final deflection due to all loads including the effect of temperature, creep and shrinkage and measured from the as cast level of the support of floors, roofs and all other horizontal members
δ ≤ Span/350 or 20mm → Final deflection ... Erection of partition & Application of finishes
δ ≤ Span /300 → Applied to prestressed concrete member
IS 800:2000 → δ ≤ L/325
Span/Depth Ratio (Span ≤ 10m)
To satisfy vertical deflection limits
L/D ratio Depends on → Span, Ast, fy & Area in compression.
Cantilever Beam ≤ 7
SSB ≤ 20
Continuous Beam ≤ 26
Span > 10m → Multiply above values by (10/Span) factor, & calculate actual deflection for cantilever beam
Use of HYSD results in increase in depth from point of limiting deflection
Slenderness Limit
To ensure lateral stability
SSB or Continuous beam L ≤ min. of (60b, 250b²/d)
Cantilever beam ≤ min of (25b, 100b²/d) = 2/5 of ssb
Steel Reinforcement
Astmin= 0.85bd/fy → Astmin/bd ≥ 0.85/fy
Astmax ≤ 0.04bD or 4% → Astmax/bD ≤ 0.04 → Both Tension and compression
d = D - Eff cover
Ast depends on → fck, fy, Geometry of the section
Min reinforcement → Prevent surface hair cracking, prevent sudden failure, in the form of stirrups to resist principal tension
Torsional longitudinal reinforcement → Placed at each corner of the beam
Ast = (0.5fck/fy) (1 - 1 - 4.6Mu/fckbd² ) bd
Max dia bar
ϕ ≤ 1/8 of Least dimension of beam
Side Face Reinforcement
SFR → Ast = 0.1% of web area = 0.1bD/100 → Equally distributed on both face
Max spacing = min of ( 300mm, Width of beam)
SFR provided → D > 750mm & D > 450mm (Beam subjected to torsion)
Horizontal distance b/w two parallel bar
> Dia of thicker Bar if unequal
> Dia of bar if equal
> 5mm more than nominal max size of coarse aggregate
Vertical distance b/w main reinforcement
> 15mm
> Dia of larger bar
> 2/3 rd of Nominal max size of coarse aggregate
Doubly Reinforcement Beam
Provided → When to ↓es Deflection, ↓es Torsion, Size is restricted, EQ zone, MOR of singly reinforced section < Design moment
Use of compression steel → Reduce long term deflection, Increase ductility of beam, used as anchor bars to hold shear stirrups
Advantage of doubly RB → Reduction in long term deflection due to shrinkage and creep, Prevents beam in reversal of moments
Doubly is less economical than single Reinforced beam → Bcz Compression steel is under stress and unutilised
εc= 0.0035 (1 - d'/Xulim) → Strain at level of compression reinforcement
Asc = (Mu-Mulim)/fsc(d-d’)
Max compression reinforcement < 2% of gross c/s area of beam
Equivalent area of concrete = 0.5mAsc
When a beam section span over several supports continuously then the beam at supports should be designed as Doubly reinforced beam
Allowable stress in compression steel < permissible stress in tension steel → Rectangular DRB
T-Beam
Takes span moment
Breadth of rib = (1/3 - 2/3) of Depth of rib
depth = (1/10 - 1/20) of Span
Leff = 20 x D
deff = Span/12 → Top of Flange to centre of tensile reinforcement
Heavy loads → deff = Span/10
T beam behaves like rectangular beam of width equal to its flange → if Neutral axis remains within the flange
Max shear stress → at neutral axis
Effective flange width
Lo → Dist b/w points of zero moment's in the beam
bf → Actual width of flange
Continuous beam → Lo = 0.7Leff
SSB, Series of beam supported on masonry wall → Lo = Leff
SLAB
Purely simply supported slab is not possible
max agg size depends on → Clear cover, spacing & t of sectⁿ
Deflectⁿ of main reinforcement→ fⁿ of Short Span
Shear & bond stress are very low
Yield line theory → Results in Upper bound solution
Floor slabs in buildings → Thickness > 7.5 cm
i). One way slab (ly/lx > 2)
Main reinforcement → Along shorter span → Most of the load is carried on the shortest span
Bend in one direction only along shorter span
Max BM at a support next to end support
RCC stairs → The tread slabs are designed as one-way slab
One way continuous slab → Max B.M. = At a support next to end support
ii). Two way slab (ly/lx ≤ 2)
Main reinforcement (cranked bar) → Both side or both direction
Shear ↓es → ↑es t of slab
For fixed condition → -ve moment/+ve moment = 2.5
Reduction in BM = 5/6 x (r²/1+r⁴) x BM, r = Ly/Lx.
There is no(zero) need of torsion reinforcement if edges are continuous
Short term deflection depends on the short span
For +ve moment → 100% reinforcement must be provided up to a distance = 0.15L from the simply supported edge
Width of each edge strip = 1/8 th of Panel width(Lx or Ly), Width of middle strip = 3/4 the Pane width(Lx or Ly)
Effective Length
Clear Span + effective depth or width = c/c distance b/w supports = dist b/w centre of the bearing
Leff = min of(Lo + d) or (Lo + w1/2 + w2/2)
Effective width
Cantilever slab < L/3 → L = Length of cantilever slab measured parallel to the fixed edge
Design purpose → Width = 1000mm = 1 meter
Span to depth ratio (leff/d)
To satisfy vertical deflection limit
d = depth or thickness, leff = shorter length
Cantilever slab → leff/d = 12
leff/d → Fe250 = 35, HYSD = 28 → SSB 1D(1way), SSB 2D(2way) & Continuous slab spanning in one direction or 1 way c slab
leff/d → Fe250 = 40, HYSD = 32 → Continuous slab Spanning in two direction or 2 way c slab
for HYSD multiply by 0.8 → HYSD ↑es d
Note → Basic Span/deff = 20 → Simply supported slab spanning in one direction
Ast min
Ast min → HYSD = 0.12% Ag, Mild steel bar = 0.15% Ag
Ast max = 4 % = 0.04bD
Amount of reinforcement for main bars in a slab is based upon max BM
Ast = (0.5fck/fy) (1 - 1 - 4.6Mu/fckbd² ) bd
Transverse reinforcement
Provided right angle to span → 60 % of main reinforcement at midspan
Min and Transverse/distribution reinforcement → To distribute shrinkage stress, Temp stresses and load, Also assist in keeping the main bars in exact position
Torsional reinforcement/Corner steel
To counter uplift of corner edges, to resist torsional moment
Corner steel = 0.75 x (the area of steel provided at mid span in the same direction)
in the form of a mesh or grid → Both at the top and bottom faces of the slab → If corner of slab are not allowed to lift
Mesh Size=0.25lx0.25ly
Corners held down → Provided at discontinuous edges
Meeting ages are continuous → No torsional reinforcement
Max dia bar
ϕ ≤ 1/8 of total thickness of slab → t = 8ϕ
Max distance b/w bars
Main or bottom bars ≤ (3d or 300mm)
Secondary/distⁿ bar ≤ (5d or 450mm)
Shrinkage and temperature bar ≤ (5d or 450mm)
Spacing = (Bar Area / Ast)x1000 ∝ (dia of bar)² → S2/S1 = (ϕ2 / ϕ1)²
Cover
Max of → main bar dia or 15mm
Torsional Reinforcement:
Provided at both Top & Bottom faces
Bent Up bar in slab
Alternative bars are curtailed → At a distance of 1/7 from centre of slab bearing
To resist -ve BM at support, Resist SF which is higher at support
Rankine-Grashoff
Design of two way slab simply-supported on edges and having no provision to prevent the corners from lifting
wx = Ly⁴ / Lx⁴ + Ly⁴ & wy = Lx⁴ / Ly⁴ + Lx⁴
wx ∝ Ly⁴ & wy ∝ Lx⁴
wx/wy = (ly/lx)⁴
Mx = αxwlx², My = αywlx²
Marcus correction factor to moment < 1 ← For a slab supported on its four edges with corners held down and loaded uniformly
Types of slab
Flat slab
Beam less floor slab supported directly by columns → Columns built monolithically with slab
If the beams in slabs are restricted as per the Architects plan than flat slabs are used
Eff width of column strip = 1/2 of panel width
Eff width of middle strip = 1/2 of panel width
Design of column strip and Middle strip is carried out separately
Critical section for shear → d/2 from Periphery of column/capital/drop panel
Minimum thickness = 125 mm, End panel with drop = L/32, End panel without drop = L/36
Interior Span → (-ve) design moment = 0.65 x Total design moment, (+ve) design moment = 0.35 x Total design moment
(-ve) moment → At interior support = 75 % of total, At exterior support = 100 % of total
(+ve) Moment → 60 % of total moment
Parts → Drop panel, Capital, Slab, Column
Capital → Enlarged head of supporting column
Drop panel → increase shear strength, Thickened part over its supporting column
Dia of column head = 0.25 x Span length
Non-Cellular Non-Ribbed flat Slab → Spacing of reinforcing bar < 2 x Thickness of slab
Continuous Floor Slab
Length end span = 0.9 x intermediate
t ≥ 9cm floor slab
BM = WL²/12 → Near middle of end span due to dead load
Ribbed Slab
Bar dia ≤ 22mm
Agg size = 10mm
t = 5 - 8cm←topping of ribbed slab
Clear spacing between rib ≤ 4.5cm
Width of rib ≥ 7.5cm
Overall depth of slab ≤ 4 x breadth of rib.
Plain ceiling, Thermal insulation, Acoustic insulation, fire resistance
Circular Slab
i. Fixed at ends & UDL
Max +ve radial moment = wR²/16 at centre
Max -ve radial moment = 2wR²/16
-ve/+ve = 2
ii. Point load W
Max Radial or Circumferential moment = 3WR²/16 → Simply supported slab
Circular slab subjected to external loading deflects to form Paraboloid
COLUMN
Best section in Compression → Thin Hollow circular cylinder
Secant formula → Allowable stress in Axial compression (BIS)
Short column → Crushing Failure
Long column → Buckling, large lateral deflection
STEEL → λ = leff/r → Short column ≤ 32, Medium column = 32 - 120, Long column ≥ 120
RCC → λ = Leff/LLD → Pedestal ≤ 3, Short column = 3-12, Long column ≥ 12
Pedestal/Strut → leff > 3 x LLD
Width → wall > 4t, Column < 4t
Composite sectⁿ is best for economically loaded Strut
Built-up section → λ = 50
Tolerance limit of deviation of distance b/w adjacent columns = ± 5mm
Leff
Fix-Hinge = L/√2 (.8L)
Fix-Fix = L/2 (.65L)
Hinge-Hinge = L (L)
Fix-Free = 2L (2L)
Fix-Roller = 1.2L
Fix-Partially fix = 1.5L
Eccentricity (e)
Rectangular or Square ≥ (Leff/500 + LLD/30 or 20mm)
Non rectangular & Non Circular ≥ (Leff/300 or 20mm)
emin ≤ 0.05LLD → Short axially loaded column
Longitudinal/Main Reinforcement
Ast min ≥ 0.8% Ag
Ast max ≤ 6% Ag → Lapped Splices ≤ 4% Ag
Dia ≥ 12mm
No of bar → Rectangular/Square = 4, Circular = 6, Octagonal = 8
c/c spacing ≤ 300mm → Longitudinal bars
Nominal/Clear Cover ≥ 40mm → Small column(or if bar dia < 12mm) = 25mm, Contact with fire or water = 50mm
Slenderness ratio(λ) →End restrains ≤ 60LLD, One end unrestrained ≤ 100B²/D
Lateral Ties/Stirrups/Transverse Reinforcement
independent of grade of steel
To resists buckling of longitudinal steel bar → Binding steel & Proper distⁿ of concrete
Dia (ϕ) ≥ (Largest long dia/4 or 6mm)
Spacing (Pitch) ≤ (LLD, 300mm, 16 x smallest longitudinal dia, 48 x transverse bar dia)
Spacing (Pitch) ≥ 60mm
Helical Reinforcement
5% more Strength
No of bars ≥ 6
Dia(ϕ) → Same as Tie
Pitch < (Core dia/6 or 75mm)
Pitch > (3 x dia of tie or 25mm)
Core dia = Dia of column - 2 x Clear cover
Concrete inside core → Subjected to triaxial compression stress → for short column
HR Columns are very much suitable for EQ resistant structure
Formula(LSM)
Pu = 0.45fckAc + 0.75fyAsc → Truly axially loaded columns (e = 0)
Pu = 0.45fckAc + 0.67fyAsc → Short column axially loaded column
Pu = 1.05 x (0.45fckAc + 0.67fyAsc) → Short axially loaded column with Helical reinforcement
Ac = Ag - Asc
Asc = 0 or -ve → Provide min Asc = 0.8% Ag = 0.8bD/100
Ultimate or collapse load = 1.5 x Load carrying capacity
Short column (WSM)
Pu= σccAc + σscAsc
σsc=mσcc
Long column (WSM)
Pu= Cr(σccAc + σscAsc) → if leff/LLD > 12
Reduction coeff → Cr = 1.25 - leff/48LLD= 1.25 - leff/160rmin
r = least radius of gyration
FOOTINGS
Depth of footing → Calculated for Bending moment & checked for Shear
Design of rcc footing → A flexible base and Linearly Pressure distribution is assumed
Footing Area = Total load / Safe bearing capacity
Foundations of all the columns of a structure are design on the total live and dead load basis → The settlement of exterior columns will be more than interior columns
Points of suspension from ends for lifting Pile → 0.207L
During erection, the pile of length L is supported by a crane at a distance of 0.707L From the driving end of pile which rests on the ground
Designing the pile as a column, the end conditions are → One end fixed and other end hinged
Degree of freedom of a rigid block Foundation = 6
FOS in foundation design = 2 - 3
Specification
Min depth of foundation = 500 mm
Min nominal cover = 50mm → Flexural reinforcement, Durability criteria,
Min thickness at edge of footing for RCC & PCC → Rest on piles top ≥ 300mm, Rests on soil ≥ 150mm
Depth of footing
Rankine formula → Df = qKa²/γ =(q/)(1-sin(ϕ)/1+sin(ϕ))2
Punching shear consideration → D = W(a²-b²) / 4a²bq → Square footing
Isolated footing → depth govern by shear force, punching shear and max bending moment
Net upward pressure = pb(b-a)2/8 → on a square footing(side = b) and column (side = a)
Critical Section
2 way shear or Punching shear → d/2 from face of wall
1 way shear & Rest on piles → d/2 from face of wall
1 way shear & Rest on soil → d from face of wall
Bending in footing → Face of the column
isolated footing supporting a concrete column → Face of the column
Footing under masonry wall → Halfway/midway b/w the middle and edge/face of the wall
D/B > 2/3 → Share failure happens between the base of footing and the first soil reinforcement layer
Combined Footing
τ > 5kg/cm² ← 12legged
τ < 5kg/cm² ← 8 legged stirrups
Allowable/Permissible/Working stress
Shear stress (τc) → LSM = 0.25√fck, WSM = 0.16√fck ≈ fck/30 → LSM/WSM = 25 : 16
Bearing stress (br) → LSM = 0.45fck, WSM = 0.25fck
Max settlement
isolated foundation on clay soil = 75mm
isolated foundation on sand & hard clay = 50mm
Raft on sand & Hard clay = 75mm
For design purpose of Rcc footing , Pressure Distⁿ is assumed to be Linear.
To minimise the effect of differential settlement, the area of a footing should be designed for → Dead load + Fraction of live
Two way reinforced footing
Ex → Combined f, Continuous f & isolated column f.
Isolated column footing → Depth governed by max BM, SF, Punching shear
WALLS
If the storey height is equal to length of RCC wall → 20 % increase in strength
Bulkhead → Also serve as a pier
t ≥ 100mm
H/t ≤ 30
SF = ½ KaγH² = Pa
BM = SF x (H/2) = ⅙ KaγH³
Minimum Reinforcement in wall
Vertical Reinforcement → Deformed bar or Welded ≥ 0.0012, Other bar ≥ 0.0015
Horizontal Reinforcement → Deformed bar or Welded ≥ 0.0020, Other bar ≥ 0.0025
VR/HR = 0.0012/0.0020 = 3/5
Spacing ≤ (3 x Wall thickness, 450mm) → For both HR and VR
Retaining Wall/Horizontal/Lateral load/Overturning
Stem top width = 200 - 400 mm
Base slab width = (0.4 - 0.6)H → For surcharge = (0.6 - 0.75)H
Thickness of base slab = H/12 = 8 % of Height of wall
Toe projection= (1/3 - 1/4) of base width
If 0.9FResisting force/FSliding force 1.40 → Safe against sliding
FOS → Sliding = 1.5, Overturning = 2, floatation = 1.25
If effect of WL/EQ is not consider → FOS = 1.75 for sliding
Shear key → To avoid Sliding/Lateral pressure
Lateral earth pressure → Pa=kaH2 → From Rankine theory
Cantilever Retaining wall
Resist the Earth pressure horizontal and another → By cantilever bending action
Height = 3 - 8 m(10 - 25 feet)
Width of steam = 200 mm generally
Main reinforcement → in Heel slab at Top of the slab, Backfill side in the vertical direction
Max Base pressure → at Toe of the wall
Heel slab → Act as horizontal cantilever under the action of resulting soil pressure acting …?
Counterfort Retaining wall
Stem and Heel slab → Designed and Act as a continuous slab
Economical → Height > 6m
Spacing of counterfort = (1/3 - 1/2) of Height of wall
Spacing –. Depends on height, unit pressure of soil, steel and concrete cost,
Main reinforcement → inner face in Stem, Bottom face in front counterfort, inclined face in back counterfort
Stem at support → Reinforcement only on inner face
Stem at mid span → main reinforcement Front face only
Front counterfort main reinforcement → Bottom face near counterfort and Top face near centre of span
Max (-ve) BM near counterfort = pl2/12 → p = net downwards pressure per unit area
T-Shape Retaining wall
Consists of Three cantilevers
Main reinforcement in Stem → inner face in one direction
Toe → Bottom face Perpendicular to wall
Heel → Top face Perpendicular to wall
Temperature reinforcement → on the face of stem (more on front face than on inner) at the rate of 0.15 % of gross cross sectional area
WATER TANK
IS:3370 → Liquid retaining structure, Water tanks
Nominal cover Rcc tank ≥ 45mm
320 ≤ Cement content ≤ 400 kg/m³
W/C Ratio ≤ 0.45
Min Grade ≥ M30 → Considering durability for Liquid retaining structures
Crack width ≤ 0.20 mm → LSM
Reinforcement → Mild = 0.64 %, HYSD = 0.4%
Reinforcement → Mild = 0.35 %, HYSD = 0.24% → if No dimension > 15
Meridional bars → Reinforcement bars provided in domes surrounding an opening
i) Hoop Stress/tangential/circumferential
A tensile stress → Resist by steel alone
σh = pd/2 = γhD/2
ii) Longitudinal stress
Permissible stress in water tanks
Mild steel = 115MPa, HYSD = 130 MPa
Direct tension → M25 = 1.3, M30 = 1.5 MPa
Bending Tension → M25 = 1.8, M30 = 2.0 MPa
PRESTRESSED
High tensile strength steel wires/cables/rod are used → To impart Compressive stress in concrete in tension zone → to resist tensile stresses, adequate bond stress
Advantage → Concrete remains uncracked, Protect Steel from corrosion, Shear resisting capacity is increased due to pre compression
Desirable → Cylindrical pipes subjected to internal fluid pressure, liquid retaining structures
Shear crack → Depends on shape of cross-section of the beam
In the conventional prestressing the diagonal tension in concrete decreases
Forces of tension and compression remains unchanged but lever arm changes → With the moment
Ultimate moment capacity of a simply supported PSC beam → Determined using force and moment equilibrium conditions
Efficiency factor = (Applied prestressing force P) /(Concrete capacity of the section in compression)
Prestressing force under thermal stressing → Can we estimated from elongation of wire and temperature rise
For estimating ultimate stress in flexure of prestressed beam the stress block in concrete → May be of any shape which provides agreement with the test data
Prestressed member
Fully prestressed member (Type-I) → No tensile stress allowed
Limited prestressed member (Type-II) → Tensile stress is limited within the cracking stress of concrete, The section remains uncracked
Partially prestressed member (Type-III) → Limited tensile stresses and controlled cracking are permitted
IS Code Recommendation for Pre stressed concrete
Min Grade of Concrete → Pre = M40, Post = M30 → To overcome bursting stress at the ends, To control prestress loss
Minimum cover → Pre ≥ 20mm, Post ≥ (30mm or Cable size)
for Pre tensioned work in aggressive environment cover shall be increased by 10mm
Design mix: only 'design mix concrete' can be used with cement content preferably < 530 kg/m³
Min Cement Content = 300 - 360 kg/m³
Minimum cube strength of concrete used for prestressed member ≥ 35MPa (350kg/cm²)
Minimum strength of concrete at transfer of prestress = 0.50fck
Concrete section area utilised = 75 %
Prestress concrete girder combination of load factor → 1.5DL + 2.5LL
Use high strength concrete and high strength/Tensile Steel → To retain enough residual strength after initial losses
Ultimate strength of the Steel used = 1500 MPa
Max tensile stress(fpi) 76% ultimate tensile strength(fpu)
Tension is applied after Jack placing in both pre and post tensioning
Transfer stage → Tandon force is released on concrete permanently
Ultimate MOR → Pretension section > Post tension bonded section > Post tension unbounded section
Bounded prestress concrete beam → Max eff reinforcement ratio = 0.40
Hoyer System (Long line method)
For the production of pretensioned member on large scale → Rail sleepers, Poles, Factory production
Hoyer effect → The end of wire swells and develops friction and wedge effect, at the end Prestress force become zero → when prestress force is transferred by releasing tendon
Post Tensioning
Stress in wire → End support = 0, and Increases to it’s final maximum value over its transmission length
Bond stress b/w wire and concrete → Max = Near end supports, and Decreases to nearly zero over its transmission length
Anchorage or End block zone → The zone between the end of the beam and the section where only longitudinal stresses exist
Bursting force → Fbst = Pk(0.32-0.3ypo/yo) → Bursting tensile force will affect the end zone reinforcement
Safe cable zone → Vertical limits within which cable is to be provided
Grouting → Protect against corrosion
Freyssinet System
Anchorage device consists of a Concrete Cylinder
High tension steel wires 5 mm - 8mm → Numbering 8,12,16 or 24 → form a group into a cable with a spiral spring inside
Advantage: Securing wires is not an expensive process,
Disadvantage: Stress in wires are not similar
Gifford - Udall
Single - wire system, each wire is stressed independently using a double - acting jack.
Magnel - Blaton
Number of wires = 2 - 64 → Two wires are stretched at a time
Anchorage device → Consist of metallic sandwich plates, flat wedges & distribution plate
Lec-McCall
The high-tension bars are threaded at the ends in anchorage system
Analysis & Bending Stress
Tensile(top) → σt = P/A + M/Z - P.e/z
Compression(bottom) → σc = P/A - M/Z + P.e/z
Find e from above expressions
Concentric tendon → e = 0
Circular section → emax=D/8
for No Tension at soffit or Bottom fibre → P/A - M/Z + Pe/z = 0
At Transfer
Top fibre → fsub+Mg/Zt ftt
Bottom fibre → finf-Mg/Zt fbt
fsub=P/A-Pe/Z, finf=P/A+Pe/Z
Lines
C-line → Pressure line or thrust line → The locus of point of application of resultant force at any given section
P-line → Tendon line
Distance b/w C & P-line = M/P = M/C
Cable/Tendon profile
Normally prestressing wires → Arranged in the lower part of the beam
Tendon → Parabolic with convexity downward → For both SSB and Cantilever
SSB(UDL) → e = 0 at Support/end, e = max at centre/midspan
Cantilever(UDL) → e = 0 at free end, e = max at fixed end
BMD diagram → Opposite sign of shape of cable provided → The profile of Wire follows bending moment diagram
Curve or Sloping profile → Increases share strength, increases flexural strength
Concordant profile → C-line coincide with P-line → No initial support reactions
Deflection
Tendon constant eccentricity → ∆ = Pel²/8EI = Ml2/8EI (Upward deflection)
Tendon parabolic eccentricity → ∆ = 5Pel²/48EI=5Ml2/48EI
To Nullify bending effect(UDL) → e or dip = wl2/8Pw=8Ph/l2
To nullify external point load effect → e or dip = WL/4P
Bent tendon → Net downward force = W-2Psin
Losses
Transioning stage → Friction, Anchorage stage → Anchorage slip, Subsequent loss → all other loss
immediately (Short term loss) → Elastic shortening, friction, anchorage slip
Time dependent (Long term loss) → Creep of concrete, shrinkage, relaxation of steel
Loss in Pre-tensioning → Elastic shortening, Relaxation of steel, Shrinkage of concrete, Creep of concrete → Primarily due to shrinkage and creep → total loss = 18%
Loss in Post tensioning → All above of pre-tensioning + Frictional loss + Anchorage slip, except elastic shortening → total loss = 15%
Loss in pre-tensioning > Post-tensioning → Due to effect of Elastic shortening
Percentage loss = (Loss/P) x 100
At the time of transfer of prestress → instantaneous losses occur
Bearing stress on concrete (0.48fciAbearing/Apunching or 0.8fck) → After accounting all losses …
Elastic shortening
Due to change in strain in Tendon → Only in pre tension system
Pre-tension → Loss=mfc=EsP/EcAc → fc=P/Ac
Post-tension → Loss = 0 → if there's only one bar and if all wires are tensioned simultaneously and anchored
Least loss → One wire post-tensioned beam
Frictional loss
Only for post tension
Loss=Po-Px → At distance = x
Px=Poe-(kx + )
Length effect → Px=Poe-kx → k = Wobble correction
Curvature effect → Px=Poe- → α = x/R
Jacking force applied from one end only → Max friction loss at anchored end (x = L)
Jacking force applied from both end only → Max friction loss at x = L/2
Anchorage slip
Only for post tension
Loss = (Slip /L) Es=(/L)Es
Es = 2 x 10⁵ N/mm²
Relaxation of steel (Creep of steel)
Decrease in stress with time under constant strength
Loss =ϕfcEs/Ec= 1-1.5% of initial stress
ϕ = Creep strain/elastic strain = (long term - short term strain)/short term strain
Creep of concrete
Considering permanent loads and prestressing force
Loss =mfc= 2 - 3% of initial prestressing force
m = Es/Ec → modular ratio
ϕ → 7 days = 2.2, 28 days = 1.6, 1 Year = 1.1 → Varies logarithm with time
Shrinkage of concrete
Loss = Esteel x Shrinkage strain of concrete
Shrinkage strain → Pre-tension = 0.0003, Post-tension = 0.0002/log10(t+2) → t = days
Pre-tension loss = 2 x 10⁵ x 0.0003 N/mm² = 60 Mpa
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