Budapest University of Technology and Economics

Department of Mechanics and Structures

English courses
Reinforced Concrete Structures
Code: BMETKEPB603

Lecture no. 5:

DEFORMATIONS AND CRACKING OF R.C.

STRUCTURES

Reinforced Concrete 2011 lecture 5/1

Content: I. Deformations
1. Why deformations should be limited?
2. Deflection limits
3. Loads to be considered when checking deformations
4. Deflection and flaxural rigidity
5. Limitation of the slenderness ratio l/d
6. Favourable effect of uncracked concrete between cracks
7. Effect of creep and shrinkage
8. Simplified check of deflections
II. Cracking
1. Reasons of cracking
2. Crack direction, characteristic crack patterns
3. Limits of the crack width
4. Restoration of cracked rc structures
5. Determination of the crack width
6. Effect of some parameters on the crack width
7. Simplified check of the crack width
Reinforced Concrete 2011 lecture 5/2
 1. Why deformations should be limited?

-aesthetical reasons
-psychological reasons
-safety of joining constructions (partition walls, windows, tiles of
the pavement)
-functionality (canalization of rainwater)
-modification of force distribution in arches, danger of loss of
stability

Reinforced Concrete 2011 lecture 5/3

 2. Deflection limits

wmax< l /250

simple sup. beams: continuous beams: cantilevers:

K tabulated in DA
here l is the span, l/K is the distance between M=0 points

Reinforced Concrete 2011 lecture 5/4

 3. Loads to be considered when checking deformations

Quasi-permanent load intensity:

characteristic value of permanent load +
long term part of variable loads:

pqp = g k + ψ 2q k

values of ψ2 see DA table in section 4

Reinforced Concrete 2011 lecture 5/5

 4. Deflection and flexural rigidity

beam made of elastic material: r.c. beam:

5 pl4 1 pl4 1 pl4 Ml2 1 l2

w max = = ≈ = =
384 EI 76,8 EI 80 EI 10EI R 10
Effect of creep and cracking should be considered in flexural rigidity EI:
E cm bd3
E = E c,eff = I = IiII = η
1 + ϕcr 12
η tabulated in DA for diff. compression and tension steel %-s
Reinforced Concrete 2011 lecture 5/6
 5. Limitation of the slenderness ratio l/d
The rate L/d characterises the rigidity of r.c. members
Example:

1 M ε + εs
Curvature: y '' = ≅ = tgα = c
R EI d
2 2
Ml 1l 1 10w max
Deflection: w max ≅ = → =
10EI R 10 R l2
Approximate curvature of an rc beam under service conditions:
εc + εs ≅ 0,85 ⋅ (2,0%o + 2,17%o ) = 3,54%o
1 0,00354 10w max l2 l l
= = 2
→ wmax= ≤ → ≤ 11,3
R d l 2825 d 250 d
Reinforced Concrete 2011 lecture 5/7
Consequently: if l/d=12 then – by considering deformations
characteristic for the service state – the deflection will approximately be
equal to l/250
x
For slightly reinforced members (for example slabs): ξc = c ≅ 0,08
d
εc,max≈ 0,08x1,85=0,15 %o
εc + εs = 2,00%o
0,002 10 w max
=
d l2
l2 l l
εs≈ 0,85x2,17%o=1,85%o wmax= ≤ → ≤ 20
5000d 250 d
l
Deflection problems of slabs amerge above slenderness ratios f 20
d
l
( ) allowable ratios on basis of more exact calculations see in DA tables!
d
Reinforced Concrete 2011 lecture 5/8
6. The favourable effect of uncracked concrete between cracks

cont. line: real behaviour

dashed. line: approximate
behaviour
difference:help ofncracked
concrete between cracks
Eurocode 2:
w = (1 − ζ ) w1 + ζw2
M
approximation ζ = 1 − 0,5( cr ) 2 ≥ 0
M
here indices 1 and 2 stand for stress states 1 (uncracked) and
2(cracked)
.

Reinforced Concrete 2011 lecture 5/9

 7. Effect of creep and shrinkage

Approximate character of the moment-curvature relationship of rc

setions:
plastic behavior
cracked
uncracked

Curvature:
1 M Ecm ε c , pl
= Effect of creep: Ec , eff = ϕcr =
RiM Ec, eff I i 1 + ϕcr ε c ,el
1
Effect of shrinkage:
Ri , sh
1 1 1
The total curvature: = +
Ri Ri , M Ri , sh
Reinforced Concrete 2011 lecture 5/10
Reinforced Concrete 2011 lecture 5/11
 The most effective tool to reduce creep deformation is to inrease
the quantity of compression steel (φcr decreases and Ii increases)

Reinforced Concrete 2011 lecture 5/12

 8. Simplified check of the deflection

l/K
≤ α (l / d)allowable
d

Values of K for checking of deflections

Simple supported beam or slab without
K=1
cantilever
Exterior span of continuous beam or slab K = 1.3
Interior span of continuous beam or slab K = 1.5
Flat slab K = 1.2
Cantilever K = 0.4

Reinforced Concrete 2011 lecture 5/13

Basic values of the allowable slenderness ratio (l/d)allowable for rectangular sections
Concrete p Ed
strength
β [kN/m2] (by beams b is the width of the beam in m, by slabs b=1,0 m)
grade
b
300 250
150 200 100 50 25 20 15 10 5
≥C40/50 13 14
15 14 17 20 25 27 30 35 47
C35/45 13 14
15 14 16 19 24 26 29 34 45
C30/37 13 13
15 14 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” ―――――
Modification factors:
1 p Ed M Rd 500 As, prov 500
α = β β= ≅
2 p qp M Ed f yk A s, requ f yk
For T-sections and flanged beams another table must be used

Reinforced Concrete 2011 lecture 5/14

 II. Cracking
1. Reasons of cracking

fct,d≈ 0,1 fcd σc,max≥ fct,d

-impeded deformation (for example: shrinkage)

-temperature effects (examples: external corridor cantilever slab,
fence plinth)
-tension provoced by internal forces (axial or eccentric tension,
flexure, shear, torsion)

R.c. structures in service conditions are generally cracked, even water

containers can be cracked.
Dilatation joints and correct support conditions (neopren pad) inhibit
unwanted cracking.

Reinforced Concrete 2011 lecture 5/15

 2. Crack direction, characteristic crack patterns

Crack direction shows the direction of principal stresses. Diagonal

cracks are called shear cracks. The most dangerous crack is is that of
the column.

Reinforced Concrete 2011 lecture 5/16

 3. Limits of the crack width

Problems caused by cracking under quasi-permanent loads and the

corresponding limits of the crack width
Aesthetical problems 0,4 mm
Corrosion in ambient variably dry and wet (XC2….XC4) or by
exposure to chlorides (XD1….XD3) 0,3 mm
-When prestressing steel is used – due to its high sensitivity to
corrosion – more strict requirements apply
-Requirement of watertightness is fulfilled if in serviceability state the
height of the compression zone reaches 50 mm
in wet ambient 0,2 mm
in agressive ambient, in soil 0,1 mm

Reinforced Concrete 2011 lecture 5/17

 4.Restoration of cracked rc structures

Possible ways of protection against cracking:

-exclusion of factors impeding shortening caused by cinematic
effects
-application of watertight flooring, over-spanning cracks

Ways of restoration:
wcr>1 mm: injection of cement milk
wcr <1 mm: injection of epoxy resin
sealing with fibrous foil, cladding

Reinforced Concrete 2011 lecture 5/18

 5. Determination of the crack width

distance between cracks:

wk = sr ,max (ε sm − ε cm ),

sr,max = 3,4c + 0,425 k1 k2 φs Ac,eff / As

σ s − 0,4 f ctm Ac,eff / As

ε sm =
Es

εcm = 0,4fctm / Ecm

c concrete cover
k1 = 0,8 for deformed barsl, 1,6 for smooth bar surfice
k2 = 0,5 for flexure, 1,0 for axial tension
Reinforced Concrete 2011 lecture 5/19
 Φs = diameter of tension reinforcement
Ac,eff for flexure see figures below
As =As,prov

A = level of the center of As

B = Ac,eff ,

x = xII,

2,5(h − d )

hc,ef = min (h − x) / 3
h/2


Reinforced Concrete 2011 lecture 5/20

 6.Effect of some parameters on the crack width

Variation of the crack width with

-diameter of tension steel:

-steel stress:

-steel ratio

Reduce ba diameter and increase steel ratio to control crack width!

Reinforced Concrete 2011 lecture 5/21

 7. Simplified check of the crack width

Determine:
p qp As ,requ
σ s ≈ f yd ⋅ ⋅
p Ed As , prov
As
ρ= (%)
bd
and find find the greatest allowable tension steel diameter for the given
limit of the crack width in the table below
When using different bar diameters:

Σφ2
φequ = .
Σφ

Reinforced Concrete 2011 lecture 5/22

Reinforced Concrete 2011 lecture 5/23
 Maximum steel bar diameter φ max (mm) in function of the steel ratio and the steel stress

Steel stress
σs (N/mm2)
To satisfy the crack with limitation condition wk≤ wk,allow , if c min,dur ≤ 20 mm .
wk,allow = 0,4 mm wk,allow = 0,3 mm wk,allow = 0,2 mm
Steel ratio (ρ=As/bd, %) Steel ratio (ρ=As/bd, %) Steel ratio (ρ=As/bd, %)
0,15 0,2 0,5 1,0 1,5 2,0 0,15 0,2 0,5 1,0 1,5 2,0 0,15 0,2 0,5 1,0 1,5 2,0
160 16 21 40 40 40 40 12 16 34 40 40 40 7 10 23 30 35 38
200 13 17 34 40 40 40 9 12 26 34 39 40 5 7 16 21 26 30
240 10 14 26 36 40 40 7 10 19 27 33 37 - 6 10 14 18 21
280 9 11 21 31 37 40 6 8 14 21 27 31 - 4 7 10 12 14
320 7 10 17 25 32 36 - 7 11 16 21 26 - - 4 6 8 9
360 6 8 14 21 28 32 - 6 8 13 17 20 - - - - 4 4

Reinforced Concrete 2011 lecture 5/24