Crack Width Calculation To BS 8007 For Combined Flexure and Direct Tension 2002 PDF

 

technical report:

crack widths

Crack width calculation to BS 8007 for combined flexure and direct tension H. G. Kruger derives and sets out the equations for determining surface crack widths for sections under combined flexure and direct tension, which, according to the author, are not covered in the BS 8007 standard

 H. G.(Erhard) G.(Erhard)  Kruger  Kru ger, is a specialist engineer associate in the

ε12

structural division of BKS (Pty) Ltd.

ε21

ε2

ε22 εm

B

S 8007 includes recommendations for the calculation of design crack widths for sections under flexure and for sections under direct direct tension. It does not provide recommendations for sections under the combined forces. This Technical Report examines the procedures given given in the code, and shows the separate equations for flexure and direct tension to be based on similar premises.. The situation where tension premises exists across the whole of the section is examined, and the limiting values values of the depth of the neutral axis are calculated. Equations are developed for surface strains and the stiffening effect of the concrete. Similar equations are developed for the second case where some compression is present on one face of the section. It is shown how the desi design gn crack widths for these cases can be determined.. Equatio determined Equations ns are developed with variables allowing for different reinforcement ratios and concrete cover at each face.

Es f c f s f s1 f s2 f stif  f stif1 f stif2 Fstif  Fstif1

h

acting at the level of the steel at face 1 portion of stiffening tensile force acting at the level of the steel at face 2 over overal alll d dep eptth o off sec secti tion on

k1

a constant =

k2

a constant =

Fstif2

K

Notation a’

a1 a2 acr

 A s  A s1  A s2 bt b cmin c1 c2 d e Ec

dista distance nce fr from om tthe he ccomp ompres ressio sion n fa face ce to the point at which crack width is being calculated distance from face 1 to centroid of  the reinforcement at face 1 distance from face 2 to centroid of  the reinforcement at face 2 distance from point considered to surface of the nearest longitudinal bar area of tension reinforcement area of reinforcement at face 1 area of reinforcement at face 2 width of section at the centroid of  the tension steel widt width h of sect sectio ion n cons consid ider ered ed (normally 1m) minimum cover to tension steel minimum cover to reinforcement at face 1 minimum cover to reinforcement at face 2 effective depth  M  eccentricity = T 

KJ   ON L P

modulus of elasticity of concrete ( / 2 the instantaneous value when used to determine αe)

18|The

modulus of elasticity of reinforcement compressive stress in concrete stress in reinforcement stress in reinforcement at face 1 stress in reinforcement at face 2 stiffening tensile stress in concrete stiffening tensile stress in concrete at face 1 stiffening tensile stress in concrete at face 2 total stiffening tensile force in concrete portion of stiffening tensile force

< <

a2 h - a1 h - a2 a1

F F

εm1 εm2 εsm εsm1 εs εs1 εs2 ∆εs

∆εs1

∆εs2

ρ1

strain at face 2 ignoring stiffening  effect of concrete strain due to stiffening effect of  concrete between cracks strain due to stiffening effect of  concrete between cracks at face 1 strain due to stiffening effect of  concrete between cracks at face 2 average strain at level where cracking is being considered average strain at face 1 average strain at face 2 average strain in tension reinforcement average strain in tension reinforcement at face 1 strain in tension reinforcement strain in reinforcement at face 1 strain in reinforcement at face 2 strain reduction in tension reinforcement due to tension stiffening  of concrete strain reduction in reinforcement at face 1 due to tension stiffening of  concrete strain reduction in reinforcement at face 2 due to tension stiffening of  concrete ratio of reinforcement at face 1

e o =

ρ2

V R  h S e+ 2 -a W S=   h W S e- +a W 2 WX the ssect STap appli plied ed mom moment ent at the ection ion

s1

ratio of reinforcement at face 2

e o =

a const constan antt fo forr a pa parti rticul cular ar sect section ion under a certain configuration of  moment and direct tension

 A bh

 A s2 bh

Note: Generally subscripts subscripts 1 and 2 refer to faces 1 and 2 of the section respectively.

2

1

M

considered  x h

n1

ratio

T

ap appl plie ied d dir direc ectt ten tensi sion on at the the section considered desi design gn su surf rfac ace e ccra rack ck widt width h design surface crack width at face 1 design surface crack width at face 2 di dist stan ance ce to to the the neut neutra rall axis axis ffro rom m face 2 dista distance nce to the centr centroid oid of the concrete stiffening force from face 2

w w1 w2 x x’

 E s

αe

modular ratio =

ε1

strain at level considered ignoring  the stiffening effect of concrete strain at face 1 ignoring stiffening  effect of concrete

1

ε11

Structural Engineer – 17 September 2002

b

 E c

l

Introduction The British Code of Practice, Practice, BS 80071 includes recommendations for the calculation of design surface crack widths for sections under flexure and for sections under direct direct tension tension.. However, However, the code does not provide recommendations for sections under combined flexure and direct tension. Furthermore, Furtherm ore, little guidance guidance is given in the literature. Neither Anchor Anchor2 nor 3 Batty provides a rational approach to allow for the effect of tension stiffening  for the case of combined flexure and direct tension. Since combined flexure and direct tension often exist in structural elements of certain water-retaining  structures structu res (e.g. in the horizontal direction of walls of rectangular or square tanks), a need for calculating design design crack widths for this case exists. exists. The Technical Report proposes a method for calculating calculat ing the design crack widths widths,, w1 and w2, at the two two faces of a section section under these loadings.

 

technical report:

Fig 1. Stiffening effect of  concrete in flexure

crack widths

The effective strain reduction at any level in the the section, section, therefore, therefore, is:

f2 =

2bt  h

....(6)

3 E s As

Similarly, it can be shown that the  value of the tensile tensile stres stresss at the the tension face for a design crack width of 0.1mm, is taken as 1MPa.

Crack width formulae The design crack width defined by eqn (1) of BS BS 8007: Appendix B, is: 3acr  f   m

w= 1+2

BS 8007 approach The procedure for the calculation of  design crack widths given in Appendix B of BS 8007 can be summarised as follows: • calculat calculate e the a average verage strain iin n the section at the level where cracking is being considered allowing for the stiffening effect of the uncracked concrete between cracks • calculat calculate e the design crack width using this value of the strain. The average steel strain may, as an approximation, be determined by calculating the steel stress on the basis of a cracked section, and reducing this this by the tensile force due to tension stiffening in the concrete (BS 8110: Part 24 Clause 3.8.3, Asses 3.8.3,  Assessment sment of cra crack ck widths widths). ). By considering the cracked concrete section in flexure as shown in Fig 1 (See BS 8110: Part 2 Fig 3. 3.1), 1), Eqn (2) of BS BS 8007: Appendix B, which defines defines the stiffening effect of the concrete for a design crack width width of 0.2mm, can be derived as follows5. The value of the tensile stress at the tension face between cracks is assumed as 2/3MPa. 2/3MPa. The stiffening force of the concrete in ttension, ension, therefore, therefore, is:  F stif  =

bt  (h - x) 3

....(1)

The effective strain reduction in the steel is:

Df s =

bt (h - x)

The average value of the steel strain, therefore, therefor e, is:

f sm = f s -

bt (h - x)

....(3)

3 E s As

w = 3a   cr f m

The average value of the strain at a distance a from the compression face, where the crack width is to be calculated, late d, is: l

fm =

=

tension.

a k d-x

a k d-x

=f1 - f2

a ka k a k

b  t  h - x -

Combined flexure and direct tension

al- x

...(4)

The procedure for calculating the design crack width for sections under combined flexure and tension, tension, can be summarised as follows:

3 E s A s d - x

a ka k a k

bt  h - x

where f2 =

al- x

• determine the position position of the the neutral neutral axis in the cracked cracked section, section, x • determine the strain strain due to to tension stiffening of concrete between cracks, ε2 • determine the average average st strain rain at the level where cracking is considered, εm • determine the crack crack width according  according  to eqn (7).

3 E  s As d - x

Similarly, by considering the cracked concrete section in direct tension as shown in Fig 2, Eqn (5) of BS 8007: 8007:  Appendix B for the stiffening effect of  concrete in direct tension for a design crack width of 0.2mm, 0.2mm, can be derived as follows: The stiffening force of the concrete in tension is:  Fstif =

2 3

bt  h

....(8)

Since this is the same as eqn (4) of BS 8007: App 8007: Appendi endix x B, it can, can, therefor therefore, e, be assumed that eqn (7) will also apply to the case of combined flexure and direct

(a  l- x) f sm

(al- x) f s

h-x

In a section under direct tension, the value of the depth of the neutral axis is x = – ∞ or x = ∞. By substituting  substituting  these values in eqn (7) the design crack width for a section under direct tension is:

....(2)

3 E s As

c m

....(7)

acr  - cmin

Two cases can be considered: • Case 1 1:: Complete Complete secti section on in ten tension sion • Case 2: Section Section partia partially lly in com compres pres-sion.

....(5)

Case 1: Complete section in tension  Determining the n  Determining neutral eutral axis axis depth Consider the cracked concrete section with a widt width, h, b, as shown shown in Fig Fig 3. The position of the neutral neutral axis x, is defined as negative when it is above face 2 of  the section, where face 2 is defined defined as the side of the section under compression when the section is subjected to a moment in an anti-clockwise direction. The complete section would be in tension when x ≤ 0 or x ≥ h. By consider-

Fig 2. Sect ion in direct direct tension

ing horizontal and moment equilibrium, and keeping in mind that x is negative as shown in Fig 3(c), 3(c), the position of the the neutral neut ral a axis xis,, x, is2: (See panel at end)

....(9)

17 September 2002 – The Structural Engineer Engineer| |19

 

technical report:

crack widths

From equation 9 the following equations can be derived for the limits of x as indicated: For  x

and K  =

#

c c

0:

 t1

$  -

 t2

h - a2 e+ 2 e-

h

+ a1

2

k1 K  where k1 =

m m 2

h - a1

....(12)

 t 1

For x ≥ h: t #    - k 2 K  2

h  - a2 a1

Fig 4. (Below) Relationship between x and ρ1/ ρ2 Fi Fig g 5. ( Bottom) Stiffening effect of  concrete, comp lete secti section on in tension

x = –∞ or x = ∞

 f s1 =

_

i i n

 M + T 0.5 h - a2

 f s2 =

 _ d

bh t 1 h - a1 - a 2 1

 t 2

T    - t 1  f s1 bh

....(14)

 t 1

the depth of the neutral axis, axis, and  t , is 2 shown in Fig 4.  Determining the sstiffening  Determining tiffening force Consider Consi der a section section with width, width, b, as shown in Fig 5(a), with the neutr neutral al axis position at x ≤ 0. The max maximum imum stiffening tensile stress in the concrete is:

....(15)

Consider for example a concrete section, with b = 1000m 1000mm, m, h = 400 400mm, mm, a1 = 5 50mm 0mm,, a2 = 60mm, M = 10kNm a and nd T = 400kN 400kN.. The relationship relationship between

 f stif 1 = 2

3

MPa for w = 0.2mm, and

 f stif 1 =1MPa for w = 0.lmm

 f stif2 = f stif1 

e ≠ (h/2) – a1

....(16)

Since x is negative, negative, it follows from the figure that:

_ i

....(13)

When the eccentricity of the tension force coincides with the centroid of the reinforcement reinforceme nt at face 1, ie when the denominator denominat or of eq 11 equals 0, it can be shown that the section is partially in compression. compressio n. The equation equationss in the following section should then be used to determine x. In summary, summary, provided that:

- x

_ i

....(17)

h-x

The total stiffening force is:    F stif  =

e

 fstif1 + f sti f2  2

o

....(18)

bh

The centroid of the stiffening force is at: l

 x =

_ _

i i

h 2f stif1 + f stif2  3  f stif1 + f stif2 

....(19)

Taking moments about the steel in face 2 yields: #

 t When - K < 1  t 2

20|The

2

When none of the above applies, 0 < x < h. The equations for steel stresses are:

a2

....(11)

 t1

When - k1 K

 t 1

When  t = - K ,

....(10)

For x = ∞ or x = –∞: t  = - K 

where k 2 =

Fi Fig g 3. Case 1 – Complete section in tension

 t 1  t 2 < - K , –∞ < x ≤ 0 # - k 2 K ,

h≤x