International Federation for Structural Concrete (fib) - Model Code for Concrete Structures 2010

fib Model Code for Concrete Structures 2010

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7.3 Verification of structural safety (ULS)...

7.3

Verification of structural safety (ULS) for predominantly static loading 7.3.1 General This subsection gives methods of verifying that, for a structure as a whole and for its component parts, the probability of an ultimate limit state exceeding the resistance of critical regions is acceptably small.

The determination of the partial safety coefficients and action effects is to be undertaken in accordance with the principles set out in chapter 4.

7.3.2 Bending with and without axial force 7.3.2.1 Beams, columns and slabs

Figure 7.3-1 shows the possible range of strain distributions for concrete, reinforcing steel and prestressing steel. In the figure, the following limits are shown: A = reinforcing strain limit; B = concrete compression limit; C = concrete pure compression strain limit.

Figure 7.3-1: Possible strain distributions in the ultimate limit state

This subsection applies to undisturbed areas of beams, slabs and similar types of members for which sections remain approximately plane before and after loading. The discontinuity regions of beams and other members, where plane sections do not remain plane, may be designed and detailed according to subsection 7.3.6.

When determining the ultimate bending resistance of reinforced or prestressed concrete cross-sections, the following assumptions are made: –– plane sections remain plane; –– the strain in bonded reinforcement or bonded prestressing tendons, whether in tension or in compression, is the same as that in the surrounding concrete; –– the tensile strength of the concrete is ignored; –– the stresses in the concrete are derived from stress–strain relations for the design of cross-sections as given in subsection 7.2.3.1.5; –– the stresses in the reinforcing and prestressing steel are derived from the design curves in subsections 7.2.3.2 and 7.2.3.3; –– the initial strain in the prestressing tendons is taken into account when assessing the stresses in the tendons.

For cross-sections with symmetrical reinforcement loaded by a compression force, the minimum eccentricity should be taken as e 0 = h/30 but not less than 20 mm, where h is the depth of the section.

7.3.2.2 Shells

Figure 7.3-2: Three-layer plate model and stress resultants

The subscript notations inf and sup refer to the inferior and superior faces of the element. The inferior face is the tensile face for an element in positive bending.

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Shell elements may be modelled as comprising three layers (Figures 7.3-2 and 7.3-3). The outer layers provide resistance to the in-plane effects of both the bending and the in-plane axial loading, while the core layer provides a shear transfer between the outer layers.

The action effects of the applied loads are expressed as eight components, three moments per unit width, three axial forces per unit width and two shear forces per unit width in directions parallel to the orthogonal reinforcement. The stress resultants mx, my, mxy, nx, ny, nxy, vx, vy produce the following forces per unit width on the element: nx inf,sup =

nx m x v x2 ± + cot θ 2 z 2vo

(7.3-1)

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ny inf,sup = The effective shear depth z may be taken as 0.9d, where d denotes the distance between the extreme compression fibre and the mean planes of the reinforcement layers at the opposite face. The effective shear depth needs not be taken as less than 0.72h.

ny

nxy inf,sup =

2

±

nxy 2

my z

±

+

m xy z

vy2

2vo +

cot θ

vx vy 2vo

(7.3-2)

cot θ

(7.3-3)

where: θ is the inclination of the compression stresses in the core layer; z is the average lever arm between the forces in the x and y directions in the top and bottom layers and the effective shear depth, respectively; vo is the principal transverse shear force per unit length and follows from: (7.3-4) vo = vx2 + v y2 For members with shear reinforcement, the angle θ is to be selected in accordance with subsection 7.3.3.3. For members without shear reinforcement, a value of cot θ = 2 may be used as is (implicitly) suggested in subsection 7.3.3.2.

Figure 7.3-3: (a) Layer forces in sandwich model and (b) transfer of transverse shear force in uncracked and cracked core

Design of outer membrane layers If at least one principal stress is in tension, the outer layers may be designed as membrane elements according to plasticity theory such that (Figure 7.3-4):

Figure 7.3-4: Stresses acting on and within a reinforced concrete element

σ sx =

1 σ x + τ cot θ pl ≤ f yd ρx

(7.3-5)

σ sy =

1 σ y + τ cot θ pl ≤ f yd ρy

(7.3-6)

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(

)

)

τ ν f ≤ ck (7.3-7) sin θ pl cos θ pl γc If no reinforcement is yielded and at least one principal stress is in tension: σc =

ν=

To ensure that the ductility demand is met, the term |qpl – qel| in Eq. (7.3-9) should not be greater than 15°, unless refined calculations are undertaken to justify a higher value.

(

1.18 ≤ 1.0 1.14 + 0.00166 σ si

(7.3-8)

where σsi is the maximum tensile stress (in MPa) in any layer of reinforcing steel (i = x, y).

If one or more layers of reinforcement yield:

(

18 ) 1.14 + 01..00166 f yd

ν = 1 − 0.032 θ pl − θel ⋅

(7.3-9)

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where: qpl is the compression field angle with respect to x-axis at the ULS qel is the first cracking angle with respect to the x-axis.

If both principal stresses are compressive:

σ 2 ≤ ν fcd (7.3-10) where s2 is the minor principal (compressive) stress and n may be taken as 1.0 or determined in accordance with subsection 5.1.6.

Design of inner core layer The shear core should be designed in accordance with subsection 7.3.3.

The models presented in this section represent an advance in philosophy to more physical based models. The change is made at this time, in recognition of the maturity of the new methods and of their capacity for further development, using a consistent framework, over the future years.

For the past 30 years, empirical approaches have formed the basis of models for design for shear and are widely adopted in national design standards. Such models have been validated for a wide range of structural applications and may continue to be used in design, including as models for beam shear and for punching shear.

Figure 7.3-5: Forces in the web of a beam

Further background information on shear provisions treated in this section is given by Sigrist, V., Bentz, E., Fernández Ruiz, M., Foster, S., Muttoni, A. (2013), Background to the fib Model Code 2010 Shear Provisions – Part I: Beams and Slabs. Structural Concrete, 14. doi: 10.1002/suco.201200066.

The depth d denotes the effective depth in flexure which is defined as the distance from the extreme compressive fibre of the concrete to the resultant tensile force in the tensile reinforcing steel and tendons.

The dimension z may also be taken as the distance between the centrelines of the top and the bottom chord, where the depth of the compression chord may be calculated for the location of maximum

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7.3.3 Shear 7.3.3.1 General Design shear force and shear resistance

The following equations are provided for the shear resistance of the webs of beams and the core layers of slabs and do not include the effects of flanges. Figures 7.3-3 and 7.3-5 show the regions of members being designed both for slabs and beams, respectively.

In beams, a minimum quantity of shear reinforcement in accordance with subsection 7.13.5.2 must be provided.

The shear resistance of a web or slab is determined according to: VRd = VRd ,c + VRd , s ≥ VEd (7.3-11) where: VRd is the design shear resistance; VRd,c is the design shear resistance attributed to the concrete; VRd,s i s the design shear resistance provided by shear reinforcement; VEd is the design shear force.

The design can be based on a stress field analysis or a strut-andtie model, as outlined in subsection 7.3.6. Such models are especially suitable for the design of discontinuity regions (D-regions) at supports or transverse applied forces.

Alternatively, a cross-sectional design procedure may be applied.

The corresponding rules are given in the following subsections.

Cross-sectional design For a cross-sectional design, the design shear force must in general be determined for control sections at a location d from the face of supports (see Figure 7.3-6) and from discontinuities of geometry or applied loads.

For the effective shear depth z a value of 0.9d can be assumed.

Other control sections may be required, for example in case of varying web widths along a span, for non-uniform or

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bending and a stress block according to Figure 7.2-12. For nonprestressed members z must not be less than 0.9d.

For members containing mild steel reinforcement as well as prestressed tendons, the effective shear depth z can be taken as:

significant concentrated loads, or at sections near points of curtailment of reinforcement.

z=

zs2 As + z 2p Ap

(7.3-12)

where zs and zp denote the distances between the centreline of the compressive chord and the reinforcement and tendon axes, respectively. zs As + z p Ap

Sections closer to supports than the distance d may be designed for the same shear force as at the control section provided that the member is directly supported.

Unless more refined modelling techniques are used to consider loads taken directly to a support through strut or arch action (see subsection 7.3.6), the following rules apply: –– the contribution of point loads applied within a distance of d < av ≤ 2d from the face of the support to the design shear force VEd may be reduced by the factor:

β = av (2d ) (7.3-13) Figure 7.3-6: Definition of control section for sectional design

The effect of redistribution of internal forces in slabs with concentrated loads can result in higher shear capacities when compared to one-way slabs or beams subjected to uniformly distributed loading. This effect may be accounted for by assuming a uniform distribution of the shear force along a control width bw, as shown in Figure 7.3-7.

–– in the case of point loads applied as close as av < d from the face of the support, the design shear force VEd must be calculated with b = 0.5 as if the load was applied at av = d.

Where a concentrated load is applied to a slab near a support line, its capacity must be checked for punching at the control perimeter around the loaded area, as described in subsection 7.3.5, and for shear at a control section taken parallel to the line of the support, as defined in Figure 7.3-7.

The control section is taken at the lesser of the distances equal to d and av/2 from the face of the support. The load distribution angle must be taken as a = 45° for the case of clamped edges and a = 60° for simply supported edges.

Figure 7.3-7: Location and length of the control section, bw, for the determination of the shear resistance of slabs with point loads located near a support-line; (b) simple edge support; (c) clamped edge support

For determining VEd, the shear force from the sectional analysis V Ed0 may be reduced by favourable contributions resulting from any inclined tension chords (VEtd), compression chords (VEcd) and prestressing tendons (V Epd) – see Figure 7.3-8. In determining V Epd, an eventual reduction in prestress due to the development length must be considered.

Any unfavourable contributions from inclined chord and prestressing tendon forces must be added to VEd0.

Figure 7.3-8: Contributions of inclined chord forces to design shear force (M Ed0 , VEd0 and N Ed0 denote bending moment, shear and normal forces resulting from sectional analysis)

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Membrane or arching action due to internal or external restraints further increases the design shear resistance and, as a consequence, the shear reinforcement may be decreased; however, the compression stresses in the concrete are enhanced and should be checked carefully. It is recommended to study such a situation with help of a strut-and-tie model (see subsection 7.3.6.).

In the design for shear in beams and in slabs, the effects of axial tension due to creep, shrinkage and thermal effects in restrained members should be considered wherever appropriate.

Design and analysis of members in shear may require the state of strain to be taken into account. Within the framework of undertaking a cross-sectional analysis, the longitudinal strain (Figure 7.3-9) is calculated at the mid-depth of the effective shear depth or core layer being considered as follows: 1  M Ed  1 ∆e   + VEd + N Ed     (7.3-16)  2 Es As  z z  2 In the use of Eq. (7.3-16), the following conditions apply: –– M Ed and VEd must be taken as positive quantities and NEd as positive for tension and negative for compression.

–– It is permissible to use a value of ex that is greater than half the yield strain of the longitudinal bars (esy/2) but a more detailed cross-sectional analysis must be undertaken. The strain ex must not exceed 0.003.

–– If the value of ex is negative it must be taken as zero.

–– For sections closer than d to the face of the support, the value of ex taken at d from the face of the support may be used.

–– For sections within a distance z/2 of a significant bar curtailment, the calculated value ex must be increased by a factor of 1.5.

–– A s comprises the main longitudinal reinforcing bars in the tensile chord; any distributed longitudinal reinforcement is neglected.

–– In calculating As (and Ap) the area of the bars that are terminated less than their development length from the section under consideration must be reduced in proportion to their lack of full development.

–– If the axial tension is large enough to crack the flexural compression face of the section, the calculated value of ex must be multiplied by a factor of 2.0.

εx =

Figure 7.3-9: Definitions

For members prestressed with bonded tendons, Eq. (7.3-16) is replaced by:

(

)

M z −e   Ed + VEd + N Ed p p   z  z  (7.3-14) εx =  zp  zs  2  Es As + E p Ap  z  z 

If the value of ex is negative it must be taken as zero.

For prestressed members the sectional forces are taken as: M Ed = M Ed 0 + M Pd

N Ed = N Ed 0 − Fp cos δ p

VEd = VEd 0 − Fp sin δ p

(7.3-15)

where M Pd denotes the design bending moment due to prestressing which includes a possible moment M P,ind resulting from static indeterminacy, that is M Pd = ±Fp cosdp(ep) + M P,ind.

Analogously, shear and normal forces are affected.

7.3.3.2 Members without shear reinforcement In case of a support that penetrates into the beam or slab, z is replaced with the effective depth, dv in accordance with subsection 7.3.5.2.

General The design shear resistance of a web or a slab without shear reinforcement is given by: fck zbw (fck in MPa) (7.3-17) γc

where the value of fck must not be taken as greater than 8 MPa. The longitudinal reinforcement in the flexural tensile chord at each section of interest must be able to resist an additional force component due to the shear of: VRd ,c = kv

∆Ftd = VEd (7.3-18) However, the total demand on longitudinal reinforcement must not exceed the demand due to maximum moment alone in the respective maximum moment region.

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The level I equation is derived from the more general level II approximation with the assumption that the mid-depth strain at the control section can be taken as ex = 0.00125, which corresponds to half the yield strain for a reinforcing bar with f yk = 500 MPa (ex ≈ f yk/ (2Es)).

In higher strength concrete and lightweight aggregate concretes, the fracture surface may go through the aggregate particles, rather than around, reducing the crack roughness.

There is evidence that the shear resistance of members without shear reinforcement is influenced by the maximum size of the aggregate dg. If concrete with a maximum size of the aggregate different from dg = 16 mm is used, the value kdg may be calculated with: kdg =

32 ≥ 0.75 16 + dg

Level I approximation For members with no significant axial load, with f yk ≤ 600 MPa, fck ≤ 70 MPa and with a maximum aggregate size of not less than 10 mm: kv =

180 (z in mm) 1000 + 1.25z

(7.3-19)

Level II approximation For the level II approximation, the design shear resistance is determined with: kv =

0.4 1300 ⋅ (z in mm) 1 + 1500ε x 1000 + kdg z

(7.3-21)

Provided that the size of the maximum aggregate particles, dg, is not less than 16 mm, kdg in Eq. (7.3-21) can be taken as kdg = 1.0.

(7.3-20)

For concrete strengths in excess of 70 MPa and for lightweight concrete, dg in Eq. (7.3-20) should be taken as zero, in order to account for the loss of aggregate interlock in the cracks due to fracture of aggregate particles.

The web reinforcement ratio given by Eq. (7.3-22) corresponds to the minimum reinforcement ratio as defined in subsection 7.13.5.

Members containing a lower reinforcement ratio are to be treated according to section 7.3.3.2.

Size effects are limited in members with web reinforcement greater than that required by Eq. (7.3-22).

The strength reduction factor kc consists of two parts: the state of strain in the webs of beams or the core layers of slabs is taken into account by k e; the effect of more brittle failure behaviour of concrete of strengths greater than 30 MPa is considered in hfc.

7.3.3.3 Members with shear reinforcement General This subsection applies to members that meet the demand for minimum shear reinforcement according to:

ρ w ≥ 0.08

fck

(fck and f yk in MPa)

The design shear resistance is then determined from: f yk

(7.3-22)

VRd = VRd ,c + VRd , s (7.3-23) but must not be taken as greater than: fck bw z sin θ cos θ (7.3-26) γc

where q denotes the inclination of the compressive stress field.

The strength reduction factor is defined as: VRd ,max = kc

kc = kε η fc (7.3-27) with ke as given in the following and: 1/ 3

 30  η fc =    fck 

≤ 1.0 (fck in MPa)

(7.3-28)

Figure 7.3-10: Geometry and definitions

In the case of stirrups that are inclined relative to the beam axis the Eqs. (7.3-26) and (7.3-29) are replaced by: VRd ,max = kc and

fck cot θ + cot α bw z γc 1 + cot 2 θ

(7.3-24)

Asw zf ywd ( cot θ + cot α ) sin α (7.3-25) sw

where a is the inclination of the stirrups as shown in Figure 7.3-10.

VRd ,s =

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The design shear resistance provided by stirrups is: Asw zf ywd cot θ (7.3-29) sw

where f ywd denotes the design yield strength of the shear reinforcement.

The design shear resistance attributed to the concrete can be taken as: VRd ,s =

The limitation on fck is provided due to the larger observed variability in shear strength of members with higher strength concrete, particularly for members without stirrups such as slabs.

Values of kD depend on the material of the duct and whether it is grouted or not. Suggested values for design are: –– grouted steel duct: kD = 0.5; –– grouted plastic duct: kD = 0.8; –– ungrouted duct: kD = 1.2.

Factor kD may be reduced in presence of reinforcement transverse to the plane of the web.

In the case of stirrups that are inclined relative to the beam axis the Eq. (7.3-34) is be replaced by: ∆Ftd =

VEd ( cot θ − cot α ) 2

(7.3-32)

In the level III approach, where a concrete contribution VRd,c ≠ 0 is considered, in the Eqs. (7.3-32) and (7.3-34) the design shear force VEd is replaced by VEd* where: * VEd = VEd + VRd ,c (7.3-33)

The value of qmin is determined by the level of approximation.

VRd ,c = kv

fck bw z γc

(7.3-30)

where the value of fck must not be taken as greater than 8 MPa. In Eq. (7.3-30), the effective web width bw must be taken as the minimum concrete web width within the effective shear depth z.

In the case of prestressing tendons with duct diameters ∅D ≥ bw/8, the ultimate resistance of the compression struts must be calculated on the basis of the nominal value of the web width: (7.3-31)

where S∅D is to be determined for the most unfavourable prestressing tendon configuration.

The longitudinal reinforcement in the flexural tensile chord at every section of interest is to be designed to resist the additional force due to shear of: bw,nom = bw − kD ∑ ∅ D

VEd cot θ (7.3-34) 2

However, the total demand on longitudinal reinforcement need not exceed the demand at the maximum moment location due to moment alone. ∆Ftd =

The limits of the compressive stress field inclination q, relative to the longitudinal axis of the member (Figure 7.3-10), are:

θ min ≤ θ ≤ 45° (7.3-35) where q may be chosen freely between these limits for design.

The level I approximation represents a variable angle truss model approach.

Level I approximation In the level I approach, the design shear resistance is given by: VRd = VRd ,s ≤ VRd ,max (7.3-36) but must not to be taken less than the resistance of the same member without shear reinforcement.

The minimum inclination of the compressive stress field is: qmin = 25° for members with significant axial compression or prestress; qmin = 30° for reinforced concrete members; qmin = 40° for members with significant axial tension.

The width of the beam or web must be checked for the selected inclinations of the compression stresses where ke is taken as: kε = 0.55 (7.3-37) Eqs. (7.3-36) and (7.3-37) apply to cross-sections where the longitudinal strain ex remains below a value of 0.001.

The level II approximation is based on a generalized stress field approach.

Level II approximation In the level II approach, the design shear resistance is given by: VRd = VRd ,s ≤ VRd ,max (7.3-38) but must not to be taken less than the resistance of the same member without shear reinforcement.

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The stress field approach allows the strut inclination q to be selected within certain limits and is confirmed by experimental observations.

For a preliminary design or analysis, ex maybe taken as 0.001.

Within the limits of Eq. (7.3-35) the inclination of the compressive stress field can be freely selected for design, or analytically determined for assessment.

The minimum inclination of the compressive stress field is:

θ min = 20° + 10000ε x

The variable value ke considers the influence of the state of strain in the web. This influence is important for prestressed members or members in compression, but less significant for reinforced elements and/or sections with higher q and ex values.

(7.3-39)

where ex represents the longitudinal strain at the mid-depth of the effective shear depth as shown in Figure 7.3-9.

The design shear resistance attributed to the concrete is neglected, that is kv = 0. The width of the beam or web should be checked for the respective inclination of the compression stresses where ke is taken as: 1 ≤ 0.65 1.2 + 55ε1

where:

(7.3-40)

kε =

ε1 = ε x + ( ε x + 0.002 ) cot 2 θ (7.3-41)

The longitudinal strain ex at the mid-depth of the effective shear depth is calculated on the basis of Eq. (7.3-14) or (7.3-16).

Level III approximation represents a general form of sectional shear equations and is based on the simplified modified compression field theory.

A comparison of the relative predictions for modelling levels I to III is shown in Figure 7.3-11.

Level III approximation In the level III approach, the design shear resistance in the range of VRd < VRd,max(qmin) is given by: VRd = VRd ,s + VRd ,c (7.3-42) where VRd,max(qmin) is calculated from Eq. (7.3-26) for q = qmin.

In the range of VRd ≥ VRd,max(qmin) the resistance is determined as in the level II approximation.

The inclination qmin is taken as given by Eq. (7.3-39).

For determining the design shear resistance VRd,c attributed to the concrete the following expression should be used: 0.4 1 + 1500ε x

  VEd (7.3-43) 1 −  ≥ 0  VRd ,max (θ min ) 

The strain ex at the mid-depth of the effective shear depth is calculated with Eq. (7.3-14) or (7.3-16). kv =

Figure 7.3-11: Comparison of levels I, II and III results for members with fck = 50 MPa (Note: for the curves shown in the figure the value fcd is defined as hfc fck / gc )

The use of tools based on advanced methods of analysis often requires extensive experience to ensure that safe and consistent results are obtained.

Level IV approximation The resistance of members in shear, or shear combined with torsion, may be determined by satisfying the applicable conditions of equilibrium and compatibility of strains and by using appropriate stress–strain models for the steel and for diagonally cracked concrete.

7.3.3.4 Hollow core slabs

For hollow core slabs and similar structural members the design shear resistance may be calculated on the basis of this subsection or alternatively of subsection 7.3.3.3; the higher of the results may be adopted as the capacity.

In single span prestressed hollow core slabs without shear reinforcement, shear failure occurs when the principal tensile stress in the web exceeds the tensile strength of the concrete.

Level I approximation The design shear resistance can be determined by: VRd ,ct = 0.8

Figure 7.3-12: Basis for derivation Eq. (7.3-44)

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I c ⋅ bw Sc

2 fctd + α l ⋅ σ cp ⋅ fctd

(7.3-44)

where: Ic is second moment of area; Sc is first moment of area above and about the centroidal axis; bw is width of the cross-section at the centroidal axis;

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i s concrete compressive stress at the centroidal axis due to prestressing, in the area where the prestressing force is fully introduced; al = lx / lbpt,95%; lx follows from Figure 7.3-12; lbpt,95% follows from Eq. (7.13-5a).

scp

Level II approximation In a level II approximation, the design shear resistance is determined by: VRd ,ct =

I c ⋅ bw ( y) 2 + α l ⋅ σ cp ( y) ⋅ fctd − τ cp ( y)] [ fctd Sc ( y)

(7.3-45)

where: Ic is second moment of area; Sc(y) is first moment of area above height y and about the centroidal axis; bw(y) is width of the cross-section at the height y; y is the height of the critical point at the line of failure; scp(y) is concrete compressive stress at height y and distance lx; τ cp ( y) is the shear stress in the concrete due to transmission of prestress at height y and distance lx.

The concrete compressive stress at height y and distance lx is determined from:  1 y − y σ cp ( y) =  + c (7.3-46)  ⋅ Fp (lx ) I   Ac

and the shear stress in the concrete due to transmission of prestress:

τ cp ( y) =

 A ( y) Sc ( y) ⋅ ( yc − y pt )  dFp (lx ) 1 ⋅ c − ⋅ bw ( y)  Ac I dx 

(7.3-47)

where: yc is height of concrete centroidal axis; Ac is area of concrete cross-section; Ac ( y) is concrete area above height y; ypt is height of centroidal axis of prestressing steel; Fp (lx) is the prestressing force at distance lx.

By varying y in the calculation, the lowest value of VRd,ct in Eq. (7.3-45) is found.

7.3.3.5 Shear between web and flanges of T-sections

Figure 7.3-13: Strut-and-tie model for force introduction into the flanges

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The introduction of tensile or compressive forces into the flanges (Figure 7.3-13) creates shear forces at the transition of web and flanges, inducing corresponding transverse tensile and compressive forces in the flanges.

The spread of the compressive forces in the flanges must be examined with the aid of stress fields. Recommended values for the angle of spread are: 25° ≤ qf ≤ 45° for compressive flanges; and 35° ≤ qf ≤ 50° for tensile flanges.

Unless a more detailed analysis is undertaken, reinforcement for the introduction of forces into the flanges is to be superimposed on that required for transverse bending.

In addition, the minimum transverse reinforcement must not be less that that required by subsection 7.13.5.

The longitudinal flange reinforcement must be anchored in accordance with the assumed stress field requirements.

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7.3.3.6 Shear at the interface between concrete cast at different times Background information on this subject is given by Randl, N. (2013), Design recommendations for interface shear transfer in fib Model Code 2010. Structural Concrete, 14. doi: 10.1002/ suco.201300003.

In addition to the requirements formulated in subsections 7.3.3.1– 7.3.3.5 the shear stress at the interface between concrete cast at different times should also satisfy the following condition:

τ Edi ≤ τ Rdi (7.3-48) where tEdi is the design value of the shear stress in the interface, given by

τ Edi = β ⋅ VEd / ( zbi ) (7.3-49) where: b is the ratio of the longitudinal force in the new concrete and the total longitudinal force either in the compression or tension zone, both calculated for the section considered; z is the inner lever arm of the composed section; bi is the width of the interface and VEd is the shear force on the composed section.

If a “rigid” bond-slip behaviour is expected and very good adhesive bonding guaranteed on the site, the adhesive bond effect should be taken into account without superimposing effects of interface reinforcement. The most important precondition for the assumption of good adhesive bond is a well prepared and very clean concrete surface at the time of casting.

Adhesive bond resistance should only be applied for design if no tensile loading perpendicular to the interface is expected.

The adhesion coefficients ca actually depend on a variety of influencing parameters (see subsection 6.3.3); nevertheless the ca factors given in Table 7.3-1 represent reasonable values on the safe side for the given roughness categories.

Special attention must be given to edge zones – see detailing rules in section 6.3.

For the definition of roughness of the classes distinguished in Table 7.3-1, see subsection 6.3.2.

Interface without reinforcement (rigid bond-slip behaviour) The design limit value tRdi for the interface shear in Eq. (7.3-31) follows from:

τ Rdi = ca ⋅ fctd + µ ⋅ σ n ≤ 0.5 ⋅ν ⋅ fcd (7.3-50) where: is the coefficient for the adhesive bond; ca m is the friction coefficient from Table 7.3-2; sn is the (lowest expected) compressive stress resulting from an eventual normal force acting on the interface.

The adhesion factor ca depends on the roughness of the interface (see Table 7.3-1; Rt is derived from the sand patch method).

Table 7.3-1: Coefficients for the adhesive bond resistance Surface characteristics of interface

ca

Very rough (including shear keys)

Rt ≥ 3.0 mm

0.5

Rough (strongly roughened surface)

Rt ≥ 1.5 mm

0.40

Smooth (concrete surface without treatment after vibration or slightly roughened when cast against formwork)

0.20

Very smooth (steel, plastic, timber formwork)

0.025

Under fatigue or dynamic loads the values for ca as found in Table 7.3-1 have to be reduced to 50%.

Interface intersected by dowels or reinforcement If strong adhesive bond cannot be guaranteed on the site or the design shear resistance provided by adhesive bond from Eq. (7.350) is lower than the design shear stress, interface connectors are required and the design limit value tRdi follows from: Figure 7.3-14: Transmission of shear forces across an interface intersected by reinforcing bars

Eq. (7.3-51) relates to interfaces intersected by dowels or reinforcement and characterized by a rather non-rigid bond-slip behaviour.

Connectors may be omitted in interface regions where the design shear stress does not exceed the resistance given in Eq. (7.3-50).

fib_MC_CS6.indb 224

1/ 3 t Rdi = cr ⋅ fck + μ ⋅ σ n + k1 ⋅ ρ ⋅ f yd ⋅ (μ ⋅ sin α + cos α ) + k 2 ⋅ ρ

⋅ f yd ⋅ fcd ≤ bc ⋅ ν ⋅ fcd

(7.3-51)

where strength values are in N/mm2 and: cr is the coefficient for aggregate interlock effects at rough interfaces; k1 is the interaction coefficient for tensile force activated in the reinforcement or the dowels;

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225

7.3 Verification of structural safety (ULS)...

Note that for a bar in tension, as explained in section 6.3, the tensile strength of the bar is reduced when the bar is also subjected to dowel action, as shown in Figure 7.3-15.

The detailing rules given in section 6.3 concerning embedment depth of connectors and minimum amount of steel cross-section must be obeyed.

k 2 is the interaction coefficient for flexural resistance; m is the friction coefficient; r is the reinforcement ratio of the reinforcing steel crossing the interface; sn is the (lowest expected) compressive stress resulting from an eventual normal force acting on the interface; a is the inclination of the reinforcement crossing the interface (see Figure 7.3-14); bc is the coefficient for the strength of the compression strut; 30 ν = 0.55( )1/ 3 < 0.55. fck

The coefficients for different surface roughness in interfaces reinforced with dowels or rebars are given in Table 7.3-2.

Figure 7.3-15: Dowel action under simultaneous tension and shear

For the background of the values in Table 7.3-2, see Randl, N., Design recommendation for interface shear transfer in MC2010 (Structural Concrete, Vol. 14, No. 3, 2013).

The roughness of a concrete surface can be measured in various ways (see subsection 6.3.2). An appropriate way is the sand patch method, as depicted in Figure 7.3-16 (Kaufmann, N., “Das Sandflächenverfahren”, Strassenbautechnik (1971), Nr. 3). A volume of sand V is spread on the rough surface in a circular area with diameter D. The roughness parameter Rt follows from: Rt [mm] =

40 ⋅ V

π D2

(7.3-52)

Table 7.3-2: Coefficients for different surface roughness Surface Roughness

cr

Very rough* Rt ≥ 3.0 mm

0.2

0.5

0.9

0.5

Rough Rt ≥ 1.5 mm

0.1

0.5

0.9

0.5

0.7

Smooth

0

0.5

1.1

0.4

0.6

Very smooth

0

0

1.5

0.3

0.5

k1

k2

bc

m fck ≥ 20

fck ≥ 35

0.8

1.0

* valid also for shear keys

Under fatigue or dynamic loads, the values for tRdi according to Eq (7.3-51) have to be reduced to 40%.

Figure 7.3-16: Principle of sand patch method for the qualification of the roughness of an interface

fib_MC_CS6.indb 225

11.09.2013 10:43:08

226

7 Design

A stepped distribution of the transverse reinforcement may be used, as indicated in Figure 7.3-17.

Figure 7.3-17: Shear diagram representing the required interface reinforcement

7.3.4 Torsion Where static equilibrium depends on the torsional resistance of elements of the structure, a full torsional design for both the ultimate and serviceability limit states must be undertaken.

Where in structures torsion arises from consideration of compatibility only, and the structure is not dependent on torsional resistance for its stability, it will normally not be necessary to consider torsion at the ultimate limit state. In such cases minimum torsional reinforcement must be provided in the form of stirrups, and longitudinal bars should be provided to prevent excessive cracking, as per the requirements of subsection 7.13.5.2.

The determination of the torsional resistance of box-girders and beams of solid cross-section is based on an ideal hollow crosssection as shown in Figure 7.3-18.

Figure 7.3-18: Definition of the ideal hollow cross-section

If the internal forces and moments, the cross-sectional dimensions and the reinforcement do not change abruptly in the longitudinal direction, it may be assumed that the shear flow due to torsion is constant over the circumference of the effective cross-sectional area. The torsional moment T Ed may then be resolved into equivalent panel forces such that: TEd zi (7.3-53) 2 Ak where A k is the area within the centre line of the thin-walled effective cross-section, including inner hollow areas.

The provisions of subsection 7.3.3 apply analogously for the dimensioning of the reinforcement and checking of the panel dimensions. The effective panel thickness of solid cross-sections (Figure 7.3-19) can thereby be taken into account as: VEd ,Ti =

Figure 7.3-19: Minimum effective panel thickness

fib_MC_CS6.indb 226

dk (7.3-54) 8 where d k denotes the diameter of the circle that might be inscribed at the most narrow part of the cross-section. tef ≤

11.09.2013 10:43:09

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