Ultimate Limit State: Bending

Presented by:

John Robberts
Design Point Consulting Engineers (Pty) Ltd
john.robberts@design-point.co.za
ULS: Flexure with and without axial force –
fundamental principles
There are three fundamental principles (that has not changed) for ULS (Cl.
6.1(2)P):
1. Plane sections remain plane:
• Unless it is a deep beam, where the span less than 3 times the overall section depth
(Cl. 5.3.1(3))
• Cracking in the tensile zone can be ignored (provided the gauge length under
consideration spans more than one crack).
2. The stress-strain relationships for the materials are known:
• Concrete stress-strain relationships are defined in Figs. 3.2 to 3.5
• Non-prestressed reinforcement stress-strain relationships are defined in Fig. 3.8
• Prestressed reinforcement stress-strain relationships are defined in Fig. 3.10
3. At each section the actions (applied forces and moments) must be in
equilibrium with the action effects (internal stresses). Although not
explicitly defined in Cl. 6.1(2)P, it is fundamental to analysis and design.
Additional principles, related to materials (as before) at ULS (Cl. 6.1(2)P):
• Strain in bonded reinforcement or prestressing tendons is the same as
that in the surrounding concrete
• Provided the gauge length under consideration spans more than one crack.
• Unbonded prestressing tendons obviously do not comply here.
• The tensile strength of the concrete is ignored.
• Initial strain in prestressing tendons needs to be taken into account.
Failure at ULS occurs when:
1. The concrete reaches it’s ultimate strain (e.g. 0.0035 for concrete class
 C50/60).
2. The prestressed or non-prestressed reinforcement ruptures when the
design strain limit (eud) is exceeded. This failure mode is highly unlikely if
the limits to the neutral axis depth are complied with.
Notation and Terminology

• Paragraphs are numbered as:

• (x)P – for Principles. No alternatives are allowed, unless specifically stated.
• (x) – for Application Rules. Generally accepted methods, which follow the principles
and satisfy their requirements. Alternatives are possible, provided that it will comply
with the principles with regard to structural safety, serviceability and durability.
• “Loads” are now referred to more generally as “Actions”, to include
imposed deformations (e.g. temperature and settlement).
• General notation:
• Permanent (G) = Dead Loads
• Variable (Q) = Live or Wind Loads
• Accidental (A)
• Prestressing (P), which is treated as a permanent action in most situations.
• “Action Effects” collectively refer to the member forces, bending moments, shears
and associated deformations.
• Characteristic values have a subscript “k”. Design values have a subscript “d”,
which takes into account the relevant partial safety factors applied to the
characteristic values.
• References to EN 1991 (Actions on Structures) should be replaced by
SANS 10160 (2011).
• All formulae and expressions in the code refers to the cylinder strength (fck).
• When used, the cube strength will be denoted by fck,cube (e.g. see Table 31.).
• The cube strength is always higher than the cylinder strength due to frictional effects at the
boundaries and differences in aspect ratios of test spcimen.
• A concrete class C50/60 refers to a cylinder strength of 50 MPa and a cube strength of
60 MPa
• For cube strengths less than approximately 80 MPa, a reasonable approximation will be:
fck = 0.8 fck,cube
• Quality control in South Africa (as in the UK) will continue to use cube strengths.
 Eurocode 2 relationship between cylinder Experimental relationship between cylinder
and cube strengths (Table 3.1) and cube strengths (Table 3.1)
100
95 110

Cylinder strength fc (MPa) 150  300 mm cylinders

90 100
85
90
80
75 80
70
Cylinder strength (MPa)

70
65
60 60
55 50
50
40 Held (Ref. 2-10)
45 Smeplas (Ref. 2-11)
40 30 fc = 0.8 fcu
35
20
30 30 50 70 90 110 130
25 Cube strength fcu (MPa) 100 mm cubes
20
15
10 EC2 values
5 Ratio = 0.8
0
0 10 20 30 40 50 60 70 80 90 100 110 120

Cube strength (MPa)

Design concrete compressive strength

• From Eurocode 2:

• It is recommended to follow the UK National Annex here:

• 𝛼𝑐𝑐 = 0,85 for compression in flexure and axial loading.
• 𝛼𝑐𝑐 = 1,0 for other phenomena.
• However,𝛼𝑐𝑐 = 0,85 may be taken conservatively for all phenomena.
• The coefficient 𝛼𝑐𝑐 takes into account:
• The long-term effects on the compressive strength.
• Unfavourable effects resulting from the way the load is applied.
• The Commentary to Eurocode 2 (Beeby et al., 2008a) justifies the use of
𝛼𝑐𝑐 = 1,0 by taking into account that:
• Under a constant load, the stress at failure approaches 80% of the short-term
capacity.
• The 28 day strength is used in design while the structure will experience the design
load at a much later stage when the concrete strength has increased (about 12%).
• Resistance equations in a design code of practice is based on full-scale tests which
occur over about 90 minutes, whereas the short-term capacity is determined in about
2 minutes. This means that the reduction in strength of about 15% is accounted for in
the full-scale test.
• The background document to the UK National Annex (PD 6687, 2006)
recommends 𝛼𝑐𝑐 = 0,85. They point out that, at higher strengths:
• The Eurocode 2 value (𝛼𝑐𝑐 = 1,0) represents an upper characteristic value.
• The UK National Annex recommendation (𝛼𝑐𝑐 = 0,85) represents a median.
Background document to the UK
National Annex (PD 6687, 2006).

For European research also see:

• fib 52 (2010) Structural
Concrete - Textbook on
behaviour, design and
performance, Volume 2, Basis
of design, fib Bulletin No. 52
Fédération Internationale du
Béton, Lausanne.

For American research also see

ACI 318 and:
• Rüsch, H. (1960). Researches
Toward a General Flexural
Theory for Structural
Concrete, ACI Journal, July,
Vol. 57 No. 1.
Concrete stress-strain relationship

Four stress-strain relationships are available for concrete in compression:

1. Parabolic: for non-linear structural analysis (Fig. 3.2).
2. Parabolic-rectangular: for design of cross-sections (Fig. 3.3).
3. Bi-linear: for design of cross-sections (Fig. 3.4).
4. Equivalent rectangular: for design of cross-sections (Fig. 3.5).
The equivalent rectangular stress block is recommended for general design:
• The parameters for the stress-strain relationships have been calibrated to
yield similar results.
• The section will be designed for flexure so that the reinforcement yields
prior to failure of the concrete. Therefore, the reinforcement stress-strain
relationship dominates the behaviour and the exact shape of the concrete
stress-strain relationship has minimal impact.
𝜆 = 0,9 in SANS 10100

𝜂 = 0,85 in SANS 10100

Redistribution of moments

• Linear elastic analysis with limited redistribution (Cl. 5.5)

Moment redistribution ratio (𝛽𝑏 in SANS 10100):
moment at section after redistribution
𝛿= ≤1
moment at section before redistribution
For concrete Classes up to C50/60:
𝑥𝑢
𝛿 ≥ 𝑘1 + 𝑘2
𝑑
From SANS 51992-1-1 C. 5.5(4):
0,0014
𝑘1 = 0,4 and 𝑘2 = 0,6 +
𝜀𝑐𝑢2
For concrete Classes up to C50/60:
0,0014
𝑘2 = 0,6 + =1
0,0035
So that
𝑥𝑢
𝛿 ≥ 0.4 +
𝑑
Rearranging to have the neutral axis depth on the left, we obtain (the same equation as before):
𝑥𝑢
≤ 𝛿 − 0.4
𝑑
Also from Cl. 5.6.3(2):
𝑥𝑢
≤ 0,45
𝑑
Which implies 𝛿 = 0.85, or 15% redistribution of moments (In SANS 10100 𝛽𝑏 = 0.9 or 10% redistribution
of moments).
Design of flexural members (without axial force)

The Eurocodes focus on defining design principles rather than providing

design “recipes”. Design equation can be found in text books such as Bond
et al. (2007):
(Bond et al. 2007)
(MPa)
Flanged beams

• Effective flange width (Cl. 5.3.2.1) is slightly different than before:

• The distance 𝑙0 between points of zero moment are defined below

For the majority of flanged sections, where the flange is in compression, the
depth of the stress block will fall inside the flange:
0,8 𝑥 < ℎ𝑓 or 𝑥 < 1,25ℎ𝑓
And the equations for a rectangular section will apply, with 𝑏 = 𝑏𝑒𝑓𝑓
Bending plus axial load at ULS

• For pure compression the strain in the concrete is limited to 𝜖𝑐3 if the
equivalent rectangular stress block is used. For concrete classes up to
C50/60 𝜖𝑐3 = 0,00175.
• With bending, but without tensile strain in the section, the strain pivots
about a hinge point at ℎ/2.
(Bond et al. 2007)
References
1. Mosley, W. B.; Bungey, J. H. & Hulse, R. (2012). Reinforced Concrete Design to Eurocode 2, 7th Ed., Palgrave Macmillan, 448 pp.
2. Narayanan, R. S. & Goodchild, C. (2006). Concise Eurocode 2: For the design of in-situ concrete framed buildings to BS EN 1992-1-1:
2004 and its UK National Annex: 2005, MPA - The Concrete Centre, Camberley, 107 pp.
3. Bond, A. J.; Harrison, T.; Brooker, O.; Moss, R.; Narayanan, R.; Webster, R. & Harris, A. J. (2007). How to Design Concrete Structures
using Eurocode 2 - The Compendium, MPA Concrete Centre, 98 pp.
4. PD 6687 (2006). Background paper to the UK National Annexes to BS EN 1992-1, British Standards Institute, London, 40 pp.
5. Beeby, A. W.; Peiretti, H. C.; Walraven, J.; Westerberg, B. & Whitman, R. V. Jacobs, J.-P. (Ed.) (2008a). Eurocode 2 Commentary,
European Concrete Platform ASBL, Brussels. http://www.europeanconcrete.eu/publications/eurocodes/114-commentarytoeurocode2
6. Beeby, A. W.; Peiretti, H. C.; Walraven, J.; Westerberg, B. & Whitman, R. V. Jacobs, J.-P. (Ed.) (2008b). Eurocode 2 Worked Examples,
European Concrete Platform ASBL, Brussels. http://www.europeanconcrete.eu/publications/eurocodes/112-worked-examples-for-
eurocode-2
7. IStructE (2010). Manual for the design of concrete building structures to Eurocode 2, The Institution of Structural Engineers, London,
246 pp.
8. IStructE (2006). Standard Method of Detailing Structural Concrete - A manual for best practice, 3rd. Ed., The Institution of Structural
Engineers, London, 188 pp.
9. Narayanan, R. S. & Beeby, A. (2005). Designers' Guide to EN 1992-1-1 and EN 1992-1-2 Eurocode 2: Design of Concrete Structures.
General Rules and Rules For Buildings and Structural Fire Design, Thomas Telford, London, 218 pp.
10. Threlfall, T. (2013). Worked Examples for the Design of Concrete Structures to Eurocode 2, CRC Press, Boca Raton,243 pp.