III. RESULTADOS
3.1. Resultados en Tablas y Figuras
5.8.1. Definitions and introduction to second-order effects
Second-order effects are additional action effects caused by the interaction of axial forces and deflections under load. First-order deflections lead to additional moments caused by the axial forces and these in turn lead to further increases in deflection. Such effects are also sometimes called P effects because additional moments are generated by the product of the axial force and element or system deflections. The simplest case is a cantilevering pier with axial and horizontal forces applied at the top, as in Fig. 5.8-1. Second-order effects can be calculated by second-order analysis, which takes into account this additional deformation.
Second-order effects apply to both ‘isolated’ members (for example, as above or in Fig. 5.8-2(a)) and to overall bridges which can sway involving several members (Fig. 5.8-2(b)). EC2 refers to two types of isolated member. These are:
a
C b
B
D
A Internal actions path
O
C’
MEd, MRd
M(qud)
qud
N
( ) /
γRdγO
qud
N
( )
γO
qud
M
( ) /
γRdγO
qud
M
( )
γO
N(qud) NEd, NRd
Fig. 5.7-2. Safety format for vector over-proportional behaviour using inequality 2-2/Expression (5.102 aN)
H P
Deflection from P and H (second order) Deflection from H alone
(first order)
Fig. 5.8-1. Deflections for an initially straight pier with transverse load
Braced members – members that are held in position at both ends and which may or may not have restraining rotational stiffness at the ends. An example is a pin-ended strut. The effective length for buckling will always be less than or equal to the actual member length.
Unbraced members – members where one end of the member can translate with respect to the other and which have restraining rotational stiffness at one or both ends. An example is the cantilevering pier above. The effective length for buckling will always be greater than or equal to the actual member length.
The compression members of complete bridges can often be broken down into equivalent isolated members that are either ‘braced’ or ‘unbraced’ by using an effective length and appropriate boundary conditions. This is discussed in section 5.8.3.
Some engineers may be unfamiliar with performing second-order analysis, which is the default analysis in the Eurocodes. A significant disadvantage of second-order analysis is that the principle of superposition is no longer valid and all actions must be applied to the bridge together with all their respective load and combination factors. Fortunately, there will mostly be no need to do second-order analysis, as alternative methods are given in this chapter and frequently second-order effects are small in any case and may therefore be neglected. Some rules for when second-order effects may be neglected are given in clauses 5.8.2 and 5.8.3 and are discussed below.
Slender compression elements are most susceptible to second-order effects. (The definition of slenderness is discussed in section 5.8.3.) The degree of slenderness and magnitude of second-order effects are related to well-known elastic buckling theory. Although elastic buckling in itself has little direct relevance to real design, it gives a good indication of suscept-ibility to second-order effects and can be used as a parameter in determining second-order effects from the results of a first-order analysis; section 5.8.7 refers.
Second-order analysis can be performed with most commercially available structural soft-ware. In addition to the problem of lack of validity of the principle of superposition, the flexural rigidity of reinforced concrete structures, EI, is not constant. For a given axial load, EI reduces with increasing moment due to cracking of the concrete and inherent non-linearity in the concrete stress–strain response. This means that second-order analysis for reinforced concrete elements is non-linear with respect to both geometry and material behaviour. This non-linearity has to be taken into account in whatever method is chosen to address second-order effects. This is dealt with in subsequent sections.
For bridges, it is slender piers that will most commonly be affected by considerations of second-order effects. Consequently, all the Worked examples in this section relate to slender piers. The provisions, however, apply equally to other slender members with significant axial load, such as pylons and decks of cable-stayed bridges.
5.8.2. General
When second-order calculations are performed, it is important that stiffnesses are accurately determined as discussed above. 2-1-1/clause 5.8.2(2)P therefore requires the analysis to consider the effects of cracking, non-linear material properties and creep. This can be done either through a materially non-linear analysis (as discussed in section 5.8.6) or by using linear material properties based on a reduced secant stiffness (as discussed in section 5.8.7). Imperfections must be included as described in section 5.2 as these lead to additional
2-1-1/clause 5.8.2(2)P
(a) Buckling of individual piers (braced) (b) Overall buckling in sway mode (unbraced)
Fig. 5.8-2. Buckling modes for braced and unbraced members
first-order moments and consequently additional second-order moments in the presence of axial compression.
Soil–structure interaction must also be taken into account (2-1-1/clause 5.8.2(3)P), as it should be in a first-order analysis. There are few rules specific to the analysis of integral bridges, but the slenderness of integral bridge piers can be determined using the general procedures discussed in section 5.8.3 of this guide.
2-1-1/clause 5.8.2(4)Prequires the structural behaviour to be considered ‘in the direction in which deformation can occur, and biaxial bending shall be taken into account when necessary’. Often, deformation in two orthogonal directions needs to be considered in bridge design under a given combination of actions, although the moments in one direction may be negligible compared to the effect of moments in the other. A related clause is 2-1-1/
clause 5.8.2(5)P which requires geometric imperfections to be considered in accordance with clause 5.2. 2-1-1/clause 5.8.9(2) states that imperfections need only be considered in one direction (the one that has the most unfavourable effect), so biaxial bending conditions need not always be produced simply due to considerations of imperfections.
It will not always be necessary to consider second-order effects. 2-1-1/clause 5.8.2(6) permits second-order effects to be ignored if they are less than 10% of the corresponding first-order effects. This is not a very useful criterion as it is first necessary to perform a second-order analysis to check compliance. As a result, 2-1-1/clause 5.8.3 provides a simplified criterion for isolated members based on a limiting slenderness. This is discussed below.
5.8.3. Simplified criteria for second-order effects 5.8.3.1. Slenderness criterion for isolated members
Where simplified methods are used to determine order effects, rather than a second-order non-linear computer analysis, the concept of effective length can be used to determine slenderness. This slenderness can then be used to determine whether or not second-order effects need to be considered. According to 2-1-1/clause 5.8.3.2(1), the slenderness ratio is defined as follows:
¼ l0=i 2-1-1/(5.14)
where l0 is the effective length and i is the radius of gyration of the uncracked concrete cross-section.
A simplified criterion for determining when second-order analysis is not required is given in 2-1-1/clause 5.8.3.1(1), based on a recommended limiting value of the slenderness as follows:
lim ¼ 20A B C= ffiffiffi pn
2-1-1/(5.13N) This limiting slenderness may be modified in the National Annex. n¼ NEd=ðAcfcdÞ is the relative normal force. The greater the axial force and thus n become, the more the section will be susceptible to the development of second-order effects and, consequently, the lower the limiting slenderness becomes. Higher limiting slenderness can be achieved where:
. there is little creep (because the stiffness of the concrete part of the member in com-pression is then higher);
. there is a high percentage of reinforcement (because the total member stiffness is then less affected by the cracking of the concrete);
. the location of the peak first-order is not the same as the location of the peak second-order moment.
These effects are accounted for by the terms A, B and C respectively where:
A¼ 1=ð1 þ 0:2efÞ (D5.8-1)
2-1-1/clause 5.8.2(3)P
2-1-1/clause 5.8.2(4)P
2-1-1/clause 5.8.2(5)P
2-1-1/clause 5.8.2(6)
2-1-1/clause 5.8.3.1(1)
where ef is the effective creep ratio according to 2-1-1/clause 5.8.4. If efis not known, A may be taken as 0.7. This corresponds approximately to ef¼ 2:0 which would be typical of a concrete loaded at relatively young age, such that 1 ¼ 2:0, and with a loading that is entirely quasi-permanent. This is therefore reasonably conservative. A is not, in any case, very sensitive to realistic variations in ef, so using the default value of 0.7 is reasonable.
B¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 2!
p ðD5.8-2)
where !¼ Asfyd=ðAcfcdÞ is the mechanical reinforcement ratio. If ! is not known, B may be taken as 1.1 which is equivalent to !¼ 0:1. This value would usually be achieved in a slender column, but is slightly generous compared to the actual minimum reinforcement required by 2-1-1/clause 9.5.2(2):
C¼ 1:7 rm ð (D5.8-3)
where rmis the moment ratio M01=M02, where M01and M02are the first-order end moments such thatjM02j jM01j. If rm is not known, C may be taken as 0.7 which corresponds to uniform moment throughout the member. C should also be taken as 0.7 where there is trans-verse loading, where first-order moments are predominantly due to imperfections or where the member is unbraced. The reasons for this are explained in section 5.8.7 of this guide.
Before carrying out a non-linear analysis or using the simplified methods of 5.8.7 and 5.8.8, it would be usual to check whether such effects can be ignored using 2-1-1/Expression (5.13N). The use of this formula is illustrated in the Worked example 5.8-2.
5.8.3.2. Slenderness and effective length of isolated members
2-1-1/clause 5.8.3.2 gives methods of calculating effective lengths for isolated members.
Typical examples of isolated members and their corresponding effective lengths are given in 2-1-1/Fig. 5.7, reproduced here as Fig. 5.8-3. Examples of application include piers with free sliding bearings at their tops (case (b)), piers with fixed bearings at their tops but where the deck (through its connection to other elements) provides no positional restraint (case (b) again), and piers with fixed (pinned) bearings at their tops which are restrained in position by connection by way of the deck to a rigid abutment or other stocky pier (case (c)).
The effective lengths given in cases (a) to (e) assume that the foundations (or other restraints) providing rotational restraint are infinitely stiff. In practice, this will never be the case and the effective length will always be somewhat greater than the theoretical value for rigid restraints. For example, BS 5400 Part 49 required the effective length for case (b) to be taken as 2.3l instead of 2l. 2-1-1/clause 5.8.3.2(3) gives a method of accounting for this rotational flexibility in the effective length using equations 2-1-1/Expression (5.15)
2-1-1/clause 5.8.3.2(3)
M
(a) l0 = l (b) l0 = 2l (c) l0 = 0.7l (d) l0 = l /2 (e) l0 = l (f) l /2 < l0 < l (g) l0 > 2l l
θ
θ
Fig. 5.8-3. Examples of different buckling modes and corresponding effective lengths for isolated members
for braced members and 2-1-1/Expression (5.16) for unbraced members:
where k1 and k2 are the flexibilities of the rotational restraints at ends 1 and 2 respectively relative to the flexural stiffness of the member itself such that:
k¼ ð=MÞ ðEI=lÞ where:
is the rotation of the restraint for bending moment M
EI is the bending stiffness of compression member – see discussion below l is the clear height of compression member between end restraints
2-1-1/Expression (5.16) can be used for unbraced members with rotational restraint at both ends. Quick inspection of 2-1-1/Expression (5.16) shows that the theoretical case of a member with ends built in rigidly for moment ðk1 ¼ k2¼ 0Þ, but free to sway in the absence of positional restraint at one end, gives an effective length lo¼ l as expected.
It is the relative rigidity of restraint to flexural stiffness of compression member that is important in determining the effective length. Consequently, using the un-cracked value of EI for the pier itself will be conservative as the restraint will have to be relatively stiffer to reduce the buckling length to a given value. It is also compatible with the definition of radius of gyration, i, in 2-1-1/clause 5.8.3.2(1) which is based on the gross cross-section.
2-1-1/clause 5.8.3.2(5), however, requires cracking to be considered in determining the stiffness of a restraint, such as a reinforced concrete pier base, if it significantly affects the overall stiffness of restraint offered to the pier. For the pier example, however, often the overall stiffness is dominated by the soil stiffness rather than that of the reinforced concrete element.
The Note to 2-1-1/clause 5.8.3.2(3) recommends that no value of k is taken less than 0.1.
For integral bridges, or other bridges where restraint is provided at the top of the pier by its connection to the deck, cracking of the deck must also be considered in producing the stiff-ness. The value of end stiffness to use for piers in integral construction can be determined from a plane frame model by deflecting the pier to give the deflection relevant to the mode of buckling and determining the moment and rotation produced in the deck at the con-nection to the pier. Alternatively, the elastic critical buckling method described below could be used to determine the effective length more directly.
It should be noted that the cases in Fig. 5.8-3 do not allow for any rigidity of positional restraint in the sway cases. If significant lateral restraint is available, as might be the case in an integral bridge where one pier is very much stiffer than the others, ignoring this restraint will be very conservative as the more flexible piers may actually be ‘braced’ by the stiffer one.
In this situation, a computer elastic critical buckling analysis will give a reduced value of effective length. (In many cases, however, it will be possible to see by inspection that a pier is braced.)
Where cases are not covered by 2-1-1/clause 5.8.3.2, effective lengths can be calculated from first principles according to 2-1-1/clause 5.8.3.2(6). This might be required, for example, for a member with varying section, and hence EI, along its length. The procedure is to calculate the buckling load, NB, from a computer elastic critical buckling analysis, using the actual varying geometry and loading. It will be conservative to assume un-cracked con-crete section for the member of interest and cracked concon-crete for the others (unless they can be shown to be un-cracked). An effective length is then calculated from:
l0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where EI may be freely chosen but a compatible value of radius of gyration, i, and therefore concrete cross-sectional area, Ac, must be used in calculating the slenderness according to 2-1-1/Expression (5.14). A ‘sensible’ choice of EI would be to base it on the actual cross-section in the middle third of the buckling half wave.
Effective lengths can also be derived for piers in integral bridges and other bridges where groups of piers of varying stiffness are connected to a common deck. In this instance, the buckling load, and hence effective length, of any one pier depends on the load and geometry of the other piers also. All piers may sway in sympathy and act as unbraced (Fig. 5.8-4(b)) or a single stiffer pier or abutment might prevent sway and give braced beha-viour for the other piers (Fig. 5.8-4(a)). The analytical method above could also be used in this situation to produce an accurate effective length by applying coexisting loads to all columns and increasing all loads proportionately until a buckling mode involving the pier of interest is found. NB is then taken as the axial load in the member of interest at buckling.
Finally, effective lengths can be taken from tables of approximate values such as was provided in Table 11 of BS 5400 Part 4.9
(a) Buckling of individual piers (braced) (b) Overall buckling in sway mode (unbraced)
Fig. 5.8-4. Typical braced and unbraced situations
Worked example 5.8-1: Effective length of cantilevering pier
A bridge pier with free-sliding bearing at the top is 27.03 m tall and has cross-section dimensions and reinforcement layout (required for later examples), as shown in Fig. 5.8-5 and Fig. 5.8-5.8-6. The pier base has a foundation flexibility (representing the rotational flexibility of the pile group and pile cap) of 6:976 109rad/kNm. The short term E for the concrete is Ecm¼ 35 103MPa. Calculate the effective length about the minor axis.
The inertia of the cross-section about the minor axis¼ 3.1774 m4so:
EI
l ¼35 106 3:1774
27:03 ¼ 4:114 106kNm=rad
At the base of the pier, k1¼ ð=MÞ ðEI=lÞ ¼ 6:976 109 4:114 106¼ 28:7 103. This is less than the lowest recommended value of 0.1 given in 2-1-1/clause 5.8.3.2(3). However, as the stiffness above was derived using lower-bound soil properties and pile cap stiffness, the stiffer calculated value of k will be used.
At the top of the pier, there is no restraint so k2¼ 1.
From 2-1-1/Expression (5.16), the effective length is then:
l0¼ l max
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 10 0:0287 1 0:0287þ 1 r
;
1þ 0:0287 1þ 0:0287
1þ 1
1þ 1
" #
¼ l max½1:13; 2:06
¼ 2:06l
The effective length is therefore close to the value of 2l for a completely rigid support.
2500 944
4000 10001000
4501788
500
450 500
50Ø pier base drainage pipe 450
734
Fig. 5.8-5. Pier dimensions for Worked example 5.8-1
Fig. 5.8-6. Pier reinforcement layout for Worked example 5.8-1
Worked example 5.8-2: Check limiting slenderness of bridge pier
The bridge pier in Worked example 5.8-1 has concrete with cylinder strength 40 MPa and carries an axial load of 31 867 kN. Calculate the slenderness about the minor axis and determine whether second-order effects may be ignored. Take the effective length as 2.1 times the height.
The inertia of the cross-section about the minor axis¼ 3.1774 m4. The area of the cross-section¼ 4.47 m2. The limiting slenderness is determined from 2-1-1/Expression (5.13N) as follows:
lim ¼ 20A B C= ffiffiffi pn
Since the split of axial load into short-term and long-term is not given, the recom-mended value of A¼ 0:7 will be conservatively used as discussed in the main text. The reinforcement ratio is also not known at this stage, so the recommended value of B¼ 1:1. Since the pier is free to sway, this is an unbraced member and benefit cannot be taken from the moment ratio at each end of the pier. Hence C¼ 0:7 (which also corresponds to equal moments at each end of a pier that is held in position at both ends).
5.8.4. Creep
Creep tends to increase deflections from those predicted using short-term properties as discussed in section 3.1.4, so 2-1-1/clause 5.8.4(1)P requires creep effects to be included in second-order analysis. To perform this calculation rigorously, different stress–strain relationships would be required for different load applications. To overcome this problem, a simplified relationship is given using an effective creep ratio, ef, which, used together with the total design load, gives a creep deformation corresponding to that from the quasi-permanent load only, as is required. The effective creep factor is given in 2-1-1/
clause 5.8.4(2)as follows:
ef¼ ð1; t0ÞM0Eqp=M0Ed 2-1-1/(5.19)
where:
ð1; t0Þ is the final creep coefficient according to 2-1-1/clause 3.1.4
M0Eqp is the first-order bending moment in the quasi-permanent load combination (SLS) M0Ed is the first-order bending moment in the design load combination (ULS) The use of an SLS value for quasi-permanent loads and a ULS value for the design combination does not appear logical and it is recommended here that either SLS or ULS values are used to calculate both moments. By way of illustration, if all the moments were due to permanent load then 2-1-1/Expression (5.19) as written would lead to
ef< ð1; t0Þ which is incorrect.
With appropriate modification as suggested, the use of 2-1-1/Expression (5.19) in second-order calculations will generally be conservative. This is because short-term increments of axial load from live load will lead proportionately to a greater increase in second-order moment than a similar increment of dead load as the increment is occurring at higher axial load. The first-order moment ratio in 2-1-1/Expression (5.19) therefore overestimates
With appropriate modification as suggested, the use of 2-1-1/Expression (5.19) in second-order calculations will generally be conservative. This is because short-term increments of axial load from live load will lead proportionately to a greater increase in second-order moment than a similar increment of dead load as the increment is occurring at higher axial load. The first-order moment ratio in 2-1-1/Expression (5.19) therefore overestimates