Structural analysis
5.2. Global analysis
5.2.2. Structural stability of frames and second-order analysis This section has been split into three sub-sections in this guide for convenience.
5.2.2.1. General
Where it is necessary to take second-order effects and imperfections into account, this may be achieved in one of three ways according to 3-1-1/clause 5.2.2(3) and 3-1-1/clause 5.2.2(7): 1. Use of second-order analysis including both ‘global’ system imperfections and ‘local’ member imperfections as discussed in section 5.3. Where a beam is susceptible to lateral torsional buckling, imperfections must also be modelled to cater for second- order effects from this mode of buckling as discussed in section 5.3.4. If this method is followed, no individual checks of member stability are required using 3-2/clause 6.3 and members are checked for cross-section resistance only. Rather than superimposing local and global imperfections, it is possible to apply a unique overall imperfection to the structure based on the shape of the lowest mode of buckling of the structure. This method is given in 3-1-1/clause 5.3.2(11) and is discussed in section 5.3.2 of the guide. 2. Use of second-order analysis including ‘global’ system imperfections only with stability
checks according to 3-2/clause 6.3 subsequently carried out for individual members using the end moments and axial loads from the analysis. Since the member end forces and moments contain second-order effects from global behaviour, the effective length of individual members is then based on the member length, rather than a greater effective length that includes the effects of global sway deformations. It should be noted that when 3-1-1/clause 6.3.3 is used for member checks, the member moments will be further multiplied by the ‘kij’ parameters. Since the second-order analysis will already
have amplified these moments (providing sufficient nodes have been included along the member in the analysis model), this is conservative and it would be permissible to limit the values of the calculated ‘kij’ parameters to unity where they exceed unity.
However, the imperfections within the members have not been considered or amplified by the second-order analysis. These are included by way of the first term in the equations in this clause
NEd
NRk=M1
3. Use of first-order analysis without modelled imperfections. Members are then checked to 3-2/clause 6.3 using appropriate effective lengths covering the lowest buckling mode of the bridge involving the element under consideration. All second-order effects are then included in the relevant resistance formulae in 3-1-1/clause 6.3. This latter method will be most familiar to UK bridge engineers, as tables of effective lengths for members with varying end conditions of rotational and positional fixity have commonly been used. The use of effective lengths for this method is discussed later.
Second-order analysis itself can be done either by direct analysis that accounts for the deformed geometry (computer programs are readily available to do this) or by amplification of the moments from a first-order analysis (including the effects of imperfections) as discussed below – 3-1-1/clause 5.2.2(4) refers. Where either approach is used, it should only be performed by, or under the guidance of, experienced engineers because the guidance on the use of imperfections in terms of shapes, combinations and directions of application are not comprehensive in EC3; judgement is required.
5.2.2.2. Second-order analysis by the use of moment magnifiers
Although the elastic critical buckling load or moment itself has little direct relevance to real member strength, it gives a good indication of susceptibility to second-order effects and can
3-1-1/clause 5.2.2(3) 3-1-1/clause 5.2.2(7) 3-1-1/clause 5.2.2(4)
also be used as a parameter in determining second-order effects from the results of a first- order analysis. The method of 3-2/clause 5.2.2(5) is based on the elastic theory that total moments in a pin-ended strut, including second-order effects, can be derived by multiplying first-order moments (including moments arising from initial imperfections) by a magnifier that depends on the axial load and the Euler buckling load of the member. The simplest example of this is a pin-ended column, length L, under axial load only with an initial sinusoidal bow imperfection of maximum displacement a0. The Euler buckling load is
given by: Fcr¼ 2EI =L2
If the axial load is FEdthen the final deflection is given by:
a¼ a0
1 1 F Ed=Fcr
(This is obtained from simple elastic theory by solving EId
2
v v0
ð Þ
dx2 þ FEdv¼ 0
where v is the lateral displacement as a function of height x up the column and v0¼ a0sin x=L.)
The corresponding final maximum moment including second-order effects, MEdII ¼ FEda, is
then given by: MIIEd¼ FEd a0 1 Fð Ed=FcrÞ ¼ MIEd 1 1 Fð Ed=FcrÞ (D5.2-4) where MEdI ¼ FEda0is the first-order moment. The magnifier here is 1=ð1 FEd=FcrÞ, which
assumes that the initial imperfection is sinusoidal. Similar results are produced for the magnification of moments in pin-ended struts with applied end moments or transverse load, but the magnifier varies depending on the distribution of the first-order moment. For uniform moment, the amplifier above is slightly unconservative, but it will generally suffice with sufficient accuracy.
The pin-ended strut case is not itself an application of great practical significance as second-order effects and imperfections for pin-ended struts are covered in the resistance formulae for flexural buckling in 3-1-1/clause 6.3. It does however illustrate the basis of expression 3-2/(5.2), which allows total moments in bridges and bridge components, includ- ing second-order effects, to be found by increasing the first-order moments (including the effects of all imperfections) as follows:
MII¼ MI
1 1 1=ð crÞ
3-2/(5.2) with cr¼ Fcr=FEd defined in section 5.2.1.1 above. For uniform isolated members,
cr¼ Fcr=FEd is safe to use for sinusoidal or triangular distribution of curvature but is
slightly unconservative for uniform curvature, although not unduly so. A similar expression is given in EN 1992-1-1 thus: MII¼ MI 1þ Fcr=FEd ð Þ 1 (D5.2-5) with ¼ 2=c0 and Fcr¼ 2EI =L2cr. c0 depends on the distribution of moment and hence
curvature in the column. For uniform curvature, c0¼ 8. For sinusoidal curvature (and
approximately for triangular curvature or parabolic curvature), c0¼ 2and the expression
for moment simplifies to the simple form of expression 3-1-1/(5.4). Lcris the effective length
for buckling which can be determined as discussed in section 5.2.2.3 below. Alternatively, Fcr=FEdcan be determined directly by computer elastic critical buckling analysis.
The above expressions all assume that the peak first-order moment occurs at the same section as the peak moment from the P– effect. Considering an integral pier, with end
3-2/clause 5.2.2(5)
rotational restraint arising from connection to the foundations at one end and the deck at the other, Fig. 5.2-4 shows that the P– moment actually reduces the peak first-order end moment at the top. EN 1992 overcomes this conservatism for concrete elements by allowing an equivalent first-order moment to be used, but only where there is no transverse load applied in the height of the column and the members cannot sway. A more detailed discussion on this is provided in the Designers’ Guide to EN 1992-2.6
The limitations on use and accuracy of this method mean that it will usually be better to perform an elastic second-order computer analysis where it is necessary to consider second- order effects, or to include them by means of appropriate effective lengths and resistance formulae.
5.2.2.3. Effective lengths
Where second-order effects need to be accounted for but it is not desired to carry out a second-order analysis, the concept of effective length can be used together with the resistance formulae and interactions in 3-1-1/clause 6.3. In this case, imperfections need not be modelled if local and global effects are included in the effective length as stated in 3-1-1/ clause 6.3.3(3) and 3-1-1/clause 5.2.2(8). This method will be most familiar to UK bridge engineers. Effective lengths can also be used in the moment magnification method described above.
3-2/Annex D gives methods of calculating effective lengths for isolated bridge members in trusses and for buckling of arch bridges. (It also gives imperfections for arches for use in second-order analysis.)
Further relevant guidance on effective lengths for axially loaded members can be found in EN 1992-1-1. Typical examples of isolated members include:
. piers with free sliding bearings at their tops (Fig. 5.2-5(b)), assuming the load moves with
the pier
. piers with fixed bearings at their tops, but where the deck itself provides no positional
restraint and moves with the pier (Fig. 5.2-5(b) again)
. piers with fixed (pinned) bearings at their tops which are restrained in position by connec-
tion via the deck to a rigid abutment or other stocky pier (Fig. 5.2-5(c)).
The effective lengths given in the cases (a) to (e) of Fig. 5.2-5 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 and 3-1-1/clause 5.2.2(8) requires any flexibility to be considered. 2- 1-1/clause 5.8.3 gives a method of accounting for this rotational flexibility in the effective
3-1-1/clause 5.2.2(8)
M2
M1
(a) (b)
Fig. 5.2-4. Amplification of applied first order moments (imperfections excluded for clarity): (a) first-order applied moment and resulting deflection; (b) additional moments from second-order effects
length using equation (D5.2-6) for braced members (Fig. 5.2-5(f )) and equation (D5.2-7) for unbraced members (Fig. 5.2-5(g)):
Lcr¼ 0:5l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ k1 0:45þ k1 1þ k2 0:45þ k2 s (D5.2-6) Lcr¼ l max ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 10 k1k2 k1þ k2 s ; 1þ k1 1þ k1 1þ k2 1þ k2 ( ) (D5.2-7) where k1 and k2are the flexibilities of the rotational restraints at ends 1 and 2 respectively
relative to the flexural stiffness of the member itself such that: k¼ =Mð Þ EI=lð Þ where:
k ¼ ð=MÞðEI=lÞ
is the rotation of the restraint for a bending moment M; EI is the bending stiffness of the compression member;
l is the clear height of compression member between end restraints.
As can be seen from the formulae, equation (D5.2-7) can also be used for members with different rotational restraints at both ends but no lateral restraint at the top. This is useful for piers which are integral with a deck where deck and pier can sway. Quick inspection of equation (D5.2-7) shows that the theoretical case of a member with ends built in rigidly for moment (k1¼ k2 ¼ 0), but free to sway in the absence of positional restraint at one
end, gives an effective length Lcr¼ l as expected. 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 connection to the pier. Alternatively, the analytical method described below could be used. Cracking of concrete should be considered in deriv- ing the stiffness of the foundation or other members if relevant. The Note to 2-1-1/clause 5.8.3.2(3) recommends that no value of k is taken less than 0.1.
It should be noted that the cases in Fig. 5.2-5 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.)
For more complex situations (such as for a member with varying section along its length), it is preferable to work directly from Fcr. Fcrcan be calculated from a computer elastic critical
M
(a) (b) (c) (d) (e) (f) (g)
l
θ
θ
Fig. 5.2-5. Examples of different buckling modes and corresponding effective lengths for isolated members: (a) Lcr¼ l; (b) Lcr¼ 2l; (c) Lcr¼ 0:7l; (d) Lcr¼ l=2; (e) Lcr¼ l; (f ) l=2 < Lcr< l; (g) L > 2l
buckling analysis and then used either to perform a moment magnification calculation using expression 3-2/(5.2) or to determine the slenderness from expression 3-1-1/(6.50) for use with the member resistance curves in 3-2/clause 6.3.1.
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. 5.2-6(b)) or a single stiffer pier or abutment might prevent sway and give braced behaviour for the other piers (Fig. 5.2-6(a)). The analytical method above could also be used in this situation to produce an accurate effective length by applying coexisting loads to all piers and increasing all loads proportionately until a buckling mode involving the pier of interest is found. Pcris
then taken as the axial load in the member of interest at buckling.