Structural analysis
5.2. Geometric imperfections
5.2.1. General (additional sub-section)
The term ‘geometric imperfections’ is used to describe the departures from the exact centreline, setting out dimensions specified on drawings that occur during construction.
This is inevitable as all construction work can only be executed to certain tolerances.
2-1-1/clause
2-1-1/clause 5.2(1)Prequires these imperfections to be considered in analysis. The term does not apply to tolerances on cross-section dimensions, which are accounted for separately in the material factors, but does apply to load position. 2-1-1/clause 6.1(4) gives minimum requirements for the latter. Geometric imperfections can apply both to overall structure geo-metry and locally to members.
Geometric imperfections can give rise to additional moments from the eccentricity of axial loads generated. They are therefore particularly important to consider when a bridge or its elements are sensitive to second-order effects. 2-1-1/clause 5.2(2)P, however, requires imperfections to be considered for ultimate limit states even when second-order effects can be ignored in accordance with 2-1-1/clause 5.8.2(6). For short bridge elements, the additional moments caused by imperfections will often be negligible and the effects of imperfections could then be ignored in such cases with experience. Imperfections need not be considered for serviceability limit states (2-1-1/clause 5.2(3)).
2-2/clause 5.2(104) states that the values of imperfections used within 2-2/clause 5.2 assume that workmanship is in accordance with deviation class 1 in EN 13670. If other levels of workmanship are to be used during construction, then the imperfections used in design should be modified accordingly.
In general, imperfections can be modelled as either bows or angular departures in members. EC2 generally uses angular departures as a simplification, but sinusoidal bows will often be slightly more critical and better reflect the elastic critical buckling mode shape. For this reason, sinusoidal imperfections have to be used in the design of arches to 2-2/clause 5.2(106). The type of imperfection relevant for member design will depend on the mode of buckling. An overall lean to a pier will suffice where buckling is in a sway mode, which EC2 describes as ‘unbraced’ conditions in section 5.8. This is because the moments generated by the imperfection will add to those from the additional deflections under load. An overall lean would not, however, suffice for buckling within a member when its ends are held in position, which EC2 describes as ‘braced’ conditions. In this latter case, a lean of the entire column alone would not induce any moments within the column length. It would, however, induce forces in the positional restraints, so an imperfec-tion of this type would be relevant for the design of the restraints. A local eccentricity within the member is therefore required for buckling of braced members. This illustrates the need to choose the type of imperfection carefully depending on the effect being investigated. Further discussion on ‘braced’ and ‘unbraced’ conditions is given in section 5.8 of this guide and Fig. 5.2-1 illustrates the difference.
A basic lean imperfection, l, is defined for bridges in 2-2/clause 5.2(105) as follows:
l¼ 0h 2-2/(5.101)
where:
0 is the basic value of angular departure
h is the reduction factor for height with h¼ 2= ffiffi pl
; h 1 l is the length or height being considered in metres
2-1-1/clause
Fig. 5.2-1. Effect of geometric imperfections in isolated members
The lower limit for hgiven in EC2-1-1 was removed in EC2-2 to avoid excessive imper-fections in tall bridge piers. The value of 0 is a nationally determined parameter but the recommended value is 1/200, which is the same as previously used in Model Code 90.6
2-1-1/clause 5.2(7) allows imperfections in isolated members to be taken into account either by modelling them directly in the structural system or by replacing them with equiva-lent forces. The latter is a useful alternative, as the same model can be used to apply different imperfections, but the disadvantage is that the axial forces in members must first be known before the equivalent forces can be calculated. This can become an iterative procedure. These alternatives are illustrated in Fig. 5.2-1 for the two simple cases of a pin-ended strut and a cantilever. They are:
(a) Application of an eccentricity, ei
2-1-1/clause 5.2(7) gives the following formula for the imperfection eccentricity:
ei¼ ll0=2 2-1-1/(5.2)
where l0is the effective length.
For the unbraced cantilever in Fig. 5.2-1(a), the angle of lean from 2-2/(5.101) leads directly to the top eccentricity of ei¼ ll¼ ll0=2, when l0 = 2l (noting that l0>2l for cantilever piers with real foundations as discussed in section 5.8.3 of this guide).
For the braced pin-ended pier in Fig. 5.2-1(b), partially reproducing 2-1-1/Fig. 5.1(a2), the eccentricity is shown to be applied predominantly as an end eccentricity. This is not in keeping with the general philosophy of applying imperfections as angular deviations. An alternative, therefore, is to apply the imperfection for the pin-ended case as a kink over the half wavelength of buckling, based on two angular deviations, l, as shown in Fig. 5.2-2. This is then consistent with the equivalent force system shown in Fig. 5.2-1(b).
It is also the basis of the additional guidance given in EC2-2 for arched bridges where a devia-tion a¼ ll=2 has to be attributed to the lowest symmetric modes as discussed below. This method of application is slightly less conservative.
2-1-1/Expression (5.2) can be misleading for effective lengths less than the height of the member, as the eccentricity ei should really apply over the half wavelength of buckling, l0. This interpretation is shown in Fig. 5.2-3 for a pier rigidly built in at each end. It leads to the same peak imperfection as for the pin-ended case, despite the fact the effective length for the built in case is half that of the pinned case. This illustrates the need to be guided by the buckling mode shape when choosing imperfections.
(b) Application of a transverse force, Hi, in the position that gives maximum moment The following formulae for the imperfection forces to apply are given in 2-1-1/clause 5.2(7):
Hi¼ lN for unbraced members (see Fig. 5.2-1(a)) 2-1-1/(5.3a) Hi¼ 2lN for braced members (see Fig. 5.2-1(b)) 2-1-1/(5.3b) where N is the axial load.
2-1-1/clause 5.2(7)
N
L/2
L/2
ei = θlL/2 θl
θl
ei
Fig. 5.2-2. Alternative imperfection for pin-ended strut as an angular deviation
Where the imperfections are applied geometrically as a kink or lean as discussed above, these forces are directly equivalent to the imperfection.
5.2.2. Arches (additional sub-section)
2-2/clause 5.2(106)covers imperfections for arches for buckling in plane and out of plane.
In-plane buckling
For in-plane buckling cases where a symmetric buckling mode is critical, for example from arch spreading, a sinusoidal imperfection of a¼ 1L=2 has to be applied as shown in Fig. 5.2-4. This magnitude is derived by idealizing the actual buckling mode as a kink made up from angular deviations, l, despite the clause’s recommendation that the imperfection be distributed sinusoidally for arch cases as discussed above.
Where an arch does not spread significantly, the lowest mode of buckling is usually anti-symmetric, as shown in Fig. 5.2-5. In this case, the mode shape, and thus imperfection, can be idealized as a saw tooth using the same basic angular deviation in conjunction with the reduced length L=2 relevant to the buckling mode. The imperfection therefore becomes a¼ lL=4. Once again, EC2 requires the imperfection to be distributed sinusoidally.
Out-of-plane buckling
For out-of-plane buckling, the same shape of imperfection as in Fig. 5.2-4 is suitable but in the horizontal plane.
2-2/clause 5.2(106)
l0 = l/2
θl
ei ei
e = 2ei = θll/2
Sinusoidal imperfection Angular imperfection
Fig. 5.2-3. Imperfection for pier built in at both ends
L Initial shape
Kink imperfection a
a Sinusoidal
imperfection
θl
Fig. 5.2-4. Imperfection for in-plane buckling with spreading foundations