second order effects

Chapter 1 : Second-order effects - who needs them?

1Chapter

Second-order effects - who needs them?

The Structural Engineer, 5th November 2002, Volume 80, number 21, page 19.

Introduction Whether you think you need them or not, second-order effects in structures will always occur and always need to be considered. Whether you need to do anything about them depends upon the type of structure with which you are dealing. In a cable stayed, membrane structure, second-order effects are obviously important. This article asks the question in relation to 'normal rectilinear' multi-storey buildings in steelwork.

What second-or

effecarederts?

There are, no doubt, many definitions of second-order effects. That given below is my own invention,

The effects that changes in the geometry of the structure under load (the deformations) have on the stiffness of the frame and, the magnitude and distribution of the forces within the frame.

Within the context of this article, the second-order effects fall into two categories, effects of deflections within the length of members - usually termed P-

effects (P-little delta), effects of displacements of the structure as a whole - usually termed P-

effects (P-big delta).


These are illustrated in figure 1. The effect of the axial load P acting at the displaced position can introduce significant additional bending moments in the member or frame.

Why second-or

effeimportant noarederctsw?

The answer is that they always were important. Unfortunately, previous editions of BS 5950-1 (1985 and 1990) were, arguably, unclear and ambiguous. There were also a number of common misconceptions such as, braced frames are non-sway (i.e. second-order effects are small enough to be ignored). Simple (hand) analysis of simple structures of the past was probably not awry in ignoring such effects. The world of structural engineering has become more complex and more 3D which we can either lament, or amend our design procedures to suit.

Irrespective of our views on the matter, the structure itself will generate and then respond to second-order effects. Our job should be to identify whether these effects are important and then design them out or design them in.

Figure 1: P-delta effects


What does theCode say?

Whilst there are references elsewhere in BS 5950-1: 2000, the main clause dealing with this topic is 2.4.2 Stability limit states. There are two main requirements,

To provide a practical level of robustness against the effects of incidental loading, all structures, including portions between expansion joints, should have adequate resistance to horizontal forces. This is accommodated by ensuring a minimum level of wind load of not less than 1% of the applied dead loads and by adopting a set of (usually) small lateral loads for inclusion with the 'gravity only' loadcase. The latter are termed Notional Horizontal Forces.

All structures (including portions between expansion joints) should have sufficient sway stiffness, so that the vertical loads acting with the lateral

The useNotio

HorizonFordisplacements of the structure do not induce excessive secondary forces and moments in the members or connections The Code goes on to say that where such effects are too large to ignore then allowance should be made for them.

The second requirement raises two questions, how do I determine whether these effects are significant and if they are, how do I allow for them? The answer to both of these questions is provided in the remainder of this article. Importantly, the answer to the first question is also tied up with the first of the two requirements above - the use of Notional Horizontal Forces (NHF's). The purpose of NHF's needs some clarification and this is given in the text immediately following.

ofnaltal

ces

The NHF's are taken as 0.5% of the factored vertical dead and imposed loads. They are calculated once but used twice. This is for convenience but is unfortunate. The two uses are diverse but the single calculation of their value was one of the main contributors to the mis-application or non-application of the 'sway' rules in the earlier versions of the Code. Incidentally, this is particularly unfortunate in today's computer age since a clearer approach to calculating two different sets of numbers for two different purposes is no effort for a computer!


The two uses are, NHF's plus other loading - to allow for 'frame imperfections' such as lack

of verticality NHF's alone - to test for sway susceptibility.

NHF's plus otherloading

When applied with other loading the NHF's allow for practical imperfections in the frame such as lack of verticality i.e. structures cannot be built entirely plumb and square. They are applied in two orthogonal directions separately and, depending upon the asymmetry of the structure, may need to be applied in each of the positive and negative directions. They are applied with gravity loads only -

NHF's alothat is they are not included with other lateral loads. Thus as well as the gravity only combination of dead and imposed loads, there can be up to four other load combinations that always need to be considered,

dead + imposed + NHF (+X) dead + imposed + NHF (-X) dead + imposed + NHF (+Y) dead + imposed + NHF (-Y)

ne As an alternative to carrying out a buckling analysis on all combinations (including those listed above), NHF's can be applied with no other loading to provide a simple test for the susceptibility of the structure to second-order effects. If such susceptibility is found, the structure is deemed to be sway sensitive. Conversely the structure can be found to be non-sway. The NHF's are applied in two orthogonal directions separately to determine sway sensitivity in the X and Y directions. It is not necessary to apply them in the positive and negative directions. A set of NHF's will exist for each combination including those containing normal lateral loads but based on the gravity load component of those combinations. This will indicate the susceptibility of the structure to second-order effects when loaded by each combination.


Separating these two uses of NHF's clearly in your mind is essential for a full understanding of the stability requirements of BS 5950-1: 2000.

Determiningsusceptibility to

second-ordereffects

The simplest test to determine a structure's susceptibility to second-order effects is to use the NHF's alone, as described above. In the context of this article, there is an assumption that the structure is clad but that the stiffening effects of the cladding are ignored - see Clause 2.4.2.6 of BS 5950-1:2000.

From the analyses of the NHF loadcases, the differential deflection between two adjacent floor levels can be determined - the storey height deflection. If this is less than column height divided by 2000 then the structure is non-sway for that

combination in that direction. (The Code actually determines the elastic critical load factor, cr, based on height over 200 times the storey height deflection and limits this factor to 10 for non-sway structures. The effect is the same.)

There may be other directions of application of NHF's or other load combinations that result in the opposite classification i.e. sway sensitive. This simple test is based on research carried out in the 1970's by Prof. M R Horne and reported in the Journal in June 1974. Interestingly the approximate method described in the paper for determining the sway of a plane frame used 1% of the factored dead and imposed loading but twice the deflection limit.

For multi-storey buildings of 'simple construction' it is recommended that the storey height deflections of all the columns, the simple columns as well as those in braced or moment resisting frames, are assessed. This can generate significant data to sift through in order to identify the most critical in each of the two orthogonal directions.

In addition where the structure is significantly asymmetric, the NHF's in one direction will not only produce storey height deflection in that same direction but can produce significant out-of-plane deflections. These need to be calculated and


taken into consideration. This is exemplified in the figure from Fastrak Multi-storey in which the most critical columns in the X and Y directions are identified in the 'tree-view' on the left hand side. The figure also shows that for this structure (which is an extreme case) the out-of-plane deflections (Sway XY) are significant. It can be seen that for NHF's applied in the X direction the structure deflects further in the Y direction (out-of-plane) than in the X direction (in the plane of the load). The Code gives no guidance on this matter and so designers need to apply their own engineering judgement. Figure 2: Critical columns for sway and out-of-plane deflections


The simple test for sway using NHF's is an approximation for the determination of the elastic critical buckling load factor, cr. (The elastic critical buckling load factor is the factor that would need to be applied to the design loads to bring about elastic buckling of the structure or part thereof. It is an entirely theoretical buckling load that is similar in concept to the Euler buckling load of a strut.) This factor can of course be determined directly and more accurately by using the buckling analysis component that is included with many general analysis programs, such as CSC's S-Frame. For many multi-storey structures, particularly those using braced frames to provide lateral load resistance, this is not as straightforward as it appears. All structures will have a number of 'buckling modes'. In multi-storey structures where the columns are axial load predicated, many of the lower buckling modes will be individual or combined column buckling modes. It is the buckling mode of the whole structure that is important if a simple allowance for second-order effects, as described in the next section, is to be adopted. These two types of buckling mode are illustrated in the figures below.

Chapter 1 : Second-order effects - who needs them?Figure 3: Second order column buckling mode from CSC's S-Frame


Designingsecond-or

effe fordercts

Having determined the susceptibility of the structure to second-order effects i.e. whether it is non-sway or sway sensitive, there are a number of options open to you. Firstly, if it is non-sway (cr 10 but see Clause 2.4.2.6 of BS 5950-1:2000), you need do nothing - second-order effects are small enough for your existing (first-order) analysis results to be acceptable. If the structure is classed as sway sensitive then you could choose to stiffen up the structure by providing more or stiffer lateral load resisting frames until the analysis indicates that your structure is non-sway. This might seem the best approach since you can then 'ignore' second-order effects but it may not be the most efficient structurally. Allowing for second-order effects is not as complicated or as onerous as you may think.

Figure 4: Second order building buckling mode from CSC's S-Frame


If the structure is sway sensitive, If 10 > cr 4.0 then second-order effects are significant but still small

enough to be accommodated, either approximately, by using the sway mode in-plane effective lengths for

columns and designing the beams elastically or approximately, by multiplying the sway effects by the amplification

factor kamp determined from Clause 2.4.2.7 (b)

or more accurately, by carrying out a second-order elastic analysis to directly determine the design forces and moments cr < 4.0 second-order effects are large and need to be accounted for accurately by using second-order elastic analysis to determine the design forces and moments

In the first condition, using the effective length method is not preferred since it does not increase the connection moments due to the second-order effects. In any event, it cannot be applied to braced frames in simple construction.

The amplified sway effects method is simple to understand and simple to apply. To allow for the second-order effects in a particular direction, the loads causing sway in that direction are increased. The simplest method of achieving this is to increase the load factor associated with the horizontal loading included in a particular design combination by the amplification factor in the appropriate direction. For other design combinations the critical crit and hence kamp should be identified and applied to the constituent horizontal loadcases. Two simplifications can be made in applying these kamp factors but at the penalty of increased levels of conservatism. An example is given in the table below.

Note that, if you chose to use second-order elastic analysis (or are required to do so because cr is less than 4.0) then this will regularly give better answers than those from using the amplified sway effects method.


The final four columns of the table below show the load factor and, the product of the load factor and the amplification factor, kamp. Three levels of conservatism are given,

the least conservative is the product of the load factor and the amplification factor calculated for each individual loadcase and direction,

the more conservative is the product of the worse amplification factor from both directions for each individual loadcase,

the most conservative is the product of the worst amplification factor from all loadcases and both directions.

DeComb

D+I+

D+I+

D+I+W

D+I+W

D+W

D+WThe determining factor in deciding to use the increased levels of conservatism is the tendency for the building to deflect out of the plane in which the principal horizontal loads have been applied. This judgment may also be influenced by the proximity of the value of crit to the lower limit of 4.

sign ination

crit AmplificationFactor LoadcaseOriginal

LoadFactor

Sway L/F &conservatism

X Y X Y Least More Most

NHF X 5.0 6.5 1.176 1.088 NHF X 1.0 1.176 1.176 1.176

NHF Y NHF Y 1.0 1.088 1.176 1.176

ind X 8.5 10.5 1.027 1.0 Wind X 1.2 1.232 1.232 1.411

ind Y Wind Y 1.2 1.2 1.232 1.411

ind X 12.0 13.5 1.0 1.0 Wind X 1.4 1.4 1.4 1.4

ind Y Wind Y 1.4 1.4 1.4 1.4


In a multi-storey structure, it is often the Dead + Wind case that is most critical for the design of the braces or moment frames. It can be seen from the table that for this combination the structure is non-sway and hence the amplifier is 1.0 i.e. no change to the design forces. Whereas, despite having been increased by nearly 18% it was found that the Dead + Imposed + NHF case was still not critical. Hence, there has been a slight increase in design effort (even less so when using software) yet no actual change to section sizes. For those who ask, So why should I bother? this will not always be the case. For long, thin buildings with reasonably heavy floor loads e.g. 5-6 kN/m2 the lateral loads from the NHF's can swamp those due to wind loads in the longitudinal direction.

Current afuture practndice

In all this we must not lose sight of the fact that we are creating 'models' that are only an approximation to the real world. It is important that we address the underlying structural behaviour and not get caught up in (or even try to manipulate) the mathematics.

The traditional approach of isolating 2D frames from a structure that behaves in 3D was probably not unconservative for simple structures but did ignore some of the 3D effects. Not explicitly taking account of second-order effects was probably not often unconservative either. A more accurate and fully defensible approach, in terms of the current code requirements, is to treat the structure in 3D but consider the second-order effects in two orthogonal directions (2 x 2D).

Many of the multi-storey buildings we see today are anything but simple - they may be curved on plan, multi-faceted and have severe restrictions on where we can place bracing systems. For these structures, the interpretation, application and accuracy of the code rules as 2 x 2D can become more uncertain.

The future is almost undoubtedly 3D. Whether we like it or not the theory and the code rules (particularly when one looks towards Eurocodes) are becoming increasingly sophisticated. The aim is a more accurate mathematical prediction


of the real world such that our ability to predict the behaviour of structures advances, and hence (hopefully) structures become more efficient with fewer constraints on layout. The use of 3D in analysis and design matches the trend towards building 3D models for visualisation, drawing production, and prototyping using programs such as CSC's 3D+. Here all elements of the building, not just the main structure, can be modelled; sub-sets can be exported to the analysis/design software and the whole model passed down the construction chain and form the basis of the, now common, 3D steelwork detailing models.

This transition from simple traditional to full 3D will, rightly, be evolutionary. It

Concluswill be aided and abetted by the provision of well-designed robust software. Training in the use of such tools and the provision of Continuing Professional Development courses on the subject, such as those offered by the Steel Construction Institute, are paramount to the correct and efficient adoption of such new approaches.

ion Whether you think you need them or not, second-order effects exist. For many structures, ignoring them would make it difficult to prove that the structure is safe, and so you 'need' them. Whatever your view on Eurocodes, they are approaching and (relatively) quickly. The approach in mainland Europe is to include second-order effects and only ignore them by exception - the opposite of the current approach in the UK. The upside is that getting to grips with second-order effects now will,

improve the transition to Eurocodes and enable more straightforward export into mainland Europe

allow more accurate modelling of structures (the closer we get to predicting reality, we, as a profession, should demand lower load factors)

allow us to tackle more 'slender' structures, for example those where bracing is undesirable.


However, we must take care that the capabilities placed at the designer's fingertips must never be allowed to outstrip his understanding.

Second-order effects - who needs them?IntroductionWhat are second-order effects?Why are second-order effects important now?What does the Code say?The use of Notional Horizontal ForcesNHF's plus other loadingNHF's aloneDetermining susceptibility to second-order effectsDesigning for second-order effectsCurrent and future practiceConclusion

second order effects

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