Assessing the Stability of Frames

For every structure, frame stability is an important area of consideration. Designers of structural steel-work were the first to recognise the importance of considering the stability of steel frames in BS-5950. Even though the requirement to ensure that frames are made sufficiently stiff was included in earlier codes, however no guidance on assessing frame stability was given. Thus, this post attempts to give a background and then focuses on the practicalities of structural steel-work design to Eurocode 3. However, designers working to different code of practice with different materials will no doubt gain useful guidance on assessment of frame stability.

Frame stability can be defined as the effect of displaced vertical loads that are no longer concentric with their normal positions. This effect usually manifest in the form of lateral displacement which can either be caused by externally applied load such as wind, or due to the out of plumb of the frame by some degree. The latter is mostly the case and leads to the vertical loads applied on the frame being displaced. The displaced vertical load in turn causes further lateral displacement. This is classified as Second-Order Effect and most design codes require that the magnitude of this effect be assessed and allowed for within the design if necessary.

Although second-order effect described above can be ignored in some frames where their effects are small enough, they’re always present and thus must be always assessed.

Sway Sensitive Frames Vs Non-Sway Sensitive Frames

Structural steel-work designers most often refer to sway sensitive frames and non sway frames. The latter description is unfortunate since all frames under the application of loads will sway. The distinction between the two, however, is the significance of those sway effects. Some designers also refer to second-order effects as P-delta effects. To be clear P-delta effects are due to the likely initial imperfections within the length of members. These are usually allowed for automatically in the actual design, and would not be discussed further in this post.

Some designers also refer to “sway frames” when a proper terminology would probably be “unbraced frame”. In an unbraced frame, resistance to lateral loads is provided by the continuity of structural elements (Moment resisting frames). Whereas, braced frames in contrast to unbraced frames derive their resistance to lateral forces from the disposition of steel beacings and diagonal steel members or perhaps by the provision of a concrete core. This distinction as well as this terminology is very important and should be given utmost importance as a braced frame can be “sway sensitive” just as an unbraced frame can be sufficiently stiff such that second order effects are small enough to be ignored.

The Elastic Critical Load

Before the stability of a steel frame can be assessed, one very important parameter must be determined. This is known as the elastic critical load. The elastic critical load can be described as the load at which the entire frame will collapse under the application of vertical loads only. The elastic critical load is also a function of the frames property and shape of loading. Consider a frame subjected to vertical loads only (Figure 2). If the vertical loads is gradually increased, at some point the frame will collapse. Now consider the frame was initially out of plumb by some degree whilst the vertical loads is increased. The additional deformations due to the vertical loads just before reaching the elastic critical load would be significant. Hence the ratio between the elastic critical load and the applied load is an important pointer towards second order effects.

shows  frames out of plumb
Figure 1: Simple out-of-plumb frame

In EC3 this ratio is known as Fcr/FEd. Thus ratio shows that, as FEd increases alpha reduces, indicating increased sensitivity to second order effects.

The Eurocode Approach to Frame Stability

Ideally, designers would have to calculate the magnitude of Fcr for the frame and shape of loading, but this would be a tedious task for manual analysis. Thus software can be used to determine Fcr. As an alternative design standard provides a simplified but conservative method of determining alpha critical. In the Eurocode assessment of frame stability is explicitly dealt with in section 5.2.

The expression can be written as

{ \alpha }_{ cr }=\left[ \frac { { H }_{ Ed } }{ { V }_{ Ed } } \right] \left( \frac { h }{ { \delta }_{ h,Ed } } \right)

h is the storey height
HEd is the horizontal shear at the base of the storey. This is equal to the sum of the lateral loads applied at all floor and roof levels above the storey under consideration. In general, these lateral loads will be the Equivalent Horizontal Forces (EHF) prescribed in Clause 5.3.2 together with any wind forces ( if the wind is part of the combination of actions being considered).
VEd is the total vertical load at the base of the storey. This is equal to the sum of the vertical loads from all the floors and roof, above the storey under consideration.
δH.Ed is the lateral displacement over the storey i.e. the displacement between levels due to the lateral loads.

This expression is evaluated for each storey, starting from the lowest storey to the top of the frame.

In simple steel frames containing similar several bracings, the magnitude of δH.Ed can be determined by analysing just one bracing. If one bracing is analysed, the horizontal and vertical actions applied should be in proportion of stiffnesses.

It is also important to state that, the approximate formula given in section 5.2 of EC3 has certain limitations. It can not be applied to irregular frames or portal frames with significant axial forces in the rafters.

Allowing for Second-Order Effects

The Eurocode defines when second order effects are small enough to be ignored. For frames designed elastically, second order effect may be neglected if αcr is greater than 10. If αcr is less than 10, the frame is susceptible to buckling failure and second order analysis needs to be carried out. However, second order effects can be allowed for by amplifying the horizontal actions (wind, EHF’s etc.).

\quad load\quad amplifier\quad =\quad \frac { 1 }{ 1-\frac { 1 }{ { \alpha }_{ cr } } }

Note that the amplification factor can only be applied if alpha is greater than 3. Where the reverse is the case, a full blown second order analysis must be carried out. The simple amplification is only one way to allow for second-order effects. There are other approaches, including the use of software.

In conclusion, steel frames are typically lightweight, so sensitivity to second order effects is not something that must be avoided, there is nothing absolutely wrong with a structure having αcr less than 10. It is indeed expected that many frames will fall into this category, hence the provisions. The use of software which will allow for these effects is one convenient approach. For straightforward frames, the Eurocode contains a simple method to assess the significance of second order effects and how to allow for them if necessary.

Worked Example

Figure 5.0 shows the typical braced bay of an office building consisting of UKB’s, UKC’s and diagonal bracings. Assess the sway sensitivity of the structure. Are second order effects significant? if so what are the amplification factors.

worked example of a frames out of plumb
Figure 2: Worked Example

The design actions as been apportioned according to the stiffnesses of the braced bays and shown as Table 1 & 2. Table one shows the design action for the imposed load as leading variable action and wind as accompanying variable action, while combination two shown as table two has wind as leading variable action with the imposed load as accompanying variable action. In each case, EHF has been considered and included in the horizontal actions.

See: Application of Notional Horizontal Loads on Structures

FloorsVertical Actions (kN)Horizontal Actions (kN)Total deflection(mm)
Roof - 1st Floor403434.19.3
1st floor - Ground Floor917673.65.1
FloorsVertical Actions (kN)Horizontal Actions (kN)Total deflection (mm)
Roof - 1st Floor351945.311.9
1st Floor - Ground Floor733196.76.5

The last column in both tables shows the total displacement at each storey, this has been obtained by analyzing the frame for the horizontal loads only per requirement.

Load Combination One
{ \alpha }_{ cr }=\left[ \frac { { H }_{ Ed } }{ { V }_{ Ed } } \right] \left( \frac { h }{ { \delta }_{ h,Ed } } \right)

a) Roof – 1st Floor

{ H }_{ Ed }=34.1kN\quad { V }_{ Ed }=4034kN;\quad h\quad =3000mm\\ { \delta }_{ H,Ed }=9.3-5.1=4.2mm
{ \alpha }_{ cr }=\frac { 34.1 }{ 4034 } \times \frac { 3000 }{ 4.2 } =6.04<10

b) 1st – Ground Floor

{ H }_{ Ed }=34.1+73.6=107.7kN\quad { V }_{ Ed }=4034+9176=13210kN
h\quad =3000mm\quad { \delta }_{ H,Ed }=5.1mm
{ \alpha }_{ cr }=\frac { 107.7 }{ 13210 } \times \frac { 3500 }{ 5.1 } =5.6<10

Therefore the worst case of is = 5.6

\quad load\quad amplifier\quad =\quad \frac { 1 }{ 1-\frac { 1 }{ { \alpha }_{ cr } } } =\frac { 1 }{ 1-\frac { 1 }{ 5.6 } } =1.22

Thus, all horizontal actions acting on the frame for combination one must be increased by 22% in order to allow for second-order effects.

Load Combination Two

a) Roof – 1st Floor

{ H }_{ Ed }=45.3kN\quad { V }_{ Ed }=3519kN;\quad h\quad =3000mm\\ { \delta }_{ H,Ed }=11.9-6.5=5.4mm
{ \alpha }_{ cr }=\frac { 45.3 }{ 3519 } \times \frac { 3000 }{ 5.4 } =7.15<10

b) 1st – Ground Floor

{ H }_{ Ed }=45.3+96.7=142.0kN\quad { V }_{ Ed }=3519+7331=10850kN
h\quad =3500mm\quad { \delta }_{ H,Ed }=6.5mm
{ \alpha }_{ cr }=\frac { 142.0 }{ 10850 } \times \frac { 3500 }{ 6.5 } =7.05<10

Therefore the worst case of is = 7.05

\quad load\quad amplifier\quad =\quad \frac { 1 }{ 1-\frac { 1 }{ { \alpha }_{ cr } } } =\frac { 1 }{ 1-\frac { 1 }{ 7.05 } } =1.17

Thus, all horizontal actions acting on the frame for combination one must be increased by 17% in order to allow for second-order effects.

We can conclude that this frame is sensitive to second-order effects and thus allowances must be made in it analysis and design.

See: Derivation of Wind Actions on Buildings

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