### In-Plane Stability - Structural Forms.

The somewhat simple procedures of Section 2 of BS 5950: Part 1 cannot be utilized when considering the in-plane stability of portal frames since they do not consider axial compression within the rafters.Axial compression in the rafters has a signiﬁcant effect on the stability of the frame as a whole.

In-plane stability checks are required to ensure that the load that would cause buckling of the frame as a whole is greater than the sum of the applied forces.

Unlike a beam and column structure, a single-storey, moment-resisting portal frame does not generally have any bracing in the plane of the frame. As such, restraint afforded to individual columns and rafters is a function of the stiffness of the members to which they connect. In simplistic terms, rafters rely on columns, which in turn rely on rafters. The stability check for the frame must therefore account for the stiffness of the frame itself.

When any structure is loaded it deﬂects. To this end, the deﬂected shape is different from the idealized representation of the frame when it is deemed unloaded or ‘at rest’. If a frame is relatively stiff, such deﬂections are minimized. If, however, a frame is of such a ‘small’ stiffness as to induce signiﬁcant deﬂections when loaded, ‘second-order’ effects impact on both the frame’s stiffness and the individual members’ ability to withstand the applied load.

Consider the example of a horizontal, axially loaded strut as shown in

**Fig. 1.8**. Prior to application of the axial force, the strut would deﬂect under its self-weight.

Fig. 1.8 Horizontal, axially loaded strut |

If, however, the strut were of a stiffness such that the initial self-weight deﬂection was relatively large, the induced second-order effect, P.Δ, would signiﬁcantly increase, and it is possible that the bending moment would play a greater part in the design of the member, since a greater deﬂection would induce a larger second order bending moment, which in turn would induce a further deﬂection and a further second-order moment, etc.

In relation to the deﬂections of portal frames under load, the above phenomenon is more than sufﬁcient to reduce the frame’s ability to withstand the applied loads.

BS 5950: Part 1 addresses the problem of in-plane stability of portal frames by the use of one of the following methods:

**(1)**the sway check method with the snap-through check

**(2)**the ampliﬁed moments method

**(3)**second-order analysis.

**1 The sway check method**This particular check is the simplest of those referred to above, in so far as elastic analysis can be used to determine the deﬂections at the top of the columns due to the application of the NHFs previously mentioned.

If, under the NHFs of the gravity load combination, the horizontal deﬂections at the tops of the columns are less than height/1000, the load factor for frame stability to be satisﬁed, γr, can be taken as 1.0.

The code also offers the possibility that for frames not subject to loads from either cranes, valley beams or other concentrated loads, the height/1000 criterion can be satisﬁed by reference to the Lb/D ratio, where Lb is the effective span of the frame, and D the cross-sectional depth of the rafter.

It should be noted, however, that this particular check can only be applied to frames that fall within certain geometric limits.

**2 The ampliﬁed moments method**Where the frame does not meet the criteria of the sway check method, the ampliﬁed moments method referred to in BS 5950: Part 1 Section 5.5.4.4 may be used.

The basis of this method revolves around the calculation of a parameter known

*as the lowest elastic critical load factor, γcr, for a particular load combination. (N.B. BS 5950: Part 1 does not give a method for calculating γcr.)*

A detailed treatment of the calculation of γcr is outside the scope of this particular chapter, and the reader is again referred to reference [5]. Fortunately, the industry software to carry out the analysis of portal frames is capable of calculating γcr to BS 5950: Part 1.Accordingly, the task is not as daunting as ﬁrst appears.However, it is imperative that the reader understands the background to this particular methodology.

On determining γcr, the required load factor, γr, is calculated as follows:

If γcr < 4.6, the ampliﬁed moments method cannot be used and a second-order analysis should be carried out.

The application of lr in the design process is as follows:

• For plastic design, ensure that the plastic collapse factor, γp >= γr, and check the member capacities at this value of γr.

• For elastic design, if γr > 1.0,multiply the ultimate limit state moments and forces arrived at by elastic analysis by lr, and check the member capacities for these ‘ampliﬁed’ forces.ç

**3 Second-order analysis**Second-order analysis, brieﬂy referred to above, accounts for additional forces induced in the frame due to the axial forces acting eccentrically to the assumed member centroids as the frame deﬂects under load.

These secondary effects, often referred to as ‘P-Delta’ effects, can be best illustrated by reference to

**Fig.1.9**of a simple cantilever.

Fig. 1.9 Second-order effects in a vertical cantilever |

As can be seen, the second-order effects comprise an additional moment of PΔ due to the movement of the top of the cantilever, Δ, induced by the horizontal force, H, in addition to a moment within the member of Pδ due to deﬂection of the member itself between its end points. (It should be noted that, in certain instances, second-order effects can be beneﬁcial. Should the force P, above, have been tensile, the bending moment at the base would have been reduced by PΔ kNm.)

In the case of a portal frame, there are joint deﬂections at each eaves and apex, together with member deﬂections in each column and rafter. In the case of the columns and rafters, it is standard practice to divide these members into several subelements between their start and end nodes to arrive at an accurate representation of the secondary effects due to member deﬂections.

As a consequence of these effects, the stiffness of the portal frame is reduced below that arrived at from a ﬁrst-order elastic analysis.Again, the industry software packages include modules for the rapid calculation of these second-order effects, and utilize either the matrix or energy methods of analysis to arrive at the required solution, in most cases employing an iterative solution.

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