By Amanuel Gebremeskel,

Development Engineer, SAISC

Stability is one of those topics in structural engineering that most of us would rather avoid. There are at least two reasons for this. One is that it requires solving differential equations. The second is that even if one knows how to solve eigenproblems, real structures are simply too complex to solve comprehensively.

This article attempts to explain the more relevant aspects of column and frame stability without solving complicated equations. It addresses second order effects and discusses how they are commonly solved in practice. Finally it scans SANS 10162-1 in regards to stability of steel structures and illustrates how the tools within it can be used to comply with the law, carry out preliminary design and check software output.

**BUT WHY DO WE CARE ABOUT STABILITY?**

For one, checking for stability related limit states is more important than checking for yield limit states because stability failure is unforgiving in its suddenness. Structural failure modes that contain no warning to occupants, such as buckling and non-ductile rupture, must be avoided by the designer at all cost.

Secondly, the South African building regulations state that structures must be stable enough to transfer loads to foundations. Standards such as SANS 10162-1 contain sections 8.7 and 9 that deal exclusively with the stability of steel structures and provide tools that are deemed to satisfy South African law if applied correctly. This means that understanding the requirements of SANS 10162-1 and how they relate to software output is effectively a legal requirement.

**SO WHAT IS MEANT BY STABILITY?**

For most structural designs, providing stability has much to do with limiting compressive loads in members and frames that have eccentricities which are caused by geometric deformations. The source of the geometric deformations can be manufacturing imperfections or loads imposed during use. Therefore if a column is supplied bowed then it is important to limit the axial loads in it in order to prevent it from excessive bowing. Of course a straight column would have the same problem if a lateral load were to be applied along its length, or moments at its ends, resulting in a bowed shape.

There are of course instances when an engineer has to solve stability problems that involve members without geometric deformations, but these are rare. Figure 1 plots the two column behaviours on the same graph. In one case a straight column is loaded until it reaches a critical load, at which point it suddenly buckles. This is called linear buckling. If initial bowing is present however, because of the resulting eccentricity between P and the column centre line, the column bows even further as the axial

load is increased. This results in nonlinear buckling. We are typically concerned with nonlinear buckling of columns and frames

because initial imperfections are quite common and excessive deformations must be avoided.

The same phenomenon is observed for a column, or even moment frame, that is cantilevered up from the ground. If a lateral load is imposed at its top or the frame is not exactly vertical when supplied then adding compressive loads is bound to increase its lateral drift. Limiting the compressive loads in such cases, either to limit excessive drift or prevent total collapse, achieves the goal of providing stability.

**AND SECOND ORDER EFFECTS?**

The increased frame drifts, or bowing in the case of the columns, are called second order effects because they result from the interaction of the compressive loads with the initial deformations that arise from lateral loads or imperfections. Figure 2 illustrates how P and Δ or δ can interact to give even greater values of Δ or δ until equilibrium is reached.

Second order analysis is somewhat more difficult than linear analysis because one has to account for first order effects – the initial drifts – and then iterate until the compressive loads times the accentuated drifts are equal to the flexural resistance at the centre of the column or the base of the cantilever. Such iterations can of course be time consuming, even for software programs. However iterative methods are popular with structural analysis software and inhouse programs because they are relatively accurate and straightforward.

There is yet another way to see the problem. Just as cables can be made laterally stiffer by introducing tension in them, columns can be made laterally softer by introducing compression in them. In fact, it is possible to solve a distinct stiffness reduction for columns and frames for a given compressive load. This is of course a reduction in stiffness that results not from changes in the modulus of the material but from the geometric effects of the compressive load.

One can then solve for the second order drifts directly by applying lateral loads on columns and frames that have this reduced lateral stiffness. There is no need to add the effects on drift of the compressive load because it is accounted for in the reduced stiffness. This turns the problem back into a linear first order problem while it still accounts for second order effects.

**THEN HOW DO STRUCTURAL PROGRAMS WORK?**

Structural software like Prokon and Fastrak make use of such stiffness reductions to account for second order effects. At times they iterate if initial deflections change the compressive loads in members but their solutions typically require no iteration for building type structures where the compressive loads – gravity loads – remain constant. Programs such as SAP allow the user to input specific stiffness reductions if desired.

**HOW ABOUT SANS 10162-1?**

SANS 10162-1 has three simple tools that are especially relevant for evaluating stability and checking software output. The first involves accounting for initial frame out of plumbness and softening of columns and beams due to yielding from high axial and bending loads. This material stiffness reduction is of course different from the geometric one that we have been discussing above.

According to section 8.7 these effects must be accounted for by applying a lateral load at each storey that is equal to 0.005 times the factored gravity loads on the storey. This is in addition to other lateral loads such as wind and seismic. After applying all lateral loads a second order analysis must be carried out to determine final forces, drifts and deflections.

Section 8.7 provides a second tool in the form of a simple equation that can approximate second order effects. Software output can be checked by multiplying first order drifts and forces by U2 to obtain approximate second order results. Moreover by keeping the value of U2 below certain limits, say 1.5, it is possible to develop a rule of thumb for securing storey stability. This is equivalent to limiting factored gravity loads on the storey to less than a third of the theoretical critical buckling load of the storey.

A third tool helps to answer a question that keeps coming up involving the design of bracing members. It is never clear how much capacity or stiffness is required from members that are used to brace columns or the compression flanges of flexural members. Section 9.2.7.2 provides a direct method of evaluating how much load is imposed in such members if the initial geometric imperfection in the braced member is known.

Of course iteration is required since evaluation of Δb is dependent on knowing Pb. Moreover it is important to note that Δb includes not only the deflection of the bracing member but also all the connections involved. If this sounds like too much work, a simplified method is provided in section 9.2.6 where bracing members are designed for 2% of the compressive load in the braced member or flange.

In either case a stiffness limit on the bracing member and its connections is set in sections 9.2.6 and 9.2.7 by limiting Δb to a maximum value of Δo. These initial imperfections can either come from the requirements of SANS 2001:CS1 as per section 9.2.2 or from actual measurements for existing structures.

**READY FOR STABILITY…**

This article has attempted to shed light on three areas. First, it has provided a qualitative description of what is meant by column and frame stability for commonly encountered problems. Second, it has addressed the causes and implications of second order effects while providing insight into how structural programs solve them using either iterative or stiffness methods. Finally it has shown how complying with SANS 10162-1 not only satisfies a legal requirement for the structural engineer but also provides quick and easy tools to check for stability and verify computer output