### Portal Frame Analysis - Steel Structures.

**1 Methods of analysis BS 5950: Part 1allows two main methods of analysis of a structure:**

**(1)**Linear elastic. The frame is analysed either by hand or by computer assuming linear elastic behaviour. Once the forces, moments and shears have been derived by elastic analysis the ultimate capacity of each section is checked using the rules given in Section 4 of the Code.

**(2)**Simple plastic theory. The frame is analysed using the basic principles of simple plastic theory.Once the forces,moments and shears have been derived by analysis the member capacities are checked. Those containing plastic hinges are checked in accordance with Section 4.

**Elastic analysis**

Using elastic analysis it is to be expected that the structure will be heavier than that designed by plastic methods, but less stability bracing will be needed. It may well be that the ﬁnal details will also be more simple.

It will remain the engineer’s responsibility to ensure that stability is provided both locally and in the overall condition. BS 5950: Part 12 provides no speciﬁc rules regarding the stability of the frame as a whole; this means that the engineer must ensure that the stability is checked using the general rules for all frames. He must also check that the movement of the frame under all loading cases is not sufﬁcient to cause damage to adjacent construction, i.e. brick walls or cladding, the serviceability limit state of deﬂection.

**Plastic analysis**

The method of calculating the ultimate load of a portal frame is described in many publications.The main essence of the method is to assume that plastic ‘hinges’ occur at points in the frame where the value of M/Mp is at its highest value, the load being considered as increasing proportionally until the failure or ultimate state is reached.

Because of the straining at the hinge points it is essential that the local buckling and lateral distortion do not occur before failure. Failure is deemed to have taken place when sufﬁcient hinges have formed to create a mechanism.

The member capacities are calculated using the rules given in Section 4 of the Code but with the additional restrictions applied to hinge positions. In addition, positive requirements are put on checking frame stability for both single-bay and multi-bay frames. Plastic designed frames are lighter than elastic designed frames, providing deﬂection is not a governing point; however, additional bracing may well be required.

**2 Stability**

With the use of lighter frames, various aspects of stability take a more prominent part in the design procedures.As far as portal frames are concerned the following areas are important:

**(1)**overall frame stability, in that the strength of the frame should not be affected by changes in geometry during loading (PD effect)

**(2)**snap-through stability, in multi-bay frames (three or more), where the effects of continuity can result in slender rafters

**(3)**plastic hinge stability, where the member must be prevented from moving out of plane or rotating at plastic hinges

**(4)**rafter stability, ensuring that the rafter is stable in bending as an unrestrained beam

**(5)**leg stability, where the leg below the plastic hinge must be stable

**(6)**haunch stability, where the tapered member is checked to ensure that the inner (compression ﬂange) is stable.

**3 Selecting suitable members for a trial design**

The design of a portal frame structure is in reality a process of selecting suitable members and then proving their ability to perform in a satisfactory manner. Inex- perienced engineers can be given some guidance to estimate initial member sizes.

In order to speed the initial selection of members, three graphs have been produced to enable simple pin-based frames to be sized quickly.

These graphs have been prepared making the following assumptions:

**(1)**plastic hinges are formed at the bottom of the haunch in the leg and near the apex in the rafter, the exact position being determined by the frame geometry

**(2)**the depth of the rafter is approximately span/55 and the depth of the haunch below the eaves intersection is 1.5 times rafter depth

**(3)**the haunch length is 10% of the span of the frame, a limit generally regarded as providing a balance between economy and stability

**(4)**the moment in the rafter at the top of the haunch is 0.87Mp, i.e. it is assumed that the haunch area remains elastic

**(5)**the calculations assume that the calculated values of Mp are provided exactly by the sections and that there are no stability problems. Clearly these conditions will not be met, and it is the engineer’s responsibility to ensure that the chosen sections are fully checked for all aspects of behaviour.

The graphs cover the range of span/eaves height between 2 and 5 and rise/span of 0 to 0.2 (where 0 is a ﬂat roof). Interpolation is permissible but extrapolation is not.

The three graphs give:

**Figure 11.5:**the horizontal force at the feet of the frame as a proportion of the totalfactored load wL, where w is the load/unit length of rafter and L is the span of the frame

**Figure 11.6**: the value of the moment capacity required in the rafters as a proportion of the load times span wL^2

**Figure 11.7**: the value of the moment capacity required in the legs as a proportion of the load times span wL^2.

Fig. 11.5 Rise/span against horizontal force at base for various span/eaves heights |

Fig. 11.6 Rise/span against required Mp of rafter for various span/eaves heights |

Fig. 11.7 Rise/span against required Mp of leg for various span/eaves heights |

In the worked example kilonewtons and metres are used.

Method of use of the graphs

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