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Lateral Stability of Steel Buildings




1.1 Background BS5950/ EC3 Steel frame

1.2 Steel frames

1.3 Braced frame

1.4 Unbraced frame

1.5 Problem statement

1.6 Objective

1.7 Scope of the present study



2.1 Introduction

2.2 Practice code EURO-CODE

2.3 Practice code BS 5950

2.4 Wind-moment frame

2.5 Necessary checks

2.6 Bracing system

2.7 Internal moments and forces









3.6 Frame Deflection

3.7 Frame Stability

3.8 Frame analysis software method



1.1 Background BS5950/ EC3 Steel frame

In the past, in the UK has been provided multiple publications by SCI, through technical information issued by material and product suppliers and by specialist software the regulation on the design of structural elements in multi-storey steel framed buildings in relation to BS 5950 Structural use of steel frame. Some in the UK Standards, part of BS 5950, have been replaced by the Structural Euro codes in 2010. The Euro codes are coordinated design standards that are appropriate and oriented to limited national alterations, throughout the European Union. That doesn’t mean structures designed according to the Euro code will be significantly heavier or lighter compared to the structures designed based on BS 5950 but only detailed rules do differ. SCI publication P334 Design of multi-storey braced frames was published in 2004. Other than numerous publications giving regulation about the design of structural elements and connections, this publication have given a guidance on the particular aspect of the strength of braced frames. Following up a publication was made to replace the P334, for design in accordance with the Euro codes. Its purpose was similar to that of P334 but, at the time of publications was not updated in accordance with the Euro codes. Those publications are still generally important and the references to them have been retained but designers will need to consider carefully the use of regulation provided in relation to BS 5950 when designing to the Euro codes. There is a continuing programme to update the design regulation in line with the Euro codes. (BROWN, 2004)

1.2Steel frames 

This dissertation is subject to a Comparison of braced and unbraced frames in steel structures evaluated through the British standard BS5950 and Euro code EC3.

Nowadays, Steel structure is the most prevailing material in the construction industry. (Mahmood Md. Tahir BSc, 1997)

The main advantages of steel Frame are:

  • Steel construction allow easily a large span up to 20m.
  • Increase speed, accuracy, and quality of the construction as it can be prefabricated.
  • Steel can be combined with concrete to provide composite element.
  • With steel can be achieved any architectural shape. (Mahmood Md. Tahir BSc, 1997)

Structural stiffness can be increased in Steel moment resisting frames by introducing steel bracings. There are different types of steel bracings, such as cross bracing ‘X’, diagonal bracing

‘D’, and ‘V’ type bracing etc… (BISWAL, 2014)

Steel frames without bracing, failure will generally occurs at beam and column connections. They have lateral stability by flexure and shear force in beams and columns i.e. by frame action. Under severe lateral loading, ductile fracture appears at beams and columns connections. Unbraced frames have low elastic stiffness. P-Δ effect is a problem associated with such structures. So, to increase the structure stability to lateral loading and good ductility fractures to perform well under seismic loading, bracings system can be provided. Hence, allow the structure to obtain a great lateral stiffness with minimal added weight and to ensure that the displacement demand of a building is to be kept below its limited displacement. The bracings are installed in the columns along the perimeter.

In this dissertation the lateral stability of steel buildings under braced and unbraced system is investigated. A Six story building is analysed for lateral loading (wind load). A simple computer based modelling in Strand7, Robot Software is performed for Equivalent static analysis, stability spectrum analysis. The stiffness of different types of bracing frames has been examined. Steel bracing is used to minimize the displacement demand. Bracing can either be installed from inside the frame or outside of the system.

1.3 Braced frame

Generally, a braced frame provides an efficient lateral stability of at least five times stiffened than the frame itself. Bracing system reduces the horizontal displacement by at least 80%. To meet this requirement, the stiffness of the two systems (unbraced frames and braced frames) have to be compared and the following relationship has to be satisfied: Kb ≥ 5Ka where Ka and Kb are lateral stiffness spring constant for unbraced and braced frame respectively. The bracing frame is a system to transfer the factored loads down to the foundations. (Mahmood Md. Tahir BSc, 1997)

Figure 1.3.1 Type of bracings (Malik, 2016)

  1. A column may be considered braced if bracing or walls are installed.
  2. Braced columns are not designed to resist lateral load
  3. There is no sway force
  4. Bracing is most used for high buildings.

1.4 Unbraced frame

Steel frame which does not meet the requirements for a braced frame is classified as

Unbraced. In practice frames are braced against horizontal displacements to simplify the behaviour and to avoid as much as possible bending action about the minor axes of the column. Unbraced frames also known as “sway” frames in which need to be accounted a secondary moment action the “P-Δ effect. (Mahmood Md. Tahir BSc, 1997)

Figure1.4.1 Type of unbraced frame (Malik, 2016)

  1. Is called unbraced if lateral stability of the structure as a whole is provided by columns only.
  2. In unbraced columns the lateral load is resisting by itself
  3. Unbraced columns are subjected to sway
  4. Second moment is taken in account for the analysis

1.5 Problem statement

A brace must be designed to satisfy two things:

  1.               It must have enough stiffness to strengthen the braced members in order to resist secondary effects;
  2.               It must have sufficient strength itself, reduced bracing stiffness lead to a greater deformation in the frame structure; causing larger forces in the bracing system. If the bracing strength is too small, the requested bracing forces can be high.

Other problem encountered in this study is the following:

Given the number of stories, bays, and dimensions of a braced and an unbraced multi-storey frame, and the working loads which the frame must carry together with appropriate load factors; find suitable distributions of bending moment, shear force and axial force throughout the frame for appropriate load combinations. A prime concern in multi-story frame design is the sway (or the physiological response to sway acceleration and vibration) induced by wind at the working load level. (D G BROWN BEng, 2009)

1.6 Objective

The objective of this dissertation is to evaluate the stiffness of different type braced and unbraced frame of a multi-storey structure subjected to lateral loads and to identify the suitable bracing system to resist lateral load efficiently.  

1.7 Scope of the present study

In this dissertation a present study is taken by modelling of different steel frame under the analysis mentioned above using Strand7 software, robot software and hand calculation. Then results obtained are compared. Conclusions are based on the tables and graphs obtained.

A design of multi-storey is carried out with braced steel frame buildings up to about 6 storeys, where the beam-to-column connections are assumed to be pinned connections and the horizontal loads effect is prevented by a system of vertical bracing, in the other hand an unbraced steel frame building of 6 stories is implemented, whereas the beam-to-column connection is assumed as a rigid.

The dissertation briefly reviews the overall design basis, according to the Structural Euro codes and British-standard and gives advice on the actions (mainly vertical loads) that this type of building should be designed to resist. It covers the design of the vertical bracing system, which, as well as providing resistance to lateral forces caused by wind, provides stiffness against horizontal sway. The stability is a key factor in determining the resistance of the frame to second order effects. The requirements, in relation to the Euro codes and the UK Building Regulations, are discussed.



2.1 Introduction

The following chapter provides an overview of the different type of braced frames and non-braced frame respectively to their stiffness performance advantages and modelling considerations. The historical background is outlined shortly in this review and the correspondent difference on the practice code based on EC3 and BS5059.

2.2 Practice code EURO-CODE 

2.2.1 Background of Euro code 3 (EC3)

European Code, or better known as Euro code, was initiated by the Commission of European Communities as a standard structural design guide. It was intended to smooth the trading activities among the European countries. Euro code is separated by the use of different construction materials. Euro code 1 covers loading situations; Euro code covers concrete construction; Euro code 3 covers steel construction; while Euro code 4 covers for composite construction. (HAN, NOVEMBER, 2006)

2.2.2 Scope of Euro code 3: Part 1.1 (EC3)

EC3, “Design of Steel Structures: Part 1.1 General rules and rules for buildings” covers the general rules for designing all types of structural steel. It also covers specific rules for building structures. EC3 stresses the need for durability, serviceability and resistance of a structure. It also covers other construction aspects only if they are necessary for design. Principles and application rules are also clearly stated. Principles should be typed in Roman wordings. Application rules must be written in italic style. The use of local application rules are allowed only if they have similar principles as EC3 and their resistance, durability and serviceability design does not differ too much. EC3 stresses the need for durability, serviceability and resistance of structure (Taylor, 2001). It also covers other construction aspects only if they are necessary for design. (HAN, NOVEMBER, 2006)

2.2.3 Design Concept of EC3

All designs are based on limit state design. EC3 covers two limit states, which are ultimate limit state and serviceability limit state. Partial safety factor is applied to loadings and design for durability. Safety factor values are recommended in EC3. Every European country using EC3 has different loading and material standard to accommodate safety limit that is set by respective countries. (HAN, NOVEMBER, 2006)

2.2.4 Application Rules of EC3

A structure should be designed and constructed in such a way that: with acceptable probability, it will remain fit for the use for which it is required, having due regard to its intended life and its cost; and with appropriate degrees of reliability, it will sustain all actions and other influences likely to occur during execution and use and have adequate durability in relation to maintenance costs. It should also be designed in such a way that it will not be damaged by events like explosions, impact or consequences of human errors, to an extent disproportionate to the original cause. (HAN, NOVEMBER, 2006)

Potential damage should be limited or avoided by appropriate choice of one or more of the following criteria: Avoiding, eliminating or reducing the hazards which the structure is to sustain; selecting a structural form which has low sensitivity to the hazards considered; selecting a structural form and design that can survive adequately the accidental removal of an individual element; and tying the structure together. (HAN, NOVEMBER, 2006)

2.2.5 Ultimate Limit State

Ultimate limit states are those associated with collapse, or with other forms of structural failure which may endanger the safety of people. Partial or whole of structure will suffer from failure. This failure may be caused by excessive deformation, rupture, or loss of stability of the structure or any part of it, including supports and foundations, and loss of equilibrium of the structure or any part of it, considered as a rigid body. (HAN, NOVEMBER, 2006)

2.2.6 Serviceability Limit State

Serviceability limit states correspond to states beyond which specified service criteria are no longer met. It may require certain consideration, including: deformations or deflections which adversely affect the appearance or effective use of the structure (including the proper functioning of machines or services) or cause damage to finishes or non-structural elements; and vibration, which causes discomfort to people, damage to the building or its contents, or which limits its functional effectiveness. (HAN, NOVEMBER, 2006)

2.2.7 Actions of EC3

An action (F) is a force (load) applied to the structure in direct action, or an imposed deformation in indirect action; for example, temperature effects or settlement. Actions are classified by variation in time and by their spatial variation.

In time variation classification, actions can be grouped into permanent actions (G), e.g. self-weight of structures, fittings, ancillaries and fixed equipment; variable actions (Q), e.g. imposed loads, wind loads or snow loads; and accidental loads (A), e.g. explosions or impact from vehicles. Meanwhile, in spatial variation classification, actions are defined as fixed actions, e.g. self-weight; and free actions, which result in different arrangements of actions, e.g. movable imposed loads, wind loads, snow loads. (HAN, NOVEMBER, 2006)

2.3 Practice code BS 5950

2.3.1 Background of BS 5950

BS 5950 was prepared to supersede BS 5950: Part 1: 1990, which was withdrawn. Several clauses were technically updated for topics such as sway stability, avoidance of disproportionate collapse, local buckling, lateral-torsional buckling, shear resistance, members subject to combined axial force and bending moment, etc. Changes were due to structural safety, but offsetting potential reductions in economy was also one of the reasons.

BS 5950 comprises of nine parts. Part 1 covers the code of practice for design of rolled and welded sections; Part 2 and 7 deal with specification for materials, fabrication and erected for rolled, welded sections and cold formed sections, sheeting respectively; Part 3 and Part 4 focus mainly on composite design and construction; Part 5 concerns design of cold formed thin gauge sections; Part 6 covers design for light gauge profiled steel sheeting; Part 8 comprises of code of practice for fire resistance design; and Part 9 covers the code of practice for stressed skin design (HAN, NOVEMBER, 2006).

2.3.2 Scope of BS 5950

Part 1 of BS 5950 provides recommendations for the design of structural steelwork using hot rolled steel sections, flats, plates, hot finished structural hollow sections and cold formed structural hollow sections. They are being used in buildings and allied structures not specifically covered by other standards. (HAN, NOVEMBER, 2006)

2.3.3 Design Concept of BS 5950

There are several methods of design, namely simple design, continuous design, semi-continuous design, and experimental verification. The fundamental of the methods are different joints for different methods. Meanwhile, in the design for limiting states, BS 5950 covers two types of states – ultimate limit states and serviceability limit states. (HAN, NOVEMBER, 2006)

2.3.4 Ultimate Limit States

Several elements are considered in ultimate limit states. They are: strength, inclusive of general yielding, rupture, buckling and mechanism formation; stability against overturning and sway sensitivity; fracture due to fatigue; and brittle fracture. Generally, in checking, the specified loads should be multiplied by the relevant partial factors γgiven in Table 2. The load carrying capacity of each member should be such that the factored loads will not cause failure. (HAN, NOVEMBER, 2006)

2.3.5 Serviceability Limit States

There are several elements to be considered in serviceability limit states – Deflection, vibration, wind induced oscillation, and durability. Generally, serviceability loads should be taken as the unfactored specified values. In the case of combined imposed load and wind load, only 80% of the full specified values need to be considered when checking for serviceability. In the case of combined horizontal crane loads and wind load, only the greater effect needs to be considered when checking for serviceability. (HAN, NOVEMBER, 2006)

2.3.6 Loading

BS 5950 had identified and classified several loads that act on the structure. There are dead, imposed and wind loading; overhead traveling cranes; earth and ground-water loading. All relevant loads should be separately considered and combined realistically as to compromise the most critical effects on the elements and the structure as a whole. Loading conditions during erection should be given particular attention. Where necessary, the settlement of supports should be taken into account as well. (HAN, NOVEMBER, 2006)

2.4 Wind-moment frame

On an Unbraced steel frame, a rotational stiffness of the beam-to-column connections is more of the concern. Under vertical load the connections are assumed to be pinned. This design is well known as the wind-moment method or wind-connection method and is used widely in the UK and North America. The wind-moment method is described as of low rise frames design. The procedures for the design are given in the publication of BS 5950. (J S HENSMAN BEng, 20/12/2016), (P R SALTER BSc, 1999)

2.4.1 Benefits of the wind-moment method

One of the main advantages of the wind-moment method is its simplicity. There is no relation between internal moments/forces with the relative stiffness of the frame members, because the frame is considered as statically determinate. From a construction viewpoint, the main advantage of wind-moment frames is the simplicity of the steelwork compared to a fully rigid construction. Steelwork contractors are more concerned with making the connections, and it is estimated that the fabrication and workshop handling costs related with the connections can be as high as 50% of the total cost of the erected steelwork. (J S HENSMAN BEng, 20/12/2016), (P R SALTER BSc, 1999)

2.4.2 Choice of wind-moment method

The designer can quickly determine whether the wind-moment method will be adequate for a given frame by using the flowchart provided in Figure 2.4.2. An alternative design method should be considered if the frame is dominated by SLS Sway deflection. (J S HENSMAN BEng, 2016)

Figure 2.4.2  (J S HENSMAN BEng, 20/12/2016)

2.4.3 Principles of wind-moment method design

The assumption made at a design stage are the following:

– Under vertical loads the connections are assumed to be pinned (Figure 2.4.3(a)),

– Under lateral loads the connections are assumed to behave as rigid joints, with points of contra flexure appearing at the mid-height of the columns and at the mid-span of the beams (Figure 2.4.3(b)).

Figure 2.4.3  (J S HENSMAN BEng, 20/12/2016)

2.5 Necessary checks

The publication in BS 5950-1:2000 gives an explanation of the three below methods of checking in plane stability:

(a) The Sway-check method

(b) The Amplified Moment method

(c) Second-order analysis

2.5.1 The Sway-check method

The Sway-check method mainly is used for portals which comply the following geometrical limitations: 

-Span/height to eaves is not more than 5.

-Rise of apex above column tops is not more than span/4 for symmetrical spans or a value given by a formula for a symmetric rafters.

-Either the notional sway deflection from notional forces (calculated by first order analysis) is not more than h/2000, or the span/depth ratio of the rafters is within a limit given by a formula. (MlStructE, 20/12/2016) Advantages and disadvantages

The Sway-check method is the easiest method and provide cheaper designs if the frame is stiff enough to comply either the h/2000 check or the formula check based on the section sizes selected either to induce the required stiffness or to satisfy the Serviceability Limit State (SLS) requirements. This method will often provide the most economical designs. When using the Sway-check method, the steel strength (e.g. S275 or S355) has no effect on the in-plane stability calculation. (MlStructE, 20/12/2016)

2.5.2 The Amplified Moment method Range of application

If the design does not comply the limitation of the sway check method then The Amplified Moment method comes into roll. It may be used for portals that are not tied portals and which have an elastic critical buckling ratio,


Figure 2.5.3 (Doshi, 2016) Advantages and disadvantages

Second-order analysis is simple to apply in software. Flexible frames such as multi-span frames provide a lower cost designs. It can provide less cost effective designs than the other methods for stiffer frames because it will always calculate a reduction of frame strength due to the second-order (P-delta) effects. The Second-order method does allow the idea of higher strength steel (grade S355 steel). (MlStructE, 20/12/2016)

2.6 Bracing system

Two type of bracing frame are identified in multi-storey design: Horizontal bracing and vertical bracing.

-Vertical bracing resists the lateral loads and produce great stability against sway to the structure. Vertical bracing transfer the loads carried out to the ground and some part of it to the roof which these act as horizontal diaphragms.

– Horizontal bracing, generally transfer the horizontal forces implemented by floor plate action, caused due to wind pressure on the cladding to the planes of vertical bracing.

2.6.1 Horizontal bracing

A horizontal bracing system is necessary at each floor, in order to transfer lateral loads (mainly the forces transferred from the perimeter columns) to the vertical bracing that deliver lateral resistance. There are two horizontal bracing systems that can be installed in multi-storey braced frames:  Diaphragms  Discrete triangulated bracing.

Typically, horizontal bracing systems span between columns, which are connected to the vertical bracing. This type of Brace arrangement often known as truss, with a depth ratio equal span/20 or equal to the bay centres (D G BROWN BEng, 2009)

In terms of design, simply the truss is considered as a simple supported and the reaction at the end of the truss can be calculated, consecutively used to design the critical bracing member,. most of the time is suggested to use same member size throughout the bracing system.

2.6.2 Vertical bracing

In order to avoid any torsional effects (twist on plan) due to the wind or lateral forces it is preferable to introduce vertical bracing at or near the perimeter of the structure. Mainly bracing is installed around service cores, lift shafts, stair. Some of the time bracing arrangements may have to be modified due to the architectural arrangements to facilitate the instalment and location bracing members. In some cases intermediate columns may have to be installed to act as part of the bracing system. (D G BROWN BEng, 2009)

In a multi-storey building most common vertical bracings are installed diagonally between two columns (inclined at approximately 45° are suggested). In which case they are considered the most efficient system compared to other arrangements. In the inclined bracing the connection between the bracing itself and beam/column (junction) are compact. Wide bracing systems will result in more stable structures.

The minimum brace is a single diagonal, as shown, where has to be designed either for tension or compression. . (D G BROWN BEng, 2009) (Fattorini, 1995)

The design conducted for vertical bracing frame must be able to resist to the following:

– Wind loads

– Equivalent horizontal forces, representing the effect of initial imperfections

– Second order effects due to sway (if the frame is flexible). Definition of second order effects in Section 2.5.3.

2.7 Internal moments and forces

  Internal forces and moments are determined using statics. In a statically indeterminate structure are determined using either:

  1. Elastic global analysis
  2. Plastic global analysis

Internal forces can be generally be determined using: first order theory based on the structure initial geometry, or can be used second order theory due to the deformation of the structure.

As continuing first order may be used in the following cases:

  1. Braced frames
  2. Non-sway frames

And the second order may be used in all cases. (PR SALTER BSc, 2004)

2.7.1 Elastic global analysis

1. The relation between STRESS-STRAIN is assumed to be linear

2. The calculated bending moment is modified by redistributing up to 15% of the peak calculated in any member based on:

a) The applied load is in equilibrium with the internal forces and moments in the frame,

b) All the members have a CLASS1 or CLASS2 cross-section. (PR SALTER BSc, 2004)

2.7.2 Plastic global analysis

Plastic global analysis can be induced using RIGID-PLASTIC method, and lateral restraint shall be provided at all plastic hinge location where hinge rotation can occur under any load case. This method does not apply the second order theory.  (PR SALTER BSc, 2004)




This project compare braced and unbraced frames applied to euro code EC3 and British standard BS5950.

Method used to compare are hand calculation, and software such as strand7 and robot analysis. As an engineer we know that one only trial wouldn’t be satisfactory to make a point, so it has been created 6 floors building for each procedure and each procedure contain designing of EC3 and BS 5950 with two different dimensions as dimensions can affect the buckling and bending of the member, and P-delta factor has been analysed.

Drawing are been made on Revit (figure3.1) then transferred to robot for the analysis.

After each member design, a type of cross sectional area is been chosen from the Blue book based on their resistance formulated on the calculation.


3.2.1 Dead Load

Dead load as well as known as the permanent load. All the members taking part on the structure and keeping together the structure are called dead Loads, as each has its own weight and effects the stiffens of the next permanent member. Good examples are: slab, beam column, roof, cladding etc… (PR SALTER BSc, 2004)

In order to have a secure and durable member a safety factor is used. As of the Euro-code imply a factor of 1.35, and BS standard imply 1.4. (PR SALTER BSc, 2004)

3.2.2 Live load

Live load known as a variable Load. This load changes from second to seconds depending on how many and what type of activities are induced. Anything that can be moved out or into the structure during its life time. Good example are: wind load, a person walking by or a small machinery installed for couple hours or days.

Due to the impossible prediction that can’t be made on the future loads a safety factor is used:

Euro-code imply a factor equal to 1.5 and BS Standard Imply 1.6. (PR SALTER BSc, 2004)


3.2.1 Load combinations BS 5950 

The combination loads considered in this project are as follow:

1.4 × Dead load + 1.6 × Imposed load + 1.0 × Notional horizontal forces    Combination 1

1.2 × Dead load + 1.2 × Imposed load + 1.2 × Wind load                             Combination 2

1.4 × Dead load + 1.4 × Wind load                                                              Combination 3

The notional horizontal forces should be taken as 0.5% of the factored dead plus imposed loads (BS 5950-1:1990 Clauses 5.6.3,

(King, 20/12/2016), (PR SALTER BSc, 2004)

3.2.2 Lad Combinations Euro code

1.35 x GK+1.5 x QK


1.35 x GK+1.5 x QK1+1.5 x 0.5 x QK2


1.35 x GK+1.5 x QK1+1.5 x 0.7 x QK2



1.35 GK=considered the dead Load

1.5 QK=considered the as the Imposed load


1.35 GK=considered the dead Load

1.5 x QK1=considered the as the Imposed load main variable

1.5 x 0.5 x QK2=Considered as the wind load with the combination factor of 0.5


1.35 GK=considered the dead Load

1.5 x QK1=considered the as the Wind load main variable

1.5 x 0.5 x QK2=Considered as the Imposed Load wiith the combination factor of 0.7


This dissertation show how to design Bracing system for non-sway frames and sway frames, but as an engineer is well know that column and beam are members of a frame so to start designing first it has been determined type of beam used and column, therefore the following topics show what approach taken to design: column, beam and mainly bracing system.

3.3.1 Design of beams

A beam is a member that takes role as part of the frame, its bending and deflection could cause effect on the frame system, hence it has to be classified and designed as follow:

The floor system chosen in this dissertation is a situ concrete floor of 225 mm depth, with an impose load of 5


, one way spanning.

Load combination has been calculated based on EURO-CODE (3.2.2) and British standard (3.2.1)

Shear force has been determined:


W= distributed load due to the dead load and imposed load acting on the beam.

L= length of the beam

Then maximum moment which appears to be in the middle:


Next step is to identify weather the beam is restraint or unrestraint, therefore an adequate section is chosen from the blue book table section. (southbank, 2016)

3.3.2 Design of columns

Colum is a member part of the frame as well. Buckling takes, place so it could affect the frame assisting on horizontal deflection or a sway due to the vertical and horizontal loads. Hence the stiffer and adequate section the less buckling appear.

A Column is subjected to compression due to that a buckling resistance is needed as follow:

  1. Verified buckling



NED=compration force acting on the column

NbRd=Design Buckling resistance

  1. Design buckling resistance should be taken as:

NbRd=X A fyΥM1

For class 1, 2, 3

NbRd=X Aeff fyΥM1

For class 4

  1. χ =1ф+ф2-⅄2<1
  2. ф=0.5[1+α⅄-0.2+⅄2]
  3. ⅄=Afy NCr                            For class 1, 2, 3

⅄=Aefffy NCr

For class 4

Where;   χ= Reduction factor

⅄=non dimensional slenderness

α=Imperfection factor

NCr=is the elastic critical force for the relevant buckling mode

Even though the cross section chosen are adequate, a slight of deformation will appear and this will lead as to verify and determine the Serviceability limit state.


3.4.1 Sway prediction

Once a frame designed for the ultimate limit state should, the next step should be analysing the frame as an elastic rigidly-jointed to determine the sway displacements.

 The magnitude of sway displacements is caused by a number of factors: relative member stiffnesses, connection characteristics, the ratio of horizontal to vertical loading, and column behaviour. All these factors have a significant impact on the sway response of an unbraced frame.

Generally a common sway limit is h/300. For frames with a wider span bays, the sway displacement is likely to be below the above limit even under high wind loads. (MIStrucE, 2016), (J S HENSMAN BEng, 2016)

3.4.2 Sway limit

It is important to check that the sway on each storey is less than h/2000 cladding neglected (where is the storey height) (MIStrucE, 2016) (Fattorini, 1995).The bottom storey sway is likely to be the most critical.

3.4.3 Redesign for stiffness

If sway deflections are higher than the limit, the frame may be modified to increase its stiffness, by increasing member size, such as: the beam depth, the column flange thickness or the connection stiffness (although they may not necessarily increase its moment resistance), or redesign as a braced frame.

3.4.4 First-order Elastic Analysis (Portal Method)

This method is used for the unbraced frame.

Every member designed is assumed to be under elastic phase in accordance to all design loads for limit states. (PR SALTER BSc, 2004)

Below method allow us to calculate the approximate axial shear and moment in each member.

Calculation has been carried twice:

Wind load on X direction (4 bays)

Wind load on Y direction (3 bays)

Please see below figure 3.4.4

Figure 3.4.4 Plan view

Portal method allow as to solve indeterminate structure. In order to simplify the calculation, the structure is changed into determinate in order to use equilibrium equations by adding Pinned hinges were axial moment is assumed to be Zero positioned at the centre of each member. (Vary, 2016)

Below figures 3.4.41 show the fraction of shear force applied on members and the pinned hinges known as points of contra-flexure.








Figure 3.4.41 side view with pinned joints

On figure 3.4.41 the load applied in the interior columns are double that the loads on the exterior columns. Usually interior columns are stiffer because the carry more gravity load.

The Lateral loads are determined as follow:

w=wind load x width bay=kNm      UDL all along the hieght of the building

F=w x h2=KN  ( total shear acting on points of contraflaxuere)

Where h= Height of one floor

w= UDL                                                                                        (Vary, 2016)

The above shear calculation will be repeated for each floor taking in consideration the deference in height.

The next step is to determine the moment on each member as follow for:

Wind load on X direction (4 bays)

M= F8xh2=KNm  ( moment due to the shear on the external column)

M= 2F8xh2=KNm  ( moment due to the shear on the internal column)

Wind load on Y direction (3 bays)

M= F6xh2=KNm  ( moment due to the shear on the external column)

M= 2F6xh2=KNm  ( moment due to the shear on the internal column)

 Moment on the beams:

M= M1+M2 =KNm  ( moment on the beam)


M1=moment induced from the upper floor

M2=moment induced from the lower floor

(Vary, 2016)


3.5.1 General

Using a triangulated system, all the members of a bracing system need to be verified and designed based on the design forces due to different combinations of actions. It is recommended to make all diagonal members the same size, adequate to resist the largest design force.

  • Verification must be carried out by checking the resistance of the attachment column to beam, resistance of 1% of the vertical force on the column.
  • Check all the horizontal forces in the column if been transferred into the vertical bracing system.

3.5.2 Bracing design

Once the wind load is determined, calculation as follow:

W=wind load x height of building (total wind load)

w=W2height of building distributed wind load

F=w x storey hieght ( load acting on each node)

(Vary, 2016)

Once determined the forces, each node has to be resolved relatively to the forces applied, therefore a design size is chosen from the blue book depending on the force acting on each bracing.

For consistency two size are taken into consideration.

3.6 Frame Deflection

Deflection for each frame has been calculated, because on these dissertation the aim is to compare the behaviour into sway of braced and unbraced frame.

Deflection can be determined by 2 methods:

-slope deflection method

-unit load method.

In this dissertation the deflection of frame has been carried out by using Unit load method which practically quicker way to do it by using the volume integration table, please see Appendix xxx.

3.6.1 Unit load Method

One of the easiest and more precise way to analyse deflection is the principle of virtual work, great analytical tool to analyse determinate and indeterminate structures. This principle states that the sum of the external work done is equal to the sum of the internal word done.

The Unit load method apply the principle of virtual work to determine the displacement of the structure as follow (Datoo, 2015):

  1. The deformation of the structure has been determined using the externally applied loads.
  2. Virtual unit load applied at the point and into the direction where displacement is assumed to appear.
  3. Determine the internal work due to the internally force and unit load.
  4. Use the principle of virtual work by equating the sum of the external equal to the sum of the internal relative to the specific point chosen.

1x∆ = ∑∫0LM0 M1EIds




M0=Moment due to internal work         M1=moment due to unit load

3.6.12 Volume integration

Volume integration table is used by comparing the bending diagram due to internal work of each member with its correspondent bending diagram formulate from the unit load applied. If the bending moments due to the internal work and the unit load are on the same side then the product is positive but if they are on the opposite side then the product of the two is negative (Datoo, 2015). Please find Volume integral Table on Appendix xxx.

3.7 Frame Stability

Any frame structure should be examined for susceptibility to sway instability into second order effect. (Eng, 2009)

3.7.1 Second order

P∆ effect



> 10 any second order effect are good enough to be neglected. This check is an UlS check (PR SALTER BSc, 2004), (Eng, 2009), the combination considered is the equation with wind as a leading action as follow:

Euro code:  COMBINATION 3 on section 3.2.2

UK standard: COMBINATION 3 on section 3.2.1

Calculation has been processed as follow:

  1. Design value of wind load as leading action
  2. Design value of vertical Loads in combination with wind Load as leading action.

The bracing carry also horizontal loads induced by the imperfections. This loads are equivalent to 1/200 (0.5%) of the total permanent and variable factored loads acting on each floors and roof. Each bracing bay will be distributed a value equivalent of ½ of the total horizontal load. (Eng, 2009), (D G BROWN BEng, 2009)

  1. Each load above on point 2 has been added to the loads on point 1.
  2. Sway analysis is carried in accordance to the deflection values obtained on section 3.6.
  3. Frame stability verified using the following formula:

acr=HEdVEd x h∆ >10        CITATION MEB091 l 2057 (Eng, 2009)

3.8 Frame analysis software method

Different software has been used to determine the behaviour of the frame.

3 buildings are designed on Revit:

  • Unbraced frame
  • K-Braced frame
  • X-Braced frame.

Each drawing is transferred into robot for analysis applying euro-code factors and British standard factors.

On the other hand the frame systems above are analysed on strand 7.

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