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Department of Civil and
Structural Engineering
Plastic Design of Portal
frame to Eurocode 3 Supporting Notes
University of
Sheffield
2
CONTENTS
CONTENTS ............................................................................................................................... 2
List of Figures ............................................................................................................................ 5
1 Introduction ........................................................................................................................ 1
1.1 Outline of Portal Frames ............................................................................................ 1
1.2 Pitched Roof characteristics and parts ....................................................................... 2
2 Plastic analysis of portal frames ......................................................................................... 3
2.1 Plastic Design of Portal Frames ................................................................................. 3
2.2 Elastic vs. plastic analysis .......................................................................................... 3
3 Differnces Between EC3 and BS5950 ............................................................................... 4
3.1 Load Combinations .................................................................................................... 4
3.2 Consideration of second-order effects, ....................................................................... 4
3.3 Base fixity .................................................................................................................. 4
3.4 Separate checks for cross-section and buckling, ........................................................ 4
4 Worked example ................................................................................................................ 5
5 Material .............................................................................................................................. 7
5.1 Scopes ......................................................................................................................... 7
5.2 Design values of material coefficient ......................................................................... 8
6 Ultimate Limit State ........................................................................................................... 8
6.1 Outline ........................................................................................................................ 8
6.2 Design main concept .................................................................................................. 8
6.3 Partial Safety factors .................................................................................................. 9
6.3.1 Partial Safety factor of resistance ........................................................................... 9
6.3.2 Partial Safety factor for loads ................................................................................. 9
6.4 Load combination ..................................................................................................... 10
7 Ultimate limit state Analysis (First Order Vs Second Order) .......................................... 11
7.1 P.Ξ΄ effects ................................................................................................................. 11
7.2 P. Ξ effects ............................................................................................................... 12
7.3 How EC3 account for Second Order effects ............................................................ 13
7.3.1 Checking if second order affects significance ...................................................... 13
7.3.2 Determination of Ξ±cr ............................................................................................. 13
7.3.3 Accounting second order effects .......................................................................... 16
3
8 Serviceability limit state ................................................................................................... 18
8.1 Outlines .................................................................................................................... 18
8.2 Load Combinations .................................................................................................. 18
8.3 Deflections limiting values: ..................................................................................... 18
8.4 Procedures ................................................................................................................ 19
9 Frame Imperfections ........................................................................................................ 20
10 Connection of members ................................................................................................... 22
10.1 Eaves and apex connections ..................................................................................... 22
10.2 Lateral restraint to portal frames .............................................................................. 23
10.3 Bracing to compression flanges of rafters ................................................................ 23
10.4 Purlins ....................................................................................................................... 24
10.5 Column base ............................................................................................................. 24
11 Bracing ............................................................................................................................. 25
11.1 Overall column bracing ............................................................................................ 25
11.1.1 Portal bracing ................................................................................................... 25
11.2 Bracing to inner flanges ........................................................................................... 26
11.3 Bracing at the plastic hinge ...................................................................................... 27
12 Cross section resistance .................................................................................................... 28
12.1 The classification of cross-section ........................................................................... 28
12.2 Bending moment ...................................................................................................... 29
12.3 Shear ......................................................................................................................... 29
12.3.1 For low shear (see EN 1993-1-1 Section 6.2.8), .............................................. 30
12.3.2 For high shear ................................................................................................... 30
12.4 Axial force ................................................................................................................ 30
12.4.1 For low axial force ........................................................................................... 30
12.4.2 For high axial force .......................................................................................... 31
13 Stability checks, ............................................................................................................... 31
13.1 Restraint of plastic hinge .......................................................................................... 31
13.2 Rafter Stability ......................................................................................................... 32
13.3 Stability of haunch ................................................................................................... 33
13.4 Stability of the column ............................................................................................. 34
14 Fire Boundary Condition .................................................................................................. 35
14.1 Outline ...................................................................................................................... 35
4
14.2 Building Regulation requirements in UK................................................................. 35
14.3 Alternative approaches to boundary fire .................................................................. 35
14.4 Effects of fire on Portal frames ................................................................................ 36
14.5 Mathematical Model ................................................................................................ 37
15 Appendix .......................................................................................................................... 39
16 Reference .......................................................................................................................... 47
5
LIST OF FIGURES
1 multiple span frames
2 Single-span symmetrical portal frame (Malik SN051a-EN-EU)
3 Side Cladding (Malik SN051a-EN-EU)
4- P.Ξ΄ effects
5- Magnification of sway displacements
6- Snap-through failure
7- Parameters required
8- Examples of Category A
9- Definition of horizontal displacements
10- Frame imperfections as column slopes
11- Force equivalent to frame imperfections
12- Apex and Eaves Connections
13- Stabilising of the column
14- Braces to rafter bottom flange
15- Braced portals
16- Effects of purlin flexibility on bracing
17- Bracing to inner flanges
18- Typical stiff torsional restraint
19- Bracing at plastic hinge
20- Bending moment diagram for uniform temperature rise
21- Probable positions of fire hinges in portal frame
22- Rafter collapse to eaves level
23- Moment Ratios
24- Dimensions defining taper factor
25- Value of Bt
26 Forces and moments acting on the column and rafters
1
1 INTRODUCTION
1.1 Outline of Portal Frames
Portal frame construction is a method of building and designing simple structures, primarily
using steel. The connections between the columns and the rafters are designed to be moment-
resistant, i.e. they can carry bending forces. Because of these very strong and rigid joints
some of the bending moment in the rafters is transferred to the columns. This means that the
size of the rafters can be reduced or the span can be increased for the same size rafters. This
makes portal frames a very efficient construction technique especially in steel being able to
make use of plastic methods. Portal frames construction is popular due to the following
reasons;
1) It provides large clear floor areas offering maximization of the site for
example in; retail stores, distribution warehouse, manufacturing
facilities, and leisure centers.
2) They are capable of being extended in the future if this is accounted
within design.
3) Multiple bays are possible where internal column can be eliminated
where required by using valley beam as shown if figure .1.
Figure 1 multiple span frames
Portal frames can be clad with all sorts of material but the most popular solution, for reasons
of economy and speed, is some form of lightweight insulated metal cladding with cavity
masonry work to the bottom 2m of the wall to provide security and impact resistance (Morris,
1981). The lightweight cladding would be carried on sheeting rails spanning between the
columns of the portal frames.
2
1.2 Pitched Roof characteristics and parts
Figure 2 Single-span symmetrical portal frame (Malik SN051a-EN-EU)
A single-span pitched roof portal frame (figure2) will typically have (Malik SN051a-EN-EU):
1) Span between 15 meters to 50 meters.
2) Eaves 5 meters to 10 meters.
3) A roof pitched between 5 degrees and 10 degrees. (6 degrees is commonly
adopted).
4) A frame spacing between 5 meters and 8 meters (the greater the spacing being
associated with the longer span portal frames).
5) Haunches in the rafters at the eaves and apex used to reduce the depth of the rafter
and achieve efficient moment connection at these points.
6) The portal frame supports side cladding by means of cold formed purlins and
sheeting.
Figure 3 Side Cladding (Malik SN051a-EN-EU)
Most of these characteristics are prescribed by the economics of portal frames relative to other
forms of constructions.
3
2 PLASTIC ANALYSIS OF PORTAL FRAMES
2.1 Plastic Design of Portal Frames
Plastic theory has been used in the UK for the basis of the design of low-rise frames,
particularly portal frames, since its use was first allowed in BS449 in 1948. In many
countries the use of plastic theory is permitted by their own national codes. But plastic
methods of analysis are mainly used to allow engineers to more easily do the analysis
and more economical design portal frames than typical elastic methods. The concept
of plastic analysis method is to calculate the loads that can be applied to cause the
frame to fail as a failure mechanism occur e.g. enough plastic hinges occur (Morris,
1981).
2.2 Elastic vs. plastic analysis
The methods of analysis fall into either elastic analysis or plastic analysis. Elastic
analysis often produces less economical structures because it does not allow large-
scale plastic redistribution of bending moments, although these will occur in an
adequately braced structure and will enable the structure to carry higher loads than
calculated by elastic analysis. Generally, plastic analysis results in more economical
structures because plastic redistribution allows slightly smaller members to carry the
same loads. On the other hand in some cases elastic analysis can be more economical
as plastic analysis depends on bracing system, because plastic redistribution imposes
extra requirements on bracing of members. The overall economy of the frame might
therefore depend on the ease with which the frame can be braced.
4
3 DIFFERNCES BETWEEN EC3 AND BS5950
There are a number of differences between EC3 and previous UK practice, which need to be
clearly understood by the designers that used to BS5950 but intend to use EC3. The main
differences affecting the portal frame designing noted below (King, Technical Report P164).
3.1 Load Combinations
BS5950 uses one set of load factors for a combination of (dead + live) loads, but a
lower load factor for (dead + live + wind) loads. In principal, EC3 requires that all
variable actions (live loads, wind loads, ect.) are considered in the same load
combination, but includes a reduction factor on all variable actions, except the most
unfavorable.
3.2 Consideration of second-order effects,
EC3 requires that the second order effects are explicitly considered in analysis, either
by second-order analysis or by modifications to classic first-order analysis. For single-
storey portal frame design, the sway check method in BS 5950 uses deflection checks
to assess the stiffness. If the stiffness is too low, it has to be increased. If the stiffness
is above a certain value, second order can be ignored.
3.3 Base fixity
It is common practice in the UK to assume that column bases are truly pinned for
ultimate limit state bending moment diagrams, where they are actually nominally
connected. EC3 requires consideration of the actual flexibility. In the case of
nominally pinned bases, the assumption of a truly pinned base is conservative, so it is
acceptable. However, for fixed bases the actual flexibility must be considered.
3.4 Separate checks for cross-section and buckling,
EC3 has entirely separate checks for the cross-section resistance and buckling
resistance. This is unfamiliar, but very helpful for checking elements of varying
sections along the fram
5
4 WORKED EXAMPLE
The worked example is aimed to provide guidance on method of designing a portal frame to
Eurocode for maximum vertical load combination; which is been used due to the fact that
most portal frame design is governed by gravity (dead + snow) loading. The required
designing steps where identified from (EN2005-1-1) and the following procedure where
developed for the worked example (see Appendix A for the flow chart);
(1) Define frame geometry.
(2) Calculate the Loading on the portal frame accurately based on the initial sizing of the
sections as follow (Gravity Load);
Snow loading.
Gravity loading.
(3) Choose a trial sections and a trial haunch length by selecting universal beam section that
have plastic modules as given by Wellerβs charts (The institutionof Structural Engineers,
TP/08/43 EC3/08/16). The haunch length should be chosen to optimize the overall portal
structure. A length of L/10 from the column face is a reasonable initial choice, but the
proportions of the haunch generally depend of the characteristics of each individual
building. A haunch length of L/10 will normally place the first hinge in the top of the
column.
(4) Calculate frame imperfection equivalent forces, referring to section 5.3.2 of EN1993-1-
1:2005.
(5) Perform plastic analysis of the frame, assuming no reduction in plastic moment of
resistance from coexistent axial and shear force. This approach may need modifying,
where axial load are high.
(6) Determine sensitivity of portal frame to sway, by calculating Ξ±cr, s, EST.
6
(7) Determine sensitivity of portal frame to snap-through rafter buckling, by calculating
Ξ±cr,s,est.
(8) Determine Ξ±cr,est = min (Ξ±cr,s,est , Ξ±cr,s,est).
(9) If Ξ±cr,est < 15 the member design forces is calculated due to second order effects; this is
done by using amplified partial load factors method (section 5.3.2 of EN1993-1-1:2005).
(10) For the column check the following:
(A) The classification is Class 1 or Class 2.
(B) The cross-section resistance is adequate, and checks if axial and shear force has
effect on plastic resistance of section (section 5.5.2 of EN1993-1-1:2005).
(C) The resistance of the member to minor axis buckling between lateral restraints
is adequate; this might be between plastic hinge and lateral restraint or between
two lateral restraints. To prevent this spacing is limited below Lm (section
BB.3.1.1of EN1993-1-1:2005).
(D) The resistance of member to minor axis buckling between torsional restraints is
adequate, it checked for by same criteria as lateral restraint where maximum
spacing between torsional restraint are limited to ( Lk , Ls ,Ls ) (section
BB.3.2.2of EN1993-1-1:2005).
(11) For the rafter check that,
(A) The classification is Class 1 and Class 2 as step 7A.
(B) The cross-section resistance is adequate, as a step 7B
(C) The resistance of the member to minor axis buckling between lateral restraints is
adequate as step 7C.
(D) The resistance of the member to minor axis buckling between torsional restraints
is adequate as step 7D.
7
(12) For the Haunches check that:
(A) T he classification is Class 1 and Class 2 as step 7A.
(B) The cross-sectional resistance is adequate, as step 7B above, but checking at
several cross-sections within the length of the haunch (ends, quarter, and mid-
span and three quarter points).
(C) The resistance of the member to minor axis buckling between lateral restraints is
adequate as step 7C.
5 MATERIAL
5.1 Scopes
EC3 covers design in steel to BSEN 10025 (grades Fe430 Fe510 only). For members, the
hunches, the end plates, the base-plated and any stiffeners an S275 steel to EN10025 with
yield 275N/mm2 and ultimate strength of 430N/mm2 is adopted
Table 1-Nominal values of yield Strength and ultimate tensile Strength (Baddoo, 1993 )
8
5.2 Design values of material coefficient (BS EN 1993-1-1:2005)
Modulus of elasticity (E) = 210 000 N/mm2.
Shear modulus (G) = E/2(1+v) = 81 000N/mm2.
Poissonβs ratio (v) = 0.3.
Coefficient of linear thermal expansion (Ξ±) = 12 x10-6 per Co.
Density (Ο) = 7 850 kg/m3.
6 ULTIMATE LIMIT STATE
6.1 Outline
According to EC3 all actions that could occur at the same time are considered together, so
frame imperfection equivalent forces (which are discussed in section 9) as well as wind loads
should be considered as additives to permanent actions and other imposed loads with an
appropriate combination factor applied to them. Although several combinations exist, only the
least favourable combination of loads should be considered. Most general example is wind
load can act on both sides of portal frame but only combination need to be considered is the
least favourable direction with respect to each load combination.
6.2 Design main concept
A structure must have adequate resistance to internal forces and bending moments, known as
structural resistance. In addition it must exist in a state of static equilibrium under the action
of the applied loads, i.e. it must have adequate resistance to overturning, uplift ECT. Due to
fact that static equilibrium is rarely critical for portal framed buildings, so not considered in
details. The following are cases where static equilibrium might be critical:
1. In high wind speed areas.
2. In high but narrow buildings, result in overturning.
3. Where there is large opening allowing generation of high positive wind pressure inside
the building at the same time as high suction over the roof.
9
6.3 Partial Safety factors
EC3 uses partial safety factor format for checking ULS, in which the partial safety factors
obtain a considerable safety margin against uncertainties of load on structure and resistance of
the material. So that it reduces the resistance of the material and increase the applied load on
the structure.
οΏ½ππβππππππππππππππππππππππππ ππππππππππππππππππππ
πΎπΎπποΏ½ > ππππππππππππ ππππ β(ππβππππππππππππππππππππππππ ππππππππππππ) Γ πΎπΎπΉπΉ
6.3.1 Partial Safety factor of resistance
πΎπΎππππ 1.1 For the resistance of the cross-section
πΎπΎππ1 1.1 For the resistance buckling of the member
πΎπΎππππ 1.25 For the resistance of the net sections
πΎπΎππππ 1.25 For the resistance of the bolts
πΎπΎππππ 1.25 For the resistance of welds
Table 2- Partial Safety Factors of Steel Resistance (BS EN 1993-1-1:2005)
6.3.2 Partial Safety factor for loads
Partial factors for actions are either
β’ Permanent action dead load πΎπΎπΊπΊ
β’ Variable action live load πΎπΎππ
Those will have different values depending on wither the action is favourable or
unfavourable for the structure stability.
Table 3-Partial safety factors for actions in building structure (Baddoo, 1993 )
10
6.4 Load combination
EC3 normally consider all possible coincident actions but with the less unfavourable variable
action reduced by combination reduction factor (Ο). The βcombination factorβ reduces the
intensity of loading, when several variable loads are considered to be acting together. There is
only low probability of all variable action occurring at full intensity at the same time, so the
combination factor is used.
All ULS load combinations must include the effect if any of imperfection forces (horizontal
forces) from any direction particularly in frame analysis. The Single expression for ULS load
combination is as follow (Baddoo, 1993 )
Note that there are different Ο factors for different types of variable actions. The table below
from EN1990:2002 shows different factors.
Table 4- Recommended Values of Ο for buildings (Baddoo, 1993 )
Characteristic value of the permanent action (πΊπΊππ ππ ) X πΎπΎπΊπΊ
+ Full characteristic value of the most unfavorable variable action (ππππ 1) X πΎπΎππ 1
+ Reduced values of all other unfavorable variable action (ππππ ,ππ Γ Ο0,ππ) X πΎπΎππ,ππ
11
7 ULTIMATE LIMIT STATE ANALYSIS (FIRST ORDER VS SECOND ORDER)
Normally first-order analysis is acceptable unless when the frame or frame member is
exceedingly slender. The limitations are given below; where structures exceed these
limitations second order analysis must be used.
7.1 P.Ξ΄ effects
P.Ξ΄ effects are the effects on member behaviour due to displacements at right-angles to the
axial force in the member. These displacements may be enforced (by external load or
moments), most notably in the case of axial loading along a member with initial bow, or
maybe the result of the natural tendency to buckle under pure axial load. The displacements
are the sum if the initial deformation of the member and deflection due loading. The result of
such deflections is to reduce the effective stiffness when axial load is compressive, due to
increased bending moment see below figure 4. Conversely, when axial load is tensile, it
increases the effective stiffness, though the effect will generally be minimal in common single
storey portal frames.
Figure 4- P.Ξ΄ effects (King, Technical Report P164)
12
7.2 P. Ξ effects
P. Ξ effects are the effects on overall frame behaviour due to the displacement at right-angles
to the applied forces, directly due to those same forces. The displacements are the sum of the
initial deformation of the frame and deflection due loading. For pitched roof portals, the prime
example is the effect of gravity loads on sway deflections. The effect is to magnify the sway
deflection due to overall inclination of the column which produces a reduction in effective
stiffness, see the figure 5,
Figure 5- Magnification of sway displacements (King, Technical Report P164)
Another possible mode of failure which is sensitive to P. Ξ effects is βarching failureβ or
βsnap-throughβ of pair of rafters (see fig below). This has to be checked because it is possible
to design portals of 3 spans or more with stiff outer bays that provide horizontal support to the
rafters of the inner spans. Then the rafters of the inner spans can act as arches with the
horizontal reaction provided by the outer bays. Where this arching action works, the rafters
will support more vertical load than if they were acting only as beams. This check is used to
ensure that the rafters are not so flexible that they Β΄snap throughΒ΄.
Figure 6- Snap-through failure (King, Technical Report P147)
13
7.3 How EC3 account for Second Order effects
7.3.1 Checking if second order affects significance
First order analysis may be used for the structure where the increase of the relevant internal
forces or moments or any other change of structural behaviour caused by deformations can be
neglected. This condition may be assumed to be fulfilled, if the following criterion is satisfied
(BS EN 1993-1-1:2005):
β’ Ξ±cr β₯ 10 for elastic analysis
β’ Ξ±cr β₯ 15for plastic analysis
- Where:
Ξ±cr Is the factor by which the design loading would have to be increased to
cause elastic instability in a global mode.
7.3.2 Determination of Ξ±cr
For frames with pitched rafters,
Ξ±cr,est= min (Ξ±cr,r,est , Ξ±cr,s,est)
- Where
Ξ±cr,s,est is the estimate of Ξ±cr for sway buckling mode (see Section 3.3.1)
Ξ±cr,r,est is the estimate of Ξ±cr for rafter snap-through buckling mode
(see Section 3.3.2)
14
7.3.2.1 Sway buckling load factor (Oppe, SN033a-EN-EU)
The parameters required to calculate Ξ±cr,s,est, for portal frames are shown in Figure 7 . As
can be seen, Ξ΄HEF is the lateral deflection at the top of each column when subjected to an
arbitrary lateral load HEHF. (The magnitude of the total lateral load is arbitrary, as it is simply
used to calculate the sway stiffness HEHF/Ξ΄EHF.) The horizontal load applied at the top of each
column should be proportional to the vertical reaction. Thus, for an individual column:
π»π»πΈπΈπ»π»πΉπΉ ,ππ
ππππππππ ,ππ=π»π»πΈπΈπ»π»πΉπΉππππππππ
- Where
HEHF is the sum of all the equivalent horizontal forces at column tops
(see figure 7)
VULS is the sum of all factored vertical reactions at ULS calculated from
first-order plastic analysis
HEHFi is equivalent horizontal force at top of column i (there are two
columns in a single-span portal, three in a two-span portal, etc.)
VULS,i is factored vertical reaction at ULS at column i, calculated from first-
order plastic analysis
Figure 7- Parameters required to estimate alpha critical
for sway mode failure (J.B.P.Lim, November 2005)
15
An estimate of Ξ±cr can then be obtained from
Ξ±cr,s,est = 0.8 οΏ½1 β οΏ½πππ π ,πππππππππ π ,ππππ
οΏ½πππππποΏ½ οΏ½οΏ½ βππππππππππ ,ππ
οΏ½ οΏ½π»π»πΈπΈπ»π»πΉπΉ ,πππΏπΏπΈπΈπ»π»πΉπΉ ,ππ
οΏ½πππππποΏ½
Where:
οΏ½πππ π ,πππππππππ π ,ππππ
οΏ½ππππππ Is the maximum ratio in any rafter.
πππ π ,ππππππ Is the axial force in rafter (see figure 7 (b)).
πππ π ,ππππ = ππ2 πΈπΈ πΌπΌππππ2 Is the Euler load of the rafter on full span (assumed pinned)
Ir Is the in-plane second moment of area of rafter
Ξ΄EHF,i Is the horizontal deflection of column top (see figure 7 (c))
οΏ½ βππππππππππ ,ππ
οΏ½ οΏ½π»π»πΈπΈπ»π»πΉπΉ ,πππΏπΏπΈπΈπ»π»πΉπΉ ,ππ
οΏ½ππππππ Is the minimum value for columns 1 to n (n = the number of columns)
7.3.2.2 Rafter Snap-through buckling load factor (Oppe, SN033a-EN-EU)
For frames with rafter slopes not steeper than 1:2 (26Β°), Ξ±cr,r may be taken as:
Ξ±cr,r,est = οΏ½π·π·πποΏ½ οΏ½
55.7οΏ½4+ππβοΏ½
πΊπΊβ1οΏ½οΏ½πΌπΌππ+πΌπΌππ
πΌπΌπποΏ½ οΏ½275
πππ¦π¦πποΏ½ tan 2ππππ
D cross-section depth of rafter
L span of the bay
h means height of the column
πΌπΌππ in-plane second moment of area of column
πΌπΌππ in-plane second moment of area of rafter
πππ¦π¦ππ nominal yield strength of the rafter
ππππ roof slope if roof is symmetrical
πΊπΊ Fr/Fo the ratio of the arching effect of the frame where
Fr= factored vertical load on the rafter
F0 = maximum uniformly distributed load for plastic failure of the rafter treated as a
fixed end beam of span L οΏ½πΉπΉππ = 16 ππππππ ,π¦π¦ ,π π πππ¦π¦ππ
οΏ½
But where Ξ© β€ 1, Ξ±cr,r = β
16
7.3.3 Accounting second order effects (J.B.P.Lim, November 2005)
Eurocode account to second order effects in the absence of elastic-plastic second order
analysis software by deriving loads that are amplified to account for the effects of deformed
geometry (second order effects). Application of these amplified loads through first order
analysis gives bending moment, axial forces and shear forces that include the second-order
effects approximately. The magnification factor is calculated by method called Merchant-
Rankin method, which is as following:
7.3.3.1 Category A: Regular, symmetric and asymmetric pitched and mono-pitched frames
Regular, symmetrical and mono pitched frames are either single-span or multi span frames in
which there is only a small variation in height (h) or span (L) between different spans;
variation in height and span of the order of 10% may be considered sufficiently small. 1
1 β οΏ½ 1πΌπΌππππ
οΏ½
Provided that πΌπΌππππ β₯3
Where πΌπΌππππ can be calculated as shown above in section 7.3.2
Figure 8- Examples of Category A (Oppe, SN033a-EN-EU)
17
7.3.3.2 Category B: Frames that fall outside of Category A and excluding tied portals
Frames that are outside of category A and excluding tied portals 1.1
1 β οΏ½ 1πΌπΌππππ
οΏ½
7.3.3.3 Procedure of Plastic analysis
The method, as currently given in EN 1993-1-1, implies the following procedure:
(i) Choose initial portal sections. (ii) Calculate Ξ±cr factor by which the design loading would have to be increased to
cause elastic instability in a global mode (iii) Comply with Merchant Rankin by multiplying the partial safety factors by the
following magnification factor;
1
1 β οΏ½ 1πΌπΌππππ
οΏ½
(iv) Perform the rigid plastic analysis of portal frame using the Partial safety factors in step (iii).
(v) Check that collapse load factor of step (iv) analysis is greater one (πΆπΆππππππππππππππ ππππππππ ππππππππππππ β₯1.0) where,
πΆπΆππππππππππππππ ππππππππ ππππππππππππ =πΆπΆππππππππππππππ ππππππππππππππ ππππππππ
(vi) If the Collapse load factor in step (v) is less than one then increase the section size and re-check if the increased section size can withstand second-order effects.
18
8 SERVICEABILITY LIMIT STATE
8.1 Outlines
The serviceability limit state (SLS) analysis should be performed using SLS load cases
mentioned below, to ensure that deflections are within acceptable limit at characteristic
loading. This is normally first-order analysis, where plasticity occurs under SLS loads, the
deflections from plastic deformations should be included in the analysis.
It is important to ensure that the deflections etc. are acceptable for the cladding and use of the
building, for example for brittle cladding such as brickwork, the limit of h/150 on horizontal
deflection may be unsuitable.
8.2 Load Combinations
Structural imperfections should not normally be considered. No partial safety factors
applied to actions for the SLS. The load combination as follow (BS EN 1991-1-3:2003):
Note that there are different Ο factors for different types of variable actions, see section ULS
8.3 Deflections limiting values (BUREAU, SN035a-EN-EU):
No specific deflection limits are set in Eurocode 1993-1-1 [1]. According to EN 1993-1-1, Β§
7.2 and EN 1990 β Annex A1.4 [2], deflection limits should be specified for each project and
agreed with the client. The National Annex to EN 1993-1-1 may specify limits for application
in individual countries. Where limits are specified they have to be satisfied. Where limits are
not specified, the following might be helpful when deciding relevant deflection limits.
Characteristic values of permanent actions (βGk j)
+Full characteristic value of the most unfavorable variable action Qk,1
+Reduced values of all other unfavorable variable actions Ο0,i Qk,i
19
β’ Portal frames without gantry cranes h/150 β’ Other single storey buildings h/300
Figure 9- Definition of horizontal displacements (BUREAU, SN035a-EN-EU)
8.4 Suggested Procedure to check serviceability limit state
1) Determine SLS load combinations and Ο factors.
2) Perform first order analysis.
3) Check that Ξ±cr β₯ 15 to allow for second-order effects to be ignored
4) If Ξ±cr β€ 15to, re-analyze including magnification factor described in section (7.3.3)
5) Check that the deflections do not impair the function of the building, including the
cladding.
20
9 FRAME IMPERFECTIONS
Frame imperfections are assumed in EC3 as a slope of the columns and can be induced in
the analysis model as shown below;
Figure 10- Frame imperfections as column slopes (BS EN 1993-1-1:2005)
EC3 allows the use of the equivalent horizontal forces, which are probably easier to use on
sloping roof structures such as portals. These forces may be calculated from the column axial
force using ββsimply supportedββ assumptions as below (BS EN 1993-1-1:2005);
Ο = Οo Ξ±h Ξ±m
- With Οo for single storey frame it is 1/200
Ξ±h is the reduction factor for the height h applicable to columns:
Ξ±β = 2ββ
But 23β€ Ξ±h β€ 1.0
h is the height if the structure in meters.
Ξ±m is the reduction factor for the number of columns in the row
Ξ±ππ = οΏ½0.5(1 + 1ππ
)
m is the number of columns in a row.
Columns which carry a vertical load NEd of less than 50% of the mean value of the
vertical load per column in the plane considered shall not be considered in m.
21
Frame imperfections induce internal forces and moments that will increase or
decrease the effect of other actions depending on the direction. Therefore if the frame
imperfections is to be induced in geometry model of EC3 (discussed earlier) it is
require two versions to be considered; from each side, but are only considered in one
direction at a time.
It must be remembered that the horizontal forces due imperfections do not increase the
total horizontal load on the building as shown if figure below,
Figure 11- Force equivalent to frame imperfections (BS EN 1993-1-1:2005)
The following simplified approach to frame imperfection can be used for regular portal frame;
1) Calculate total axial load on structure, NEd.
2) Calculate Π€ NEd
3) Divide this Π€ NEd into series of equal horizontal forces applies at top of each column.
Frame imperfections induce internal forces and moments that will increase or decrease the
effects of actions, depending on the direction.
22
10 CONNECTION OF MEMBERS
The rafter and columns of I-section portal frames consist of rolled I-sections, with rafter end
being haunched and sit-bolted to the columns. The column section is heavier than the rafter
section, so that the relative column-haunch rafter strengths roughly follow the shape of the
bending moment diagram up the column and across the rafter. As a scope of plastic analysis,
moments referred to are plastic moments under gravity loading, with plastic hinges
developing either in the column top or in the rafter at the haunch and in the rafter near apex
(Position of maximum bending moments).
10.1 Eaves and apex connections
The types of eaves and apex haunches shown in figure 12 are the ones almost universally
used because of their relative simplicity and the ease with which the frame can be erected.
The critical design condition is usually gravity loading with the rafter-to-column connection
having to sustain a high negative moment and apex connection smaller positive moment. The
moment at the eaves produces a high tensile force in the upper flange of the rafter that is
transmitted through the upper tension bolts and the end plate to the inner flange of the
column. The compressive force in the lower flange of the haunch is transferred in bearing
through the end plate onto the column flange and into the web.
Figure 12- Apex and Eaves Connections ( The Southern African Institute of Steel
Construction,)
23
The transfer of moment at the apex is similar, except that here the moment is positive so the
forces are reversed. The haunch and apex regions are vitally important parts of the frame and
must be carefully proportioned. It is possible to achieve economy through simplification of
the connections, but only when every aspect of the transfer of direct, moment and shear forces
has been carefully considered.
10.2 Lateral restraint to portal frames
As there are high bending moments at the column-to-rafter junction it is necessary to provide
adequate lateral-torsional restraint in this region. This may be done by means of a strut
connected into the column web within the depth of the rafter haunch. A typical detail is
shown in figure 13, where a circular hollow section with an end plate is used. This section is
light, as well as strong in both tension and compression; because of the butt-welded end plate
it is also able to offer torsional restraint to the column in this highly stressed area.
Figure 13- Stabilising of the column ( The Southern African Institute of Steel
Construction,)
10.3 Bracing to compression flanges of rafters
Where the lower flanges of the rafters are in compression, either near the columns when under
gravity loading or further up the slope when under wind uplift loading, they require to be
restrained laterally to prevent buckling. This is usually done by fitting angle braces to the
purlins, as shown in figure 14.
24
Figure 14- Braces to rafter bottom flange (King, Technical Report P164)
10.4 Purlins (Chaffotec, SS049a-EN-EU)
In a typical portal-frame building the main structural components are the portal frames
themselves and the purlins. The principal function of roofing purlins is to transfer the forces
on the roof of a building to its main structure. The wall rails perform the same role on the
facades. Purlins and wall rails are important components in the secondary structure of a
building. The purlin structure of a building is designed in accordance with the type of roofing
to be used. The nature of the roofing, in particular, directly influences the spacing between
purlins; to achieve maximum economy it is necessary to optimize the combined cost of these
two items by choosing the correct spacing for the frames. Various spanning arrangements can
be used for the purlins, e.g. single, double or multi-span or Z-sections overlapped over the
rafters.
10.5 Column base (King, Technical Report P147)
The great majority of portal frames are designed with nominally pinned bases. This is for
reasons of economy and simple design. Not only are fixed bases more expensive because of
the need for thicker and larger base plates and the stiffening that is necessary, but the
foundations require being much larger to resist the base moments. Only in cases of large
lateral deflection, or possibly where brick walls are built into the columns, is it necessary to
resort to fixed bases. These should be kept as simple as possible.
25
11 BRACING
Bracing is required to resist lateral loads, principally wind loads, and the destabilising effects
of the imperfections defined in Section 5.2.3 of EN 1993-1-1:2005. This bracing must be
correctly positioned and have adequate strength and stiffness to justify the assumptions made
in the analysis and member checks.
EN 1993-1-1:2005 Section 5.2.3 of allows the imperfections to be described either as
geometrical imperfections or as equivalent horizontal forces. The equivalent horizontal forces,
which cause the forces in the bracing, do not increase the total load on the whole structure,
because they form a self equilibrating load case.
11.1 Overall column bracing (King, Technical Report P164)
Bracing must be provided to hold the columns upright and resist loads, such as wind loading,
occurring at right angles to the frame. Columns are assumed to be built slightly out of
vertical, and the simplest method to account for this effect is by means of the equivalent
horizontal forces shown in Figure imperfection section. These forces can occur in any
direction, but are considered to act in one direction at a time.
The bracing is designed to resist wind loads plus equivalent horizontal forces, which amounts
to approximately [wind forces 0.5% vertical forces causing axial compression]. Care must be
taken when detailing the bracing, as details that reduce the stiffness (e.g. by deflection from
local bending of plates, rings or angles) might increase the imperfection loads due to second-
order effects. If the columns carry a net tensile force, as in an uplift load case due to wind,
then this loading does not destabilise the structure, so may be neglected.
11.1.1 Portal bracing
The term βportal bracingβ is commonly used to describe a system of bracing comprising
portals instead of cross-bracing to provide the restraint normal to the main frames. An
example of portal bracing is shown in Figure 15. It is frequently used to provide lateral
stability to the top of the internal columns, because the use of cross-bracing would result in an
unacceptable restriction to freedom of use. It is also used at exterior walls, where cross-
bracing might obstruct windows, doors, etc.
26
Figure 15- Braced portals (King, Technical Report P147)
The bracing portals are designed to resist the total equivalent horizontal forces from all the
columns that rely on those portals for stability, plus the relevant wind load.
11.2 Bracing to inner flanges
Bracing to the inner flanges is often most conveniently formed by diagonal struts from the
purlins or sheeting rails or sheeting rails to gluts welded to the inner flange and web, as shown
in figure 17. The effectiveness of such bracing depends on the stiffness of the system,
especially the stiffness of the purlins. The effect of purlins flexibility on the bracing is shown
in figure 16.
Figure 16- Effects of purlin flexibility on bracing (King, Technical Report P164)
Figure 17- Bracing to inner flanges (King, Technical Report P147)
The compression flange of eaves and valley haunches must be braced at the column-haunches
connection, unless stability analysis test data prove it to be unnecessary. This restraint is
needed, because the haunch stability checks are based on full torsional restraint at this point.
27
11.3 Bracing at the plastic hinge (King, Technical Report P164)
Due to the fact that at plastic hinges, all material of the cross-section is at yield; this result in
cross-section unable to resist any twisting action. Hence it is recommended by EN1993-1-
1:2005 that a restraint should be provided at plastic hinge position in order to resist against
lateral torsion buckling.
Figure 18- Typical stiff torsional restraint (King, Technical Report P164)
Section 6.3.5.2 of EN 1993-1-1:2005 recommends that bracing should be provided to both
tension and compression flanges at or within 0.5h of the calculated plastic hinges, where h is
the depth of the member (see fig below)
Figure 19- Bracing at plastic hinge (King, Technical Report P147)
At each rotated plastic hinge location the cross section should have an effective lateral and
torsional restraint with appropriate resistance to lateral forces and torsion induced by local
plastic deformations of the member at this location. At each plastic hinge location, the
connection (e.g. bolts) of the compression flange to the resisting element at that point (e.g.
purlin), and any intermediate element (e.g. diagonal brace) should be designed to in its plane
and perpendicular to the web plane, without any combination with other loads.
28
12 CROSS SECTION RESISTANCE
12.1 The classification of cross-section
EC3 (BS EN 1993-1-1:2005) defines four classes of cross section. The class into which a
particular cross section falls depends upon the slenderness of each element (defined by a
width-to-thickness ratio) and the compressive stress distribution i.e. uniform or linear. The
classes are defined in terms of performance requirements for resistance of bending moments:
1) Class 1 cross-sections are those which can form a plastic hinge with the required
rotational capacity for plastic analysis.
2) Class 2 cross-sections are those which, although able to develop a plastic moment,
have limited rotational capacity and are therefore unsuitable for structures designed by
plastic analysis.
3) Class 3 cross-sections are those in which the calculated stress in the extreme
compression fibre can reach yield but local buckling prevents the development of the
plastic moment resistance.
4) Class 4 cross-sections are those in which local buckling limits the moment resistance
(or compression resistance for axially loaded members). Explicit allowance for the
effects of local buckling is necessary.
The Eurocode classifications according to slenderness of flange or web elements is given in
EN 1993-1-1:2005 (E) Table 5.2.In portal frame designed by plastic methods, the columns
and rafters must be Class 1 or Class 2. These two classes have the same rule for checking
resistance. The only element which may be different is the haunch web, which may be Class
3, according to EN1993-1-1:2005 (E).
29
12.2 Bending moment (BS EN 1993-1-1:2005)
The design value of the bending moments MEd at each cross-section shall satisfy: πππΈπΈππ
ππππ ,π π ππ β€ 1.0
Where Mc, Rd is determined considering fastener holes (see EN 1993-1-1:2005
Section 6.2.5.)
The design resistance for bending about one principal axis of a cross-section is determined as
follow:
ππππ ,π π ππ = ππππππ ,π π ππ = ππππππ πππ¦π¦πΎπΎππππ
for class 1 and 2 cross-section.
ππππ ,π π ππ = ππππππ ,π π ππ = ππππππ ,ππππππ πππ¦π¦πΎπΎππππ
for class 3 cross-section.
ππππ ,π π ππ = ππππππππ ,ππππππ πππ¦π¦πΎπΎππππ
for class 4 cross-section.
12.3 Shear (BS EN 1993-1-1:2005)
The design value of the shear force πππΈπΈππ at each cross section shall satisfy:
πππΈπΈππππππ,π π ππ
β€ 1.0
- Where ππππ ,π π ππ is the design shear resistance.
For plastic design ππππ ,π π ππ is the design plastic shear resistance ππππππ ,π π ππ which is given by;
ππππππ ,π π ππ = π΄π΄π£π£ (πππ¦π¦ /β3)πΎπΎππ0
- Where π΄π΄π£π£ is the shear area which is for rolled I and H sections given by;
Av = Aβ2bππππ + (ππππ + 2ππ)ππππ
Where
A is the cross-sectional area
b is the overall breadth
tf is the flange thickness
r is the root radius
Where the shear force is present allowance should be made for it, effects on the moment resistance are as following.
30
12.3.1 For low shear (see EN 1993-1-1 Section 6.2.8),
Where the shear force is less than the half the plastic shear resistance its effect on the
moment resistance may be neglected.
VEd <0.5 Vpl.Rd
12.3.2 For high shear
Where high shear force exists reduced moment resistance should be taken as the design
resistance of the cross-section, calculated using reduced yield strength
VEd > 0.5 Vpl.Rd
(1 β ππ)πππ¦π¦
ππ = οΏ½οΏ½2πππΈπΈππππππππ ,π π ππ
οΏ½ β 1οΏ½ 2
12.4 Axial force (BS EN 1993-1-1:2005)
Where axial force is present, allowance should be made for its effects on the plastic moment
resistance of the cross-section. For doubly symmetrical I and H sections or other flanged
sections,
12.4.1 For low axial force
Allowance need to not be made for the effect of the axial force on the plastic resistance
moment about y-y axis when both the following criteria a satisfied:
πππΈπΈππ β€ 0.25 ππππππ ,π π ππ and
πππΈπΈππ β€ (0.5 βπππππππππ¦π¦)/πΎπΎππππ
Provided that the design value of shear force VEd does not exceed 50% of the design plastic
shear resistance Vpl,Rd no reduction of the resistance defined for bending and axial force need
to be made.
31
12.4.2 For high axial force
Where VEd exceeds 50% of Vpl,Rd the design resistance of the cross-section to combinations of
moment and axial force should be calculated using reduced yield strength.
(1 β ππ)πππ¦π¦
ππ = οΏ½οΏ½2πππΈπΈππππππππ ,π π ππ
οΏ½ β 1οΏ½ 2
13 STABILITY CHECKS,
The following stability checks should be carried out
β’ Restraint of plastic hinge β’ Stability of the rafter β’ Stability of haunch β’ Stability of columns
13.1 Restraint of plastic hinge
At a plastic hinge location both flanges should location both flanges should be provided with
lateral and torsional restraint by means of diagonal stays. If is not practical to provide such
restraint at actual hinge location, it is permissible to provide it within a distance along the
member equal to h/2. (See section 11.3) (King, Technical Report P147)
At the location of plastic hinges, all of the material of the cross-section is at yield. It is
therefore unable to resist any twisting action and therefore the restraint against lateral torsion
buckling is reduced. It is thus necessary to impose a more stringent limitation on the distance
to the next restraint than would otherwise be the case.
For members with restraints to the compression flange but without intermediate restraints to
the tension flange, the maximum distance to the next compression flange restraint Lm may be
found using either a conservative method based on the uniform moment equal to the most
32
adverse moment in the segment or using a method which takes some account of moment
gradient.
Lm, the distance from a plastic hinge to the next lateral restraint may conservatively be
calculated using the expression (BS EN 1993-1-1:2005)
Lm =38iz
157.4
NEd
A
+
1756C1
2
W pl,y2
AIt
fy
235
2
In this equation
NEd is the design value of the compression force in the member
A is the cross section area (mm2) of the member
Wpl is the plastic section modulus of the member
fy is the yield strength of the material (N/mm2 )
It is the torsion constant of the member
C1 is a factor which depends on moment distribution and can safely be taken as 1.
If the member is restrained on the flange then the maximum distance to then the
maximum distance to the nearest restraint to both flanges should not be greater than Lk
(assuming constant uniform moment) or it may be taken as Ls calculated as for the
stability of the haunch but with taper factor of 1.0 ( see clause 8.8.3).
13.2 Rafter Stability (The institutionof Structural Engineers, TP/08/43
EC3/08/16)
The rafter should be checked to ensure that the stability is maintained in all load cases.
Unless there is wind uplift, the following checks should be made;
a) At the plastic hinge location near the ridge both flanges should be laterally restraint in
order to provide torsional restraint.
b) A purlin or other restraint is needed on compression flange at a distance determined in
the same way as mentioned above for plastic hinge (Spacing restraint < Lm).
33
c) In areas where there is a compression on the bottom flange the procedure given for
haunches in section 13.3 should be applied using constants applicable to haunch/depth
= 1
13.3 Stability of haunch
Provided that the tension flange of the haunch is restrained, then the maximum length
between restraints to the compression flange of the haunch should be limited to Ls
obtained as shown below, provide that;
a) The rafter is a UB section
b) The haunch flange is not smaller than the rafter flange.
c) The depth of the haunch (hh) is not greater than twice the depth of the rafter
allowing for its slope (hh)
d) Adjacent to plastic hinge locations the spacing of the tension restraint complies
with Lm.
Where Ls is given by;
1) For three flanged haunches (BS EN 1993-1-1:2005)
ππππ =οΏ½οΏ½πΆπΆππ πππποΏ½
ππ
2) For two flanged hunches (BS EN 1993-1-1:2005)
ππππ =οΏ½οΏ½πΆπΆππ πππποΏ½
ππ
Where
- C is the taper factor to take account of the haunch shape
- Cn is a non-linear moment gradient factor given as below;
- Lk is the length derived for a uniform member with a cross-section equal to the shallowest section
34
14.5
6004.5
2
β
+
=
f
y
zf
y
k
th
Ef
ith
Ef
L
13.4 Stability of the column
Near the top of the column a restraint should be provided at the location of the plastic hinge,
together with a further restraint at a distance below the position of the hinge restraint, based
on that given for the spacing of restraints at plastic hinges (Lm). If the column is restrained on
the tension flange then the distance to the nearest restraint on the compression flange may be
taken as, (The institutionof Structural Engineers, TP/08/43 EC3/08/16)
+=
EdRkyN
Rkyplkms aNM
MLCL
,,
,,
Where
Cm is the modification factor for linear moment gradient.
a is the distance between the centroid of the member with the plastic hinge
and the centroid of the restraint members.
Mpl,y,Rk is the characteristic plastic moment of resistance in the strong direction.
MN,y,Rk is the characteristic plastic moment of resistance in the strong direction
allowing for the effects of axial load.
35
14 FIRE BOUNDARY CONDITION
14.1 Outline
If the building is constructed close to the site boundary, the external wall of the building
will require a degree of fire resistance. It is generally accepted that the structural members
that support these wall will also require fire resistance to ensure that the walls remain stable
for a reasonable period during fire.
14.2 Building Regulation requirements in UK (Newman, Technical Report P313)
The requirement for external walls to have fire resistance has resulted in portal framed
building being treated as a special case. The argument put forward by building authorities was
that columns and rafters of portal frames are designed as a single continuous element thus;
this means that the whole elements of the portal frames requires same level if fire resistance.
As a result of providing this fire protection to rafter and columns as well, represented a
significant increase in the cost and therefore an alternative solutions where investigated to
keep portal frames building economical.
14.3 Alternative approaches to boundary fire (Newman, Technical Report P313)
Following a study (Constrado 1979) of portal frames behavior in actual fire it was
concluded that the most viable alternative to fire protecting rafters is to ensure that, even if the
roof collapsed, the stability of the external walls would be maintained. Guidance on how to
achieve this conclusion was initially published in 1979 by Constrado. The guide was later
published by SCI, as a second edition the behavior of steel portal frames in boundary
conditions (P087). That publication has been referenced in Approved documents B for a
number of years. The designing of boundary walls to resist the forces and moments as
recommended in SCI P087, is generally a more economical solution than fire protecting the
whole portal frame. This engineering solution is considered to meet the βreasonablenessβ test
of the Building Regulations.
36
14.4 Effects of fire on Portal frames
In the early stages of fire, the portal rafter starts to heat up and expands which results in a
small outward deflection of the eaves and small upward deflection of the apex. As fire
continues to burn, the rafter temperature continues to raise the deflection and moments due to
thermal expansion increase. The bending moments induced by thermal expansion are shown
in figure 20.
Figure 20- Bending moment diagram for uniform temperature rise (Newman, Technical
Report P313)
The temperature raising causes a reduction in the yield strength of the steel and although the
loading is constant, the reduction in moment capacity causes the formation of plastic hinge at
a high temperature. Plastic hinges starts to form at in the rafter where the term βfire hingeβ is
used to distinguish this type of plastic hinge from plastic hinges which can form at normal
temperatures. The moment resistance of fire hinge is slightly less corresponding value at
normal temperature, so the fire hinges tends to forms at the end of hunches and near to the
ridge see figure below.
Figure 21- Probable positions of fire hinges in portal frame (Newman, Technical Report
P313)
37
At this stage, the frame maintains its basic shape where the loading on the frame is itβs self-
weight and purlins but with only proportion of the weight of cladding. The rafter continues to
collapse and falls to the eaves level but remains reasonably straight between fire hinges.
Torsion stability may occur as the purlins lose their strength. The rafter is acting partially as
centenary, creating a tension load, which pulls on the top of the column. The columns are still
upright and shows sign of distress.
The rafter continues to collapse as it loses stiffness and the section may rotate so that it sags
with the web horizontal. The moment at the end of the hunch still withstand an applicable
value. As the rafter further losses strength; it continue to sag to below eaves level and begin to
pull inward the tops of columns.
Figure 22- Rafter collapse to eaves level (Newman, Technical Report P313)
The mathematical model of the rafter collapse mechanism at this stage is given in Appendix.
It is the forces and moments at this stage that are used to determine the overturning moment.
14.5 Mathematical Model (Newman, Technical Report P313)
A full mathematical model of the collapse mechanism for symmetrical pitched portal frames
has been developed and presented in Appendix. The use of the full model is tedious to hand
calculations. Thus a more simplified version was recommended in SCI P087 which applies to
symmetrical pitched rafter of single or multi-bay. Also the columns on the boundary should
be adequately restrained in longitudinal direction. The requirement is for fire protection of
columns and requirement to resist the calculated forces and moments on the column bases are
applicable only to columns that support protected area of boundary wall. Columns which do
not support protected areas will not require fire protection or moment resisting column bases.
38
So the foundations, columns and column beams base should be designed to resist the
following moment s and reactions.
Vertical Reaction
πππ π =12
ππππ Γ ππ Γ ππ + πππ·π·
Horizontal Reaction
π»π»π π = πΎπΎ οΏ½ππππππ πΊπΊ π΄π΄ βπΆπΆ ππππ
πΊπΊοΏ½ ππππππ ππππππ ππππππππ ππβππππ
ππππ
10ππ
Overturning moment:
ππππππ = πΎπΎ οΏ½ππππππ πΊπΊ ππ οΏ½π΄π΄ +π΅π΅πποΏ½ βππππ οΏ½
πΆπΆπππΊπΊβ 0.065οΏ½οΏ½ ππππππ ππππππ ππππππππ ππβππππ
ππππ
10
In which
β’ π΅π΅ = ππ2βπΊπΊ2
8πΊπΊ
Where
β’ Wf is the load at the time of collapse (KN/m2) given in table 7 in appendix.
β’ WD is the dead load of the wall cladding (KN).
β’ S is the distance between frame centers (m).
β’ G is the distance between ends of hunches (m).
β’ Y is the vertical height of the end of hunch (m).
β’ Mp is the plastic moment of resistance of the rafter (KNm).
β’ Mc is the plastic moment of resistance of column (KNm).
β’ K =1 for single bay frames or is taken from table 5
β’ L is the span (m).
β’ A and C are the frame geometry parameters, given in appendix table 6.
39
15 APPENDIX
A)
Calculation of non-linear moment factor
Cn = 12[R1 + 3R2 + 4R3 + 3R4 + R5 + 2(RsβRe)]
- In which the values of R are given by R = (My,Ed + a NEd ) fy Wy,pl
- Where a is the distance between the centroid of the member and the centroid of the
restraining members, such as purlin restraining rafter. - R1 to R5 are the values at the ends quarter points and centre numbered sequentially
along the member. Only positive values should be included.
- In addition, only positive values of (Rs-RE) should be included, where
β’ RE is the greater value of R1 and R5
β’ RS is the maximum numeric value of R anywhere along the length being
considered.
Figure 23- Moment Ratios (BS EN 1993-1-1:2005)
40
B)
For a non-uniform member with constant flanges, provided that h β₯ 1,2b and h /tf β₯ 20 then
for haunched members or segments,
Calculation of taper factor,
c =1+3
ht f
β 9
hmax
hmin
β1
2 / 3
Where
hh is the additional depth of the haunch or taper
hmax is the maximum depth of the cross-section within the length Ly
hmin is the minimum depth of the cross-section within the length Ly
hs is the vertical depth of the unhaunched section
Lh is the length of the haunch within the length Ly
Ly is the length between points at which the compression flange is laterally
Restrained
Figure 24- Dimensions defining taper factor (BS EN 1993-1-1:2005)
41
C)
πΆπΆππ =1
π΅π΅0 + π΅π΅1π½π½ππ + π΅π΅2π½π½ππ2
Calculation of (Cm) modification factor for linear moment gradient.
- In which
π΅π΅0 = 1+10ππ1+20ππ
π΅π΅1 = 5οΏ½ππππ+10οΏ½ππ
π΅π΅2 = 0.51+πποΏ½ππ
β 0.51+20ππ
- Where:
a) ππ = πππππππΈπΈππππππππ
β’ πππππππΈπΈ = ππ2πΈπΈπΌπΌπ§π§ππππ2
β’ ππππ is the distance between the torsional restraints
β’ ππππππππ is the elastic critical torsional buckling force for an I-section between restraint of both flanges at spacing ππππ with intermediate lateral restraint to the tension flange.
ππππππππ = οΏ½ 1ππππ
2οΏ½ οΏ½ππ2πΈπΈπΌπΌπ§π§ππ2
ππππ2 + ππ2πΈπΈπΌπΌππππππ2 + πΊπΊπΌπΌπποΏ½
ππππ2 = ππππ2 + ππππ2 + ππ2
β’ A is the distance between the centroid of the member and the centroid of the restraining members, such as purlin restraining rafter
b) π½π½ππ
It is the ratio of the algebraically smaller end moment to the larger end moment.
moments that produce compression in the non-restrained flange should be taken
ms positive. If the ratio is less than -1.0 the value of π½π½ππ should be taken as -1.0
see figure
42
Figure 25- Value of Bt (BS EN 1993-1-1:2005)
43
D)
D1) Derivation of overturning moments (Newman, Technical Report P313):
The model assumes a worst-case scenario in which the rafter is inverted and acts like a
centenary. However, a small allowance is made for the residual bending resistance of the
rafter. The geometry and the forces acting on the collapsing rafter are shown in figure 26.
Fire boundary
Figure 26 Forces and moments acting on the column and rafters (Newman, Technical
Report P313)
Considering vertical equilibrium, the vertical reaction on the base of column base is given as
follow:
VR=F1+F2
Considering rafter equilibrium, taking moments about apex:
ππππ1 + ππππ2 + π»π»π π 1 sinππ +πΉπΉ1π π 1 cos ππ
2= πΉπΉ1π π 1 cos ππ
Giving:
π»π» =πΉπΉ1π π 1 cos ππ β 2(ππππ1 + ππππ2)
2π π 1 sinππ
And
π»π»π π = π»π»
44
For the column equilibrium, the base overturning moment given by:
ππππππ = π»π»ππ + πΉπΉ1ππ1 + π»π»1ππ1 + ππππ1
Where:
R1 is the rafter length from end of hunch to apex including allowance for elongation R2 is the haunch length from center line of column. Y is the height of end of haunch
πΈπΈ cosπΌπΌ + π π 2 sin(ππππ β πΌπΌ) L is the span. E is the column height. ππππ Is the initial rafter angle. X1 is the horizontal distance from column base to end of haunch
πΈπΈ sinπΌπΌ + π π 2 cosππππ β πΌπΌ X2 is the horizontal distance from the column base to the end of haunch
πΈπΈ sinπΌπΌ + 12π π 2 cosππππ β πΌπΌ.
Ξ is the column deflection angle. Ξ is rafter sag angle
cosβ1 οΏ½ππβ2ππ12π π 1
οΏ½ For single bay frames
cosβ1 οΏ½ππβ2ππ1+πΈπΈ sin πΌπΌ2π π 1
οΏ½ For multi bay frames
F1 is the vertical load on the rafter length R1 F2 is the vertical load on the rafter length R2 VR is the vertical reaction on the column base. HR is the horizontal reaction on column base. H is the resultant horizontal load in rafter. MP1 is the fire hinge moment at end of haunch. MP2 is the fire hinge moment at apex
45
D2) Tables
Table 5- Multiplication factors for multi-bay frame (BS EN 1993-1-1:2005)
Table 6- Frame geometry parameters A and C (Newman, Technical Report P313)
46
Table 7- Percentage dead weight of roof cladding systems remaining at the time of the
rafter collapse (Newman, Technical Report P313)
47
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48
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