1. Lecture 2
STRUCTURAL ANALYSIS UNDER STATIC LOADING
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2. Imperfections of the steel elements and structures
The structural analysis of the steel frames takes into account the effects of imperfections,
local or global.
Local imperfections of the individual compressed member are:
 residual stresses;
 geometrical imperfections.
The lack of verticality, of straightness, of flatness, of fit and other eccentricities present in
joints of the unloaded structure are considered local imperfections. Some the
imperfections are taken care of by EN 1090 and limited at specific allowed tolerances.
In the process of analysis and design the local imperfections are considered by using
equivalent geometric imperfections unless their effects are already included in the resistance
formulae used for the design of the individual members.
The equivalent imperfections that should be taken into account are:
• a) global imperfections for frames and bracing systems (P- effect)
• b) local imperfections for individual members ( P- effect).
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Elastic instability of a framed structure: a) sway frame ; b) non sway frame
a b

3. Imperfections considered in the global analysis of frames
In the analysis of the frame the elastic buckling mode of a structure is considered for every
plane of buckling so both in and out of plane buckling including torsional buckling with
symmetric and asymmetric buckling shapes should be taken into account in the most un-
favourable direction and form.
Imperfections of the sway frames
The effect of the imperfections is inserted in the frame analysis by means of an equivalent
imperfection:
- initial sway imperfection,
- individual bow imperfections of members.
The imperfections may be determined from:
a) global initial sway imperfections:
mh0   0 – basic value of the imperfection, 0=1/200;
h – reduction factor depending on the height of the columns:
132
h
2
h
h




h – height of the structure (m) ;
m – reduction factor for the number of columns in a row:







m
1
15.0m
m – the number of columns in a row including only those columns
which carry a vertical load NEd not less than 50% of the average
value of the column in the vertical plane considered.
For structures with a dominant sway buckling mode, the effects of global and local imperfections
are considered as a deviation from verticality to which a bow is added. The initial sway
imperfections should apply in all relevant horizontal directions, but will be considered in one
direction at a time.
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4. Equivalent sway imperfections
Imperfections considered in the global analysis of frames
 For building frames sway imperfections
may be disregarded when:
EdEd V15.0H 
 For the determination of horizontal
forces applied to floor diaphragms
the configuration of imperfections
should be applied, where  is a
sway imperfection obtained from
assuming one storey with height h.
 Global imperfections are
represented by lateral equivalent
forces acting at each floor level,
much easy to be considered in the
analysis than to incline the
structure.
 The equivalent forces are
determined from the
multiplication of the
gravitational loads at every level
with the initial imperfection
angle . The equilibrium on the
height of the structure imposes a
reaction at the base of every
column.
Sway imperfections  applied to the horizontal forces
acting on floor diaphragms
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5. b) relative initial local bow imperfections e0/L of the members in flexural buckling with
a length L. The values e0 / L may be chosen in the National Annex.
Imperfections considered in the global analysis of frames
Buckling curve
Elastic analysis Plastic analysis
a0 1/350 1/300
a 1/300 1/250
b 1/250 1/200
c 1/200 1/150
d 1/150 1/100
Le0
Design values of initial local bow imperfection e0/L
Le0
Local bow imperfections may be neglected during the global analysis for determining
end forces and moments for members checking;
For frames sensitive to II order effects local bow imperfections of members additionally
to global sway imperfections should be introduced in the structural analysis of the
frame for each compressed member if the following conditions are met:
 at least one moment resistant joint at one member end;
 the reduces slenderness is increased:
Ed
y
N
fA
5.0


NEd – the design value of the compression force;
- in-plan reduced slenderness for the member considered as hinged at its ends.
.
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6. Local bow imperfections are taken into account in the members verifications
considering the buckling curves.
The effects of initial sway imperfection and local bow imperfections may be replaced
by systems of equivalent horizontal forces, introduced for each column.
Imperfections considered in the global analysis of frames
Replacement of initial imperfections by equivalent horizontal forces:
a)- sway imperfections; b)- initial bow imperfections
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7. Imperfections considered in the analysis of bracing systems
The structural bracing systems of framed structures are required to provide lateral
stability within the length of beams or of the members in compression (columns). The
effects of imperfections are included by means of an equivalent geometric imperfection
of the members to be restrained, in the form of an initial bow imperfection:
500
L
e m0   L – the span of the bracing system;







m
1
15.0m m – the number of members to be stabilized.
The effects of the initial bow imperfections of the members to be stabilized by the
bracing system may be replaced by the equivalent stabilizing force:


 2
q0
Edd
L
e
8Nq

δq - in plane deflection of the bracing system due to the load q to
which any external loads calculated from first order analysis is
added; if second order theory is used then δq may be considered 0.
e0 – imperfection;
qd – equivalent force per unit length.
Equivalent stabilizing force for the bracing system
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8. Imperfections considered in the analysis of bracing systems
Where the bracing system is required to stabilize the compression flange of a beam of
constant height, the force NEd may be obtained from:
h
M
N Ed
Ed 
MEd – the maximum moment in the beam;
h - the overall depth of the beam.
When a beam is in compression under NEd , this force should include a part of the
compression force resulted from the imperfections of the bracing system. At points where
beams or compression members are spliced, it should also be verified that the bracing
system is able to resist a local force equal to:
%NS EdmEd  
Force which is applied to it by each beam or compression member which is spliced at that
point, and to transmit this force to the adjacent points at which that beam or compression
member is restrained.
For checking to the local force, any external loads acting on bracing systems should also be
included, but the forces arising from the imperfection may be omitted.
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9. Imperfections considered in the analysis of bracing systems
Forces at splices in compression elements
The vertical bracing system may have continuity connections which are spliced. Global
imperfection is transferred in the most un-favourable way to the splices and must be
consequently taken into account when designing the connection.
The rotation  is determined identically as the previous value of the global imperfections:
100NN2
;2001
;
EdmEd
0
0m





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10. The effects of local bow imperfections of members are considered within the
relationships used for the determination of the buckling resistance for members.
A second order analysis may be developed considering the imperfection of one element as
a bow with the deflection in the middle.
In order to simplify the computation process, this imperfection (a variation in a parabola
shape along the element) may be introduced in the equation as a uniform distributed
loading and the reactions at both ends of the element:
Imperfections of individual members
2
d0
eq
L
e
8Nq 
L
e
4NR d0

Individual imperfections of the structural elements
For the verification of the lateral torsional buckling of a member in bending, the equivalent
initial bow imperfection of the weak axis of the profile, the eccentricity e0,d is considered by
adopting the value k∙e0,d, where for k the value recommended is 0.5.
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