• The steel structures behaviour is the consequence of the characteristics of the steel:
• The homogeneity of the material allows an easy global evaluation of the resistance
based on the uniformity of the cross sections,
reduced specific weight (weight-to-cross section ratio) of the steel elements
compared to the reinforced concrete,
increased slenderness of the elements (length-to-radii of gyration ratio) compared
to reinforced concrete,
Increased resistance of the cross sections compared to the reinforced concrete,
very small tolerances in the connections.
Steel structures are light, slender and sensitive to all forms of instability
But, in the same time the structural steel shows a dual behaviour: elastic up to the
limit of proportionality (0.2%) but also ductile (plastic). Important reserves of
resistance on the cross section may be used due to the development of plastic
deformations long before its failure.
Plastic hinges are easy to be estimated and controlled on the whole structure because
of the homogeneity of the steel sections.
Steel structures may be designed either in elastic or in plastic domain
due to their elastic-plastic reserves.
• The evaluation of the behaviour of a structure under static and further under
dynamic loads is important because it provides us with information needed for the
identification of the options we have in order to optimize and in the same time to
secure this structure.
• Generally called as “the global analysis” it is conducted based on several methods
that vary with respect to different factors of influence: structural ability to deform,
geometrical characteristics, material properties, all of these having personal
contribution to the structural capacity of resisting to loads while deformations
manifest at the level of element and of the whole structure.
Current structures are modeled based on planar bi-axial frames which are made of
linear elements that respond to external loading on the whole building by
developing stresses up to a proportionality limit, in parallel developing also
The deformations are infinite small at first, but the alteration of the axial internal
forces increases these deformations up to a limit.
The magnitude of deformations divides the steel frames into characteristic
categories: sway or non-sway frames; rigid or flexible; braced or un-braced.
Rigid frames- pinned and fixed (fully restraint) Braced frame
Multi-storey structures with rigid connections
Multi-storey braced frames: a- vertical
bracing placed on the whole height; b-
vertical bracing placed in both
The classification of the steel frames in braced and un-braced is not the same with the II
order sensitivity of structures which divides them into sway frames and non sway frames. A
non-sway frame is sufficiently rigid under actions in its plane if we may neglect the
supplementary moments and internal forces due to the horizontal translations of the
connections. If this condition is not fulfilled the frame becomes a sway frame and II order
sway effects must be considered in the design (P - effects).
Generally, we refer to the un-braced frames as sway frames; as for the braced frames,
although they may have simple (hinged) connections the presence of braces stiffens them
and they become non-sway frames.
There are exceptions from this generally concept:
- bracing system adopted in order to avoid torsion in the horizontal plan, is not able to
prevent the horizontal translations due to the flexibility of the structure, which remains a
- the structure is very stiff, non sensitive to II order effects, being classified as non sway
frame; for this structure a I order analysis is realistic.
In the case of the braced frames the horizontal effects of the horizontal inclination are
transferred to the bracing system and in a simplified concept both the horizontal inclination
and the horizontal forces affect only the braced frame. This frame is designed as to resist to
the following loads:
• - horizontal loads applied to the frame;
• - horizontal or vertical loads directly applied to the bracing system;
• - effect of the initial inclination (or the effect of the system of equivalent horizontal
forces) afferent to the frame and to the bracing system.
Secondary moments induced in frames due to P- effects: a)- effects of the lateral deflection; negligible moment
restraint from columns and girders; c) negligible shear restraint from diagonals and other bracing ( - drift)
a b c
Case a) Moment and shear restraints (braced frame + rigid structural joints) M2hVN
Case b) Little moment resistance: hVN;0M
Case c) Little shear resistance:
• Braced frame - frames that resist to lateral drift (from wind actions and/or other)
by developing shear internal forces in the bracing system)
• The sidesway buckling is prevented by bracing elements other than the structural
frame itself; the theoretical elastic stability analysis assumes no relative joint
displacements (considering infinitely stiff bracing).
• In the design of braced frames it is assumed that there is a negligible moment
resistance and the sidesway mode of instability is prevented.
Braced frame: 1-bracing preventing the sidesway instability; 2- drift
Conditions imposed for the bracing system in order to insure a braced frame:
a) un-braced frame; b) braced frame; a, s displacements of the braced frame and of the un-braced frame, Ra, Rs
– rigidity of the braced frame and of the un-braced frame
R = V/ Ra = 5/ Rs
If by using braces the system becomes rigid enough, the displacements may diminish to at
least 80% from those of an un braced frame. Any other version of situations corresponds to
• Un- braced frame - frames without braces, which under constant horizontal force
and vertical increasing compressive forces loading up to critical values, will fail by
sidesway (lateral) buckling.
• In such cases lateral deflection will become suddenly greater than the drift. If
horizontal forces are missing (or they are negligible) there will be no initial
deflection but a sudden sidesway will occur for vertical loads reaching the critical
Un-braced frame: -drift; -sidesway buckling
• Elastic analysis is used on the largest scale for the structural design and may be applied
for any kind of structure because no supplementary conditions must be imposed in
order to insure the ductility of the elements or of the connections.
• The structures are conservative systems because they are not in the situation to
dissipate the energy.
• The scope of the verifications of the elements to the internal forces and moments
resulted is to keep the stresses under the level of plastic capacity in the cross
• The fundamentals:
Structural element analysis:
Global (P-), and local (P-) second order effects
Member imperfections (local)
or initial bow
Global deformations hHhM;xHxM
In the I order
• Elastic analysis of I order
• This analysis is based on a linear response of the elements and connections under
external loads. The equilibrium equations are written on the un-deformed
structure and the following simplifying hypotheses are admitted:
- the material is continuous, homogeneous and isotropic;
- the stress-strain relationship is linear variable;
- also the force-displacement relationship is linear variable;
- the strain-displacement relationship is linear variable.
Model of the first order elastic internal forces and displacements:
1- linear variation of the displacements under external loading
Elastic deformation of linear elements :
du<<dx and it is neglected; when deformations
increase then they are no longer neglected (II order)
The displacements are small with respect to the dimensions of the elements and of the
structure, hence the following consequences:
- equilibrium equations are written with respect to the initial positions of the elements of
the unloaded structure;
- the principle of superposition of the effects (internal forces, displacement, etc.) is
applied being called also the principle of independence of actions;
- the structure represents a conservative system;
- the internal forces and the displacements are linear variable with respect to the
variation of the external loads;
- both rigidity and flexibility of the structure do not depend on the level of external
forces, they depend only on the structural characteristics and the nature of the
According to EN 1993-1-1 first order analysis may be used if the increase of the
relevant internal forces or moments or any other change of structural behaviour
caused by deformations may be neglected.
This condition may be assumed to be fulfilled, if the following criterion is satisfied:
Fcr – elastic critical buckling load for global instability mode based on initial elastic stiffness;
FEd – design loading on the structure;
αcr – factor by which the design loading would have to be increased to cause elastic
instability in a global mode.
• The following verifications are necessary after the determination of the internal
forces and moments:
- evaluation of II order effects;
- strength verification of the cross section;
- strength verification of the joints;
- verification of the global stability of the elements;
- verification of the local stability of the elements;
- verification of the conditions imposed by the S.L.S.
• The first order analysis has the following advantages:
- it is simple, well known and easy to perform;
- may be simplified in order to reduce the computation effort.
• There are still, several disadvantages :
- it does not include the geometric effects of the II order stage of computation; the I
order analysis must be assisted by supplementary determination of the critical
buckling force Fcr.
Approximated values of the critical buckling force Fcr may be determined with
simplified methods (i.e., the method of amplified moments).
- the flexible structures being sensitive to II order effects will not be economically
designed because the internal redistribution of the forces on the cross section is
neglected; in reality, this effect will increase the stiffness (based on plastic properties);
- the I order analysis is not recommended for structures for which the values of critical
forces Fcr are close to the values of external loading, FEd.
• Steel structures are made of elements with greater slenderness than other types of
structures so extensive specific verifications of stability must be run.
• The second order analysis has important advantages:
- it may include from the first stage of computation the effects of the global P- and
element deformations P-;
- it fits a flexible structure that may be more economically designed;
- preliminary conditions are not imposed regarding the slenderness of the structural
• The disadvantages of the II order analysis are:
- the analysis is run with increased computation efforts and cannot be done by hand;
- the calculations are sufficiently intricate as to be used on a smaller scale.
• From the fundamentals:
Non-linear relationship force-displacements
in II order analysis: V,H – design external loading
(including the effects of the global imperfections)
0 – the displacement in elastic first order analysis;
V/Vcr = cr.
If the deformations are such as to have a significant influence upon the values of the stresses
or the behaviour of the structure as a whole, the II-nd order effects must be considered, this
being verified with the relationship:
The global analysis relies on three methods that take into account the II order effects and the
presence of the imperfections:
I The geometric and material imperfections together with the II order effects on the
whole structure are considered integrally. There is no need for stability verifications of the
II Only the global imperfections and the II order effects on the structure (P- effect) are
taken into account, the local imperfections being considered for the stability verifications (local
bow imperfections of the slender members or of the internal members of the trusses) for which
the II order local effects are also considered (P-).
This is the basic current procedure; since the II order (P- effect) and local effects of the
imperfections are already included in the verifications relationships according to EN 1993-1-1,
only the global II order effects are considered explicitly in the analysis (P- effects) with the
global imperfections. In this method the buckling lengths are the length of the elements
(conservative values) as the real buckling lengths are smaller and fixed ends may be adopted for
III The imperfections are limited only for the individual verifications of equivalent
elements by using the buckling lengths corresponding to the global instability of the
structure. This method is adopted for the simple situations for which the ideal buckling
lengths may be used. The verifications relationships take into account all the imperfections.
In the II order analysis the computation is linear elastic and non-linear geometric:
between the stress and the strain (-) is a linear proportion in every fiber of the cross
section, while between the forces acting on the structure and the displacements in the
relevant direction the relationship is not linear, the displacements having cumulative
Still, because the individual elements are considered rigid, the relative rotations
must be small.
Elastic second order analysis, (linear elastic and
non linear geometric): a) stress – strain; b) force-
Under the assumption of small displacements with respect to the lengths of the elements:
- the static equilibrium equations are written with respect to the final position of the deformed
- the principle of superposition of the effects of actions on structure is applied only for the
transversal (lateral horizontal) forces while the axial loads remain constant;
- internal forces and displacements are non-linear functions with respect to axial external forces;
- the stiffness of the elements and of the structure depends on the magnitude of external loads.
A secant and a tangent stiffness are defined based on which, the relationships between force
and displacement are written, also between variation of force and variation of displacement;
- as the final deformed shape is not initially known, the solution is obtained from cyclic
Tangent and secant stiffness in the II order analysis:
1 Secant stiffness
2 Tangent stiffness
Equilibrium equations for the II order elastic analysis are expressed on the deformed
structure by considering the P- effect in case of increased axial loads. Similar with the I
order elastic analysis both the cross sections of the elements and the connections must not
verify the conditions imposed by a ductile behaviour (class of resistance of the cross
section, class of ductility of the connection).
The curve force-displacement is influenced by the geometric non-linearity and in the
vicinity of instability manifestations it becomes asymptotic to the horizontal critical line
that represents cr, corresponding to the critical elastic buckling force, Pcr.
Critical elastic buckling force, Pcr is a reference value because it represents the maximum
theoretic value of the load that the structure is capable of sustaining in the case when the
steel has not reached the yield limit.
Domain of behaviour in the elastic II order analysis:
I order elastic analysis; 2- II order linear elastic analysis;
3- limit of linear behaviour; 4- instability level
The sensitivity of a steel structure to II order effects is put in evidence by the elastic critical
Critical elastic buckling force, Pcr is a reference value because it represents the maximum
theoretic value of the load that the structure is capable of sustaining in the case when the
steel has not reached the yield limit.
The internal forces in the II order analysis contain the effects of the non linear behaviour
(stresses and deformations). The analysis is run in the same way as the I order analysis and
increasing the loads step by step will put in evidence eventually the sections or the
connections most stressed in such way as to identify the limits of the elastic stability.
The design of the structure should insure in plan stability of the frames. As most
frequently the local imperfections are not considered, the verification of the stability of
the elements and of the frame may lead to smaller values of the magnifying factor prop.
The value of this factor should not be under 1.
Importance of the analysis of the critical buckling force, Pcr of the structural compressed
once determined Pcr, the factor cr is determined and the II order effects are identified;
by using the value of Pcr we may replace a II order analysis which is more laborious;
the value of Pcr may be used in the approximating methods of evaluation of II order
it shows which combination of loads is “critical” for the structure considering the II
Steel structures have many modes of buckling instability and the superior modes may be
relevant in the evaluation of the II order effects; it is then recommended the calculation
of a sufficiently big number of modes until it reaches to a buckling mode with sway joints
and one with non sway joints.
Elastic instability of a framed structure:
a) sway frame ; b) non sway frame
• RELEVANT CASES
I. Portal frames with shallow roof slopes and beam-and-column type plane frames in
buildings may be checked for sway mode failure with first order analysis if the following
criterion is satisfied for each storey. In these structures αcr may be calculated using the
following approximated formula, provided that the axial compression in the beams or rafters
is not significant:
HEd – the design value of the horizontal reaction at the top of the storey to the horizontal loads
and fictitious horizontal loads transferred to the structure from above the current level in
VEd – total design vertical load on the structure at the bottom of the current level acting at the
bottom part of the storey height;
H,Ed – horizontal displacement at the top of the storey, relative to the bottom of the storey
when the frame is loaded with horizontal loads (e.g. wind) and fictitious horizontal loads which
are applied at each floor level;
h – the storey height.
II. Calculation of the critical load that acts upon an equivalent beam-to-column system
with the help of the distribution coefficients 1 and 2, determined from Wood’s curves
with the following relationships:
Kc – stiffness coefficient of the column represented by the ratio between the moment of
inertia about the relevant axis and the length of the column I/L;
K1, K2 – stiffness coefficients of the adjacent columns (from top and bottom of the
current column ends), also represented by the stiffness-to-length ratio, I/L;
Kij – the effective stiffness coefficients of the adjacent beams in the joints at the top and
bottom respectively of the ends of the column.
The curves are graphically represented as functions of the type of the connections of
the framed structure: with non sway joints or with sway joints.
22Equivalent frame and Wood’s curves for the determination
the critical length coefficient lf /L (slenderness) in the case of
non sway frames
Equivalent frame and Wood’s curves for the determination
the critical length coefficient lf /L (slenderness) in the case
of sway frames
Critical load for the current column is determined with the equation: 2
This is a current method for the approximation of the II order effects and consists in
the I order analysis of a non sway frame and afterwards the amplification of the
internal forces and displacements with a factor applied to the horizontal external
actions HEd (wind, for instance), and to the equivalent vertical actions affected by the
Simulation of the II order elastic analysis by the method of
a) global rotation and equivalent force on the element;
b) equivalent force with the II order effect on the element
(local bow of the element is neglected)
“Step by step method” of the equivalent lateral forces: convergence reached for step “n” when n n+1 The structure is stabilized
On the stabilized structure we may write:
The global value of n may be determined with the value of cr:
From equations above we determine directly the amplification coefficient for the lateral
forces in order to take into account of the II order effects after running a I order analysis
on single storey frames:
The following conditions are imposed:
2) for αcr < 3,0 a more accurate, second order analysis is applied;
3) the slope of the rafters is small (under 12°) and the compression in the rafter is not
relevant (under 15% from the plastic axial resistance of the cross section of the rafter).
For multi storey frames second order sway effects are carried out with the same
method, provided that all storeys have a similar:
– distribution of vertical loads;
– distribution of horizontal loads;
– distribution of frame stiffness with respect to the applied storey shear forces.
Lateral equivalent forces transmitted to a multi-storey frame
Tj – shear force at the base of the column “j”;
Nj – axial force in the column “j”;
- inter-storey drift due to horizontal forces.
Equivalent forces and reactions on the structure after
the evaluation of the global rotation
Method and application conditions of the global elastic analysis (EN 1993-1-1)
Verifications of the structural elements in elastic analysis
• Steel structures are able to sustain plastic deformations up to a point when they fail by braking
in the cross section of the elements. They are capable of developing plastic design capacities so
the plastic analysis may be approached.
• Structures with increased redundancy are able to redistribute the internal forces and
deformations and create a new state of equilibrium corresponding to a substantial increasing of
the resistance to external loading. For the steel frames the limits of stresses and strains from
the elastic domain may be exceeded without risks because the ultimate resistance is signaled
by the formation of the first plastic hinge.
• Plastic design may be used only when the minimum conditions referring to the structural
redundancy, ductility of the material, elements and connections are fulfilled:
• a) The material respects the following conditions:
• - the ratio between the minimum tensile strength fu and the minimum yield limit, fy
satisfies the relationship: fu/fy 1.2 ;
• - strains must respect the same ductility conditions: the value of the strain u corresponding
to the ultimate limit tensile stress fu is 20% y, strain corresponding to yield stress fy.
• - the limit elongation corresponding to ultimate tensile strength of a sample with a standard
length 5,65 A0 is at least 15% from this length (A0 = initial cross section of the sample);
• b) In the critical regions where plastic hinges may develop the out of plan stability of the
element must be insured, being prevented from displacement on this direction;
• c) Sections where plastic hinges occur must be of class 1 of resistance in order to have
sufficient rotation capacity for developing the plastic moments;
• d) Sections where the plastic hinges are intended to develop shall have a symmetric cross
section regarding the loading plane;
• e) in the course of plastic analysis loads are applied static or quasi-static, definitely not
Global plastic analysis compliances and necessary verifications in the design according to EN 1993-1-1.
• Plastic analysis of I order (rigid-plastic or ideal)
As we already know the conditions imposed for a I order analysis according to EN
1993-1- 1 are that there is no need for the calculation of internal forces and moments
on the deformed structure. The first order plastic analysis is considered appropriate
if the following criterion is satisfied:
Idealized curve - for rigid plastic analysis
Supplementary conditions to those imposed for the elastic analysis:
the stresses are not allowed to exceed yield limit, fy;
internal bending moments must not exceed the limit value Mp corresponding to the
moment when the whole section of the element is in plastic domain;
The lost of stability will not occur until the structure changes into a mechanism with
one degree of freedom, keeping the displacements small.
• - static equilibrium equations are expressed with respect to the initial shape of
• - in the cross sections where the bending moment reaches the limit value, Mp, a
plastic hinge will develop, the rest of the element behaving elastic;
• - the stiffness of the elements vary as a result of the modifications of end restraints
due to the plastic hinges entraining the alteration (diminish) of the stiffness of the
• - the analysis is performed in cyclic stages the scope being to determine the
response of the structure corresponding to a certain level of external forces or to
determine the strength capacity;
• - the rigid plastic analysis does not consider the elastic deformations in elements
or in the connections because they are much too small in comparison with the
• - strain hardening phenomenon is neglected and plastic deformations are
concentrated in cross sections and connections where the plastic hinges may
develop and in these particular critical regions, the capacity of rotation of the
cross sections is considered infinite.
• The analysis consists in the identification of the plastic mechanism that governs
the collapse of the structure which will be produced if the following conditions
are simultaneously met:
• - formation of a kinematic mechanism, due to the existence of a sufficient number
of plastic hinges or real articulations in the structure;
• - equilibrium between the bending moments distributed on the structure in the
stage of plastic hinges and the external loads including the reactions;
• - the plastic capacity of the cross sections and of the connections is not exceeded.
• Elastic perfect plastic analysis
It is assumed that all the relationships between stresses and strains, moment and rotation
and finally, axial loading and the structural displacements are all non-linear and the
structural instability is possible at any time of loading. Forces may vary with respect to a
single parameter or they may have their own different laws of variation. Plastic moment is
corrected as the level of axial force varies also the displacements of the structure may be
small and big also.
Loads are applied incrementally on the modified frame, for which the equilibrium and
deformation conditions are altered due to the increase step by step of the external loading
and the successive formation of the plastic hinges according to the hierarchy or
resistance of the structural components. As the process continues, the structure will
progressively alter until a full plastic mechanism is formed.
These assumptions impose important consequences:
- static equilibrium conditions are expressed with respect to the deformed shape of the
- for this specific analysis the principle of super-positions is not valid;
- the stiffness of the elements and of the structure are functions of the level of the external
forces and the magnitude of displacements;
- the computation may put into light the response of the structure for a certain level of
external forces, or the magnitude of the external forces for which the structure looses either
the stability or the resistance.
In this analysis we assume the sections and the connections remain in elastic until the
plastic bending moment resistance is reached, after which the behaviour becomes
• Elastic-plastic analysis
This analysis further on consider non linear the relationship between: stress –
strain, bending moment-rotation, axial load- lateral displacements. The progressive
alteration is described by the theory of plasticized zones applied to the section
The models for the ultimate limit state may be the following:
• - rigid: collapse is produced by successive formation of plastic hinges until the
plastic mechanism is reached ( in a similar way that the I order plastic analysis is
• - flexibile: it is possible that after a certain number of plastic hinges are formed
the structure looses its stiffness and fails due to the diminish of the stiffness of a
member in compression or to the loose of stability of a member in compression
Characteristic curve (-) for structural steel in
a) elastic-perfect plastic; b) elasto-plastic