Lecture 1

STRUCTURAL ANALYSIS UNDER STATIC LOADING

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THE ANALYSIS OF THE STEEL STRUCTURES

• 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.

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• 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.

THE ANALYSIS OF THE STEEL STRUCTURES

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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

deformations.

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.

THE ANALYSIS OF THE STEEL STRUCTURES

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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 directions

a

b

CLASSIFICATION OF STEEL FRAMES 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 sway frame;

- 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.

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BRACED AND UNBRACED FRAMES

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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 h

NV

Case c) Little shear resistance: M2N;0V

2

NM

EQUILIBRIUM EQUATIONS

• 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.

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BRACED FRAMES

Braced frame: 1-bracing preventing the sidesway instability; 2- drift

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BRACED FRAMES

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

V V

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 frames.

• 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 value.

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UNBRACED FRAMES

Un-braced frame: -drift; -sidesway buckling

ELASTIC ANALYSIS

• 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 section.

• The fundamentals:

Structural element analysis: Global (P-), and local (P-) second order effects

Member imperfections (local) or initial bow

Global deformations

hHhM;xHxM

VhHLM

h

xVVxHxM

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Local imperfections are neglected In the I order

analysis

• 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.

ELASTIC ANALYSIS

Model of the first order elastic internal forces and displacements: 1- linear variation of the displacements under external loading

EI3

hH

EI3

hM

hHM

32

Elastic deformation of linear elements : du<<dx and it is neglected; when deformations increase then they are no longer neglected (II order)

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ELASTIC FIRST ORDER ANALYSIS 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 material.

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:

10F

F

Ed

cr

cr

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.

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• 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.

ELASTIC I ORDER ANALYSIS

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ELASTIC ANALYSIS OF II ORDER

• 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 elements.

• 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)

hh

VHVhHM

EI3

hVhH

EI3

LM

22

cr

o2

3

V

V1

1

EI3

Vh1

1

EI3

Hh

2cr

h

EI3V

0 – the displacement in elastic first order analysis; V/Vcr = cr.

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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:

ELASTIC ANALYSIS OF II ORDER

10F

F

Ed

cr

cr

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 elements. 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 buckling. 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.

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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 values.

Still, because the individual elements are considered rigid, the relative rotations must be small.

ELASTIC ANALYSIS OF II ORDER

Elastic second order analysis, (linear elastic and non linear geometric): a) stress – strain; b) force-

displacement

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 structure; - 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 computation.

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a b

ELASTIC ANALYSIS OF II ORDER

Tangent and secant stiffness in the II order analysis:

1

11sec

EI

P)K(

1 Secant stiffness

2 Tangent stiffness )EI(d

dP)K(

1

11tan

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.

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STABILITY ANALYSIS

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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 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. 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 members:

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 effects;

it shows which combination of loads is “critical” for the structure considering the II order effects.

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.

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STABILITY ANALYSIS

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:

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STABILITY ANALYSIS

Ed,HEd

Edcr

h

V

H

where: 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 question; 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:

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STABILITY ANALYSIS

22212C

2C2

12111C

1C1

KKKK

KK

KKKK

KK

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.

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22 Equivalent 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

c

2

crL

EIN

Ed

crcr

N

Nthen:

METHOD OF THE AMPLIFIED MOMENTS

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 imperfections VEd∙.

Simulation of the II order elastic analysis by the method of equivalent forces: 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

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On the stabilized structure we may write: V

hH

H

h

VH

h

HR

1

n

n

n

1

METHOD OF THE AMPLIFIED MOMENTS

hRhH

VV1

cr

Then if: 1

cr

crV

hH

V

V

The global value of n may be determined with the value of cr:

cr

1n 11

1

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:

cr

11

1

The following conditions are imposed:

0,3cr 1)

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).

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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.

METHOD OF THE AMPLIFIED MOMENTS

Lateral equivalent forces transmitted to a multi-storey frame

jj NV;TH

where: Tj – shear force at the base of the column “j”; Nj – axial force in the column “j”; - inter-storey drift due to horizontal forces.

H

V

hV

V

cr

V

hHicr

icrcr V

Vmax

V

V

icrcr min

Similarly:

If: then:

Equivalent forces and reactions on the structure after the evaluation of the global rotation

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Method and application conditions of the global elastic analysis (EN 1993-1-1)

Verifications of the structural elements in elastic analysis

ELASTIC ANALYSIS OF I AND II ORDER

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PLASTIC GLOBAL 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 dynamic.

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PLASTIC GLOBAL ANALYSIS

Global plastic analysis compliances and necessary verifications in the design according to EN 1993-1-1.

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• 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:

PLASTIC GLOBAL ANALYSIS

15F

F

Ed

crcr

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.

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PLASTIC ANALYSIS OF I ORDER (RIGID-PLASTIC OR IDEAL)

• Consequences:

• - static equilibrium equations are expressed with respect to the initial shape of the structure;

• - 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 whole structure;

• - 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 plastic deformations;

• - 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.

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• 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.

PLASTIC ANALYSIS OF I ORDER (RIGID-PLASTIC OR IDEAL)

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SECOND ORDER PLASTIC ANALYSIS

• 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 structure;

- 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 perfect plastic.

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• 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 developed);

• - 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 and bending.

SECOND ORDER PLASTIC ANALYSIS

Characteristic curve (-) for structural steel in elastic-plastic analysis

a) elastic-perfect plastic; b) elasto-plastic

C.T

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re 1

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