Lecture 2

STRUCTURAL ANALYSIS UNDER STATIC LOADING

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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

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

115.0m

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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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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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

fA5.0

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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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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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

Le m0 L – the span of the bracing system;

m

115.0m 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

EddL

e8Nq

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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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

MN 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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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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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

d0eq

L

e8Nq

L

e4NR 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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