MRd MSd1 NRd Nud e2 ea ee eo eo1 eo2 etot ey ez kA or kB
lcol lot a
(yd crit
Curvature at the critical section at the base of a model column Moment of inertia (gross section) of a beam Moment of inertia (gross-section) of a column
Reduction factor for the calculation of the second order eccentricity e2 (Equation 4.68)
Coefficient, taking account of decrease in curvature (1/r) due to increasing axial force (Equation 4.71)
Design resisting moment
First order applied moment
Resisting design axial compression force
Design ultimate capacity of the section subjected to axial load only Second order eccentricity
Additional eccentricity covering the effects of geometrical imperfections Equivalent eccentricity (Equations 4.65 and 4.66 and Figure 4.29) First order eccentricity
Values of the first order eccentricity of the axial load at the ends of the member, denoted so that |eo11 r |eo2 |
Total eccentricity
Eccentricity in the y direction
Eccentricity in the z direction
Coefficients describing the rigidity of restraint at the column ends
Height of column measured between idealised centres of restraint
Length of a compression flange measured between lateral supports (Equation 4.77)
Factor taking into account the conditions of restraint of the beam at the opposite end lo/1col
Design yield strain of the steel reinforcement Slenderness ratio Critical slenderness ratio Longitudinal force coefficient for an element
P(1) This section refers to slender structures or slender members mainly subjected to compression whose load carrying capacity is significantly influenced by their deformations (second order effects). P(2) The principles given in this section apply to linear reinforced and prestressed concrete members subjected to axial compression, with or without bending, for which the effects of torsion can be neglected.
P(3) These principles may also be applied to other types of structural member, such as walls, shells, slender beams in which lateral buckling of the compression zone may occur, deep beams or other exceptional structures or members in which significant local deformations may arise.
(4) Rules are given in 4.3.5.2 to 4.3.5.6 and Appendix 3 for slender compression members and for lateral buckling of slender beams in 4.3.5.7.
(5) In compression members, the influence of second order effects should be considered if the increase above the first order bending moments due to deflections exceeds | 10 % |. This may be assumed to be the case where the slenderness of the structure or structural members considered exceed the limits given in 4.3.5.3 below or in Appendix 3, clause A3.2.
P(1) Design for structural stability taking account of second order effects shall ensure that, for the most unfavourable combinations of actions at the ultimate limit state, loss of static equilbrium (locally or for the structure as a whole) does not occur or the resistance of individual cross-sections subjected to bending and longitudinal force are not exceeded.
P(2) The structural behaviour shall be considered in any direction in which failure due to second order effects may occur.
P(3) Possible uncertainties in the restraints at connections should be considered. Material properties shall be assumed to have their design values (see 2.3.3.2) and corresponding deformation properties shall be used throughout.
(4) For normal building structures, the design procedures set out in the following clauses consist of the following three stages.
a) The structure or members are classified
— as braced or unbraced and — as sway or non-sway (see 4.3.5.3.1 to 4.3.5.3.4)
b) For the classification of structure, the necessity to consider second order effects is established by comparing the slenderness with limits set out in the appropriate clauses below.
This applies:
— for the structure as a whole if it is a sway structure
— for the individual columns considered as isolated columns. (see 4.3.5.5.3)
c) Where it is established that second order effects should be considered, design rules are given for columns in 4.3.5.4, 4.3.5.5 and 4.3.5.6. Rules for slender beams are given in 4.3.5.7.
For more detailed information on the design procedures, see the flow charts in Appendix 3.
(5) More rigorous design approaches are set out in Appendix 3.
4.3.5.3 Classification of structures and structural elements
P(1) For the purpose of design calculations structures or structural members may be classified as braced or unbraced depending on the provision or not of bracing elements and as non-sway or sway depending on their sensitivity to second order effects due to lateral displacements. P(2) Similarly, isolated columns are classified as slender or non-slender.
(1) A bracing element is a structural element which has a high flexural and/or shear stiffness and which is completely or partially fixed (restrained) to the foundation. A bracing element or a system of bracing elements should be sufficiently stiff to attract and transmit to the foundations all horizontal loads acting on the structure and to ensure the stability of the braced sub-assembly.
(2) In general, the design of bracing elements may be based on first order analysis. However, a second order analysis may be necessary where the bracing elements are relatively flexible. [see 4.3.5.1(5)].
(3) Structures with bracing elements which satisfy the requirements in (1) above are classified as braced.
(1) Structures or structural elements, with or without bracing elements, for which the influence of displacements of the connections upon the design moments and forces may be neglected are classified as non-sway. Otherwise they are classified as sway.
(2) Braced building structures, where the bracing is provided by substantial shear walls or core structures, may be assumed to be non-sway. In other cases, structures may be classified by application of the provisions of Appendix 3, clause A3.2.
(3) Frames may be classified as non-sway if the first order displacements of the connections do not increase the effects of actions calculated without considering these displacements by more than | 10 % |. Generally it is sufficient to consider only the relevant bending moments. [See 2.5.1.4].
4.3.5.3.4 Isolated columns
— isolated compression members [see Figure 4.26 a) and Figure 4.26 b)
— compression members which are integral parts of a structure but which are considered to be isolated for design purposes (see e.g. 4.3.5.5.1). See Figure 4.26 c) and Figure 4.26 d).
a) Individual isolated column b) columns with articulations in a non-sway structure c) slender bracing element considered as isolated column d) columns with restrained ends in a non-sway structure a) Individual isolated column b) columns with articulations in a non-sway structure c) slender bracing element considered as isolated column d) columns with restrained ends in a non-sway structure
Figure 4.26 — Types of isolated columns
(1) For buildings, the effective height or length of a column lo = ".lcoi can be determined by means of Figure 4.27 below in which the coefficients k^ and kB denote the rigidity of restraint at the column ends:
Figure 4.27 — Nomograms for the calculation of the effective length
Figure 4.27 — Nomograms for the calculation of the effective length
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