GLOBAL STABILITY RESISTANCE OF THE FRAME
7.2 Method of equivalent members (simple method) .1 Interaction design formulae
In the design practice mostly the method of reduction factor is used. The method of equivalent geometric imperfection is suggested using for senior engineers who have the experience. The method of partial equivalent geometric imperfection is normally used in preliminary design.
Within the method of reduction factor the following two methods can be used:
Method of equivalent members (simple method)
The design forces are computed on in-plane or spatial design model, where the instability responses are neglected. The stability resistance formulas (ex. interaction formulae) are applied on equivalent structural members isolated from the design model and supported appropriately. The details of this method are discussed in the Paragraph 7.2.
General method
The calculation of the design forces and the elastic stability analysis are performed on spatial static model which includes the effect of torsion (warping). Any software can be applied which uses general beam-column finite element method (14 DOF FE method).
The support system of the design model should be based on the real conditions of the structure (see Paragraph 5.1.2.5). The general design formulae examines the structure as a ‘super member’. The details of this method are discussed in the Paragraph 7.3.
7.2 Method of equivalent members (simple method) 7.2.1 Interaction design formulae
The stability resistance of uniform members with double symmetric cross sections should be checked by the interaction design formulae, where a distinction is made for:
– members that are not susceptible to torsional deformations, e.g. circular hollow sections or sections restraint from torsion;
– members that are susceptible to torsional deformations, e.g. members with open cross- sections and not restraint from torsion.
The interaction formulae is based on the modelling of simply supported single span members with end fork conditions and with or without continuous lateral restraints, which are subjected to compression forces and end moment in-plane:
(1) 1
NEd is the design compression force;
My,Ed is the maximum design moment about y-y axis (including the moment due to the shift of the centroidal axis of Class 4 cross-sections);
χy , χz , χLT are the reduction factors due to flexural buckling about y-y and z-z axes, and due to lateral torsional buckling;
kyy , kzy are the interaction factors;
A , Wy are the cross-sectional properties due to the class of the cross-section (plastic, elastic or effective);
fy is the characteristic design strength;
γM1 is the partial safety factor.
The kyy and kzy interaction factors have been derived from two alternative approaches. Method 1 was developed by the so called ‘French-Belgian’ group. The method provides a continuous design curve between the cross-sectional and the stability resistances but the expressions are basically complicated and no understandable. Method 2 was developed by the so called
‘German-Austrian’ group. The method provides simple and understandable expressions but the results are less sophisticated. Method 2 is used for simple ‘hand’ design, while Method 1 may be used by software tools. The expressions of Method 2 can be found in the Annex 15.
It is easy to realise that no one of the beams or columns of the frame satisfies the conditions of the interaction formulae. For example, if the column of the frame is taken as an equivalent member with simple fork supports at the ends, at least two basic conditions is not satisfied:
(i) the column is elastically supported at top;
(ii) the column is supported intermediately by wall beams (or bracing members).
The problem may be solved by the wider meaning of the interaction formulae. The formulae consists of three pure buckling modes, such as the flexural buckling about y-y axis, the flexural buckling about z-z axis, and the lateral torsional buckling. It is allowed to take different equivalent members for these pure buckling modes. The engineer is responsible for the use of the interaction formulae in wide meaning. Determination of the χy , χz and χLT frame. The slenderness can be calculated using buckling length factor or critical force:
- Slenderness due to buckling length factor
The buckling length factor of the columns of any simple frame may be found in the literature, see Figure 45. The duopitch roof structure may be modelled by the simple portal frame model shown in the figure.
Fig.45 Slenderness due to buckling length factor
χy
- Slenderness due to critical force
The slenderness of the column may be determined by global stability analysis of the appropriate model. The numerical procedure is illustrated in the Figure 46.
(a) design model (b) analysis (design forces)
(c) analysis (buckling) (d) slenderness
7.2.3 Out-of-plane buckling (χz)
The out-of-plane stability responses of the columns and the beams are similar. The members are supported laterally by purlins or wall beams, optionally by bracing members. Out-of plane flexural buckling mode may develop between two neighbouring lateral supports in form of between two lateral supports. This conservative design method is for the safe. The examples of Paragraphs 7.2.5.1 and 7.2.5.2 illustrate the design method.
7.2.4 Lateral torsional buckling (χLT)
Lateral torsional buckling mode may develop between two neighbouring lateral supports where the cross-sections are restrained in rotation around the axis of the member (fork support). Fork support can be taken into consideration in two cases:
- the lateral support is placed at the compressed flange, see Figure 47;
- knee bars support the compressed flange, see Figure 34.
NEd
Fig.46 Slenderness due to critical force
Fig.47 Equivalent member for the beam at the frame corner: the second lateral support can be considered as fork support since the support is placed at the compressed flange;
(a) structural model with lateral supports (b) equivalent member for the beam
7.2.5 How to determine the equivalent members for columns (buckling lengths)
Figure 48 shows the column structure which has pinned supports at the ends. The column is supported laterally at the middle cross-section and loaded by vertical concentrated force and concentrated moment at top. Let’s determine the equivalent members for the global stability check of the column with different cross-sections.
7.2.5.1 HEA300 section
The lateral support at the middle cross-section of the column may be considered as fork support due to the following reasons: (i) the torsional stiffness of the HEA300 profile is relatively high; (ii) the eccentricity of the lateral support is relatively low.
Fig.48 Equivalent members of the column made from HEA300 (example for fork lateral support)
(a)
(b)
Structural and load model Analysis Equivalent members Checking
However, the column is not sensitive to the flexural torsional and the lateral-torsional buckling. In this case the global stability of the column may be examined by two equivalent members: (1) part of the column between the top and the middle supports; (2) part of the column between the middle of the column and the column base. It is noted, that in this example the structural models of the two equivalent members coincide and the normal force is uniform, but the moments are higher on the upper part. Consequently it is enough to examine the upper equivalent member, see Figure 48..
7.2.5.2 IPE600 section
The lateral support at the middle cross-section of the column can not be considered as fork support due to the following reasons: (i) the torsional stiffness of the IPE600 profile is relatively low; (ii) the eccentricity of the lateral support is relatively high. Consequently the column is sensitive to the flexural torsional and the lateral-torsional buckling modes. In this case the global stability resistance of the column may be examined on the equivalent member as the whole member (the equivalent member is the member itself), see Figure 49. It is noted that a knee bar construction at the middle of the column may lead to previous example, see the Paragraph 7.5.2.1.
Fig.49 Equivalent member of the column made form IPE600 (example for no fork lateral support)
7.2.5.3 Frames
The columns and beams of the frame structure may have more equivalent members. Figure 50 shows a frame model where the lateral supports are fork supports due to knee bars construction. Let’s determine the equivalent members.
The parts of the structural members denoted by yellow colour are the equivalent members for the global stability check of the frame. The solution may be explained as follows:
- rotation of cross-sections at the lateral supports are restrained due to the knee bars constructions;
Structural and load model Analysis Equivalent member Checking
Fig.50 Equivalent members for global stability check of the frame supported by fork lateral supports
Fig.51 Reduction factors for the examination of the O1 equivalent member O1
- maximum design bending moment of the columns is found on the O1 equivalent member;
- maximum design bending moment of the beams is found on the G1 equivalent member;
- design bending moment diagram closed to constant is found on the G2 equivalent member.
It is noted that in case of fix column bases the bottom part of the column (O2) should be examined too. Figure 51 illustrates the main steps of the examination of the O1 equivalent member.
7.2.6 Members with changing cross-section
Structural members with changing cross-sections may not be checked correctly by the interaction formula and the equivalent members. Approximate check may be performed following the design idea described below. The cross-section of the member is changing due to the following structural reasons:
Short haunched members
Long haunched members
Tapered members
7.2.6.1 Short haunched members
The effect of the short haunch of beam to the critical forces can be neglected, see Figure 52b.
Assuming that the increasing of the stiffened part of the beam is relatively higher than the increasing of the design moment, the interaction design formula may be evaluated at the end of the haunch, see Figure 52a.
Fig.52 Global stability check of haunched beam
7.2.6.2 Long haunched and tapered members
The effect of the long haunch should be taken into consideration in the global stability member, the depth of the equivalent cross-section may be equal to the cross-section at the 1/3 haunch depth, see Figure 53.b.
The intermediate flange of the equivalent I cross-section may be neglected. The design interaction formula should be performed at the real cross-section which has the maximum utilization for cross-sectional resistance. The tapered members may be examined by the concept of mean depth of cross-section.
Adequate cross-section to check the global stability (b) (a)
Fig.53 Equivalent cross-section for global stability analysis of the haunched equivalent member: (a) mean depth concept; (b) 1/3 haunch depth concept
7.2.7 Application
The example below shows the global stability examination of the frame using the design interaction formulae and the equivalent member concept. This examination may be preformed by the ConSteel software too. The application guide can be found in the Annex 16.
4.5 Global stabilty resistance
The global stability resistance of the frame is examined with the design interaction formula as well as the general method. The examination based on the interaction design formula is presented for the columns only. It is noted that the examionation should be extended to the whole structure in the actual design project.
4.5.1 Examination of resistance of columns with design interaction formula 4.5.1.1 Basic assumptions
In the global stability analysis of columns the following assuptions are followed:
- the reduced slenderness for the in-plane buckling is determined due to the global stability analysis of the main frame;
- at the inermediate points of the columns offset lateral supports are applied;
- rotation of the column shape at the supports is restrained by the offset
supports, therefore the out-of-plane buckling and the LTB can be examined for the O1 upper part and the O2 top part.
critical forcce [N] N cr.y
π2.E.I c.y
cross-section at 1/3 depth haunch (b)
(a)
reduction factor α y 0.34
- Examination of the O1 column part (upper part of the column) equivalent structural length [mm] L z.1 3650
buckling length factor νz.1 1.0
buckling length [mm] L cr.z.1 νz.1 L z.1. L cr.z.1 3.65 10= . 3
- Examination of the O2 column part (buttom part of the column) equivalent structural length [mm] L z.2 3650
buckling length factor νz.2 0.7
buckling length [mm] L cr.z.2 νz.2 L z.2. L cr.z.2 2.55510= . 3
4.5.1.4 Lateral torsional buckling (LTB) Examination of the O1 column part
equivalent structual length [mm] L LT 3650
LTB length factor νLT 1.0
LTB length[mm] L cr.LT νLT L LT. L cr.LT 3.65 10= . 3
ctritical moment [Nmm]
relevant load combination: LC 4
design bending moments [kNm] M max.1 492 M min.1 72
A csökkentõ tényezõ számításánál feltételezzük, hogy az oszlopszelvény a "hengrelt szelvényekkel egyenértékû" kategóriába sorolható:
λ LT.0 0.4
β 0.75
reduced slenderness for LTB λ LT
W c.y.pl f y.
Examination of O2 column part
structural length [mm] L LT 3650
LTB length factor νLT 0.7
LTB length [mm] L cr.LT ν LT L LT. L cr.LT 2.55510= . 3
critical bending moment [Nmm]
relevant load combination: LC 4
design bending moment [kNm] M max.2 354 M min.2 72
design normal force [kN] N Ed 175
interaction factors (cross-section Class 1 and 2) k yy.1 C my 1 λ y 0.2
used capacity
The examination of the columns was preformed by the Member Designer Module of the ConSteel 6.0 design software too:
7.3 General method