5.2 Design values of material coefficient (BS EN :2005)

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1 5.2 Design values of material coefficient (BS EN :2005) Modulus of elasticity (E) = N/mm2. Shear modulus (G) = E/2(1+v) = N/mm2. Poisson s ratio (v) = 0.3. Coefficient of linear thermal expansion (α) = 12 x10-6 per Co. Density (ρ) = kg/m3. 6 ULTIMATE LIMIT STATE 6.1 Outline According to EC3 all actions that could occur at the same time are considered together, so frame imperfection equivalent forces (which are discussed in section 9) as well as wind loads should be considered as additives to permanent actions and other imposed loads with an appropriate combination factor applied to them. Although several combinations exist, only the least favourable combination of loads should be considered. Most general example is wind load can act on both sides of portal frame but only combination need to be considered is the least favourable direction with respect to each load combination. 6.2 Design main concept A structure must have adequate resistance to internal forces and bending moments, known as structural resistance. In addition it must exist in a state of static equilibrium under the action of the applied loads, i.e. it must have adequate resistance to overturning, uplift ECT. Due to fact that static equilibrium is rarely critical for portal framed buildings, so not considered in details. The following are cases where static equilibrium might be critical: 1. In high wind speed areas. 2. In high but narrow buildings, result in overturning. 3. Where there is large opening allowing generation of high positive wind pressure inside the building at the same time as high suction over the roof. 8

2 6.3 Partial Safety factors EC3 uses partial safety factor format for checking ULS, in which the partial safety factors obtain a considerable safety margin against uncertainties of load on structure and resistance of the material. So that it reduces the resistance of the material and increase the applied load on the structure. cchaaaaaaaaaaaaaaaaaaaaaaaa rrrrrrrrrrrrrrrrrrrr > eeeeeeeeeeee oooo (cchaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaa) γγ FF γγ MM Partial Safety factor of resistance γγ MMMM 1.1 For the resistance of the cross-section γγ MM1 1.1 For the resistance buckling of the member γγ MMMM 1.25 For the resistance of the net sections γγ MMMM 1.25 For the resistance of the bolts γγ MMMM 1.25 For the resistance of welds Table 2- Partial Safety Factors of Steel Resistance (BS EN :2005) Partial Safety factor for loads Partial factors for actions are either Permanent action dead load γγ GG Variable action live load γγ QQ Those will have different values depending on wither the action is favourable or unfavourable for the structure stability. Table 3-Partial safety factors for actions in building structure (Baddoo, 1993 ) 9

3 6.4 Load combination EC3 normally consider all possible coincident actions but with the less unfavourable variable action reduced by combination reduction factor (ψ). The combination factor reduces the intensity of loading, when several variable loads are considered to be acting together. There is only low probability of all variable action occurring at full intensity at the same time, so the combination factor is used. All ULS load combinations must include the effect if any of imperfection forces (horizontal forces) from any direction particularly in frame analysis. The Single expression for ULS load combination is as follow (Baddoo, 1993 ) Characteristic value of the permanent action (GG kk jj ) X γγ GG + Full characteristic value of the most unfavorable variable action (QQ kk 1 ) X γγ QQ 1 + Reduced values of all other unfavorable variable action (QQ kk,ii ψ 0,ii ) X γγ QQ,ii Note that there are different ψ factors for different types of variable actions. The table below from EN1990:2002 shows different factors. Table 4- Recommended Values of ψ for buildings (Baddoo, 1993 ) 10

4 7 ULTIMATE LIMIT STATE ANALYSIS (FIRST ORDER VS SECOND ORDER) Normally first-order analysis is acceptable unless when the frame or frame member is exceedingly slender. The limitations are given below; where structures exceed these limitations second order analysis must be used. 7.1 P.δ effects P.δ effects are the effects on member behaviour due to displacements at right-angles to the axial force in the member. These displacements may be enforced (by external load or moments), most notably in the case of axial loading along a member with initial bow, or maybe the result of the natural tendency to buckle under pure axial load. The displacements are the sum if the initial deformation of the member and deflection due loading. The result of such deflections is to reduce the effective stiffness when axial load is compressive, due to increased bending moment see below figure 4. Conversely, when axial load is tensile, it increases the effective stiffness, though the effect will generally be minimal in common single storey portal frames. Figure 4- P.δ effects (King, Technical Report P164) 11

5 7.2 P. Δ effects P. Δ effects are the effects on overall frame behaviour due to the displacement at right-angles to the applied forces, directly due to those same forces. The displacements are the sum of the initial deformation of the frame and deflection due loading. For pitched roof portals, the prime example is the effect of gravity loads on sway deflections. The effect is to magnify the sway deflection due to overall inclination of the column which produces a reduction in effective stiffness, see the figure 5, Figure 5- Magnification of sway displacements (King, Technical Report P164) Another possible mode of failure which is sensitive to P. Δ effects is arching failure or snap-through of pair of rafters (see fig below). This has to be checked because it is possible to design portals of 3 spans or more with stiff outer bays that provide horizontal support to the rafters of the inner spans. Then the rafters of the inner spans can act as arches with the horizontal reaction provided by the outer bays. Where this arching action works, the rafters will support more vertical load than if they were acting only as beams. This check is used to ensure that the rafters are not so flexible that they snap through. Figure 6- Snap-through failure (King, Technical Report P147) 12

6 7.3 How EC3 account for Second Order effects Checking if second order affects significance First order analysis may be used for the structure where the increase of the relevant internal forces or moments or any other change of structural behaviour caused by deformations can be neglected. This condition may be assumed to be fulfilled, if the following criterion is satisfied (BS EN :2005): α cr 10 for elastic analysis α cr 15for plastic analysis - Where: α cr Is the factor by which the design loading would have to be increased to cause elastic instability in a global mode Determination of α cr For frames with pitched rafters, α cr,est = min (α cr,r,est, α cr,s,est ) - Where α cr,s,est is the estimate of α cr for sway buckling mode (see Section 3.3.1) α cr,r,est is the estimate of α cr for rafter snap-through buckling mode (see Section 3.3.2) 13

7 Sway buckling load factor (Oppe, SN033a-EN-EU) The parameters required to calculate α cr,s,est, for portal frames are shown in Figure 7. As can be seen, δ HEF is the lateral deflection at the top of each column when subjected to an arbitrary lateral load H EHF. (The magnitude of the total lateral load is arbitrary, as it is simply used to calculate the sway stiffness H EHF /δ EHF.) The horizontal load applied at the top of each column should be proportional to the vertical reaction. Thus, for an individual column: - Where HH EEEEEE,ii VV UUUUUU,ii = HH EEEEEE VV UUUUUU H EHF V ULS H EHFi V ULS,i is the sum of all the equivalent horizontal forces at column tops (see figure 7) is the sum of all factored vertical reactions at ULS calculated from first-order plastic analysis is equivalent horizontal force at top of column i (there are two columns in a single-span portal, three in a two-span portal, etc.) is factored vertical reaction at ULS at column i, calculated from firstorder plastic analysis Figure 7- Parameters required to estimate alpha critical for sway mode failure (J.B.P.Lim, November 2005) 14

8 An estimate of α cr can then be obtained from Where: α cr,s,est = NN RR,UUUUUU NN RR,cccc mmmmmm h ii VV UUUUUU,ii HH EEEEEE,ii δδ EEEEEE,ii mmmmmm NN RR,UUUUUU NN RR,cccc mmmmmm NN RR,UUUUUU Is the maximum ratio in any rafter. Is the axial force in rafter (see figure 7 (b)). NN RR,cccc = ππ 2 EE II rr LL 2 Is the Euler load of the rafter on full span (assumed pinned) I r Is the in-plane second moment of area of rafter Is the horizontal deflection of column top (see figure 7 (c)) δ EHF,i h ii VV UUUUUU,ii HH EEEEEE,ii δδ EEEEEE,ii mmmmmm Is the minimum value for columns 1 to n (n = the number of columns) Rafter Snap-through buckling load factor (Oppe, SN033a-EN-EU) For frames with rafter slopes not steeper than 1:2 (26 ), α cr,r may be taken as: D cross-section depth of rafter L span of the bay h means height of the column II cc in-plane second moment of area of column II rr in-plane second moment of area of rafter ff yyyy nominal yield strength of the rafter θθ rr roof slope if roof is symmetrical α cr,r,est = DD LL LL h II cc+ii rr 275 tan 2θθ ΩΩ 1 II rr ff rr yyyy ΩΩ Fr/Fo the ratio of the arching effect of the frame where Fr= factored vertical load on the rafter F0 = maximum uniformly distributed load for plastic failure of the rafter treated as a fixed end beam of span L FF oo = 16 WW pppp,yy,rr ff yy LL But where Ω 1, αcr,r = 15

9 7.3.3 Accounting second order effects (J.B.P.Lim, November 2005) Eurocode account to second order effects in the absence of elastic-plastic second order analysis software by deriving loads that are amplified to account for the effects of deformed geometry (second order effects). Application of these amplified loads through first order analysis gives bending moment, axial forces and shear forces that include the second-order effects approximately. The magnification factor is calculated by method called Merchant- Rankin method, which is as following: Category A: Regular, symmetric and asymmetric pitched and mono-pitched frames Regular, symmetrical and mono pitched frames are either single-span or multi span frames in which there is only a small variation in height (h) or span (L) between different spans; variation in height and span of the order of 10% may be considered sufficiently small αα cccc Provided that αα cccc 3 Where αα cccc can be calculated as shown above in section Figure 8- Examples of Category A (Oppe, SN033a-EN-EU) 16

10 Category B: Frames that fall outside of Category A and excluding tied portals Frames that are outside of category A and excluding tied portals αα cccc Procedure of Plastic analysis The method, as currently given in EN , implies the following procedure: (i) (ii) (iii) Choose initial portal sections. Calculate α cr factor by which the design loading would have to be increased to cause elastic instability in a global mode Comply with Merchant Rankin by multiplying the partial safety factors by the following magnification factor; αα cccc (iv) (v) (vi) Perform the rigid plastic analysis of portal frame using the Partial safety factors in step (iii). Check that collapse load factor of step (iv) analysis is greater one (CCCCCCCCCCCCCCCC llllllll ffffffffffff 1.0) where, CCCCCCCCCCCCCCCC llllllll CCCCCCCCCCCCCCCC llllllll ffffffffffff = UUUUUU llllllll If the Collapse load factor in step (v) is less than one then increase the section size and re-check if the increased section size can withstand second-order effects. 17