Rating the seismic performance of hollow-core floors 4.1 Introduction
4.4 Example 1: Frame building designed to NZS4203: 1992 and NZS 3101:
A perimeter frame in a building with an importance classification of 2 has a fundament period of less than 2 seconds and it is found to have the design shear strength in the critical storey of 1,000kN. The inter-storey height is 3,550mm. The corresponding modal response spectrum storey shear force for an elastic response is 6,600kN. On the basis of a structural ductility factor, µ, is 6. the relevant critical values of storey shear and deformation found in the analyses for the inertial forces and P-delta forces are listed in bold in the second to bottom row in Table 4-1. The required inertial storey shear force is 6,600/µ, which equals 1,100kN. The critical drift limit associated with the particular hollow-core floor that is being assessed has been found from the criteria in Chapter 6 to be 1.5%.
For an importance classification of 2 the design earthquake return period is 500 years and the earthquake return factor, R, is equal to 1.0. To calculate the %NBSb for a limiting inter-storey drift it is necessary to adjust the analytical values obtained in the structural analysis so that the storey shear required by inertial and P-delta actions is equal to the provided design storey shear strength of 1,000kN.
The basic set of P-delta forces is multiplied by βK to assess the P-delta deflections. The value of β is constant for µ greater than 3.5 but it reduces with smaller µ values (NZS1170.5: 2004, clause 6.5.4.2). Provided the structural ductility factor is equal to or more than 3.5 changing its value does not change the magnitude of the storey drifts associated with the inertial forces and hence it does not change the P-delta storey shear force. The storey shear force available to resist the inertial storey shear is equal to the provided design storey strength, 1,000kN, minus the P-delta storey shear, 321kN, which gives a value of 679kN. As the elastic response for inertial actions was 6,600kN the revised structural ductility factor is 6,600 divided by 679, giving a value of 9.72. The storey drift due to P-delta actions increases in proportion to the structural ductility factor. Hence with a ductility of 9.72 the P-delta storey drift increases from 22mm (see Table 4.1) to22×9.726=36mm. The design inter-storey drift consisting of the sum of inertial and P-delta drifts multiplied by the drift modification factor to give a value of 109mm, which corresponds to a storey drift divided by storey height of 3.07%.
Table 4-1: Calculation the 100%NBSb for a limiting drift
Design
shear Factor ** Ductility Inertial response P-delta actions Total drift Total drift x DMF* Storey Drift (kN) Μ Shear (kN) (mm) Drift Shear (kN) (mm) Drift (mm) (mm) %
1,421 6 1,100 37 321 22 59 88.5 2.49
1,000 9.72 679 37 321 36 73 109 3.07
* DMF drift modification factor equal to 1.5 for µ > 3.5. ** Structural ductility factor
Hence in terms of NBSb the percentages are;
• for strength 70%
(
1.000 ,1421)
;• for ductility 62%
(
6 9.72)
;Calculation of ratio of design earthquake return factors %NBSa
As noted previously a change in the design level earthquake magnitude is achieved in NZS1170.5: 2004 by multiplying the design response spectrum by the Return Factor R (see clauses 3.1.1 and 3.1.5). Hence to find a limiting condition a series of analyses could be carried out by varying R until the critical condition is reached. However, by carrying out separate analyses for the inertia forces and P-delta actions further analyses are not required as each component can be modified to allow for the change in structural ductility and associated return factor.
The process to determine the %NBSa values conveniently starts from the 100%NBSb values, which are listed in Table 4.1 and shown in bold at the top of Table 4.2. In the table a range of return periods are shown so that trends can be seen. However, in practice the critical return period can be obtained using a simple spread sheet. In this case the critical value is an inter- storey drift divided by storey height of 1.5%.
The design response spectrum is proportional to the earthquake return factor. Hence a change in this factor leads to linear changes in the inertial shear force, the storey drifts and the P-delta storey shear force (as this is calculated from the deformed shape induced by the inertial actions). The storey shear available to resist the inertial forces is equal to the provided storey shear strength minus the P-delta storey shear force. Thus a change of R from 1 to 0.9 results in the elastic inertial response storey shear decreasing to 0.9 times the R=1 (assuming this is a category 2 building) elastic response of 6,600kN giving a value of 5,940kN. The P-delta storey shear decreases to 0.9 x 321, giving 289kN, which leaves the design storey strength of 1000kN minus 289kN giving 711kN to resist the inertial forces. The structural ductility factor is equal to the elastic inertial storey shear divided by 711, which gives a value of 8.35. The corresponding P- delta drift due to inertial forces is proportional to the R factor and the ratio structural ductility factors. Hence the P-delta drift of 22mm for R=1 reduces to 0.9 x 22 = 19.8 due to the change in R from 1 to 0.9 and the corresponding change due to different structural ductility factors is equal to 19.8×8.359.72= 17.0mm. When the structural ductility factor drops below 3.5 the β factor used to assess P-delta actions changes (see NZS1170.5, clause 6.5.4) and it is necessary to allow for this change by multiplying by the ratio of β values (shown as β/βi in the table). The resultant
storey drift is found by multiplying the sum of the inertial and P-delta storey drifts by the drift modification factor (see Section 3.3.3).
In terms of the ratio of earthquake return factors the %NBSa are;
• for strength of 1000kN (sum of inertial and P-delta shears) at maximum ductility R is 0.71, giving 71%NBSa;
• for structural ductility factor of 6, R equals 0.71 (same as for strength) giving 71%NBSa
• for drift limited by hollow-core floor of 1.5%, R equals 0.63 giving 63%NBSa..
Comparison of %NBSa and NBSb values found by options “a” and “b”
For the example above there is appreciable difference between the critical limit measures given by the %NBSa and %NBSb values. However, this is not always the case. The main cause of the difference arises from the non-linearity of drift with return period factor, which principally arises from the drift due to P-delta actions. This becomes significant where the provided design strength is low compared with the 100%NBS strength and for the cases where the lateral force resistance is provided by ductile moment resisting frames. With structures where the lateral resistance is provided by walls P-delta actions are smaller and the differences between the “a” and “b” options is smaller. Option NBSa gives a more consistent measure of potential seismic performance than NBSb.
Table 4-2: Calculation of earthquake return factor
R ductility Inertia inertia P-delta P-delta β/βi DMF Storey % drift/ Stable
shear drift shear drift drift storey ht
(kN) (kN) (kN) (mm) (mm) 1 9.72 679 37.0 321 35.6 1 1.50 109.0 3.07 OK 0.9 8.35 711 33.3 289 27.6 1 1.5 91.3 2.57 OK 0.8 7.10 743 29.6 257 20.8 1 1.5 75.7 2.13 OK 0.7 5.96 775 25.9 225 15.3 1 1.5 61.8 1.74 OK 0.6 4.90 807 22.2 193 10.8 1 1.5 49.5 1.39 OK 0.5 3.93 840 18.5 161 7.2 1 1.5 38.6 1.09 OK 0.4 3.03 872 14.8 128 4.4 0.87 1.5 28.9 0.81 OK 0.3 2.19 904 11.1 96 2.4 0.63 1.32 17.8 0.50 OK 0.2 1.41 936 7.4 64 1.0 0.40 1.14 9.6 0.27 OK 0.1 0.68 968 3.7 32 0.3 0.19 0.97 3.8 0.11 OK 0.53 4.22 830 19.6 170 8.2 1 1.5 27.8 1.17 OK
β/β1 is the ratio of β used in calculating P-delta actions OK indicates that µ<10
DMF is the drift modification factor Ductility = structural ductility factor