Comparison of multistorey frame and dual systems
The behaviour of the most common structural systems will be now examined. The structural system comprising only columns is called a frame system, while the one comprising both columns and walls is called a dual system. The comparison of the two systems will be performed by means of examples. The structures under consideration are those used in the two previous paragraphs while the corresponding number of storeys is 4, 8 and 15. Two main quantities are compared, an actual one and an idealised one. The displacement of each level is the actual quantity and the equivalent column crosssection of each storey is the idealised one. The term equivalent column defines a theoretical column which develops a displacement equal to that developed by all storey columns. Infinite types of such idealised equivalent columns could be considered, however the fixedended one with 300 mm width is selected here. Alternatively, the rectangular fixedended column could be considered as equivalent. The three examples under consideration in both frame and dual versions are included in the related software. The modelling accounts for the elasticstiffnesses (Beam & Column Inertia Factor=1), the rigid bodies effect (Rigid Body=1) and the shear effect ( Shear Effect=1). Stiffness parameter values and values for seismic forces are listed below.
Figure 5.3.31 Figure 5.3.31

Figure 5.3.32 Figure 5.3.32
 The results derived from the software analysis are listed in the following tables and figures. Fourstorey structures (frame and dual type) under the same seismic loading    Frame Structural System (project <B_533a>)  Storey  Force H  Shear force V  Displacement δ_{total}  Displacement δ_{relative}  Σ(K) x 10^{6}  Equivalent rectangular column  Equivalent column with b=300   kN  kN  mm  mm  N/m  mm  mm  0  5  50  0.801  0.801  62  483  569  1  10  45  1.766  0.965  47  449  517  2  15  35  2.532  0.766  46  447  513  3  20  20  2.983  0.451  44  442  505     Dual Structural System (project <B_533b>)  Storey  Force H  Shear force V  Displacement δ_{total}  Displacement δ_{relative}  Σ(K) x 10^{6}  Equivalent rectangular column  Equivalent column with b=300   kN  kN  mm  mm  N/m  mm  mm  0  5  50  0.117  0.117  427  801  1169  1  10  45  0.341  0.224  201  655  872  2  15  35  0.580  0.239  146  603  775  3  20  20  0.792  0.212  94  537  660  Eightstorey structures (frame and dual type) under the same seismic loading    Frame Structural System (project <B_533c>)  Storey  Force H  Shear force V  Displacement δ_{total}  Displacement δ_{relative}  Σ(K) x 10^{6}  Equivalent rectangular column  Equivalent column with b=300           kN  kN  mm  mm  N/m  mm  mm  0  5  180  2.941  2.94  61  480  566  1  10  175  6.780  3.85  45  444  509  2  15  165  10.538  3.74  44  442  505  3  20  150  13.991  3.45  43  439  501  4  25  130  17.031  3.04  43  439  501  5  30  105  19.539  2.51  42  437  497  6  35  75  21.397  1.86  40  431  489  7  40  40  22.511  1.11  36  420  471     Dual Structural System (project <B_533d>)  Storey  Force H  Shear force V  Displacement δ_{total}  Displacement δ_{relative}  Σ(K) x 10^{6}  Equivalent rectangular column  Equivalent column with b=300           kN  kN  mm  mm  N/m  mm  mm  0  5  180  0.488  0.49  367  769  1100  1  10  175  1.511  1.02  172  629  823  2  15  165  2.760  1.25  132  587  747  3  20  150  4.053  1.29  116  567  712  4  25  130  5.280  1.23  106  554  689  5  30  105  6.374  1.09  96  540  665  6  35  75  7.311  0.93  81  517  626  7  40  40  8.079  0.78  51  459  532  Fifteenstorey structures (frame and dual type) under the same seismic loading    Frame Structural System (project <B_533e>)  Storey  Force H  Shear force V  Displacement δ_{total}  Displacement δ_{relative}  Σ(K) x 10^{6}  Equivalent rectangular column  Equivalent column with b=300   kN  kN  mm  mm  N/m  mm  mm  0  5  600  9.939  9.9  60  478  563  1  10  595  23.384  13.4  44  442  505  2  15  585  37.166  13.8  42  437  497  3  20  570  50.950  13.8  41  434  493  4  25  550  64.571  13.6  40  431  489  5  30  525  77.880  13.3  39  428  484  6  35  495  90.728  12.8  39  428  484  7  40  460  102.971  12.2  38  426  480  8  45  420  114.467  11.5  37  423  476  9  50  375  125.075  10.6  35  417  467  10  55  325  134.658  9.6  34  414  462  11  60  270  143.085  8.4  32  407  452  12  65  210  150.226  7.1  29  397  437  13  70  145  155.958  5.7  25  382  416  14  75  75  160.198  4.2  18  352  371   600  6200          Dual Structural System (project <B_533f>)  Storey  Force H  Shear force V  Displacement δ_{total}  Displacement δ_{relative}  Σ(K) x 10^{6}  Equivalent rectangular column  Equivalent column with b=300   kN  kN  mm  mm  N/m  mm  mm  0  5  600  1.797  1.8  333  749  1059  1  10  595  5.737  3.9  153  610  788  2  15  585  10.851  5.1  115  566  710  3  20  570  16.605  5.8  98  543  670  4  25  550  22.687  6.1  90  531  650  5  30  525  28.899  6.2  85  523  637  6  35  495  35.101  6.2  80  515  623  7  40  460  41.186  6.1  75  507  609  8  45  420  47.064  5.9  71  500  598  9  50  375  52.659  5.6  67  492  585  10  55  325  57.904  5.2  63  485  573  11  60  270  62.750  4.8  56  470  550  12  65  210  67.165  4.4  48  452  521  13  70  145  71.160  4.0  36  420  471  14  75  75  74.816  3.7  20  361  385   600  6200      
The storey displacements in case of beams having infinite moment of inertia [*] NoteIn practice very high moment of inertia only approaches infinite value. High moment of inertia is met in cases of beams with large height e.g. 800 mm or even larger. However such cases oppose the capacity design rule, described in volume A’ §1.4.2. , using the values of §5.1.1 are: Frame type structure: δ_{tot}= Σ(V)_{storeys}/ Σ(K)_{storey}=6200kN/(3·K_{400/400})=(6200·10^{3}Ν)/(3 ·29.82·10^{6}N/m)=69 mm compared to 160 mm of the actual storey. Dual type structure: δ_{tot}= Σ(V)_{storeys}/ Σ(K)_{storey}=6200kN/(2·K_{400/400}+1·K _{2000/300})= =(6200·10^{3}Ν)/[(2·29.82+1410.75)·10^{6}N/m)=4 mm only, compared to 75 mm of the actual storey. Fourstorey structures under the same seismic loading
Figure 5.3.33: FRAME type structure comprising three columns with crosssection 400/400 and equivalent structure of one fixedended column per storey Figure 5.3.33: FRAME type structure comprising three columns with crosssection 400/400 and equivalent structure of one fixedended column per storey
Figure 5.3.34: DUAL type structure comprising one wall with a crosssection of 2000/300 and two columns a with a crosssection of 400/400 and Equivalent structure of one fixedended column per storey Figure 5.3.34: DUAL type structure comprising one wall with a crosssection of 2000/300 and two columns a with a crosssection of 400/400 and Equivalent structure of one fixedended column per storey
The interstorey stiffness variation of the frame type structure is small (equivalent crosssection 570/300 at level 1 and 505/300 at level 4), compared to that corresponding to dual type structure, which is significant (1170/300 and 660/300 respectively). Eightstorey frame type structure under triangular seismic loading
Figure 5.3.35: FRAME type structure comprising three column with crosssection 400/400 and equivalent structure of one fixedended column per storey Figure 5.3.35: FRAME type structure comprising three column with crosssection 400/400 and equivalent structure of one fixedended column per storey
 In all types of structures, frame or dual, the sum of column shear forces of a storey is equal to the sum of the seismic forces of all the above storeys. Indicatively, for the first storey the sum is 54.3+71.4+54.3=180, while for the last 10.7+18.6+10.7=40. The middle column of the first storey carries the 71.4/180=40% of the total shear force, while each of the end columns carries 30%. In the last storey the middle column carries the 18.6/40=46%, while each of the end columns carries 27%.
 In both frame and dual systems, for each column M_{o}M_{u}=V·h, where M_{o} is the moment at the top, M_{u} is the moment at the base, V is the shear force and h is the height of the column. For instance, for the middle column of the previous structure 98.7(115.6)=71.4·3.0 (214.3≈214.2).
Eightstorey dual type structure under triangular seismic loading
Figure 5.3.36: DUAL type structure comprising two columns and one wall with crosssections 400/400 and 2000/300 respectively and equivalent structure of one fixedended column per storey Figure 5.3.36: DUAL type structure comprising two columns and one wall with crosssections 400/400 and 2000/300 respectively and equivalent structure of one fixedended column per storey
 In the first storey, the sum is 10.6+158.9+10.6=180. The wall carries 158.9/180=88% of the total shear force, while each column carries 11%. In the last storey, the sum is13.8+12.5+13.8=40. The wall carries 12.5/40=32% of the total shear force, while each column 34%. It is concluded that the wall has a favourable effect on the first storey columns, in contrast to that corresponding to the last storey.
 The expression M_{o}M_{u}=V·h, applies for both columns and wall. Indicatively, for the first storey wall 309.9(786.4)=158.9·3.0 (476.5≈476.7), while for that of the last storey 89.852.3=12.5·3.0 (37.5=37.5).
 The maximum displacement of the dual type structure is equal to 8.08 mm, i.e. almost three times smaller than the one corresponding to the frame type structure (22.51 mm).
Fifteenstorey frame type structure under triangular seismic loading
Figure 5.3.37: FRAME type structure comprising three columns with crosssection 400/400 and equivalent structure of one fixedended column per storey Figure 5.3.37: FRAME type structure comprising three columns with crosssection 400/400 and equivalent structure of one fixedended column per storey
It should be emphasised that in all previous examples, the comparison of the two structural systems is important rather than the absolute quantities, which after all derive from specific values of the seismic forces. These values have been selected arbitrarily, yet satisfying the triangular distribution rule. Fifteenstorey dual type structure under triangular seismic loading
Figure 5.3.38: DUAL type structure comprising one wall with a cross section of 2000/300 and two columns with a cross section of 400/400 with crosssections 400/400 and 2000/300 respectively and equivalent structure of one fixedended column per storey Figure 5.3.38: DUAL type structure comprising one wall with a cross section of 2000/300 and two columns with a cross section of 400/400 with crosssections 400/400 and 2000/300 respectively and equivalent structure of one fixedended column per storey
The maximum displacement of the frame type structure is equal to 160 mm, i.e. almost twice of that corresponding to the dual type structure (75 mm). Comparative table of Frame and Dual structural systems         Need for wide seismic joint between buildings in frame systems.  Earthquake resistance of brick walls    EC8 prohibits in general using brittle walls e.g. brick walls for High Ductility Class (DCH) (see§6.1.5.4).  Fluctuation of interstorey relative displacements and stiffnesses    In dual systems, walls develop a favourable behaviour in lower while frames develop a favourable behaviour in upper storeys. To this end, the coexistence of both frames and walls in a structure is necessary (dual system).     Walls should be founded on foundation beams of high stiffness. The more efficient foundation for walls way is on the basement walls ensuring fixity at their base.     The shear effect should be taken into account in the design of dual systems in contrast to the design of frame systems where it can be omitted.     The rigid bodies effect should be taken into account in the design of dual systems while in the design of frame systems it can be omitted.  The behaviour of dual systems is more favourable than frame systems under seismic actions.
