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 cross-section 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 fixed-ended one with 300 mm width is selected here. Alternatively, the rectangular fixed-ended column could be considered as equivalent.

In the comparison of the two systems, the distribution and not the magnitude of the seismic forces plays an important role. Consequently, the triangular distribution is used, approximating to the extent possible the actual distribution [*] NoteThe distribution of seismic forces derives from the modal response spectrum analysis method, described in chapter 6. of the majority of buildings designed by an engineer. In the related software the seismic forces are imposed on the considered building by selecting Apply Seismic Forces=ON in the dialog located in the central menu: "Horizontal Forces" à "Seismic Forces".

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 5.3.3-1 Figure 5.3.3-1

Figure 5.3.3-2 Figure 5.3.3-2

The results derived from the software analysis are listed in the following tables and figures.

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)=(6200·10³Ν)/(3 ·29.82·10⁶N/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)= =(6200·10³Ν)/[(2·29.82+1410.75)·10⁶N/m)=4 mm

only, compared to 75 mm of the actual storey.

Four-storey structures under the same seismic loading

Figure 5.3.3-3: FRAME type structure comprising three columns with cross-section 400/400 and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-3: FRAME type structure comprising three columns with cross-section 400/400 and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-4: DUAL type structure comprising one wall with a cross-section of 2000/300 and two columns a with a cross-section of 400/400 and Equivalent structure of one fixed-ended column per storey

Figure 5.3.3-4: DUAL type structure comprising one wall with a cross-section of 2000/300 and two columns a with a cross-section of 400/400 and Equivalent structure of one fixed-ended column per storey

Figure 5.3.3-5: FRAME type structure comprising three column with cross-section 400/400 and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-5: FRAME type structure comprising three column with cross-section 400/400 and equivalent structure of one fixed-ended 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).

Figure 5.3.3-6: DUAL type structure comprising two columns and one wall with cross-sections 400/400 and 2000/300 respectively and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-6: DUAL type structure comprising two columns and one wall with cross-sections 400/400 and 2000/300 respectively and equivalent structure of one fixed-ended 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 89.8-52.3=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).

Figure 5.3.3-7: FRAME type structure comprising three columns with cross-section 400/400 and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-7: FRAME type structure comprising three columns with cross-section 400/400 and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-8: DUAL type structure comprising one wall with a cross section of 2000/300 and two columns with a cross section of 400/400 with cross-sections 400/400 and 2000/300 respectively and equivalent structure of one fixed-ended column per storey

Figure 5.3.3-8: DUAL type structure comprising one wall with a cross section of 2000/300 and two columns with a cross section of 400/400 with cross-sections 400/400 and 2000/300 respectively and equivalent structure of one fixed-ended column per storey