Approximate analysis of the frame and wall type structures
On the basis of the method of §6.4.3, perform the approximate analysis of both the frame and the wall type structures of the two previous paragraphs.
Frame type structure (§6.4.1)
The building height is H=6 × 3.0=18.0 m.
The structure is classified as frame system in both directions, therefore C_{t,x}=C_{t,y}=C_{t}=0.075 The structure is also regular in both plan and elevation, thus q=3.60. (The accurate value is 3.90, but for the comparability of the results we use the same value as in §6.4.1).
T_{1,x}= T_{1,y}= Τ _{1} =C_{t} × H^{3/4}=0.075 × 18.0^{3/4}=0.7 sec
Since 0.50=T_{C}≤0.70=T_{1}<2.0=T_{D} (see §6.1.6),
S_{d}(T_{1})=max[ γ _{I} · a_{gR} · S · ( 2.5/q) · ( T_{c}/T_{1}), β · γ _{I} · a_{gR}]=
=max(1.0 × 0.15 × 1.2 × (2.5/3.6) × (0.50/0.70), 0.2 × 1.0 × 0.15) · g=0.089g.
Since T_{1}=0.70≤1.0=2T_{C} → λ=0.85, thus a_{CM}= λ · S_{d}(T_{1})=0.85 × 0.089g=0.076g
The mass of each floor is M=177.4 t except from the mass of the last floor which is M_{6}=166.9 t, therefore Σ (M_{i})=166.9+5 × 177.4=1053.9 t.
The elevation of the centre of mass is
Z_{CM}= Σ (M_{j} · Z _{j} )/ Σ (M_{j})= =(177.4 × 3.0+ 177.4 × 6.0+ 177.4 × 9.0+ 177.4 × 12.0+ 177.4 × 15.0+ 166.9 × 18.0)/1053.9 =10987.2/1053.9 → Z_{CM}=10.4 m
The triangular distribution gives the accelerations by means of the expression
a_{j}=(Z_{j}/Z_{CM}) · a_{CM} =(0.076g/10.4) · Z_{j} → a_{j}=0.0073g × Z_{j}
a_{1}=0.0073g × 3.0= 0.022g, a_{2}=0.044g, a_{3}=0.066g, a_{4}=0.088g, a_{5}=0.110g, a_{6}=0.131g
H_{1}=a_{1} · M_{1}= 0.023 · 10m/sec^{2} · 177.4 · 10^{3}kg=39 kN, H_{2}=78, H_{3}=117, H_{4}=156, H_{5}=195 και H_{6}=0.131 · 10m/sec^{2} · 166.9 · 10^{3}kg =219 kN .
According to the application of the modal response spectrum analysis carried-out in §6.4.2, the average fundamental natural period of the structure, in the various cases, is equal to T_{1x}=T_{1y}=1.0 sec taking into account a stiffness reduction factor equal to 0.50 due to cracking of structural elements, while T_{1x}=T_{1y}=0.65 sec taking into account full stiffnesses. It therefore follows that the value range of the fundamental natural period of the structure is quite wide depending on its type, e.g. cross-sectional dimensions, foundation type, etc., as well as on the assumption of the stiffnesses reduction due to cracking.
Wall type structure (§6.4.2)
The building height is H=6 × 3.0=18.0 m.
The structure is classified as wall system in both directions, thus C_{t,x}=C_{t,y}=C_{t}=0.050. The structure is also regular in both plan and elevation, thus q=3.60.
T_{1,x}= T_{1,y}= Τ _{1} =C_{t} · H^{3/4}=0.050 × 18.0^{3/4}=0.44 sec
Since 0.15=T_{B}≤0.44=T_{1}<0.50=T_{C} (see §6.1.6)
S_{d}(T_{1})= γ _{I} · a_{gR} · S · ( 2.5/q)=1.0 × 0.15 × 1.2 × 2.5/3.6= 0.125g.
Since T_{1}=0.44≤1.00=2T_{C} → λ=0.85, thus a_{CM}= λ × S_{d}(T_{1})=0.85 × 0.125g=0.10625g
The mass of each floor is M=186.3 t, except from the mass of the last floor which is M_{6}=165.0 t, therefore Σ (M_{i})=165.0+5 × 186.3=1096.5 t.
The elevation of the centre of mass is
Z_{CM} =Σ( M_{j} · Z _{j} )/Σ( M_{j} )= =(186.3 · 3.0+ 186.3 · 6.0+ 186.3 · 9.0+ 186.3 · 12.0+ 186.3 · 15.0+ 165.0 · 18.0)/1096.5=11353.5/1096.5 → Z_{CM}=10.4 m
The triangular distribution gives the accelerations by means of the expression
a_{j}=(Z_{j}/Z_{CM}) · a_{CM} =(0.10625g/10.4) · Z_{j} → a_{j}=0.0102g · Z_{j}
which yields a_{1}=0.031g, a_{2}=0.061g, a_{3}=0.092g, a_{4}=0.122g, a_{5}=0.153g, a_{6}=0.184g and seismic forces H_{1}=a_{1} · M_{1}=0.031 · 10m/sec^{2} · 186.3 · 10^{3}kg=58 kN, H_{2}=114, H_{3}=171, H_{4}=227, H_{5}=285 andH_{6}=0.184 · 10m/sec^{2} · 165 · 10^{3}kg =304 kN.
According to the application of the modal response spectrum analysis carried-out in §6.4.2, the average fundamental natural period of the structure, in the various cases, is equal to T_{1x}=T_{1y}=0.7 sec taking into account a stiffness reduction factor equal to 0.50 due to cracking of structural elements, while T_{1x}=T_{1y}=0.5 sec taking into account full stiffnesses. It therefore follows that the value range of the fundamental natural period of the structure is quite wide depending on its type, e.g. cross-sectional dimensions, foundation type, etc., as well as on the assumption of the stiffnesses reduction due to cracking.
a) The value of the approximate calculation of the fundamental natural period and the design seismic acceleration lies in checking the correctness of the corresponding quantities derived from the algorithms of the modal response spectrum analysis.
In the related software, the fundamental natural period is obtained from the modal response spectrum analysis and is shown as a red arrow in the seismic acceleration graph at the elevation of the centre of mass.
b) The value of the approximate method for the triangular distribution of seismic accelerations consists in the evaluation of the accelerations obtained by the algorithms of the modal response spectrum analysis, which should follow the triangular distribution to some extent.
In the related software, the triangular distribution is shown using a red dashed line.