Seismic stresses

The seismic accelerations a_XX, a_XY, a_XZ of each diaphragm are obtained under the X horizontal component of the seismic action using appropriate methods (e.g. the CQC method applied by the related software) from the combination of the modal analysis results. Their application point of the seismic accelerations is the diaphragm’s centre of mass. For nodes not belonging to a diaphragm, such as the nodes of the corner column of the building illustrated in the figure, the application point of the seismic accelerations is the node position.

Structure comprising only columns, project <B_641-1>

Four cases are considered:

1. Fixed condition at the ground level
2. Foundation with footing beams
3. Foundation with simple footings
4. Basement with perimeter walls

The stiffnesses of elements are taken being equal to 50% of elastic. 12 mode shapes are taken into account.

1)    In all cases only the x horizontal component of the seismic action is considered, the y component produces almost identical results.

2)    The natural period of the first mode shape is of the order of 1 sec, while, if the stiffnesses of the elements are taken as being 100% of the elastic, the value of periods is of order of 0.65 sec (see §6.3.3).

3)    Apparently the more economical solution of simple footings without strap (connecting) beams eventually becomes the most unfavourable one, even for economic reasons due to the need for additional reinforcement and increased dimensions of beams and columns cross-sections.

4)    The best solution is the basement with perimeter walls, i.e. along the 75% of the perimeter, as providing behaviour that better approximates the fixed condition.

Figure 6.4.1-2: 1st mode in X: T=0.975 sec, participation 84%	Figure 6.4.1-3: 2nd mode in X: T=0.321 sec, participation 10%
Figure 6.4.1-4: 3rd mode in X: T=0.189 sec, participation 3.5%	Figure 6.4.1-5: 4th mode in X: T=0.134 sec, participation 1.6%

The sum of the effective modal masses of the first four modes amounts to 99% of the total mass. The first mode is the fundamental one as its effective mass is the 84% of the total mass.

Figure 6.4.1-11: Structure and model Frame system with q=3.60	Figure 6.4.1-12: Seismic acceleration-forces-shear forces 1st fundamental period:T1=1.012 sec, participation 85%
	Bending moments of ground floor columns
Figure 6.4.1-13: Displacements under seismic action in x δmax=25.7 mm	Figure 6.4.1-14: Ground floor column 0c2 (400/400)	Figure 6.4.1-15: Ground floor column 0c6 (500/500)

Due to stronger cross-sections of footing beams compared to those of columns, the displacement of the structure is slightly larger than that of assumed fixity at the base (25.7 against 24.5).

Figure 6.4.1-16: Structure and model Torsionally flexible system with q=2.00	Figure 6.4.1-17: Seismic acceleration-forces-shear forces 1st fundamental period:T1=1.075 sec, participation 88%
	Bending moments of ground floor columns
Figure 6.4.1-18: Displacements under seismic action in x δmax=47.6 mm	Figure 6.4.1-19: Ground floor column 0c2 (400/400)	Figure 6.4.1-20: Ground floor column 0c6 (500/500)

In the absence of strap beams, the structure is classified as torsionally flexible system with q=2.00.

Figure 6.4.1-21: Structure and model Frame system with q=3.60	Figure 6.4.1-22: Seismic acceleration-forces-shear forces 1st fundamental period:T1=0.999 sec, participation 71%
	Bending moments of ground floor columns
Figure 6.4.1-23: Displacements under seismic action in x δmax=25.3 mm	Figure 6.4.1-24: Ground floor column 0c2 (400/400)	Figure 6.4.1-25: Ground floor column 0c6 (500/500)

The overall behaviour of the structure is better than the one corresponding to the strong foundation at the ground floor, approaching that for assumed fixity at the base.

Structure comprising only columns and perimeter walls, project <B_642-1>

 The structure derives from the frame type structure by replacing columns c2, c5, c8 and c11 of cross-section 400/400 with four perimeter walls of cross-section 2000/300.

It is extremely useful to compare between the variants of the wall system, but also between the frame and wall systems.

1)    In all cases, the natural period of the first mode shape is of the order of 0.70 sec. If however the stiffnesses of the elements are taken as being 100% of the elastic, the value of periods is of order of 0.50 sec.

2)     The wall system behaviour is clearly better than the behaviour of the frame system, particularly when in the presence of a basement with perimeter walls.

In the absence of symmetry in y direction, the fundamental mode shape in x (the 2^nd) has also a component in y, meaning that both translational and torsional responses are developed.

The sum of the effective modal masses of the 12 first modes amounts to 99.5% and 98.1% of the total mass of the structure in x and in y directions respectively.

The displacement of the wall type structure is significantly smaller than the one of the frame type structure of the previous paragraph (16.3 mm against 24.5 mm).

The cross-sections of the footing beams are not strong enough to carry the high bending moments of the perimeter walls, resulting in the system along y to shift to a wall-equivalent dual system one with overall q=3.00. Moreover, the displacement of this system, along x, is equal to 21.3 mm compared to 25.7 mm of the corresponding frame system. Compared to case 1, the bending moments of the walls are smaller, while of columns are higher.

In both directions the structure functions as a wall-equivalent dual system, due to attenuation of wall behaviour caused by the ground’s low ability to oppose rotation.

The situation would be considerably improved using strong strap beams.

Step 1:

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=Τ₁=C_t·H^3/4=0.050·18.0^3/4=0.44 sec

Step 2:

Since 0.15=T_B≤0.44=T₁<0.50=T_C (see §6.1.6)
S_d(T₁)=γ_I·a_gR·S·(2.5/q)=1.0·0.15·1.2·2.5/3.6= 0.125g.

Since T₁=0.44≤1.00=2T_C à λ=0.85, thus a_CM=λ·S_d(T₁)=0.85·0.125g=0.10625g

Step 3:

The mass of each floor is M=186.3 t, except from the mass of the last floor which is M₆=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 à 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 à a_j=0.0102g·Z_j

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.