# Modal Response Spectrum Analysis

**ICONS**

* f _{ixn,max }^{(X)}* = (X) the largest modal earthquake load acting on the i th floor in the nth natural vibration mode in the x axis direction of the building

*= The upper part of the i th floor in the upper part above the basement floors of the building*

**H**_{i }

**M**

_{oxn}*= Variation of modal base overturning moment with respect to time in nth vibration mode under the joint effect of (X) and (Y) earthquake ground motion components at the same time*

_{ ,max }^{(X)}*= i th floor total mass*

**m**_{i}*= mass moment of inertia of the i th floor*

**m**_{iθ}

**m**_{ixn }^{(X)}**= (X) i'th floor modal effective mass of the building in the x-axis direction of the building in the x-axis direction**

*= (X) of the nth natural vibration mode of the building in the y-axis direction of the building for the (X) earthquake direction i' th floor modal effective mass*

**m**_{iyn }^{(X) }*= (X) i th floor modal effective mass moment of inertia pertaining to the nth natural vibration mode of the building around the z axis for the earthquake direction*

**m**_{iθn }^{(X) }*= Typical finite element nodal point j'*

**m**_{j }^{(S) }*of the building above the basement floors*

**m**_{t = total mass of the upper section}**m _{ txn}**

*= (X) base shear modal effective mass of the building in the x-axis direction of the building for the earthquake direction*

^{(X)}

**m**_{tyn }^{(Y)}**= (Y) the base shear modal effective mass of the building in the y-axis direction for the earthquake direction**

*corresponding to any response magnitude (displacement, relative story drift, internal force component) for the earthquake direction Typical unit modal behavior quantity corresponding to a behavior quantity (displacement, relative story translation, internal force component)*

**r**) combined typical largest modal response magnitude_{max }^{(X) }= (^{X }* r _{n,max }^{(X) }* = typical largest modal response magnitude corresponding to any response magnitude (displacement, relative story drift, internal force component) for the earthquake direction in the nth natural vibration mode (X)

*= n Reduced design spectral acceleration of the 'th vibration mode*

**S**_{aR}(T_{n})*= the natural vibration period of the nth mode*

**T**_{n }*of the nth vibration mode in the x-axis direction of the building for the earthquake direction = Sufficient number of vibration modes*

**V**_{txn,max }^{(X)}^{ }= (X) the largest modal base shear force YM_{ }*= ratio of m'th and nth natural vibration periods*

**β**_{mn }**Φ**

*= nth natural vibration mode shape amplitude in the direction of earthquake (X) at i'th floor*

_{i(X)n }**Φ**

*= nth natural vibration mode shape amplitude in x-axis direction at i'th floor vibration mode shape amplitude*

_{ixn }**Φ**

*= nth natural vibration mode shape amplitude in the y-axis direction at i'th floor Φ iθn = nth natural vibration mode shape amplitude*

_{iyn }**as**

*around z axis at i'th floor*

_{rotation }*= (X ) for the earthquake direction, the modal contribution factor of the nth vibrational mode*

**Γ**_{n }^{(X) }*= modal damping ratio of the nth vibrational mode*

**ξ**_{n }*= natural vibrational angular frequency of the nth vibrational mode*

**ω**_{n }*= Cross correlation coefficient of the mth and nth natural vibrational modes in the Perfect Quadratic Coupling Rule*

**ρ**_{mn }**4.8.1. Modal Calculation Methods**

**4.8.1.1 -** the modal behavior of the structural system under seismic action based *Modal Analysis Methods* , **4.8.2** from the computational with the seismic spectrum *combine mode method* and **4.8.3** 'in the account in the time domain based *mode summation method* ' is. Detailed explanations for these methods are given in **ANNEX 4B** . In buildings with and without basement, **3.3.1** will be used for the definition of building base and building height .

**4.8.1.2 - In** modal calculation methods, *sufficient number of vibration modes* to be taken into account , YM,

**(a)** According to **Annex 4B** , the *base shear force* calculated for each mode in earthquake directions (X) and (Y) shall be determined according to the rule that the sum of the *modal effective masses* is not less than 95% of the total building mass .

However, all modes with a contribution greater than 3% will be considered.

**(b) **The largest of the YMs calculated for both directions will be taken into account in the three dimensional calculation.

**4B.1. MODAL ACCOUNT PARAMETERS**

**The modal calculation parameters defined in 4B.1.1 –4B.1.4, 4B.1.5** and **4B.1.6** , regardless of the earthquake data, are calculated only according to the earthquake direction taken into account and the information obtained from the free vibration calculation of the carrier system, and **4B.2** and **4B** below. They are the quantities used in both modal calculation methods explained in **.3 .**

**4B.1.2** – Modal calculation parameters are defined below only for (X) horizontal earthquake direction. The same parameters can be defined similarly for the earthquake direction (Y) perpendicular to (X).

**4B.1.3** – As the degrees of freedom of the carrier system in the definition of the modal calculation parameters:

**(a)** In case the floor slabs are modeled as a rigid diaphragm, the displacements defined in the x and y horizontal directions at the center of mass of any ith floor slab and the rotation around the vertical axis passing through the floor mass center are taken into account, and the floor mass *mi* corresponding to these degrees of freedom is taken into account. The mass moment of inertia *m _{iθ}* is defined.

**(b)** If floor slabs are not taken as rigid diaphragms and are modeled with two-dimensional slab (membrane) finite elements to include degrees of freedom for displacements in their plane according to **4.5.6.2 , ***m _{j }^{(s)}* at finite element nodes instead of

*m*will be taken into account.

_{i storey masses }^{)}**4B.1.4 – *** Modal Additive Factor and Base Shear Modal Effective Mass:* For the given earthquake direction (X), the modal contribution factor of the n th vibration mode

*Γ*of the building in the x-axis direction

_{n }^{(X)}and the base shear modal effective mass*is defined by*

**m**_{txn }^{(X)}**Eq.(4B.1)**:

**4B.1.5 – *** Floor Modal Effective Masses: In* a typical n'th vibration mode for a given earthquake direction (X), the storey modal effective masses for the degrees of freedom defined in

**4B.1.3 above are defined by Equation (4B.2)**:

**4B.1.6 – *** Unit Modal Response Magnitude: Typical unit modal response magnitude* corresponding to any response magnitude (displacement, relative story drift, internal force component) in the typical nth vibration mode for a given earthquake direction (X) It is obtained by a static calculation in which the

*floor modal effective masses*defined by

**Eq.(4B.2) are acted as a load in their own direction.**

**4B.2. EARTHQUAKE ACCOUNT WITH MOD JOINING METHOD**

**4B.2.1 – ***Mode Combination Method* is explained below for (X) earthquake direction. A similar calculation will be made for the earthquake direction (Y) perpendicular to (X).

**4B.2.2 – Strike combination according to 4.4.2** shall be applied to the maximum response magnitudes obtained separately for the horizontal (X) and (Y) earthquake directions .

**4B.2.3 – Typical largest modal response magnitude *** r _{n,max }^{(X)}* corresponding to any response magnitude (displacement, relative story translation, internal force component) in a typical nth vibration mode for a given earthquake direction (X) ,

**Eq**. It is calculated by

**.(4B.3) :**

Here It represents the typical *unit modal response magnitude defined in ***4B.1.6** , and *S _{aR} ( Tn ) represents _{the} reduced design spectral acceleration* obtained from

**Eq.(4.8)**

*for*the

*n'th natural vibration period*

_{typical}*Tn*.

**4B.2.4 – Asynchronous maximum modal contributions calculated according to 4B.2.3** for each vibration mode, *to be applied separately for each of the* behavior quantities such as internal force components, displacement and relative story translation, shall be combined statistically as described below:

**(a)** As the most general mode coupling rule, Perfect *Quadratic Combination (TKB) Rule is given in ***Equation (4B.4)** .

Here , *r _{m,max }^{(X)}* and

*r*calculated with 4B.2.3 for

_{n,max }^{(X)}are the largest modal response quantities**typical m'th**and n'th vibration modes , and

*ρ*of these modes . shows the

_{mn}is the cross correlation coefficient**(b)** The cross-correlation coefficient in **Equation (4B.4) above** is given in Equation (4B.5a):

Here , *β _{mn represents the ratio of the m 'th and n}* 'th natural vibration periods considered, and

*ξ*and

_{m}*ξn*represent the modal damping ratios that belong to the same modes and can be taken different from each other.

**(c)** Assuming the modal damping ratios to be the same in all modes, the cross-correlation coefficient can be simplified as given in **Eq.(4B.5b)** :

**(d) If the condition ***β _{mn}* < 0.8 is satisfied for all modes considered , the

**Square**

*Root of Sum of Squares Rule (KTKK)*given in

**Eq.(4B.6)**can be used instead of the combination rule given in Eq.(4B.4) .

This coupling rule corresponds to the special case of taking *ρ _{mn}* = 0 (m ≠ n ) and

*ρ*=1 (m = n) in

_{mn}**Eq.(4B.4) .**

**4B.2.5 – In a typical n'th vibration mode for a given earthquake direction (X), the ***maximum modal base shear force V _{txn,max }^{(X)}* of the carrier system in the x-axis direction and the corresponding

*largest base overturning moment M*

_{oxn,max }^{(X) )}**is calculated by Eq.(4B.7)**:

Combining the mod contributions of these magnitudes will also be done according to **4B.2.4** .