# Determination of Modal Pseudo-Acceleration, Modal Displacement Increment (5B.1.2 , 5B.1.3)

Modal Pseudo-Acceleration and Modal Displacements are calculated automatically.

**ICONS**

* a _{1 }^{(X,k)}* = modal pseudo-acceleration [m/s2] of the first mode modal single degree of freedom system at the kth push step for earthquake direction [m/s2]

*= (X) earthquake modal displacement of the modal single degree of freedom system belonging to the first mode in the kth thrust step for the kth thrust for the earthquake direction[m]*

**d**_{1 }^{(X,k)}*total mass of the i th floor*

**m**_{i }= the*= (X) the first thrust in the x-axis direction for the earthquake directionith floor*

**m**_{ix1 }^{(X,1)}*modal effective mass*[t]

* m _{tx1 }^{(X,1)}* calculated accordingto the

*constant mode shape*determined in step and never changed during the thrust calculation

= (X)

*modal effective mass of the base shear force*calculated according to the

*constant mode shape*determined in the first thrust step in the x-axis direction for the earthquake direction and never changed during the thrust calculation [t]

* m _{iy1 }^{(X,1)}* = (X) for the earthquake direction i'th floor

*modal effective mass*calculated according to the

*constant mode shape*determined in the first thrust step in the y-axis direction and never changed during the thrust calculation [t]

* m _{iθ1 }^{(X,1)}* = (X) for the earthquake direction in the first thrust step around the z-axis The ith floor calculated according to the

*fixed mode shape*determined and never changed during the thrust calculation

*modal effective mass moment of inertia* [tm2] * u _{ix1 }^{(X,k)}* = (X) displacement calculated in the x-axis direction at the ith floor at the kth thrust step for the earthquake direction [m]

*= (X) k earthquake directions lactide pushing step in the Nth floor (top of the building) the displacement, calculated according to the axis X [m]*

**u**_{Nx1 }^{(X,k)}*= (X) k earthquake directions' th, calculated according to the x-axis in step push*

**V**_{TX1 }^{(X, k)}*base shear*[kN]

*=*

**Δa**_{1 }^{(X,k)}*modal pseudo-acceleration increment of the*first mode modal single degree

*-of-*freedom system at the kth push step for the earthquake direction (X)[m / s

^{2}]

*= (X) k earthquake direction of the 'th pushing step the first mode of*

_{Δd1 }^{(X, k)}*modal single degree of freedom system*' s

*modal displacement of*[m]

*= (X ) k earthquake direction of the 'th thrust step i' th floor, the x-direction acting*

**Δf**_{ix1 }^{(X, k)}_{ }*seismic load increment*[kN]

*= (X) k earthquake direction of the 'th thrust step i' th floor acting along the Y axis*

**Δf**_{iy1 }^{(X, k)}_{ }*earthquake load increment*[kN]

*= (X) earthquake*

**Δf**_{iθ1 }^{(X,k)}_{ }*load increase*acting in the z axis direction at the i th floor at the k th thrust step for the earthquake direction [kN]

*= (X) determined in the first thrust step for the earthquake direction and never changed during the thrust calculation*

**Γ**_{1 }^{(X,1) }*modal contribution factor*calculated according to

*fixed mode shape*

**TDY Equation 5B.1** from the Δ located _{one }^{(X, k)} two successive hinge formation defined between the k 'th thrust unknown size in step second order effects the consideration to be the initial step in the modal analysis of the first mode (the dominant mode) of the *modal single degree of freedom system* 's *Modal so-called Growth Acceleration* is.

*The modal pseudo-acceleration increment* is calculated from the *yield condition* defined in **TDY 5.3.1 of** a new plastic hinge formed at the end of each step . This value also determines the iteration level of the *impulse analysis* . The resulting *modal pseudo acceleration increment, Δa _{1 }^{(X,k)}* , is added to the pseudo acceleration value found at the end of the previous step to obtain the cumulative modal pseudo acceleration

*a*step . In your cumulative modal, acceleration

_{1 }^{(X,k) at the kth}*a*value can be written as given in

_{1 }^{(X,k)}**TDY Equation 5B.3**.

In this equation, in the kth step (X) of the thrust analysis, the *base shear force in the* earthquake direction is * _{expressed as} V _{tx1 }^{(X,k)}* , and the

*base shear force*is expressed as

*modal effective mass m*.

_{tx1 }^{(X,1)}*m _{tx1 }^{(X,1)}* is the modal effective mass obtained in the first mode of the modal analysis , the value of which is taken into account in the initial step where second-order effects are taken into account, and it is found by

**TDY Equation 4B.1**.

*The value of m*is calculated only for the first step and is the sum of the

_{tx1 }^{(X,1)}*floor modal effective masses m*values taken as constant throughout the

_{tx1 }^{(X,1)}*entire thrust calculation*.

For the earthquake direction ( X ) given in **TDY Equation 5B.3** , the base shear force *V _{tx1 }^{(X,k)} *calculated in the kth step of the thrust analysis is calculated cumulatively for each thrust step. Therefore, the so-called modal acceleration

*a*is obtained cumulatively. In this case, the

_{1 }^{(X,k)}*modal pseudo-acceleration increment, Δa*, can also be calculated from the base shear forces that change at each thrust step.

_{1 }^{(X,k)}As a result of the modal analysis performed in the initial step , **TDY Equation 5B.4 is** used to calculate *the modal displacement d _{1 }^{(X,k) of the} system with one degree of freedom* belonging to the first mode .

In this equation, *the _{Nx1 }^{(X, k)} , *(X) of the push analyzed for earthquake directions k 'th calculated in step N' peak displacement of the fifth floor 'of the latter.

*Φ*the mode amplitude at the Nth floor obtained in the first mode of the

_{Nx1 }^{(1)}is*modal analysis*performed in the initial step . The values in the

**denominator of TDY Equation 5B.4**are the values obtained in the first mode of the modal analysis performed in the initial step, where second-order effects are taken into account, and these values are used in all push steps.

*With the value of u _{Nx1 }^{(X,k)} and V _{tx1 }^{(X,k)} *, a

*thrust curve*whose coordinates are

*base shear force-peak displacement*is obtained. To find the performance point, the

*V*term

_{tx1 }^{(X,k)}**TDY Equation 5B.3**and the

*u*term

_{Nx1 }^{(X,k)}**TDY Equation 5B.4**are applied.