# Performance Point of Single-Mode Pushover Analysis Methods (5B.3.5 , 5B.3.6)

**ICONS**

* a _{y1}^{ }*= Yield pseudo-acceleration for the first mode [m/s

^{ 2}]

*=*

**C**_{R}^{ }*Spectral displacement ratio*

*=*

**d**_{1,max }^{(X)}*the modal single degree-of-freedom system for the (X) earthquake direction [m]*

**displacement of**_{ Maximum}

**f**_{e}**=**Carrier Calculated linear (elastic) strength demand for the system

**f****Projected ductility capacity and period-dependent yield strength**

_{y}=*=Linear elastic spectral acceleration corresponding to thefirst natural vibration period T*

**S**_{ae}(T_{1})_{ 1}[g]

*=Linear elastic spectral displacement [m] corresponding to thefirst natural vibration period T*

**S**_{de}(T_{1})_{1}

*=Nonlinear spectral displacement corresponding to thefirst natural vibration period T*

**S**_{di}(T_{1})_{1}[m]

*= Horizontal elastic design acceleration spectrum corner period [s]*

**T**_{B}^{ }*= Natural vibration period of the first mode [s]*

**T**_{1}^{ }

**μ(R**_{y},T_{1}) =*Yield Strength Reduction Coefficient*and

*ductility demand*calculated according to the first natural vibration period

*= First mode*

**ω**_{1}^{(1)}*natural angular frequency*[rad/s]found from the free vibration calculation, which is renewed at each kth thrust step

**5.6.5. Obtaining the Modal Displacement Demand of the Earthquake in Single-Mode Pushover Analysis Methods**

**5.6.5.1 -** Earthquake *modal displacement demand* 's to obtain, under the effect of earthquake modal capacity diagram represented by *modal single degree of freedom systems to maximum displacement of* ' corresponds to the account.

**5.6.5.2 –** The modal displacement demand of the earthquake;

**(a)** Modal can be obtained as *Nonlinear Spectral Displacement* in a single degree of freedom system .

**(b) **It can be obtained from the time history calculation of a modal single degree-of-freedom system under earthquake effect. Both methods are described in **Appendix 5B** .

**5B.3. ACQUISITION OF THE EARTHQUAKE'S MODAL REPLACEMENT DEMAND AS NONLINEAR SPECTRAL DISPLACEMENT**

Earthquake *modal displacement demand* 's to obtain, under the effect of earthquake *modal capacity diagram* represented by *a single degree of freedom systems to maximum displacement of modal* ' corresponds to the account.

**5B.3.1 – **In a modal single degree of freedom system, the largest displacement is defined as a nonlinear spectral displacement:

Here, d _{1, max }^{(X) }*of the modal single degree of freedom systems to maximum displacement of* 'n, S _{di} (T _{1} ) of the conveyor system of the first natural vibration period T _{1} ' and corresponding to **Eq. (5B.13)** defined by *non-linear spectral* indicates *displacement* .

S wherein _{the} (T _{1} ), **Eq. (2.5)** defined by the *elastic displacement of spectral design* , expand C _{R} In **Eq. (5B.14)** 't defined by *the spectral displacement ratio* ' n is Impr.

**5B.3.2 - Eq. (5B.13)** 'in point *spectral displacement ratio* C _{R} , **Eq. (5B.14)** have been identified:

Wherein the *yield strength reduction factor* 'n indicating R _{y} , *Design by Solidarity* to approach **EK 4** ' unlike the definition given in predicted not a quantity which is defined depending on the ductility capacity, directly obtained from a thrust account *the resistance to flow* 's-dependent magnitude represents:

In this relation f _{e} and S _{aa} (T _{1} ) *elastic strength demand* 's and her corresponding *elastic response spectrum* ' y, f _{y} and _{y1} is *the yield strength* 's and her corresponding *flow pseudo-acceleration* ' n represents ( **Figure 5B.4** ).

**5B.3.3 –** μ(R _{y} ,T _{1} ) in **Eq.(5B.14)** is the *ductility demand* expressed depending on the *yield strength* and the natural vibration period . To account for this size **ANNEX 4** 'in **Eq. (4A.2)** with the reverse relations are obtained by writing the relation:

**(a)** Seismic ductility demand μ (R _{y} , T _{1} ), *equal displacement rule* in accordance with *the stiffness of not more* for carrier systems Yield Strength Reduction Factor R _{y} 'is taken equal to:

**(b) **For carrier systems with *more *rigidity, the equation in **Eq.(5B.16b)** is obtained from **Eq.(4A.2b)** :

**5B.3.4 - Eq. (5B.14)** 't defined by the spectral displacement ratio C _{R} , **Eq. (5B.16)** from utilizing **Eq. (5B.17)** wherein is expressed as:

**5B.3.5 – The** modal capacitance diagram belonging to the first (dominant) vibration mode in **Figure 5B.3** and **Figure 5B.4** and whose coordinates are *modal displacement–modal pseudo-acceleration* (d _{1} ,a _{1} ) and its coordinates *spectral displacement–spectral acceleration* ( The linear earthquake spectrum with S _{de} , S _{ae} ) was drawn together.

**(a) The** situation shown in **Figure 5B.3 **corresponds to the application of** Equation (5B.17a)** together with **Equation (5B.13)** . In this case, without doing anything on the modal capacitance diagram, it is sufficient only to show that the natural vibration period in the first impulse step satisfies the condition T _{1} >T _{B} or ( ^{(1) }_{1 }^{(1)} ) ^{2} ≤ ^{2 }_{B }^{2} .

**(b)** On the other hand , the situation shown in **Figure 5B.4 **corresponds to the application of** Equation (5B.17b)** together with **Equation(5B.13)** . In this case the spectral displacement ratio C _{R} will be calculated in the sequential approach. For this purpose modal capacity diagram, **figure 5b.4 A** as depicted in prior C _{R} = 1 two rectilinear taking *elastoplastic* converted to a diagram. The equality of the areas under the diagrams is taken as basis in the transformation process. From **Eq.(5B.15)** using the approximate yield pseudo-acceleration a _{y1 }^{o} found in this way, R _{y} and accordingly**Eq. (5b.17b)** from the C _{R} and **Eq. (5B.13)** 'skin S _{di} (T _{1} ) is calculated. Accordingly, the *elasto-plastic* diagram is reconstructed **(Figure 5B.4b)** and the same operations are repeated based on the re-found a _{y1} . At the step where the results are close enough, the sequential approach is terminated.

**5B.3.6 -** Earthquake *modal displacement demand* 's **in Eq. (5B.13)** and **Eq. (5B.17)** ' benefiting from **Eq. (5B.12)** according to the calculation **(a)** and **(b)** as defined in cases are not applicable.

**(a) **In cases where the distance of the nearest fault to the building is less than 15 km, *calculation* will be made *in the time *history according to **5B.4** using near-field earthquake records selected and scaled according to **2.5** .

**(b) ***In* case the post-yield slopes of the modal capacity diagram are negative due to *second order effects , the calculation* will be made *in the time *history according to **5B.4** using earthquake records selected and scaled according to **2.5** .

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