Evaluation of the Seismic Capacity of Existing Moment Resisting Frames by a Simplified Approach: Examples and Numerical Application

Abstract

The capacity of a structure can be assessed using inelastic analysis, requiring sophisticated numerical procedures such as pushover and incremental dynamic analyses. A simplified method for the evaluation of the seismic performance of steel moment resisting frames (MRFs) to be used in everyday practice has been recently proposed. This method evaluates the capacity of buildings employing an analytical trilinear model without resorting to any non−linear analysis. Despite the methodologies suggested by codes, the assessing procedure herein described is of easy application, also by hand calculation. Furthermore, it constitutes a suitable tool to check the capacity of the buildings designed with the new seismic code prescriptions. The proposed methodology has been set up through a large parametric analysis, carried out on 420frames designed according to three different approaches: the theory of plastic mechanism control (TPMC), ensuring the design of structures showing global collapse mechanism (GMRFs), the one based on the Eurocode 8 design requirements (SMRFs), and a simple design against horizontal loads (OMRFs) without specific seismic requirements. In this paper, some examples of the application of this simplified methodology are proposed with references to structures supposed to exhibit global, partial and soft storey mechanism.

1. Introduction

The safeguard of the built heritage is gaining an increasing interest in structural and seismic engineering [1,2]. The knowledge of the seismic response of the structures plays a fundamental role in the evaluation of the seismic vulnerability of existing structures [3,4,5,6] and in the prediction of the expected losses [7,8,9]. The designer, to achieve a correct safety evaluation in terms of the capacity demand ratio, must resort to non−linear models and non−linear analyses [10,11,12,13], which, even though they are described in the codes, are very prone to being misinterpreted. The implementation of a non−linear model needs adequate experience and, first of all, an adequate knowledge of the condition of the building to be modelled in terms of geometry, loads, structural details, materials and degradation [14,15,16,17,18]. In the perspective of a seismic classification of the built heritage, it is necessary to define a standardizable, unique and user−friendly methodology [19,20,21,22]; therefore, recently, a simplified assessment approach has been developed concerning steel moment resisting frames (MRFs) [23] based on a trilinear model that needs only the elastic structural analysis and the rigid-plastic analysis and does not require any static or dynamic non−linear effort. Rigid plastic analysis has generally been used as an assessment tool against vertical and horizontal loads. The novelty point of the procedure is to use the rigid plastic and elastic analysis joined with a performance-based evaluation aimed at easily assessing the seismic performance of existing buildings, also ensuring a rapid mapping of the built heritage in terms of seismic vulnerability.

This assessment procedure is herein applied to some study cases to support their ease of application. It mainly consists of the definition of a trilinear capacity curve by some characteristic points that define three branches [24,25]. These points correspond to different limit states. It is important to mention that the code assessment approach does not provide the definition of specific performance points on the pushover curve, thus entrusting the designer to the definition of required target limits which are provided by codes in terms of member chord rotation.

The method has been set up by performing several parametric analyses, in terms of pushovers, on 420 frames, designed according to three different approaches, i.e., without any seismic detailing (O−MRFs), by Eurocode 8 (S−MRFs) and by the theory of plastic mechanism control (G−MRFs). It is usually observed that structures designed without any seismic provision exhibit a soft storey mechanism while the structures designed according to the modern approach of Eurocode 8 can achieve more robust performances even if the global mechanism can be assured only by the TPMC approach. The results belonging to the pushover analyses performed on the structures have been used to calibrate the assessment relationships, ensuring a wide application of the method. The comparison in terms of capacity and demand can be made according to two alternative approaches: the one proposed by Eurocode 8 [26] and the one proposed by Nassar and Krawinkler [27]. The former exploits the concept of ADRS spectrum while the latter has the benefit of being of more easily applicable because it does not single out between low and high periods of vibration. In the following, the main model equations are reported and described.

2. Fundamental Equations of the Trilinear Model

The simplified trilinear model needs only the elastic structural analysis and the rigid-plastic analysis [28,29,30,31] not requiring any static or dynamic non-linear analyses. Therefore, the user can quickly obtain the capacity curve through the intersection of three linear branches (Figure 1), computed using simple relationships proposed in the following. Referring to the proposed model, the first branch of the curve can be approximated by the elastic response curve; the horizontal one is defined using the maximum load-bearing capacity given by the calibrated Merchant−Rankine formula, while, the softening branch is given by the collapse mechanism equilibrium curve of the structure, influenced by the second-order effects [32,33,34,35,36]. The definition of the third branch is linked to the concept of the mechanism equilibrium curve. In particular, the equation of the third branch can be obtained by equating the virtual internal work of the dissipative zones with the virtual external work of the structure, accounting for second-order effects.

The equations of the three identified branches in the α−δ plane (horizontal force multiplier—top sway displacement) are reported below:

• Elastic response curve:

$$\alpha = \frac{1}{{\delta }_1}\delta$$

(1)

• Maximum load-bearing capacity curve according to the modified Merchant−Rankine formulation [23,37,38]

$$\alpha = {\alpha }_{m\alpha x} = \frac{{\alpha }_0}{1 + \mathrm{\Psi }{\alpha }_0{\gamma }_s{\delta }_1}$$

(2)

where:

$$\mathrm{\Psi } = \alpha + b\xi$$

(3)

$$\xi = \frac{\sum \frac{E{I}_b}{L_b}}{\sum \frac{E{I}_c}{L_c}}$$

(4)

The use of Equation (3) is proposed by assuming for the coefficient Ψ with the following relation:

$$\mathrm{\Psi } = 0.28488 - 0.14042 \xi$$

(5)

• The mechanism equilibrium curve according to rigid-plastic analysis [37]:

$$\alpha = {\alpha }_0 - {\gamma }_s(\delta - {\delta }_y)$$

(6)

Characteristic performance points (points A, B, C, D of Figure 2) have been identified on the trilinear model. The points are associated with specific limit states [35], provided by codes, identifying a target performance level [34,39,40,41].

• Point A—“Fully Operational”

$${\alpha }_A = \frac{1}{{\delta }_1}{\delta }_A$$

(7)

where δ_A is the displacement corresponding to the minimum between the displacement in the service conditions and δ_y is the displacement corresponding to the formation of the first plastic hinge.

• Point B—“Operational”

$${\alpha }_B = {\alpha }_{m\alpha x} ;\hspace{1em}{\delta }_B = {\alpha }_{m\alpha x}{\delta }_y$$

(8)

where α_max is the maximum multiplier defined according to the Equation (2).

• Point C—“Life Safety”

$${\alpha }_C = {\alpha }_{max} ;\hspace{1em}{\delta }_C = {\delta }_{mecc} = \frac{{\alpha }_0 - {\alpha }_{max}}{{\gamma }_S} + {\delta }_y$$

(9)

This point derives from the intersection of the horizontal branch of equation α = α_mαx with the softening branch, representative of the collapse mechanism equilibrium curve, of equation α = α₀ − γ_S(δ − δ_y).

• Point D—“Near Collapse”

$${\alpha }_D = {\alpha }_{max} - {\gamma }_s({\delta }_D - {\delta }_C) ;\hspace{1em}{\delta }_D = {\delta }_C + ({\vartheta }_{\rho .\mu } - {\vartheta }_{\rho .mecc})H_0$$

(10)

The chord rotation at yielding ϑ_y is defined as a property of the member. For ductile elements and brittle elements involved into soft-storey mechanisms, are given by:

$${\vartheta }_y = \frac{{\gamma }_{ov}{M}_{p.m}{l}_m}{6E{I}_m}$$

(11)

while for brittle elements involved in partial and global mechanisms:

$${\vartheta }_y = \frac{{\gamma }_{ov}{M}_{p.m}{l}_m}{4E{I}_m}$$

(12)

For the evaluation of plastic rotations occurring in the critical members, an analytical formulation is proposed, based on a “shear−type” single−storey portal with different plastic moments at the top and base of the columns. The relationships for the evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism are reported as follows:

$$\frac{{\theta }_{p\mathrm{.}m\mathrm{e}c} H_0}{n_{\mathrm{s}}{\delta }_{\mathrm{y}}} = \frac{{\mathrm{\Psi }}_1}{{\mathrm{\Psi }}_2}{\mathrm{\Psi }}_3{(\frac{{\alpha }_{max}}{{\alpha }_{\mathrm{y}}} - 1)}^{{\mathrm{\Psi }}_4}\frac{1 - {\mathrm{\Psi }}_5{\gamma }_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6{\gamma }_{\mathrm{s}}}$$

(13)

$$\frac{{\theta }_{p\mathrm{.}m\mathrm{e}c} H_0}{n_{\mathrm{s}}{\delta }_{\mathrm{y}}} = \frac{{\mathrm{\Psi }\prime }_1}{{\mathrm{\Psi }\prime }_2}{\mathrm{\Psi }}_3{(\frac{{\alpha }_{max}}{{\alpha }_{\mathrm{y}}} - 1)}^{{\mathrm{\Psi }\prime }_4}\frac{1 - {\mathrm{\Psi }\prime }_5{\gamma }_{\mathrm{s}}}{1 - {\mathrm{\Psi }\prime }_6{\gamma }_{\mathrm{s}}}$$

(14)

In particular, the coefficient with the apex refers to the element achieving the collapse (i.e., the critical element), with those without the apex to the element developing the first yielding.

The Ψ_i coefficients to be used in Equations (13) and (14), are given by the following equations whose parameters are reported in

Table 1. Regression coefficients for the evaluation of plastic rotation demand for the development of the collapse mechanism.

	GMRFs	SMRFs	OMRFs
a1	2.7747755	2.982417	19.542818
b1	0.0207354	−0.14356	−1.372652
a2	1.817070	1.370201	−144.9099
b2	−0.07731	0.652663	123.8454
a3	0.0844528	0.964755	−0.028950
b3	1.616165	1.802312	0.1820582
a4	−0.112433	0.737624	−1.840828
b4	1.4966937	−0.51209	3.0361764
a5	1.0606602	0.976295	97.159963
b5	0.6787599	1.027818	25.416893
a6	1.0528759	0.975839	1.8666626
b6	0.7200734	1.030732	−0.429104
a’1	1.1674452	3.415537	19.508374
b’1	0.0575325	−0.07355	−0.637701
a’2	6.0112325	0.251316	−89.8716
b’2	0.3665074	1.394603	73.87363
a’3	1.0944684	3.860496	−0.044146
b’3	−1.169347	−0.09045	0.3181349
a’4	−2.322765	1.415893	−2.345411
b’4	7.462743	−1.18406	3.917804
a’5	0.993180	0.968454	−17.06279
b’5	0.95649	1.11087	95.899727
a’6	1.0150939	0.976968	1.5715063
b’6	0.7912074	1.069351	−0.053770

:

$${\mathrm{\Psi }}_1 = {\alpha }_1 + b_1{n}_{\mathrm{b}}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{\Psi }\prime }_1 = {\alpha \prime }_1 + {b\prime }_1{n}_b$$

(15)

$${\mathrm{\Psi }}_2 = {\alpha }_2 + b_2{n}_{\mathrm{s}}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{\Psi }\prime }_2 = {\alpha \prime }_2 + {b\prime }_2{n}_s$$

(16)

$$\mathrm{\Psi }\mathrm{i} = \alpha i + b\mathrm{i}\xi i = 3,\dots ,6\hspace{1em}\mathrm{\Psi }\prime \mathrm{i} = \alpha \prime i + b\prime i\xi i = 3,\dots ,6$$

(17)

3. Assessment Procedure in Terms of Spectral Accelerations According to ADRS Spectrum

The capacity-demand assessment procedure can be expressed through the ADRS spectrum [12,26]. For each limit state, the spectrum S_α − S_De is defined by the means of the relationship S_De(T) = S_α(T)(T/2π)². As regards the capacity, it is necessary to represent the characteristic points of the behavioral curve of the structure, in the ADRS plane. Of these points, it is necessary to know the abscissa, which is the displacement $d_{\mathrm{LS}}^{*} = d_{\mathrm{LS}}/\Gamma$.

It is necessary to distinguish between cases T* > T_C and T* < T_C. If T* > T_C 41, the capacity in terms of spectral acceleration relative to the limit state considered can be obtained through the following relationship:

$$S_{aSL} = d_{LS}^{*}{{\omega }_0^{\mathrm{*}}}^2$$

(18)

The demand is represented by the spectral acceleration provided by the code, for the specific limit state, in the case of the equivalent SDOF system with the equivalent period of vibration T*.

For the assessment procedure, the inequality S_αls ≥ S_α(T*) must be satisfied.

If T* < T_C and q > 1, according to the equality energy criterion, there is a different procedure to evaluate the capacity that leads to the anelastic spectrum whose parameters are listed below:

$$F_{lS}^{*} = \frac{m^{*}{S}_a\left(T^{*}\right)}{q_{l\mathrm{S}}}$$

(19)

$$q_{lS} = 1 + ({\mu }_{lS} - 1)\frac{T^{*}}{T_C}$$

(20)

$$S_{aSL} = q_{lS}\frac{F_{lS}^{*}}{m^{*}}$$

(21)

If T* < T_C and q ≤ 1, it results:

$$F_{SL}^{*} = m^{*}{S}_a\left(T^{*}\right)$$

(22)

$$S_{aSL} = \frac{F_{lS}^{*}}{m^{*}}$$

(23)

$$m^{*} = \sum _{k=1}^n{m}_k{ɸ}_k$$

(24)

The checking is verified when the inequality S_αls ≥ S_α(T*) is satisfied.

4. Assessment Procedure in Terms of Spectral Accelerations According to Nassar and Krawinkler

In the framework of capacity-demand checking [27,37,38,39,40,41,42,43], an equivalent SDOF system replaces the MDOF actual system by means of the modal participation factor Γ. Multiplying the multiplier α with the design base shear, the capacity curve is reported in an F_b—d_c plane. The capacity curve must be reduced through the modal participation factor and represented in an F*—d* plane. The demand is estimated through the period T* and the equivalent mass m* according to European codes. The capacity in terms of spectral acceleration for the points A, B, C, and D is given as follows:

• Point A—“Fully Operational”

$$S_{aFO}\left(T^{*}\right) = \frac{F_{FO}^{*}}{m^{*}}$$

(25)

• Point B—“Operational”

$$S_{aO}\left(T^{*}\right) = \frac{F_O^{*}}{m^{*}}$$

(26)

• Point C—“Life Safety”

$$S_{aLS}\left(T^{*}\right) = \frac{F_{LS}^{*}}{m^{*}}{q}_{LS}$$

(27)

$$q_{LS} = q_0(\mu , T, \gamma = 0) = {\left[c\right({\mu }_{LS} - 1) + 1]}^{1/c}$$

(28)

where $c = \frac{T*}{1 + T*} + \frac{0.42}{T*}$ and ${\mu }_{LS} = \frac{d_{LS}*}{d_0*}$.

• Point D—“Near Collapse”

$$S_{aNC}\left(T^{*}\right) = \frac{F_{NC}^{*}}{m^{*}}{q}_{NC}$$

(29)

$$q_{NC} = \frac{q_0}{\phi }$$

(30)

$$q_0(\mu , T, \gamma = 0) = {\left[c\right({\mu }_{LS} - 1) + 1]}^{1/c}$$

(31)

where $c = \frac{T*}{1 + T*} + \frac{0.42}{T*}$ and ${\mu }_{NC} = \frac{d_{NC}*}{d_o*}$.

$$\phi = \frac{1 + 0.62{({\mu }_{NC} - 1)}^{1.45}\gamma }{(1 - \gamma )}$$

(32)

The coefficient ϕ is a function of the ductility μ and the non-dimensional slope of the equilibrium curve γ.

5. Numeric Example

The assessment procedure described above is applied to evaluate the capacity of a seven-storey and four-span steel moment resisting frame. The permanent loads G_k are equal to 3.5 kN/m² while the live loads Q_k equal 3 kN/m². For the evaluation of gravitational loads on the beams, a frame tributary length of 6.00 m has been set. The steel used is S275. In Figure 3a, a flowchart of the procedure is reported.

5.1. Ordinary Moment Resisting Frame

Ordinary moment resisting frames are designed without any seismic prescription. The beams and column sections are reported in Figure 4.

The trilinear capacity curve is shown in Figure 5, which also shows the characteristic performance points of the model.

Parameters obtained from the elastic analysis:

• δ₁(α = 1) = 0.06305 m
• k = 15.8605 m⁻¹
• δ_A = δ_y = 0.2602 m
• α_A = α_y = kδ_y = 4.128

Parameters obtained from the rigid-plastic analysis:

• α₀ = 5.219
• γ_s = 3.729 m⁻¹
• α = α₀ − γ_S(δ − δ_y) → α = 5.219 − 3.729(δ − 0.2602)
• α(δ = 0) = α₀ + γ_sδ_y = 6.1888
• H₀ = 3.5 m(collapse mechanism Type 3, i_m = 3)

Evaluation of the maximum multiplier using the calibrated Merchant−Rankine formula:

• $${\alpha }_{m\alpha x} = \frac{{\alpha }_0}{1 + \mathrm{\Psi }{\alpha }_0{\gamma }_s{\delta }_1} = 4.2025$$

where:

$$\mathrm{\Psi } =0.28488 - 0.14042 \xi = 0.1970 \mathrm{with} \xi = \frac{\sum \frac{E{I}_b}{L_b}}{\sum \frac{E{I}_c}{L_c}} = 0.6255$$

and consequently ${\delta }_B = \frac{{\alpha }_{max}}{k} = 0.265$ and ${\delta }_c = {\delta }_{mecc} = \frac{{\alpha }_0 - {\alpha }_{max}}{{\gamma }_{\mathrm{s}}} + {\delta }_{\mathrm{y}}$ = $$\begin{gathered} \frac{5.219 - 4.2025}{3.729} + \\ 0.2602 = 0.5326 m \end{gathered}$$.

Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the first plasticized element (first storey beams):

• $${\mathrm{\theta }}_{\mathrm{p}.\mathrm{mec}} = \frac{{\mathrm{n}}_{\mathrm{s}}{\mathrm{\delta }}_{\mathrm{y}}}{{\mathrm{H}}_0}\left[\frac{{\mathrm{\Psi }}_1}{{\mathrm{\Psi }}_2}{\mathrm{\Psi }}_3{\left(\frac{{\mathrm{\alpha }}_{\max }}{{\mathrm{\alpha }}_y} - 1\right)}^{{\mathrm{\Psi }}_4}\frac{1 - {\mathrm{\Psi }}_5{\mathrm{\gamma }}_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6{\mathrm{\gamma }}_{\mathrm{s}}}\right] = 0.06612 \mathrm{rad}$$

The calculation of the corresponding capacity provides a final plastic rotation value of 8θ_y = 8 × 0.008257 = 0.06605 rad.

Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the critical element (the mechanism is partial type 3 i_m = 3, so the critical element is a third storey column):

• $${\mathrm{\theta }}_{\mathrm{p}\mathrm{.}\mathrm{mec}} = \frac{{\mathrm{n}}_{\mathrm{s}}{\mathrm{\delta }}_{\mathrm{y}}}{{\mathrm{H}}_0}\left[\frac{{\mathrm{\Psi }}_1\prime }{{\mathrm{\Psi }}_2\prime }{\mathrm{\Psi }}_3\prime {\left(\frac{{\mathrm{\alpha }}_{\max }}{{\mathrm{\alpha }}_{\mathrm{y}}} - 1\right)}^{{\mathrm{\Psi }}_4\prime }\frac{1 - {\mathrm{\Psi }}_5\prime {\mathrm{\gamma }}_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6\prime {\mathrm{\gamma }}_{\mathrm{s}}}\right] = 0.07693 \mathrm{rad}$$

The calculation of the corresponding capacity, in the case of the third storey columns, provides a value of the ultimate plastic rotation equal to 8θ_y = 8 × 0.005568 = 0.04454 rad. Therefore, the ultimate conditions are governed by the columns of the third storey.

Considering the plastic rotation capacity, the ultimate displacement is given by:

$$\begin{gathered} {\delta }_u = {\delta }_C + ({\delta }_{p.u} - {\delta }_{p.mecc})H_0 = 0.5326 + (0.04454 - 0.07693) \times 3.5 = \\ 0.4192 m \end{gathered}$$

Since the plastic rotation capacity of the third storey columns is lower than that necessary for the complete development of the kinematic mechanism, the points C and D corresponding to the limit states “Life Safety” and “Near Collapse” are coincident and correspond to the aforementioned last displacement (δ_u).

All the verification procedures considered the use of the transformation of the MDOF system into an equivalent SDOF system using the participation coefficient of the main vibration mode Γ. For this reason, it is necessary to define:

• The eigenvector ɸ = {ɸ₁, ɸ₂, ɸ₃, ɸ₄, ɸ₅, ɸ₆}that, assuming ${\mathrm{ɸ}}_k = \frac{F_k}{F_n}$, is:

$${\mathrm{ɸ}}_1 = 0.134 {\mathrm{ɸ}}_2 = 0.267 {\mathrm{ɸ}}_3 = 0.401$$

$${\mathrm{ɸ}}_4 = 0.535 {\mathrm{ɸ}}_5 = 0.669 {\mathrm{ɸ}}_6 = 0.802 {\mathrm{ɸ}}_7 = 1.000$$

The modal participation factor Γ:

$$\Gamma = \frac{{\sum }_{k=1}^n{m}_k{ɸ}_k}{{\sum }_{k=1}^n{m}_k{{ɸ}_k}^2} = 1.4381$$

being:

$$m_1 = 57.98 \times {10}^3 \mathrm{kg} m_2 = 57.98 \times {10}^3 \mathrm{kg} m_3 = 57.98 \times {10}^3 \mathrm{kg}$$

$$m_4 = 57.98 \times {10}^3 \mathrm{kg} m_5 = 57.98 \times {10}^3 \mathrm{kg} m_5 = 57.98 \times {10}^3 \mathrm{kg} m_7 = 61.94 \times {10}^3 \mathrm{kg}$$

The dynamic parameters of the equivalent SDOF system are reported in

Table 2. Dynamic parameters of the equivalent SDOF system.

m*	k*	ω*	T*
[kg 103]	[kN/m]	[rad/s]	[s]
224.76	4302.8	4.3753	1.436

.

Therefore, the characteristic points of the capacity curve are defined in the planes α − δ, F_b − d_c, F* − D*, S_α − S_D assessing the capacity in terms of accelerations for the Nassar and Krawinkler approach and ADRS spectrum approach. In

Table 3. ADRS spectrum approach.

	FO	O	LS	NC
F    [kN]	1119.90	1140.12	1140.12	1140.12
F*   [kN]	778.72	792.78	792.78	792.78
d     [m]	0.2602	0.265	0.4192	0.4192
d*    [m]	0.181	0.184	0.291	0.291
Sa*    [g]	0.353	0.359	0.569	0.569

, the results based on the use of the ADRS spectrum, and in

Table 4. Nassar and Krawinkler approach.

	FO	O	LS	NC
F    [kN]	1119.90	1140.12	1140.12	1140.12
F*   [kN]	778.72	792.78	792.78	792.78
d     [m]	0.2602	0.265	0.4192	0.4192
d*    [m]	0.181	0.184	0.291	0.291
μ     [m]	−	−	1.582	1.582
Sa*   [g]	0.353	0.359	0.575	0.575

, the results based on the use of the Nassar and Krawinkler formulation, are reported. The numbers in bold identify the values of spectral acceleration and displacement to be used in the capacity-demand assessment procedure.

Seismic performance verification requires that, for each limit state, the inequality S_α.ls(T*)_capacity ≥ S_α.ls(T*)_demand is satisfied.

5.2. Special Moment Resisting Frame

Special moment resisting frames are designed to fulfil the Eurocode 8 seismic provisions. The selected case study with the definition of the beam and column dimension is reported in Figure 6.

The trilinear capacity curve is shown in Figure 7, which also shows the characteristic performance points of the model.

Parameters obtained from the elastic analysis:

• δ₁(α = 1) = 0.03814 m
• k = 15.8605 m⁻¹
• δ_A = δ_y = 0.1802 m
• α_A = α_y = kδ_y = 4.736

Parameters obtained from the rigid-plastic analysis:

• α₀ = 7.989
• γ_s = 1.006 m⁻¹
• α = α₀ − γ_s(δ − δ_y) → α = 7.989 − 1.006(δ − 0.1802)
• α(δ = 0) = α₀ + γ_sδ_y = 8.1704
• H₀ = 14.0 m (collapse mechanism Type 1, i_m = 4)

Evaluation of the maximum multiplier using the calibrated Merchant−Rankine formula:

• $${\alpha }_{m\alpha x} = \frac{{\alpha }_0}{1 + \mathrm{\Psi }{\alpha }_0{\gamma }_s{\delta }_1} = 7.4056$$

where:

$$\mathrm{\Psi } =0.28488 - 0.14042 \xi = 0.2572 \mathrm{with} \xi = \frac{\sum \frac{E{I}_b}{L_b}}{\sum \frac{E{I}_C}{L_c}} = 0.1971$$

and consequently ${\delta }_B = \frac{{\alpha }_{max}}{k} = 0.2824$ and ${\delta }_C = {\delta }_{mecc} = \frac{{\alpha }_0 - {\alpha }_{max}}{{\gamma }_{\mathrm{s}}} + {\delta }_{\mathrm{y}}$ = $$\begin{gathered} \frac{7.989 - 7.4056}{1.006} + \\ 0.1802 = 0.7605 m \end{gathered}$$.

Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the first plasticized element (first storey beams):

• $${\mathrm{\theta }}_{\mathrm{p}\mathrm{.}\mathrm{mec}} = \frac{{\mathrm{n}}_{\mathrm{s}}{\mathrm{\delta }}_{\mathrm{y}}}{{\mathrm{H}}_0}\left[\frac{{\mathrm{\Psi }}_1}{{\mathrm{\Psi }}_2}{\mathrm{\Psi }}_3{\left(\frac{{\mathrm{\alpha }}_{\max }}{{\mathrm{\alpha }}_y} - 1\right)}^{{\mathrm{\Psi }}_4}\frac{1 - {\mathrm{\Psi }}_5{\mathrm{\gamma }}_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6{\mathrm{\gamma }}_{\mathrm{s}}}\right] = 0.03043 \mathrm{rad}$$

The calculation of the corresponding capacity provides a final plastic rotation value of 8θ_y = 8 × 0.008257 = 0.06605 rad.

Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the critical element (the mechanism is partial type 1 i_m = 4, so the critical element is a first storey column):

• $${\mathrm{\theta }}_{\mathrm{p}.\mathrm{mec}} = \frac{{\mathrm{n}}_{\mathrm{s}}{\mathrm{\delta }}_{\mathrm{y}}}{{\mathrm{H}}_0}[\frac{{\mathrm{\Psi }}_1\prime }{{\mathrm{\Psi }}_2\prime }{\mathrm{\Psi }}_3\prime {\left(\frac{{\mathrm{\alpha }}_{max}}{{\mathrm{\alpha }}_{\mathrm{y}}} - 1\right)}^{{\mathrm{\Psi }}_4\prime }\frac{1 - {\mathrm{\Psi }}_5\prime {\mathrm{\gamma }}_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6\prime {\mathrm{\gamma }}_{\mathrm{s}}}] = 0.04587 \mathrm{rad}$$

The calculation of the corresponding capacity, in the case of the third storey columns, provides a value of the ultimate plastic rotation equal to 8θ_y = 8 × 0.004212 = 0.033699 rad. Therefore, the ultimate conditions are governed by the columns of the first storey.

Considering the plastic rotation capacity, the ultimate displacement is given by:

$$\begin{gathered} {\delta }_u = {\delta }_C + ({\vartheta }_{p.u} - {\vartheta }_{p.mecc})H_0 = 0.76 + (0.033699 - 0.04587) \times 14.0 = \\ 0.5901 m \end{gathered}$$

Since the plastic rotation capacity of the third storey columns is lower than that necessary for the complete development of the kinematic mechanism, the points C and D corresponding to the limit states “Life Safety” and “Near Collapse” are coincident and correspond to the aforementioned last displacement (δ_u).

All the verification procedures considered the use of the transformation of the MDOF system by means of the participation coefficient of the main vibration mode Γ. For this reason, it is necessary to define:

The eigenvector ɸ = {ɸ₁, ɸ₂, ɸ₃, ɸ₄, ɸ₅, ɸ₆}that, assuming ${\mathrm{ɸ}}_k = \frac{F_k}{F_n}$, is:

$${\mathrm{ɸ}}_1 = 0.134 {\mathrm{ɸ}}_2 = 0.267 {\mathrm{ɸ}}_3 = 0.401$$

$${\mathrm{ɸ}}_4 = 0.535 {\mathrm{ɸ}}_5 = 0.669 {\mathrm{ɸ}}_6 = 0.802 {\mathrm{ɸ}}_7 = 1.000$$

The modal participation factor Γ:

$$\Gamma = \frac{{\sum }_{k=1}^n{m}_k{\mathrm{ɸ}}_k}{{\sum }_{k=1}^n{m}_k{{\mathrm{ɸ}}_k}^2} = 1.4381$$

being:

$$m_1 = 57.98 \times {10}^3 \mathrm{kg} m_2 = 57.98 \times {10}^3 \mathrm{kg} m_3 = 57.98 \times {10}^3 \mathrm{kg}$$

$$m_4 = 57.98 \times {10}^3 \mathrm{kg} m_5 = 57.98 \times {10}^3 \mathrm{kg} m_5 = 57.98 \times {10}^3 \mathrm{kg} m_7 = 61.94 \times {10}^3 \mathrm{kg}$$

The dynamic parameters of the equivalent SDOF system are reported in

Table 5. Dynamic parameters of the equivalent SDOF system.

m*	k*	ω*	T*
[kg 103]	[kN/m]	[rad/s]	[s]
224.76	7113.36	5.6257	1.117

.

Therefore, the characteristic points of the capacity curve are defined in the planes α − δ, F_b − d_c, F* − D*, S_α − S_D assessing the capacity in terms of accelerations for the Nassar and Krawinkler approach and ADRS spectrum approach. In particular, in

Table 6. ADRS spectrum approach.

	FO	O	LS	NC
F    [kN]	1284.84	2009.08	2009.08	2009.08
F*   [kN]	893.41	1397.01	1397.01	1397.01
d     [m]	0.1802	0.2824	0.5901	0.5901
d*    [m]	0.1253	0.1964	0.4103	0.4103
Sa*    [g]	0.404	0.634	1.324	1.324

, the results based on the use of the ADRS spectrum, and in

Table 7. Nassar and Krawinkler approach.

	FO	O	LS	NC
F    [kN]	1284.84	2009.08	2009.08	2009.08
F*   [kN]	893.41	1397.01	1397.01	1397.01
d     [m]	0.1802	0.2824	0.5901	0.5901
d*    [m]	0.1253	0.1964	0.4103	0.4103
μ    [m]	–	–	2.089	2.089
Sa*   [g]	0.405	0.634	1.353	1.353

, the results based on the use of the Nassar and Krawinkler formulation, are reported.

Seismic performance verification requires that, for each limit state, the inequality S_α.ls(T*)_capacity ≥ S_α.ls(T*)_demand is satisfied.

5.3. Global Moment Resisting Frame

Global moment resisting frames are designed according to the TPMC. The design results are reported in Figure 8.

The trilinear capacity curve is shown in Figure 9, which also shows the characteristic performance points of the model.

Parameters obtained from the elastic analysis:

• δ₁(α = 1) = 0.02684 m
• k = 37.2606 m⁻¹
• δ_A = δ_y = 0.1602 m
• α_A = α_y = kδ_y = 5.999

Parameters obtained from the rigid-plastic analysis:

• α₀ = 10.149
• γ_s = 0.53 m⁻¹
• α = α₀ − γ_s(δ − δ_y) → α = 10.149 − 0.53(δ − 0.1602)
• α(δ = 0) = α₀ + γ_sδ_y = 10.234
• H₀ = 24.5 m (Global collapse mechanism)

Evaluation of the maximum multiplier using the calibrated Merchant−Rankine formula:

• $${\alpha }_{m\alpha x} = \frac{{\alpha }_0}{1 + \mathrm{\Psi }{\alpha }_0{\gamma }_s{\delta }_1} = 9.7594$$

where:

$$\mathrm{\Psi } =0.28488 - 0.14042 \xi = 0.2763 \mathrm{with} \xi = \frac{\sum \frac{E{I}_b}{L_b}}{\sum \frac{E{I}_c}{L_c}} = 0.06129$$

and consequently ${\delta }_B = \frac{{\alpha }_{max}}{k} = 0.2619$ and ${\delta }_C = {\delta }_{mecc} = \frac{{\alpha }_0 - {\alpha }_{max}}{{\gamma }_{\mathrm{s}}} + {\delta }_{\mathrm{y}}$ = $$\begin{gathered} \frac{10.149 - 9.7594}{0.53} + \\ 0.1602 = 0.8946 m \end{gathered}$$.

Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the first plasticized element (first storey beams):

• $${\mathrm{\theta }}_{\mathrm{p}\mathrm{.}\mathrm{mec}} = \frac{{\mathrm{n}}_{\mathrm{s}}{\mathrm{\delta }}_{\mathrm{y}}}{{\mathrm{H}}_0}\left[\frac{{\mathrm{\Psi }}_1}{{\mathrm{\Psi }}_2}{\mathrm{\Psi }}_3{\left(\frac{{\mathrm{\alpha }}_{\max }}{{\mathrm{\alpha }}_y} - 1\right)}^{{\mathrm{\Psi }}_4}\frac{1 - {\mathrm{\Psi }}_5{\mathrm{\gamma }}_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6{\mathrm{\gamma }}_{\mathrm{s}}}\right] = 0.01886 \mathrm{rad}$$

The calculation of the corresponding capacity provides a final plastic rotation value of 8θ_y = 8 × 0.008257 = 0.06605 rad.

Evaluation of the plastic rotation demand corresponding to the development of the collapse mechanism for the critical element (the mechanism is global, so the critical element is one of the first storey columns):

• $${\mathrm{\theta }}_{\mathrm{p}.\mathrm{mec}} = \frac{{\mathrm{n}}_{\mathrm{s}}{\mathrm{\delta }}_{\mathrm{y}}}{{\mathrm{H}}_0}[\frac{{\mathrm{\Psi }}_1\prime }{{\mathrm{\Psi }}_2\prime }{\mathrm{\Psi }}_3\prime {\left(\frac{{\mathrm{\alpha }}_{max}}{{\mathrm{\alpha }}_{\mathrm{y}}} - 1\right)}^{{\mathrm{\Psi }}_4\prime }\frac{1 - {\mathrm{\Psi }}_5\prime {\mathrm{\gamma }}_{\mathrm{s}}}{1 - {\mathrm{\Psi }}_6\prime {\mathrm{\gamma }}_{\mathrm{s}}}] = 0.01774 \mathrm{rad}$$

The calculation of the corresponding capacity, in the case of the third storey columns, provides a value of the ultimate plastic rotation equal to 8θ_y = 8 × 0.003714 = 0.02971 rad. Therefore, the ultimate conditions are governed by the columns of the first storey.

Considering the plastic rotation capacity, the ultimate displacement is given by:

$${\delta }_u = {\delta }_C + ({\vartheta }_{p.u} - {\vartheta }_{p.mecc})H_0 = 0.8946 + (0.02971 - 0.01774) \times 24.5 = 1.188 m$$

All the verification procedures considered the use of the transformation of the MDOF system by means of the participation coefficient of the main vibration mode Γ. For this reason, it is necessary to define:

• The eigenvector ɸ = {ɸ₁, ɸ₂, ɸ₃, ɸ₄, ɸ₅, ɸ₆} that, assuming ${\mathrm{ɸ}}_k = \frac{F_k}{F_n}$, is:

$${\mathrm{ɸ}}_1 = 0.134 {\mathrm{ɸ}}_2 = 0.267 {\mathrm{ɸ}}_3 = 0.401$$

$${\mathrm{ɸ}}_4 = 0.535 {\mathrm{ɸ}}_5 = 0.669 {\mathrm{ɸ}}_6 = 0.802 {\mathrm{ɸ}}_7 = 1.000$$

• The modal participation factor Γ:

$$\Gamma = \frac{{\sum }_{k=1}^n{m}_k{\mathrm{ɸ}}_k}{{\sum }_{k=1}^n{m}_k{{\mathrm{ɸ}}_k}^2} = 1.4381$$

being:

$$m_1 = 57.98 \times {10}^3 \mathrm{kg} m_2 = 57.98 \times {10}^3 \mathrm{kg} m_3 = 57.98 \times {10}^3 \mathrm{kg}$$

$$m_4 = 57.98 \times {10}^3 \mathrm{kg} m_5 = 57.98 \times {10}^3 \mathrm{kg} m_5 = 57.98 \times {10}^3 \mathrm{kg} m_7 = 61.94 \times {10}^3 \mathrm{kg}$$

The dynamic parameters of the equivalent SDOF system are reported in

Table 8. Dynamic parameters of the equivalent SDOF system.

m*	k*	ω*	T*
[kg 103]	[kN/m]	[rad/s]	[s]
224.76	10108.5	6.7063	0.9369

.

Therefore, the characteristic points of the capacity curve are defined in the planes α − δ, F_b − d_c, F* − D*, S_α − S_D assessing the capacity in terms of accelerations for the Nassar and Krawinkler approach and ADRS spectrum approach. In particular, in

Table 9. European code approach.

	FO	O	LS	NC
F    [kN]	1627.71	2647.66	2647.66	2647.51
F*   [kN]	1131.83	1841.05	1841.05	1840.94
d     [m]	0.1602	0.2619	0.8946	1.1879
d*    [m]	0.1114	0.1821	0.6220	0.8260
Sa*    [g]	0.511	0.835	2.852	3.787

, the results based on the use of the ADRS spectrum, and in

Table 10. Nassar and Krawinkler approach.

	FO	O	LS	NC0
F    [kN]	1627.71	2647.66	2647.66	2776.3
F*   [kN]	1131.83	1841.05	1841.05	1930.53
d     [m]	0.1602	0.2619	0.8946	1.1879
d*    [m]	0.1114	0.1821	0.6220	0.8260
μ      [m]	–	–	3.415	4.535
Sa*   [g]	0.513	0.835	2.958	3.988

, results based on the use of the Nassar and Krawinkler formulation, are reported.

Seismic performance verification requires that, for each limit state, the inequality S_α.ls(T*)_capacity ≥ S_α.ls(T*)_demand is satisfied.

6. Conclusions

In this paper, some numeric examples explaining the application of a new assessment method are reported. As can be seen from the given numerical examples, the methodology is of easy and rapid application. The methodology is also completely analytical since the equations of the branches constituting the trilinear model can be obtained uniquely, given the horizontal seismic actions and the sections of beams and columns of the analyzed frame. The speed of application and the uniqueness show that this methodology is strongly indicated for the evaluation of seismic performances in the immediate post-earthquake or the large-scale assessment of the seismic vulnerability of the built heritage. Furthermore, it constitutes a suitable tool to check the capacity of the buildings designed with the new seismic code prescriptions. The reliability of the procedure is testified by the extensive regression analysis, carried out on 420 frames designed with different approaches. The feasibility of the procedure is very high and makes it suitable to be applied indiscriminately to frames belonging to different historical periods.

Table 11. Percentage scatter between the pushover analysis and the simplified approach.

	αmax	δmec	δu
	[%]	[%]	[%]
GMRF	0.9	1.9	5.3
SMRF	5.2	9.5	4.8
OMRF	1.8	5.1	7.2

shows a comparison between the results in terms of the percentage scatter between the values computed from pushover analysis and the proposed method of the maximum bearing multiplier α and the displacement corresponding to the collapse and formation of the plastic mechanism. It can be observed that the scatter is always lower than 10%, thus testifying the accuracy of the proposed formulations. Moreover, the results achieved by the simplified assessment procedure are on the safe side.

The assessment of structure performances, in terms of comparing capacity-demand, has been performed using the Nassar and Krawinkler approach, characterized by a wide generality because it does not discriminate between high and low periods of vibration and accounts for second-order effects. Finally, is important to note that the discretization of the trilinear model in characteristic performance points makes it easy to compare the capacity and demand for each limit state given by codes.