An Example-Guide for Rapid Seismic Assessment and FRP Strengthening of Substandard RC Buildings

Abstract

This paper presents a rapid seismic assessment and Fibre Reinforced Polymer (FRP) retrofit design methodology which relies on the European design guidelines recently published in Chapter 8 of fib Bulletin 90 on the use of externally applied FRP reinforcement in the seismic retrofitting of reinforced concrete (r.c.) structures. For this purpose, an example-guide is developed with step-by-step hand calculations aiming to facilitate engineers of practice and researchers working in the field to easily understand the proposed methodology. A three-storey, pilotis-type residential r.c. building is selected typical of the Mediterranean construction practice in the 1970s. The methodology followed only aims to provide preliminary results on seismic assessment and retrofitting before the implementation of more sophisticated analysis if need be (e.g., in case of irregular buildings). The assessment procedure identified that the columns of the ground storey, being the most critical structural elements for the stability of the structure, are vulnerable to brittle failure modes. To remove all the brittle failure modes attributed to inherent deficiencies and enhance the overall deformation capacity of the building, the strengthening schemes applied in the ground storey (pilotis) is a combination of local strengthening measures, such as FRP wrapping, and global interventions. The latter may refer to the addition of r.c. jacketing to the central column to remove slenderness and of metal X-braces to modify the lateral deflection shape of the building and thus moderate the interstorey displacement demand.

1. Introduction

In Europe, the use of externally applied FRPs in retrofitting of r.c. structures has been a subject of continuous research since the 1990s with numerous published papers, reports and successful research projects. The technical Bulletin on Externally bonded FRP reinforcement for r.c. structures ([1], and references therein) by the fib (International Federation of Concrete) Task Group 9.3 summarized the knowledge at that time and provided detailed design guidelines on the use of FRP, the practical execution and the quality control, based on the expertise and state-of-the-art knowledge of the task group members back in 2001. Its main purpose was to cover most of the design problems rather than tackling in detail all the aspects of r.c. strengthening with FRP composites. The first applications in construction appeared the same period, e.g., in Greece after the 1999 Athens earthquake, as a pressing need for immediate enhancement of the seismically vulnerable old structures (the problem of the seismic vulnerability of the existing building stock refers to methods to predict possible effects at the occurrence of seismic events and to develop prioritization plans for risk mitigation, see, i.e., [2,3]: a prioritization plan could include, i.e., FRP retrofitting). Since then, a significant amount of progress has been made that is reflected in the new Task Group 5.1 Bulletin 90 [4]. Especially Chapter 8, also presented in [5], summarizes design guidelines which are based on the comparative assessment of past models, requirements established from first principles and reference is made to the available experimental data ([6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] and references therein). Alternative retrofit strategies for the seismic strengthening with composite materials are presented considering the overall response of the existing structure. Apart from detailing of FRP interventions for seismic applications, in addition global interventions need to be considered provided that on first assessment it is deemed necessary to also moderate the deformation demand by increasing the effective stiffness of the structure. This approach is common in existing r.c. buildings where system deficiencies in lateral stiffness distribution along the height of the building (e.g., soft stories) are combined with lightly reinforced concrete members that possess limited deformation capacity [26,27].

The novelty of the methodology [4,5] lies in the use of performance-based design criteria for the assessment and retrofitting at both local and global level of existing structures. Based on the concepts developed, damage is expected to occur when the displacement demand imposed by the seismic excitation exceeds the displacement capacity of the individual members. The use of FRP jacketing at local level (i.e., structural element level) aims to modify the hierarchy of strengths by suppressing any premature failure mode (i.e., shear, anchorage/splice, compression reinforcement buckling) that usually occurs in old-type r.c. structural members due to improper seismic detailing and promote flexural failure. The addition of FRP jacketing at existing r.c. structural members with inherent deficiencies, however, does not affect the flexural strength, and thus the translational stiffness of the structures. Therefore, when enhancement of translational stiffness is deemed necessary, global retrofit measures should be applied.

The objective of this paper is to deliver an example-guide based on hand calculations of an existing multi-story r.c. building, representative of the older construction practice, which is going to be strengthened with the use of FRPs following the conceptual framework developed in Chapter 8 of Bulletin 90 [4,5]. The example-guide provides a step-by-step overview of the response of the building at local and global level and decisions on whether global strengthening measures need to be adopted along with measures at local level using FRPs. Furthermore, it sheds light to the detailed calculations entailed to perform assessment and design of the retrofit schemes. The ambition is that the example guide will provide a useful reference to the engineers of practice, researchers working in the field, university educators and students, and thus help in the wider application of the methodologies proposed in [4,5].

2. Synopsis of the Methodology

The methodology [4,5] comprises two stages; assessment and retrofitting at both local and global structural level. The assessment starts upon knowledge of the building’s geometry, materials properties, and reinforcing detailing of the primary elements (columns, beams). The axial loads of the ground floor columns are calculated based on the seismic combination and the elements’ slenderness is checked. Those columns subjected to high axial loads are considered susceptible to brittle failure, thus rendering them critical for the stability of the structure. The deformation capacity (in terms of drift at yielding and at ultimate) and flexural strength are assessed. The methodology suggests alternative approaches to estimate the deformation capacity aiming to demonstrate the easiness in application of each approach and the differences in the yielding values. At this stage the secant to yield translational stiffness is estimated and the effective period of the structure is calculated and directly related to demand in terms of elastic spectral displacement. The deformation capacity of the existing members is dictated by the type of deficiency identified and needs to be modified accordingly (i.e., in case of insufficient shear strength, short lap splices).

Strengthening at global level aims to modify the lateral response shape of the building by controlling the distribution of interstorey drift height-wise. This is achieved by the addition of stiffness along the height of the building as to comply with the target period value (lower than the effective period and closer to the code requirement) and the target response shape (i.e., target interstorey drift distribution). Design charts are used to facilitate the procedure of defining the required increase in stiffness per storey. In general, an exceedance of the code’s reference period value no higher than 25% and a shear-type shape leads to reduced intervention cost. The seismic displacement demand is recalculated for the strengthened building and demand at structural element level is defined. Based on the assessment outcome and if the ductility capacity is below demand, then the required FRP number of plies need to be calculated. In case of columns, the addition of FRP reinforcement also aims to alleviate brittle modes of failure (i.e., shear, lap-splice, compression bar buckling). Special attention is attributed to the external, beam-column joints which are not laterally supported by transverse beams and do not have sufficient detailing (i.e., horizontal stirrups).

3. Assessment

3.1. Description of the Building: Materials and Detailing

The 3-storey residential r.c. frame building has a plan 10 m × 12 m (Figure 1a) and total height H_tot = 9 m. The storey height is 3 m, the clear column height is H = 2.7 m, and thus the shear span length is L_V = H/2 = 1.35 m. Only the upper two storeys have masonry infills, whereas the ground storey is open (pilotis type of building, Figure 1b).

Typical columns have a rectangular cross section with h = 350 mm and b = 250 mm, reinforced internally with six longitudinal bars of diameter D_b = 14 mm, with clear cover c = 20 mm and effective depth $\mathrm{d}$ about the strong axis x being d = 350 − 20 − 0.5 × 20 = 323 mm. The reinforcing ratio is ρ_l = As/bh = 1.06%. The columns’ transverse reinforcement comprises stirrups of D_b,st = 6 mm at 150 mm centres, all anchored with 90° hooks at the ends (Figure 2a); this shear reinforcement is of poor effectiveness both due to its magnitude and anchorage detailing. The lap splicing length is developed at the bottom of the column and consists of a straight part l_o = 50 D_b = 700 mm with end hook (Figure 2a). In this case, the effective splice length is artificially increased by 12.5 D_b aiming to account for the influence of the hook (thus l_o = 62.5 D_b = 875 mm). All beams have a cross section height of h = 400 mm (including the flange thickness that is equal to the slab thickness, h_slab = 200 mm) and breadth b = 200 mm, reinforced as presented in Figure 2b. The beam transverse reinforcement comprises stirrups of D_b,st = 6 mm at 250 mm centres anchored with 90° hooks at the ends. The materials mean strength is considered for the assessment procedure [28]: the mean in situ cylindrical concrete compressive strength is f_cm = 16 MPa (by following [29]). The longitudinal reinforcement is ribbed steel StIV with an average yield strength based on tests of extracted samples f_ys = 500 MPa. Transverse reinforcement is smooth steel StI with an average yield strength based on tests of extracted samples f_yw = 240 MPa.

3.2. Assessment of Pilotis Columns

In a pilotis-type frame structure (i.e., common practice in Greece and southern countries in Europe) infill walls are absent in the ground storey bays to allow for parking or shop windows. The lateral deformation imposed by the seismic loading localizes in the ground floor due to the low stiffness compared to that of the upper floors, thus acting as a soft storey. Therefore, an assessment of ground floor columns is critical since earthquake events have revealed that ground columns’ failure precedes that of beams (for beams a similar assessment procedure may be followed, not presented here due to page limitations).

3.2.1. Slenderness

A first step is to check the slenderness of the columns. The total gravity load of each floor, W_floor, is calculated by the seismic combination, G + 0.3 Q, where G stands for the total dead load (25 kN/m³ of reinforced concrete for the slab of thickness h_f = 0.2 m and additional 2 kN/m²) and Q is the total live load (3.5 kN/m²), hence, W_floor = plan area·(Gh_f + 0.3 Q) = 120 m²·(25 × 0.2 + 2 + 0.3 × 3.5) = 966 kN whereas the total weight carried by the building’s ground floor columns is W_tot = 3 × 966 = 2900 kN (three floors). For simplicity the weight of the external brick masonry walls of the 2nd and 3rd floor was not considered in the building mass because it represents about 5–10% of the weight of the r.c. elements. For example, the masonry weight per floor is W_masonry ≈ 165 kN (see the detailed calculation in

Table A1. Detailed computations of several assessment and retrofitting indices.

Formulae	Computation
masonry weight per floor: Wmasonry = specific weight × masonry thickness × perimeter × height	Wmasonry = 1200 Kgr/m3 × 12 cm (that is the total thickness of a two-layer bricks of breadth 6 cm) × 2 × (9.95 + 11.25) m × 2.7 m ≈ 165 kN
slenderness effect: (i) the radius of gyration for the cross section of the column with respect to the weak axis y-y is i = √(Ig/Ag) ≈ 0.3b and (ii) the slenderness is λ = (βo∙H)/I and λ < λlim = max{25; 15/√vEd}	Calculations for central column with vEd = 0.52: i = √(Ig/Ag) ≈ 0.3b = 0.3 × 250 = 75 mm λ = (βo∙H)/i = (1 × 2700)/75 = 36 and λ = 36 > λlim = max{25; 15/√0.52 = 20.8} = 25
Flexural strength: My = Asl,1 × fym × jd + NEd × (0.5 h − 0.4 × 0.25∙d)	For corner columns (NEd = 181.3 kN): My = 462 × 500 × 0.85 × 323 + 181.3 × 103 × (0.5 × 350 − 0.4 × 0.25 × 323) ≈ 89 kNm
plastic hinge length lpl (three alternatives): Equation (3a) Equation (3b) Equation (3c)	lpl(3.a) = 0.1 × 1350 + 0.17 × 350 + 0.24 × (14 × 500)/√16 = 615 mm lpl(3.b) = 0.2 × 350 × [1 + 1/3 min(9,1.35/0.35)] = 160 mm lpl(3.c) = 0.5 × 323 = 161.5 mm
ultimate chord rotation θu: θu = 1/γel [θy + (ϕu − ϕy) lpl (1 − 0.5 lpl/LV)]	For corner columns: θu(lpl = 615 mm) = 1/1.5 × [0.64% + (0.05 − 0.0143) × 0.615 × (1 − 0.5 × 0.615/1.35)] = 1.6%, or θu(lpl = 160 mm) = 1/1.5 × [0.64% + (0.05 − 0.0143) × 0.16 × (1 − 0.5 × 0.16/1.35)] = 0.8%
Curvature ductility: Equation (4a) Equation (4b) μθ value shall be multiplied by 1.5 to account for the contribution of reinforcement pullout to the rotation capacity.	for corner columns (νEd = 0.13 < 0.2): μϕ = 0.45 × 0.0035/0.0025 × 350/(0.9 × 0.24 × 323) = 3.16 and μθ = 1.5 × [0.5 × (3.16 + 1)] = 3.1 and θu =3.1 × 0.64% = 2%
Equation (5b) Equation (5a)	al = (1 − 150/(2 × 210))∙(1 − 150/(2 × 310)) × 4/6 = 0.32 lou,min = 14 × 500/[1.05 + 14.5 × 0.32 × 0.00151 × 240/16 × √16] = 1513 m
Equation (6b) Equation (6c) Equation (6a)	for corner columns: VRd,c = 0.41√16 × 250 × 0.9 × 0.24 × 323/1000 ≈ 29 kN for peripheral: VRd,c = 0.41√16 × 250 × 0.9 × 0.27 × 323/1000 ≈ 32 kN VRd,s = 2 × (π × 622/4)/(150 × 210) × 210 × 310 × 240/1000 = 28 kN VRd,o|corner = 1/1.15{(350 − 0.9 × 0.24 × 323)/(2 × 1350) × min(181.25,0.55 × 250 × 350 × 16/1000) + 0.89 × (29 + 28)} ≈ 60 kN < 1.5 Vfl = 1.5 × 66 = 99 kN VRd,o|periph. = 1/1.15{(350 − 0.9 × 0.27 × 323)/(2 × 1350) × min(362.5,0.55 × 250 × 350 × 16/1000) + 0.92 × (32 + 28)} = 80 kN < 1.5Vfl = 1.5 × 85 ≈ 128 kN
translational stiffness of the ground storey (soft storey): Keff = ∑i=1nKi ΔΦi2 = K1 × 12 = K1	Keff = 4 (corner columns) × 3810 kN/m + 4 (per. columns) × 4910 kN/m + (central column) × 3810 kN/m ≈ 38690 kN/m
FRP as shear reinforcement: For simplicity: 1.5Vfl = max(1.5Vfl,corner,1.5Vfl,peripheral) VRd,o = min(VRd,o,corner,VRd,o,peripheral) VRd,f = 1.5Vfl − VRd,o = (2tf/b) × bh × Ef × εfu,h	1.5Vfl = max(99 kN, 128 kN) ≈ 128 kN VRd,o = min(60 kN, 80 kN) = 60 kN ΔV = 1.5Vfl − VRd,o = 68 kN ΔV = VRd,f = (2tf/b)·bh·Ef·εfu,h → 68,000 Nt = (2tf/250) × 250 × 350 × 165,000 × 0.0085 → tf = 0.07 mm < to = 0.12 mm → 1 layer
Equation (10)	for Db = 14 mm, μfr = 1, Nb = 3, εf,sl = 0.0015, tf = nto, Ast = 2 × π × 62/4 = 56.5 mm2, fy,st = 240 MPa and s = 150 mm: 2.3 = (2 × 1)/(3.14 × 14)(0.33 × 56.5 × 240/(3 × 150) + 2 × tf × 165,000 × 0.0015)/3) → tf = 0.25 = n × 0.12 → n = 2 layers
Equation (11b) Equation (11c) Equation (11a)	αf =1 − ((250 − 2∙33)2 + (350 – 2 × 33)2)/(3 × 250 × 350) = 0.56 ρsv= ((2 × 200 + 2 × 300) × 28.26)/(150 × 200 × 300) = 0.00314 εcu,c = 0.005 = 0.0035 + 0.075[1(0.56 × ρfv × 165,000 × 0.0085 + 0.15 × 0.00314 × 240)/16 − 0.1] → ρfv = 0.0023 and ρfv = 2tf(b + h)/(bh) → 0.0023 = 2tf (350 + 250))/(350 × 250) → tf = 0.17 mm tf = nto → 0.17 mm = n × 0.12 mm → n = 2 FRP layers
Equation (12)	µΔ = 1.3 + 12.4 × (0.5 × 0.56 × 0.0033 × 165000 × 0.0085/16 − 0.1) = 1.06 ≥ 1.3 thus µΔ = 1.3
The load from the slabs is assumed to be transferred to beam B1. Conservatively, it is assumed that the loads transferred from the slab to B1 correspond to a quarter of the area of the slab. Hence, the total linear load for the seismic combination is: g = [((1/4 × 6 × 5 × 0.2))⁄6]slab × 25 + (0.2 × 0.2)beam × 25 + [((1/4 × 6 × 5))⁄6]slab × 2 = 9.75 kN⁄m, q = [((1/4 × 6 × 5))⁄6]slab × 3.5 = 4.38 kN⁄m and g + 0.3q = 9.75 + 0.3 × 4.38 = 11.06 kN⁄m, (Vg+ψq,b)l = (g + 0.3q)l/2 = (11.06 × 6)/2 = 33.19 kN. Equation (13): Vj,v = ΣMyc(1/jdc − 1/Lb,n × Hn/H) + 1/2|(Vg+ψq,b)l − (Vg+ψq,b)r| = 101 × 103(1/((250 − 2∙30)) − 1/3000 × 3000/2700) + 1/2 × |33.19 − 0| = 510.8 kN
Equation (14)	Vj,h = 134 × 103(1/(0.9 × 370) − 1/3000 × 2500/2325) = 354.4 kN

of Appendix A—the same is valid for the next calculations where it is deemed necessary). Thus, the total masonry weight is 330 kN—this is an upper limit since the calculation does not consider any openings. Each ground column undertakes part of the total axial load after multiplying W_tot by the ratio of the tributary area (i.e., the light hatched area as per column C1 in Figure 1a) to the total floor area. The values of axial loads N_Ed along with the corresponding axial load ratios, v_Ed = N_Ed/(bhf_cm) are N_Ed = 181.3 kN—v_Ed = 0.13 for the corner columns C1/3/7/9, and N_Ed = 362.5 kN—v_Ed = 0.26 for all the peripheral columns C2/4/6/8. The central column C5, by having N_Ed = 725 kN—v_Ed = 0.52, is prone to crushing before yielding due to the high axial load. In order to reduce v_Ed below 0.4, thus securing a dominant flexural response, the increase in the cross-section dimensions is required. The slenderness effect is taken into consideration by following the procedure of ACI 318-19 [30]. The radius of gyration for the cross section of the column with respect to the weak axis y-y (Figure 1a) is i = √(I_g/A_g) ≈ 0.3 b = 75 mm and the associated slenderness is λ = (β_o∙H)/i = 36 > λ_lim = max{25; 15/√v_Ed = 20.8} = 25. The reduction in slenderness is achieved by increasing the cross section to h^new = 450 mm and b^new = 350 mm. The new calculations are: v_Ed^new = 0.29, i^new ≈ 0.3 b^new = 105 mm and λ^new = 25.7 < λ_lim = max{25; 27.9} = 27.9. Note that v_Ed^new = 0.29 is an upper value given that the increase in the cross section dimensions is usually implemented by using concrete of higher quality.

3.2.2. Deformation and Strength Indices from Flexure

The strength and deformation capacity were assessed as per the response of the building in y-y direction (strong axis: bending about x-x in Figure 1a). The corner columns are expected to yield first due to the lower axial load. The central column C5 is susceptible to crushing before yielding due to the high axial load. For lightly reinforced concrete elements, the curvature at the onset of tension reinforcement yielding ϕ_y is approximated ϕ_y = (2ε_sy)/h = 2 × 0.0025/350 = 0.0143/m, common for all columns. The moment at yielding as per the point of action of the concrete force can be approximated by Equation (1):

M_y = A_sl,1∙f_ym∙jd + N_Ed(0.5 h − 0.4 × 0.25∙d)

(1)

where jd = 0.85 d is the internal lever arm between tensile force of bottom steel reinforcement and concrete compressive force and A_sl,1 = 3 × π × 14²/4 = 462 mm² is the area of tensile reinforcement. The implementation of Equation (1) for corner columns (N_Ed = 181.3 kN) results to M_y ≈ 89 kNm and for peripheral columns (N_Ed = 362.5 kN) to M_y ≈ 115 kNm. A more precise calculation of curvature and moment at yielding can be deduced from cross section analysis by using Response2000 [31], where for corner columns ϕ_y = 0.0128/m, M_y = 90 kNm and for peripheral columns ϕ_y = 0.0148/m, M_y = 109 kNm. The two procedures for definition of ϕ_y, M_y result to similar values.

The curvature at ultimate ϕ_{u =} ε_cu/(ξd), at concrete crushing strain ε_cu = 0.0035, may be found by using the graph of Figure 3 (similar graphs were derived in [32]): for symmetrically reinforced cross section (present case), the normalized compression zone ξ = x/d (d = 323 mm) is plotted against the total longitudinal reinforcing ratio ρ_l for several axial load ratios. This graph was deduced at yielding of tensile reinforcement; aiming to address the fact that at ultimate the compressive zone x becomes shorter, a reduction factor 0.9 is also applied to ξ value. Thus, for ρ_l = 1.06% for corner columns (v_Ed = 0.13) is ξ = 0.24 and ϕ_u = 0.0035/(0.9 × 0.24 × 323) = 0.05/m and for peripheral (v_Ed = 0.26) is ξ = 0.27 and ϕ_u = 0.044/m. All useful values of the assessment are summarized in

Table 1. Results from cross section analysis and rotation capacity.

	Cross Section Analysis: Approximations				Chord Rotation from Procedures (a–c)
Column	ϕy = 2εsy/h (1/m)	My (kNm)	K (kNm−1)	ϕu (1/m) − ξ from Figure 3	θy (%)	(a): θu (%) (µθ = θu/θy)		(b): θu (%) (µθ = θu/θy)	(c): θu (%) (µθ = θu/θy)
						lpl = 160 mm	lpl = 615 mm
1/3/7/9 vEd = 0.13	0.0143	89	3810	0.05–0.24	0.64	0.8 (1.2)	1.6 (2.4)	2 (3.1)	1.7 (2.7)
2/4/6/8 vEd = 0.26		115	4910	0.044–0.27		0.73 (1.1)	1.4 (2.2)	1.7 (2.6)	1.8 (2.8)

.

The secant to yield effective stiffness of the columns is given as EI = M_y/ϕ_y. Hence, for the two groups of columns: (i) EI = 89/0.0143 ≈ 6250 kNm² for corner columns, (ii) EI = 115/0.0143 ≈ 8050 kNm² for peripheral one. For concrete grade C16/20 the Young Modulus is E = 29 GPa, the moment of inertia of uncracked section is I = bh³/12 = 8.93 × 10⁸ mm⁴ (no consideration of internal reinforcement) thus the elastic stiffness is EI_el ≈ 25,900 kNm². This is near threefold the effective values above (in pushover analysis cross sections are assumed cracked, hence 50% of the elastic value is used, EI_cr = 50%EI_el = 12,950 kNm²). The secant to yield translational stiffness of the columns is calculated by using K = 12 EI/H³. For corner columns K = 12 × 6250/2.7³ = 3810 kNm⁻¹ and for peripheral columns K ≈ 4910 kNm⁻¹ whereas the elastic and that of the cracked cross section values are 15,790 kNm⁻¹ and 7895 kNm⁻¹, respectively. By comparing values of K, the order of magnitude of the error for the estimated translational stiffness may be large, depending on the adopted approach. This difference has a great impact on the structure period of vibration, and thus on the maximum displacement that is going to be developed during the earthquake. Later, in Section 4, the secant to yield translation stiffness is used in defining the effective translational period T_eff of the building (Table 1).

The chord rotation at yielding is approximated as θ_y = (ϕ_yH)/6 = (0.0143 × 2.7)/6 = 0.64%. A more detailed expression of this property is [33]:

θ_y = 1/3ϕ_y(a_vz + L_V) + 0.0014(1 + 1.5 h/L_V) + 0.125ϕ_yD_b f_sy)/√f_cm

(2)

with a_vz = 0 (it is assumed that shear cracking is not expected to precede flexural yielding), L_V = 1350 mm (i.e., half the column clear height), h = 350 mm, D_b = 14 mm, f_sy = 500 MPa and f_cm = 16 MPa. Equation (2) results to θ_y = 1.15% that is almost double of the simplified estimation (θ_y = 0.64%). However, the most conservative value θ_y = 0.64% is used in the following calculations because it is testified in relevant tests on substandard columns as a lower bound magnitude [34].

The ultimate chord rotation θ_u is estimated by following the three alternative procedures presented in chapter 8 of [4] and are summarized in Table 1. More specifically:

(a)

From basic mechanics:

θ_u = 1/γ_el [θ_y + (ϕ_u − ϕ_y) l_pl (1 − 0.5 l_pl/L_V)]

(3)

where γ_el is equal to 1.5 for primary members, θ_y = 0.64%, ϕ_u is the ultimate curvature of the end section evaluated by assigning the concrete ultimate strain ε_cu = 0.0035 (values are shown in Table 1). The plastic hinge length l_pl can be estimated from three alternatives, as:

l_pl = 0.1 L_V + 0.17 h + 0.24 (D_bf_sy)/√f_cm

(3a)

l_pl = 0.2 h [1 + 1/3 min (9,L_V/h)]

(3b)

l_pl = 0.5 d

(3c)

Their implementation results to l_pl(3a) = 615 mm or l_pl(3b) = 160 mm or l_pl(3c) = 161.5 mm, respectively. Note that Equation (3b) considers cyclic loading and Equation (3c) is a simple definition of l_pl for members with old type detailing. The results of Equation (3b,c) coincide. The difference in results for l_pl obtained from Equation (3a–c) is considerably high and affects the estimations of the elements’ ductility. For example, for corner columns θ_u(l_pl = 615 mm) = 1.6% or θ_u(l_pl = 160 mm) = 0.8%. The corresponding rotation ductility is μ_θ = θ_u/θ_y = 1.6/0.64 = 2.4 or μ_θ = 0.8/0.64 = 1.2. Apparently, the results in Table 1 for l_pl = 160 mm suggest that all columns—except of the middle one—fail close to yielding demonstrating no ductility. For l_pl = 615 mm all columns demonstrate adequate ductility in the order of 2.

(b)

Empirically, by following the proposed Equation (4a,b):

μ_ϕ = 0.45 ε_cu,c/(ε_syν_Ed) for ν_Ed ≥ 0.2 and μ_ϕ = 0.45 ε_cu,c/ε_sy × h/ξd for ν_Ed < 0.2

(4a)

μ_θ = θ_u/θ_y = μ_Δ = 0.5(μ_ϕ + 1)

(4b)

where, term ξ from Table 1 should be multiplied with 0.9 (as previously explained), ε_cu,c = 0.0035, ε_sy = 0.0025 and θ_y = ϕ_yH/6 = ε_sy(H/h)/3 = 0.64%. The μ_θ value shall be multiplied by 1.5 to account for the contribution of reinforcement pullout to the rotation capacity. Implementing Equation (4a,b) for example for columns C1/3/7/9 (ν_Ed = 0.13 < 0.2), result in μ_ϕ = 3.16 and μ_θ = 1.5 × [0.5 × (3.16 + 1)] = 3.1 and θ_u =3.1∙0.64% = 2%. This value of θ_u [Table 1, column denoted as (b)] is close to that calculated above in procedure (a) when using l_pl = 615 mm from Equation (3a).

(c)

The value of the plastic part of the chord rotation capacity of concrete members under cyclic loading is given by the following calibrated Equation (5) (from [33]):

$${\mathsf{\theta }}_{\mathrm{u}}^{\mathrm{pl}}=\frac{1}{{\mathsf{\gamma }}_{\mathrm{el}}}0.0185\times 0.48\left(1+\frac{{\mathsf{\alpha }}_{\mathrm{sl}}}{1.6}\right){\left(0.25\right)}^{\mathsf{\nu }}{\left[\frac{\max \left(0.01,{\mathsf{\omega }}^{\prime }\right)}{\max \left(0.01,\mathsf{\omega }\right)}\right]}^{0.3}{\mathrm{f}}_{\mathrm{c}}^{0.2}{\left(\frac{{\mathrm{L}}_{\mathrm{V}}}{\mathrm{h}}\right)}^{0.35}\times {25}^{\left[\frac{{\mathsf{\alpha }}_{\mathrm{w}}{\mathsf{\rho }}_{\mathrm{wy}}{\mathrm{f}}_{\mathrm{yw}}}{{\mathrm{f}}_{\mathrm{c}}}\right]}{1.275}^{100{\mathsf{\rho }}_{\mathrm{d}}}$$

(5)

where θ_u =θ_y + ${\mathsf{\theta }}_{\mathrm{u}}^{\mathrm{pl}}$ (θ_y = 0.64%). The implementation of this expression is demonstrated for corner columns C1/3/7/9: γ_el = 1.8, α_sl = 1 because slippage of vertical bars from their anchorage or lap-splice at the lower end of the column is physically possible (presence of splices), ν = 0.13, ω = ρ_s1f_sy/f_c = 0.53% × 500/16 = 0.165, and ω’ (even if ρ_s1 = ρ_s2 = 0.5ρ_l = 0.53%) is taken double of ω due to the presence of splices, f_c = 16 MPa, f_sy = 500 MPa, L_V = 1350 mm, h = 350 mm, ρ_d = 0 (absence of diagonal reinforcement), ρ_wy = A_sy/(s_hb) = 2 × (π × 6²/4)/(150 × 250) = 0.15% (ratio of transverse steel parallel to the direction of loading, here y-direction), α_w = 0.14 (confinement effectiveness factor, limited contribution because stirrups are sparse) and f_yw = 240 MPa. This results in ${\mathsf{\theta }}_{\mathrm{u}}^{\mathrm{pl}}$ = 2.33% and θ_u = 0.64% + 2.33% ≈ 3%. For the peripheral columns the calculations lead to θ_u = 0.64% + 1.95% = 3.1%. It is important to note that in this procedure (c) the presence of splice l_o in the vicinity of plastic hinge length deteriorates the plastic rotation when l_o < l_ou,min and then the total rotation capacity should be multiplied by l_o/l_ou,min; term l_ou,min is given by Equation (5a,b):

$${\mathrm{l}}_{\mathrm{ou},\min }=\frac{{\mathrm{D}}_{\mathrm{b}}{\mathrm{f}}_{\mathrm{sy}}}{\left[\left(1.05+14.5{\mathrm{a}}_{\mathrm{l}}{\mathsf{\rho }}_{\mathrm{wy}}{\mathrm{f}}_{\mathrm{yw}}/{\mathrm{f}}_{\mathrm{c}}\right)\sqrt{{\mathrm{f}}_{\mathrm{c}}}\right]}$$

(5a)

a_l = (1 − s_h/(2b_o)) × (1 − s_h/(2h_o)) × n_restr/n_tot

(5b)

Their implementation results to a_l = 0.32 and l_ou,min =1513 m. The hooked splice may be assumed as sufficient detailing for earthquake resistance (no need to divide with 1.2). Because l_o = 875 mm < l_ou,min = 1513 mm, thus the ultimate rotation for the corner columns is θ_u = 875/1513 × 3% = 1.7% and for peripheral columns, θ_u = 875/1513 × 3.1% = 1.8%.

The calculations of θ_u deduced by the three procedures (a–c), led to µ_θ results higher than 2, however when considering a narrow plastic length, they may receive values close to 1. The demonstration of the three procedures aims at highlighting the easiness or complexity each one encompasses as per its implementation.

3.2.3. Brittle Mechanisms: Shear and Lap-Splices

The calculations in Section 3.2.2 for determining the flexural response of the ground storey columns ignored the columns’ shear capacity. The preceding calculations are meaningful provided the columns are able to develop the lateral force V_fl that corresponds to flexural capacity, i.e., V_fl = M_y/L_V. Thus, for corner columns V_fl = 89/1.35 ≈ 66 kN and for peripheral V_fl = 115/1.35 ≈ 85 kN. Note that V_fl should be multiplied by a safety factor >1 aiming to account for the over-strength of columns. More specifically, in accordance with EN1998-1 [35], for medium ductility r.c. buildings (DCM), the over-strength for beams and columns may be found by the product of the flexural capacity with (i) the material safety factor 1.15 for steel, (ii) a factor equal to 1.2 due to design-action effects at the end sections of critical regions, and (iii) a factor equal to 1 and 1.1 due to steel strain hardening for beams and columns, respectively. Thus, for columns the overstrength factor is 1.15 × 1.2 × 1.1 ≈ 1.5. The shear capacity of the columns is given by Equation (6) (from [33]):

V_Rd,o = 1/γ_el {(h − x)/(2L_V)·min(N, 0.55A_cf_c) + [1 − 0.05min(5,μ_θ^pl)](V_Rd,c + V_Rd,s)}

(6a)

V_Rd,c = 0.41√(f_c)b × x

(6b)

V_Rd,s = ρ_sw,yb_oh_of_y,st = A_sy/(s_hb_o)b_oh_of_y,st

(6c)

where γ_el = 1.15, x = ξd (ξ from Table 1, multiplied also with the correction factor 0.9 as explained before). Applying Equation (6b,c), for corner columns result to V_Rd,c ≈ 29 kN, for peripheral to V_Rd,c ≈ 32 kN and V_Rd,s = 28 kN. For the definition of μ_θ^pl = (θ_u − θ_y)/θ_{y =} μ_θ − 1, among procedures (a–c) above, procedure (b) is chosen because it results to a more conservative estimation of V_Rd,o. Thus for the corner columns the term [1 − 0.05 min(5,μ_θ^pl)] in Equation (6a) is [1 − 0.05 min(5,3.1−1)] = 0.89 and for peripheral columns is 0.92 (μ_θ^pl = 2.6 − 1 = 1.6). Next, Equation (6a) is implemented: for corner columns is V_Rd,o ≈ 60 kN < 1.5 V_fl = 1.5 × 66 = 99 kN and for peripheral V_Rd,o = 80 kN < 1.5 V_fl = 1.5 × 85 ≈ 128 kN.

It may conclude that both corner and peripheral columns will fail in shear near yielding at almost the same drift (θ_yV_Rd,o/V_fl: for corner columns is 0.64% × 60/66 = 0.59% and for peripheral 0.64% × 80/85 = 0.6%).

Another concern is whether the available splice length l_o suffices for the reinforcing bars to develop their yielding strength. The available bond strength is given by Equation (7):

τ_b = 2μ_fr/(πD_b) × [2cf_ctk + 0.33 (A_st f_y,st)/(N_bs)]

(7)

and yielding is secured when τ_b ≥ γ_elD_bf_sy/(4l_o). Values for the parameters are: μ_fr = 1, D_b = 14 mm, f_ctk = 0.33 f_cm^0.5 = 0.33 × 16^0.5 = 1.3 MPa, c = 20 mm, A_st = 2 × π × 6²/4 = 56.5 mm², f_y,st = 240 MPa, N_b = 3 bar pairs, s = 150 mm, γ_el = 1.15, f_sy = 500 MPa, thus Equation (7) yields to τ_b = 2.82 MPa. For a splice with a hook (l_o = 875 mm) is τ_b = 2.8 MPa > γ_elD_bf_sy/(4l_o) = 2.3 MPa, thus lap splice suffices longitudinal reinforcement yielding, but after that milestone (where µ_θ > 1) cover cracking is anticipated; this implication diminishes cover contribution (i.e., 2cf_ctk ≈ 0) in defining bond strength thus a splice failure is also anticipated.

From the preceding assessment analysis, it may be concluded that the ground columns may suffer shear failure before any yielding of the longitudinal reinforcement that occurs at a drift of 0.6%; the latter corresponds to lateral displacement at the top of the ground storey Δ = 0.6% × 2.7 m = 16 mm. [Note: If a straight splice without hook detailing was used, then lap splice failure is also anticipated for both the peripheral and the corner columns before yielding].

4. Global Strengthening Requirements

The strengthening of the existing building should consider whether the available effective stiffness, K_eff, suffices to resist the seismic demand or if there is need to moderate, i.e., the drift demand. In the latter case, the retrofit solution should include global intervention measures to increase K_eff along with local interventions like FRP jacketing for alleviating brittle modes of failure (Figure 4a). Note that by increasing K_eff the demand may be reduced in two different ways: (a) a higher effective stiffness results in a lower predominant period tending towards the left in the displacement spectrum, and (b) through a more uniform distribution of deformation demand in the structure, which secures that the magnitude of deformation demanded by individual members is lowered.

The necessary calculations for global measures are based on Appendix 8.2 of [4]. More specifically, the total building mass is M = 290 tonnes. The effective translational period T_eff is calculated based on the secant to yield sectional analysis for the definition of individual column translational stiffness K. By assuming that the building has a soft first storey (i.e., pilotis-type), then practically all lateral translation is concentrated in the ground storey (K_eff = ∑_i=1ⁿK_i ΔΦ_i² = K₁ × 1² = K₁). Thus, the translational stiffness against seismic sway of the ground storey is (direction y-y): K_eff ≈ 38,690 kN/m. Note that column’s C5 stiffness was taken conservatively as the least among the other two groups. Its reinforcement does not reach yielding due to the applied high axial load. Therefore, the effective period is T_eff = 2π√(M/K_eff) = 2 × 3.14 × √(290/38,690) = 0.54 s. The empirical reference value T_ref in EN 1998-1 [35] is T_ref = 0.075 H_tot^3/4 = 0.075 × 9^3/4 = 0.39 s. Because T_eff/T_ref =0.54/0.39 = 1.4, T_eff exceeds by more than 25% the empirical T_ref. Thus, K_eff should be increased to lower T_eff to an acceptable range of values, i.e., 0.4–0.45 s for a three-storey building.

The target or improved period, T_trg, of the retrofitted structure may be selected based on experience, as a value between T_ref and the initial T_eff. A note of caution is that the cost of the intervention increases as T_trg is reduced getting closer to T_ref. Besides, an increase in K_eff is required aiming at modifying the lateral deflection shape (fundamental response shape) due to localization of deformation at the ground floor columns (i.e., the assumed in Figure 5b). For the seismic assessment of the building, the peak ground acceleration was considered PGA = 0.24 g and the soil class as B. For T_eff = 0.54 s, the elastic spectral displacement demand is estimated from Equation (8) for T_C ≤ T ≤ T_D (from [35]):

S_d (T) = a_g Sηβ_o (T_CT)/40

(8)

where S = 1.2 (soil class B), T_C = 0.5 s, T_D = 2 s, a_g = 0.24 g, η = 1 (ξ = 5%) and β_o = 2.5, hence S_d = 0.048 m. The elastic displacement S_d will induce a lateral drift in the pilotis θ = S_d/H = 0.048/2.7 ≈ 1.8% which corresponds to drift or displacement ductility demand of μ_θ = μ_Δ = 1.8/0.64 = 2.8. According to Table 1, and after the implementation of only local measures to alleviate the brittle shear failure, the drift supply by considering the most conservative procedure a is lower than the demand for all columns. The reduction in the drift demand along with the improvement of the deflection shape of the pilotis-type building requires the increase in pilotis stiffness (i.e., adoption global intervention measures) before any application of FRP jacketing (local measures).

The selection of a shear-type drift distribution along the building’s height (Figure 5b) reduces the required intervention because a soft storey formation may be re-engineered towards this option for moderate improvement. By using the charts of Figure 4b,c result in the following target values for the storey stiffness ratios K_i/K₁: K₂/K₁ = 1.1 and K₃/K₁ = 1.6 (see the dashed lines in Figure 4b, K₁ stands for the ground floor or first floor). For the chosen target period T_trg = 0.45 s and by using Figure 4c, the required ground storey stiffness is K₁/m₁ = 950 kN/m, which once multiplied with storey mass m₁ = 96.6 tn results in K₁ = 91,770 kN/m, also, K₂ = 1.1 × 91,770 ≈ 100,950 kN/m and K₃ = 1.6 × 91,770 ≈ 146,830 kN/m for the 2nd and 3rd floor, respectively. The target stiffness value of the ground storey, K₁ = 91,770 kN/m, is much larger than the available K_eff = 38,690 kN/m. Such a stiffness increase in ground storey may be achieved partly by the need to increase the central column cross section due to slenderness effect and by adding metallic cross bracings in the direction of the seismic action under consideration (here the direction is y-y).

The central column of increased cross section (350 × 450 mm) is assumed as being pinned at the base with elastic stiffness K_C5,new = 3EI/H³ = 11,748 kN/m (the initial value was 3810 kN/m). The required cross braces stiffness is deduced as: ΣK_X = K₁ − K_C5,new − 4 corner columns × 3810 kN/m − 4 peripheral columns × 4910 kN/m ≈ 45,140 kN/m. The span of each brace is L = 5 m − 1.5 × 0.35 m = 4.475 m and ϕ = arctan(h_st,cl/L) = 0.6 thus ϕ = 31° (h_st,cl = 2.7 m). The length of the brace diagonals is D = √(4475² + 2700²) = 5226 mm. The stiffness of braces is (modulus of elasticity for cast steel sections, E_s = 150,000 MPa): ΣK_X = 45,140 kN/m =2E_sA_br/Dcos²ϕ = 2 × 150,000 × A_br/5226 × cos²(31°) thus A_br ≈ 1100 mm². By adding two braces symmetrically (i.e., by connecting columns C1–C4 and C6–C9), then each brace should have total cross section A_br = 650 mm².

After the increase in the pilotis stiffness under the requirement of a shear-type deflection shape using the charts of Figure 4b, one could validate this simple solution by performing the Rayleigh iterative method and considering the stiffness of the exterior infill walls of the upper floors. More specifically, in each of the upper two floors, for two exterior solid walls in y-y direction of a total cross section area A_w = 2 × (10 m − 3 × 0.35 m) × 0.12 m ≈ 2.15 m² or ρ_mw = A_w/(A_floor = 120 m²) ≈ 0.018 and for h_i = 2.7 m, f_bc = 5 MPa, f_mc = 5 MPa (each is divided by γ_m = 1.5), and μ_y^{mw =} 2.5, θ_y^mw = 0.15%, then the masonry wall stiffness in each floor is: K_mw ≈ A_floor/h_i × (0.1f_bc^0.7f_mc^0.3)/(μ_y^mwθ_y^mw) × ρ_mw,i ≈ 71,100 kN/m. The available effective stiffness of the 2nd and 3rd floor are then: K₂ = K₃ = K_eff + K_mw = 38,690 + 71,100 ≈ 109,800 kN/m (Figure 5a). Because the available magnitudes K₂ and K₃ are higher than or close to the required for the chosen shear-type response of the building (i.e., K₂ = 100,950 kN/m and K₃ = 146,830 kN/m), there is no need for global measures in these floors. For the Rayleigh iterative method, the floor mass (i.e., 96.6 tonnes) is increased by adding a share of the total masonry weight W_masonry = 33 tonnes; thus W_masonry was distributed as ¼ in each of the ground and the 3rd floor and ½ in the 2nd floor. After such analysis, the deflection shape presented in Figure 5b was obtained (the light green coloured curve titled Rayleigh-final) and is very close to the shear-type on which the addition of stiffness was based. Moreover, from this analysis the calculated period is found 0.46 s, very close to the target value T_trg = 0.45 s. In addition, the initially assumed lateral deflection shape and the initial approximated shape by using the Rayleigh method are plotted in Figure 5b for comparison reasons (the masonry mass and stiffness of the upper floors is considered).

For the selected period T_trg = 0.45 s (<T_C = 0.5 s), the elastic spectral displacement demand is estimated from Equation (9) (from [35]):

S_d (T) = a_g Sηβ_oT²/40

(9)

where a_g = 0.24 g, S = 1.2 (soil class B), η = 1 (ξ = 5%) and β_o = 2.5, thus S_d = 0.036 m. Considering that this displacement will be increased by about 20% when transferring from the spectrum (Equivalent Single Degree of Freedom) to the actual structure (Multi-Degree of Freedom), then, the relative drift demand at the ground storey (ΔΦ = 0.5, Figure 5b) is θ_dem= 1.2 × 0.5 × S_d/H = 0.8%, which corresponds to a ductility demand of μ_θ = μ_Δ= 0.80%/0.64% = 1.25. This demand is very close to the supply when a short plastic hinge length is considered [procedure (a) in Table 1]. To improve ductility and to alleviate premature shear failure, FRP jacketing is required.

5. Local Strengthening through FRP Jacketing

The objective is to remove, through FRP jacketing, brittle failure modes so that the flexural capacity of columns can be fully developed. The design methodology adopted follows the procedure described in Chapter 8 of [4,5]. FRP confinement is applied as to enhance (i) displacement ductility, (ii) shear strength, and additionally, to prevent failure due (iii) buckling of steel longitudinal bars. For the needs of the current example, a generic carbon FRP fabric with the following mechanical properties was selected for column and beam retrofitting: thickness t_o = 0.12 mm, modulus of elasticity: E_f = 165,000 MPa, ultimate tensile stress: f_fu = 2970 MPa and ultimate strain ε_fuk = 0.018. The design tensile strain ε_fu,h in the FRP layer shall not exceed the limit: ε_fu,h= η₁·η₂·η₃·ε_fuk/γ_f where γ_f = 1.5 for fully wrapped FRP arrangement (i.e., the ground floor RC frames do not accommodate any masonry). Factors η₁, η₂, η₃ are considered as follows:

-

Factor η₁ accounts for the radius of chamfer R (= c + 0.5D_b = 20 + 7 = 27 mm), at the corners of the member: η₁ = 0.25 + 2(2R + D_b)/h′ = 0.25 + 2(2 × 27 + 14)/(350 − 2 × 27) = 0.71 < 1 (h′ is the straight part of the largest cross section side).

-

Factor η₂ = 1 accounts for the sufficiency of the wrap development length: the straight parts of the cross section sides h′ ≈ 300 mm and b′ ≈ 200 mm suffice to accommodate the minimum anchorage length of the external FRP layer l_b^min = 0.5π√(E_f × t_o × s_o/τ_a)= 0.5π√(165,000 × 0.12 × 0.5/5) ≈ 70 mm (s_o and τ_a are slip and bond strength values, provided by the resin manufacturer); to this end the external layer of the FRP jacket can be anchored over the column’s shorter side.

-

Factor η₃ = 1 for fully wrapped jacket (considers the redundancy of the jacket against debonding).

Accordingly, the design tensile strain ε_fu,h in the FRP layer is ε_fu,h = η₁·η₂·η₃·ε_fu,h= 0.71 × 1 × 1 × 0.018/1.5 = 0.0085.

Given the required displacement ductility (μ_θ = 1.25) from Section 4, the associated curvature ductility is μ_ϕ = 2μ_θ − 1 = 2 × 1.25 − 1 = 1.5. The maximum compression strain demand is ε_cu,c = 2.2μ_ϕε_syν_Ed [form Equation (4a)]. For the corner columns (v_Ed = 0.13) ε_cu,c = 2.2 × 1.5 × 0.0025 × 0.13 = 0.0011 < 0.0035 and for the peripheral columns (v_Ed = 0.26) ε_cu,c = 0.0021 < 0.0035. From these calculations, it is concluded that there is no need for confinement of concrete through closed FRP jackets to meet its strain demands in the compression zone resulting from the displacement ductility calculation.

According to Section 3, flexural response dominates when the shear strength V_Rd exceeds 1.5 V_fl. For the simplicity of the following calculations it is taken as 1.5 V_fl = max(1.5 V_fl,corner,1.5 V_{fl,peripheral}) ≈ 128 kN and also the available V_Rd,o = min(V_Rd,o,corner,V_{Rd,o,peripheral}) = 60 kN. The difference ΔV = 1.5 V_fl − V_Rd,o = 68 kN will be resisted by FRP closed-type jacketing as ΔV = V_Rd,f = (2t_f/b) × bh × Ef × ε_fu,h, which it results to t_f = 0.07 mm < t_o = 0.12 mm. A single layer suffices for shear strengthening along the column’s height.

For the lap-splices, even if the available bond strength τ_b = 2.82 MPa [from Equation (7)] due to cover and stirrups contributions suffices for bar yielding [τ_b = 2.82 MPa > γ_elD_bf_sy/(4l_o) = 2.3 MPa], however it is reduced after that milestone because cover is cracked [term 2cf_ctk in Equation (7) is set equal to 0]. The contribution of FRP jacketing in restoring the bond strength to the yielding limit is given by Equation (10):

τ_b = 2μ_fr/(πD_b)[0.33A_stf_y,st/(N_b·s) + 2t_f E_fε_f,sl)/N_b]

(10)

thus, for D_b = 14 mm, μ_fr = 1, N_b = 3, ε_f,sl = 0.0015, t_f = nt_o, A_st = 2 × π × 6²/4 = 56.5 mm², f_y,st = 240 MPa and s = 150 mm Equation (10) results to t_f = 0.25 = nt_o = n × 0.12, or n = 2 layers needed to wrap the splice region of l_o = 700 mm, that is developed at the bottom of the column.

To eliminate potential buckling of compressive bars, given the required curvature ductility, μ_ϕ = 1.5, the compression strain demand is ε_cu,c < 0.0035 for both the corner and the peripheral columns (as previously was checked from ε_cu,c =2.2μ_ϕε_syν_Ed). From the buckling curve of Figure 5c (case StIV-f_y = 500 MPa and sideway buckling) and for s/D_b =150/14 ≈ 11, the strain at which the bar will become unstable is μ_εc ≈ 2 = ε_s,crit/ε_s,y thus ε_s,crit = 2 × 0.0025 = 0.005. Hence, for the estimation of the required jacket confinement, it must be ensured that ε_cu,c ≥ max(ε_cu,c = 0.0035, ε_s,crit = 0.005) = 0.005. Equation (11a–c) correlate term ε_cu,c with the FRP jacketing:

ε_cu,c = ε_cu + 0.075[ζ × (α_f ρ_fvE_f ε_fu,h + α_wρ_svf_y,st)/f_c − 0.1] ≥ ε_cu

(11a)

α_f ≈ 1 − ((b − 2R)² + (h − 2R)²)/(3bh)

(11b)

ρ_fv = 2t_f(h + b))/(hb)

(11c)

with ε_cu = 0.0035, ζ = 1 (for ε_cu,c ≤ 0.01), α_f = 0.56, α_w = 0.15, ρ_sv = 0.00314 and ε_fu,h = 0.0085 (as previously calculated); thus Equation (11a) results to ρ_fv = 0.0023 and ρ_fv = 2t_f (b + h)/(bh), which in turn corresponds to n = 2 layers along the critical regions of all columns, top and bottom, of length l_pl = 615 mm (for simplicity of the intervention this length is rounded to 600 mm).

Given the above calculations for elimination of premature buckling and splice failure, the applied jacketing (2 layers result to t_f = 0.24 mm and thus to ρ_fv = 0.0033) offers a limited displacement ductility. This is testified through Equation (12):

µ_Δ = 1.3 + 12.4 × (0.5 × α_fρ_fvE_fε_fu,h/f_c − 0.1) ≥ 1.3

(12)

Its implementation results to µ_Δ = 1.06 ≥ 1.3 thus µ_Δ = 1.3. This supply suffices against the required value (as previously calculated µ_Δ = 1.25).

The FRP strengthening of the ground storey columns (except C5) is summarized as follows: a single CFRP layer is wrapped along the height of each column to alleviate shear failure, one additional CFRP layer is wrapped at the lower 700 mm of each column to strengthen the splice region, one additional CFRP layer is wrapped at the upper 600 mm of each column to alleviate compressive bar buckling (Figure 6a). Column C5 does not require any FRP intervention because the increase in its cross-section size is implemented through reinforced concrete jacketing. (Note: For beams, by following similar procedure, FRP U-shaped jacketing of two-layer strips of width 100 mm and spacing 100 mm along the element full length is required along with special details in the ends in order to secure the jacket against debonding, e.g., adhesive anchors, see Chapter 6 in [4], Figure 6a).

Beam-Column Joint Strengthening

External beam-column joints are not laterally supported on four sides by beams, hence require horizontal shear reinforcement to prevent deterioration due to cracking induced by the high shear stress demand. This demand is the result of the steep moment gradients as they facilitate reversal of moment from one face of the member to the other. In old structures, r.c. beam-column joints rarely had in their body horizontal stirrups. So, it is deemed necessary to retrofit such a critical element of the structure.

As it is observed in the plan layout of Figure 1a, T- and L-shaped (knee or corner) joints exist. The strengthening design procedure is presented in detail for the L-shaped joint B1-C1-B7 of the ground storey; apparently all other L-shaped joints of the ground floor follow the same requirements. It is noted that in Chapter 8 of fib Bulletin 90 [4] two design approaches are presented. In this example, Approach 1 is implemented aiming to determine the required amount of horizontal FRP reinforcement through its function as added shear reinforcement in the joint panel. Approach 1 generally leads to significant amount of FRP reinforcement because, in the interest of conservatism, neglects any confining contribution, which could in turn lead to a retrofit inferior to expectations.

The first step is to calculate the sums of yield moments in the beams and in the columns framing into the joint. Hence, ΣM_yb is the sum of yield moments of the beams that frame into the joint and ΣM_yc is the sum of yield moments of the columns that frame into the joint:

x-x direction—weak axis (C1–B1): Beam B1-left has yield moment 134 kNm. The yield moments of the column below and above the joint are 53 kNm (N_Ed = 181.3 kN) and 48 kNm (N_Ed = 120.8 kN), respectively (the cross-section analysis was performed using Response2000 [11]). Hence, ΣM_yc = 53 + 48 = 101 kNm < ΣM_yb = 134 kNm. By being columns weaker than beams, they define the magnitude of shear stress in the joint. Hence, the vertical shear force V_j,v is derived by following the Greek Code for Assessment and Retrofit [36], from Equation (13) as:

V_j,v = ΣM_yc(1/jd_c − 1/L_b,n·H_n/H) + 1/2|(V_g+ψq,b)_l − (V_g+ψq,b)_r|

(13)

where jd_c is the internal lever arm of the column section, H_n and H are the theoretical and clear storey heights, and L_b,n is the theoretical half span of the beam. The beams are considered as fully fixed at their ends. The load from the slabs is assumed to be transferred to beam B1. Conservatively, it is assumed that the loads transferred from the slab to B1 correspond to a quarter of the area of the slab. Hence, the total linear load for the seismic combination is estimated as g = 9.75 kN⁄m, q = 4.38 kN⁄m and g + 0.3q = 11.06 kN⁄m, hence (V_g+ψq,b)_l = (g + 0.3q)l/2 = 33.19 kN. Then, by implementing Equation (13) results to V_j,v = 510.8 kN. The horizontal shear force V_j,h acting in the joint is obtained from: V_j,h = V_j,v × h_c/h_b =510.8 × 250/400 = 319.2 kN with upper limit of: V_j,h ≤ 80% × ηf_cm√(1 − v_Ed/η)·b_jh_jc where η = 0.6(1 − f_ck/250) = 0.58, b_j = min[max(b_c, b_b), min(b_c, b_b) + h_c/2] = 250 mm and h_jc = 250 − 2∙d₂= 250 − 2 × 27 =196 mm. Thus V_j,h = 319.2 kN < 334.4 kN.

y-y direction—strong axis (C1–B7): The yield moment of beam B7-right is 134 kNm. The yield moments of the column below and above the joint are ~90 kNm and 83 kNm: ΣM_yc = 83 + 90 = 173 kNm > ΣM_yb = 134 kNm. By being beams weaker than columns, they define the magnitude of shear stress in the joint. The horizontal shear force V_j,h is derived by following the Greek Code for Assessment and Retrofit [36], from Equation (14) as:

V_j,h = ΣM_yb(1/jd_b − 1/H_n × L_b,n/L_b)

(14)

where jd_b(=0.9 d; d = 370 mm) is the internal lever arm of the beam section, H_n is the theoretical storey height, L_b,n and L_b are the theoretical and clear half span of the beams, thus V_j,h =354.4 kN with upper limit of V_j,h ≤ 80%∙ηf_cm √(1 − v_Ed/η) × b_j h_jc, η = 0.6(1 − f_ck/250), where b_j = 250 mm, η = 0.58, h_jc = 350 − 2∙d₂ = 350 − 2 × 27 = 296 mm. Thus, V_j,h = 354.4 kN < 505 kN.

The horizontal shear force used for the calculation of the required FRP reinforcement, where fibers are oriented in the horizontal direction, is the maximum among 319.2 kN and 354.4 kN. For such a simple plan view of a building, one could deduce that the strong axis horizontal shear force apparently prevails. [Note: Alternatively, and with respect to the strong axis, the EN 1998-1 [35] for exterior beam-column joints proposes the simplified expression V_j,h = 1.20A_s1f_y − V_col, where A_s1 is the top reinforcement of beam as A_s1 = 2 × π × 10²/4 + 2 × π × 20²/4 ≈ 785 mm² (see Figure 2b) f_y = 500 MPa and V_col the shear force in the column above the joint as V_col = M_y/L_V = 83/1.35 ≈ 62 kN hence V_j,h ≈ 410 kN, which should be lower than 80%∙ηf_cm√(1 − v_Ed/η)·b_jh_jc = 505 kN. This approach increases the demand for FRP by almost 55 kN in comparison with what the most recent Code [36] deduces].

Prefabricated Carbon Fiber Reinforced Polymer (CFRP) plates will be used for the beam-column joint strengthening. An essential requirement is the proper anchorage of the FRP plates, otherwise FRP strengthening should not be considered effective. Fib Bulletin 90 [2] in its Section 9.2.3 provides details about anchorage of FRP strengthening in beam-column joints. The plates selected to be used have a thickness of t_o = 1.4 mm, modulus of elasticity E_f = 205,000 MPa, ultimate tensile stress f_fu = 3200 MPa and ultimate strain ε_fuk = 0.017. The design tensile strain ε_fd in the FRP layer shall not exceed the limit: ε_fd = ε_fuk/γ_f = 0.017/3 = 0.0056 where γ_f = 3 for FRP anchored on brittle substrate. According to fib Bulletin 90 [4] the allowable design value of FRP tensile strain shall not be taken higher than 0.004. Hence, ε_fd = 0.004. The thickness, t_f,h, is estimated from t_f,h = γ_Rd × V_j,h/(h_b∙E_f∙ε_fd) as t_f,h = 1.6 mm > t_o = 1.4 mm. Because the calculated value is very close to t_o, given the increased conservatism the Approach 1 has, one layer of CFRP L-shaped plate is required by covering the full height of the beams (detailing as shown in Figure 6b).

6. Conclusions

The methodology [4,5] implemented for the development of the detailed example –guide uses performance-based design criteria for the assessment and retrofitting at both local and global level. Damage is identified when the displacement demand due to earthquake excitation exceeds the displacement capacity of the individual members. The addition of externally bonded FRP reinforcement aims to remedy any deficiencies in the response of existing structural members due to the lack of seismic detailing (i.e., sparse shear reinforcement, short anchorage/splices). Global measures are adopted to modify the lateral response shape of the building and control interstorey drift.

The conceptual framework [4,5] is applied to a case study of a pilotis-type multi-story r.c. building, representative of the older construction practice in Southern Europe in the ‘70s. The building is assessed and retrofitted using externally applied FRP jacketing. Hand calculations are provided which cover in detail all the steps required for assessment and retrofitting.

The implementation of the methodology, as it is outlined in Section 2, revealed the following: as per the assessment stage (i) the order of magnitude of the error for the translational stiffness when cracked cross section (7895 kNm⁻¹) or secant to yield (3810 kNm⁻¹) definition is adopted may be large; this difference (here 100%) has a great impact on the estimation of the structure’s period of vibration, and thus on the maximum displacement reached during the seismic excitation; (ii) the estimated values of the deformation indices defined by using the alternative formulae included in [4,5] do not always agree with observations made on relevant experimental (the simplest formulae seem to estimate effectively the deformation indices); and (iii) the calculations of the deformation indices related to flexural response will not be realised when brittle failure modes are anticipated such as shear and lap splice which are repeatedly reported after a strong seismic event. At global level, interventions aim to lower the structure’s period close to the code reference value and to re-design the lateral response shape of the building in response to a more uniform distribution of the interstorey drift. For local interventions FRP jacketing is applied. The FRP plies required are calculated firstly to suppress all possible premature modes of failure (shear, bond, compression reinforcement buckling, exterior joints disintegration) and promote flexural response, and secondly to increase the ductility capacity in response to the seismic demand. Special attention needs to be given to the execution of the FRP application (i.e., corner rounding, special anchoring).