Rapid seismic assessment of two four-storey R.C. test buildings

Abstract

Seismic evaluation of existing reinforced concrete buildings that are classified as non-conforming to modern earthquake standards is an urgent priority, since this class of buildings represents the majority of the built environment throughout the world. To address this need a simple procedure for rapid seismic assessment (RSA) of the earthquake demand and available capacity of existing buildings has been devised and calibrated through field applications. RSA is based on first principles, considering the prevalent failure modes of the load bearing components of the structure, and easily accessible information regarding the geometric and material characteristics of the structure. In this paper the RSA method is further improved by introducing expressions for direct estimation of the local drift demands of the examined building at peak seismic response using the structure’s unique geometric and material properties. The accuracy of the RSA procedure is evaluated through application and comparison of the assessment results with the recorded seismic responses of two model experimental structures that had been tested under pseudo dynamic loads simulating earthquake effects, reported in the literature. The example structures were chosen because they were full-scale structures with relatively simple layout (planar frames), in order to develop an instructive paradigm of the RSA’s application for the interest of practitioners.

Appendix

The individual strength terms of an R.C. column can be calculated using the following expressions, which represent the present state of the art at the field. Nevertheless, these expressions may be subject to change as the knowledge base in reinforced concrete leads to improved models for the individual mechanisms of resistance.

Flexural shear demand

$$V_{flex} = \left[ {\rho_{\ell ,tot} \cdot \frac{{f_{y} }}{{f_{c} }} \cdot \left( {1 - 0.4 \cdot \xi } \right) + v \cdot \left( {\frac{h}{d} - 0.8 \cdot \xi } \right)} \right] \cdot \frac{{b \cdot d^{2} \cdot f_{c} }}{{H_{cl} }}$$

(10)

Exhaustion of shear strength

$${\text{If}}\;v < \, 0.10:\quad V_{v} = A_{tr} \cdot f_{st} \cdot \frac{d \cdot (1 - 0.4 \cdot \xi )}{s} \cdot \cot \theta_{v}$$

(11a)

$${\text{If}}\;v \ge \, 0.10:\quad V_{v} = v \cdot b \cdot d \cdot f_{c} \cdot \text{tan}\alpha + A_{tr} \cdot f_{st} \cdot \frac{d \cdot (1 - 0.4 \cdot \xi )}{s} \cdot \text{cot}\theta_{v}$$

(11b)

Anchorage failure of longitudinal reinforcement

$$V_{a} = \left[ {\rho_{\ell ,tot} \cdot \frac{{\hbox{min} \left\{ {\begin{array}{*{20}c} {\frac{{4 \cdot L_{a} \cdot f_{b} }}{{D_{b} }} + \alpha_{hook} \cdot 50 \cdot f_{b} } & ; & {f_{y} } \\ \end{array} } \right\}}}{{f_{c} }} \cdot \left( {1 - 0.4 \cdot \xi } \right) + v \cdot \left( {\frac{h}{d} - 0.8 \cdot \xi } \right)} \right] \cdot \frac{{b \cdot d^{2} \cdot f_{c} }}{{H_{cl} }}$$

(12)

Lap failure of longitudinal reinforcement

$$V_{lap} = \frac{{\left[ \begin{aligned} \hbox{min} \left\{ {\begin{array}{*{20}c} {\left( \begin{aligned} \mu_{fr} \cdot L_{lap} \cdot \left[ {\frac{{A_{tr} }}{s} \cdot f_{st} + \alpha_{b} \cdot \left( {b - N_{b} \cdot D_{b} } \right) \cdot f_{t} } \right] + \hfill \\ +\, \alpha_{hook} \cdot 50 \cdot N_{b} \cdot A_{b} \cdot f_{b} \hfill \\ \end{aligned} \right)} & ; & {N_{b} \cdot A_{b} \cdot f_{y} } \\ \end{array} } \right\} \cdot d \cdot \left( {1 - 0.4 \cdot \xi } \right) + \hfill \\ +\, v \cdot b \cdot d^{2} \cdot f_{c} \cdot \left( {0.5 \cdot {h \mathord{\left/ {\vphantom {h d}} \right. \kern-0pt} d} - 0.4 \cdot \xi } \right) \hfill \\ \end{aligned} \right]}}{{H_{cl} /2}}$$

(13)

Shear capacity of joints

$${\text{Unreinforced }}\;{\text{or}}\;{\text{ lightly }}\;{\text{reinforced}}\;{\text{ joints}}:\quad V_{j} = \gamma_{j} \cdot 0.5 \cdot \sqrt {f_{c} } \cdot \sqrt {1 + \frac{{v_{j} \cdot f_{c} }}{{0.5 \cdot \sqrt {f_{c} } }}} \cdot \frac{{b_{j} \cdot d \cdot d_{beam} }}{{H_{cl} }}$$

(14a)

$${\text{Well }}\;{\text{reinforced}}\;{\text{ joints}}:\quad V_{j} = \left[ {\gamma_{j} \cdot 0.5 \cdot \sqrt {f_{c} } \cdot \sqrt {1 + \frac{{v_{j} \cdot f_{c} }}{{0.5 \cdot \sqrt {f_{c} } }}} \cdot \frac{{b_{j} \cdot d \cdot d_{beam} }}{{H_{cl} }}} \right] \cdot \sqrt {1 + \rho_{j,horiz} \cdot \frac{{f_{st} }}{{f_{t} }}}$$

(14b)

Punching shear of slab-column connections

$$V_{pn} = \frac{{0.12 \cdot \hbox{min} \, \left\{ {\begin{array}{*{20}c} {1 + \sqrt {\frac{200}{{d_{sl} }}} } & ; & 2 \\ \end{array} } \right\} \cdot \left( {100 \cdot \rho_{\ell ,sl} \cdot f_{c} } \right)^{1/3} \cdot d_{sl} \cdot 0.25 \cdot u_{crit} \cdot \left( {h + 4 \cdot d_{sl} } \right)}}{{H_{cl} }}$$

(15)

Limiting shear due to yield of beams’ longitudinal reinforcement

$$V_{by} = \frac{{0.85 \cdot \rho_{beam} \cdot b_{beam} \cdot d_{beam}^{2} \cdot f_{y}^{beam} }}{{H_{cl} }}$$

(16)

where,

• ρ_ℓ,tot = A_s,tot/(b·d) is the total longitudinal reinforcement ratio of a column with external dimensions h × b,

• A_s,tot is the total area of the longitudinal reinforcement at the column’s critical section,

• d is the column effective depth,

• f_y is the longitudinal reinforcement yield stress,

• f_c is the concrete compressive strength,

• ξ (= x/d) is the normalized depth of compression zone,

• v is the axial load ratio acting on the cross section (N_g+0.3q/(b·d·f_c)),

• H_cl is the column’s deformable length,

• tanα = (h/d − 0.8·ξ)·d/H_cl, where a (≤ θ_v) is the angle of inclination of the diagonal strut created between the centroids of the compression zones at the top and bottom column cross sections of the column (this represents the strut forming by the axial load acting on the column according to (Priestley et al. 1996),

• θ_v (= {45° when v < 0.10, 30° when v ≥ 0.25, whereas for 0.10 ≤ v < 0.25 θ_v is calculated from linear interpolation}) is the angle of sliding plane (i.e. θ_v is the angle forming between the longitudinal member axis and a major inclined crack developing in the plastic hinge region of the column). It determines the number of stirrup legs that are intersected by the inclined sliding plane,

• h_st is the height of the stirrup legs,

• A_tr is the total area of stirrup legs in a single stirrup pattern, which are intersected by the inclined sliding plane,

• s is the stirrup spacing,

• f_st is the stirrup yield stress,

• L_α is the anchorage length of the longitudinal reinforcement,

• D_b is the diameter of longitudinal reinforcing bars,

• α_hook is a binary index (1 or 0) to account for hooked anchorages (α_hook = 0 ⟹ no hooks),

• f_b= 2· f_b,o is the concrete bond stress, where f_b,o= n₁· (f_c/20)^0.5, n₁ = {1.80 for ribbed bars; 0.90 for smooth bars}.

• μ_fr is the friction coefficient (0.2 ≤ μ_fr ≤ 0.3 for smooth bars; 1.0 ≤ μ_fr ≤ 1.5 for ribbed bars),

• L_lap is the lap-splice length,

• α_b is a binary index (1 or 0) depending on whether ribbed or smooth reinforcement has been used,

• N_b is the number of longitudinal bars in tension,

• A_b is the area of a single tension bar,

• f_t = 0.3·f ^2/3 _c is the concrete tensile strength,

• γ_j = {1.40 for interior joints; 1.00 for all other cases, whereas, for joints without stirrups these values are reduced to 0.4 and 0.3 respectively},

• v_j is the (service) axial load acting on the bottom of the column adjusted at the top of the joint (compression positive),

• b_j = (b + b_beam)/2 is the joint width, where b_beam is the web width of the adjacent beam,

• d_beam is the beam depth,

• ρ_j,horiz = A_tr/(s·b_j),

• d_sl is the slab depth,

• ρ_ℓ,sl is the total slab reinforcement ratio at the critical punching perimeter around the column, u_crit

• ρ_beam is the tension longitudinal reinforcement ratio of the beam (i.e. the total longitudinal reinforcement ratio of the beam section adjacent to the column if an interior connection is considered, or in the case of exterior connections the value of the top or bottom beam reinforcement ratio (whichever is largest)),

• f ^beam _y is the yield stress of the beam longitudinal reinforcement.