A Comparative Study: Seismic Deformability and Strength of Non-conforming Columns

Abstract

The estimation of deformation indices and strength capacities of reinforced concrete structural members is neither a simple nor a systematically unique approach. Several analytical models for such estimations have evolved over the years and some are an essential part of today’s Seismic Codes of Assessment in the form of acceptance criteria (e.g. EC8-III 2020, ASCE/SEI 41-17). However, significant dispersion still exists between experimental data and code-based analytical estimates, especially when assessing structures built prior to the consolidation of capacity-design provisions in International Codes (before the 1980’s). These structures-referred to as non-conforming—consist primarily of structural members that possess deficiencies in the quality of concrete and steel, and in the transverse and/or longitudinal reinforcement detailing. In the present work a comparative study was conducted using advanced nonlinear finite element simulations for a benchmark set of columns modelling older reinforcement detailing practices. Pushover analyses were conducted on the column models in order to determine the parametric sensitivity of the deformation indices, failure mode and associated strength of these benchmark cases to their basic design properties. It was found that deformation capacities of columns deteriorate significantly in the presence of lap splice or insufficient anchorage length in critical regions, whereas shear failure is suppressed in the case of adequate confinement and may only appear with higher axial compression. However, yield penetration and pull-out failure dominates the failure mechanism at low axial loads limiting the deformation capacity significantly in the absence of good confinement. The design expressions used in the assessment codes are calibrated against the analytical results illustrating the limits of the applicability and the discord between the different approaches used by current codes to define the so-called acceptance criteria in the seismic assessment practice.

1 Introduction

It has been observed in reconnaissance reports following previous strong ground motion events that many structural components constructed prior to the introduction of modern seismic design concepts (e.g. before the 1980’s in the Western world) exhibit premature failures which prevent these components from developing their intended full deformation capacity and strength [15]. Structural deficiencies may be associated with substandard detailing and dimensioning that was mainly based on allowable stress design with no emphasis on the confining function of adequately anchored stirrups. Therefore, substandard R.C buildings may collapse due to localization of failure in few locations of the building prior to redistribution of stresses [2, 5, 9, 11]. Premature mechanisms leading to localized failures may include buckling of compression reinforcement, slip of longitudinal reinforcement due to the presence of poorly confined lap splices in the plastic hinge zone region-which was a common practice more than 40 years ago-, crushing of concrete in the member web, etc.). Those types of structural components which do not comply to modern seismic provisions are labeled henceforth nonconforming members [7].

Nonconforming members exist in a large number of concrete structures across Canada. They are a result of older methods of construction prior to the earthquake engineering community worldwide could reach a thorough understanding of the mechanics of seismic resistance of RC; such structures are deemed unsafe according to current building codes [14]. Therefore, there is a need for improved understanding of the critical mechanisms governing the deformation capacity and strength of RC structures, along with better calibration of the prevalent assessment model.

2 Analytical Models

Using an advanced finite element software (ATENA V.5 3D Engineering) a series of benchmark columns are modeled considering different effects of detailing representing nonconforming construction [4]. All columns are subjected to monotonic loading. Using ATENA Studio (ATENA Studio ×64V.5.6.1. 17,830) the resistance curves for the columns were calculated and the deformation and strength capacities were recorded to form a data base for a comparative study. Figure 1a presents the five different cases of column models considered, with differences in longitudinal reinforcement detailing: (1) The longitudinal reinforcement bars are fully anchored into the foundation with a 90° hook, (2) Bars are lap spliced over a lap length of 15D_b, (3) Bars extend into the foundation with an anchorage length of 15D_b, (4) The column has a deep cross Secttion (700 mm depth), (5) A hinge is fabricated at the base of the column—where the longitudinal reinforcement crosses the center of the cross section and therefore produces a zero moment point. Each case of longitudinal detailing shown in Fig. 1a is modeled with different combinations of axial loading (10, 35, 50% of crushing), longitudinal reinforcement (shown in Fig. 1b) and stirrup spacing (100- and 200-mm stirrup spacing) with 8 mm stirrup diameter. The column identification code is as following: The first numeral following the letter C (for Column)-for example C35-corresponds to the normalized axial load ratio applied to the columns which in this case is 35% of the crushing load, followed by the section ID number (each section contains a different amount of longitudinal reinforcement, so as to control the hierarchy between flexural and shear demands as shown in Fig. 1b) whereas the last number digit in the numeral represents the stirrup spacing in cm along the length of the shear span of the column.

Fig. 1

a Column cases and detailing (all dimensions are in m). b Section geometry

The columns are modeled using 3D macro elements. The columns have a clear length of 3 m. The foundation is modeled as a block with dimensions (0.9 × 0.9 × 0.7) m. A symmetrical beam block is assumed at the top to enable application of lateral load. The dimensions of the foundation are increased for case 4 to (1.2 × 1.2 × 0.7) m.

2.1 Finite Element Model

The cases in Fig. 1a were modeled using a laterally swaying cantilever model with a shear span length equal to half of the column’s deformable length (3/2 = 1.5 m). Moreover, due to symmetry half the cross section was modeled. The columns were discretized into 3D brick elements (8 nodded) with a brick size ranging between 0.025 m for the bottom portion of the column (0.5 m) and 0.05 m for the rest of the column. The reinforcement is modeled through 1D reinforcement truss elements. The columns were modeled with a stiff steel plate on the top and side of the column to eliminate localized failure when applying axial and lateral loads. The axial load was applied in the first analytical step followed by monotonic steps of 0.4 mm lateral displacement until columns would completely fail. The bottom surface nodes of the foundation are restrained from movement in x, y and z directions. The section’s symmetrical plane was restrained from movement in the x-direction (the direction perpendicular to the symmetrical plane). Two monitor points were placed at the point of lateral displacement application. One for recording the reactions and one for displacements. The reported reactions represent the entire cross section. Therefore, the load-displacement (monotonic resistance envelopes) are obtained.

2.2 Material Models

2.2.1 Concrete and Reinforcement Stress-Strain Relationships

NonlinearCementitious2 User material was used to model concrete behaviour. This model uses a combination of plasticity and fracture relationships to simulate the full range of inelastic  stress-strain behavior of concrete  [3]. The cracked stiffness in this model was calculated using the help of the retention shear factor r_g as described in Eq. 1, whereas the strain value in tension or compression is calculated in this model as shown in Eq. 4 [3].

$${\text{E}}_{{{\text{ijij}}}}^{{^{\prime}{\text{cr}}}} = {\text{r}}_{{\text{g}}} .{\text{ G}}/\left( {1 - {\text{r}}_{{\text{g}}} } \right)$$

(1)

$${\text{G}} = {\text{ r}}_{{\text{g}}} .{\text{G}}_{{\text{c,}}}$$

(2)

$${\text{G}}_{{\text{c}}} = E_{c} /2 \cdot \left( {1 + v} \right)$$

(3)

$${\text{If}}\;\;\upvarepsilon _{1}^{{\text{f}}} < \,\upvarepsilon _{{{\text{loc}}}}^{{\text{f}}} ,\;\;\;\;\upvarepsilon =\upvarepsilon _{1}^{{\text{f}}} \;\;\;{\text{else}}\;\;\;\upvarepsilon = {\upvarepsilon }_{{{\text{loc}}}}^{{\text{f}}} + \left( {{\upvarepsilon }_{1}^{{\text{f}}} - {\upvarepsilon }_{{{\text{loc}}}}^{{\text{f}}} } \right)\frac{{{\text{L}}_{{\text{t}}} }}{{{\text{L}}_{{{\text{ch}}}}^{{\text{t}}} }}$$

(4)

In the above, ${\text{E}}_{{{\text{ijij}}}}^{{^{\prime}{\text{cr}}}}$ is the cracked stifness, r_g is the minimum of the shear retention factor on cracks in both directions i and j, G is the elastic shear modulus, and the retention shear factor is defined in Fig. 2a. ${\text{L}}_{{{\text{ch}}}}^{{\text{c}}}$ and ${\text{L}}_{{{\text{ch}}}}^{{\text{t}}}$ in Eq. 4 represent a size for which the diagram in tension and compression is valid and decreases the dependency on the mesh [3]. For the column models 0.03 and 0.050 m are used for ${\text{L}}_{{{\text{ch}}}}^{{\text{c}}}$ and ${\text{L}}_{{{\text{ch}}}}^{{\text{t}}}$ respectively with trial and error so as to capture an appropriate response. L_t and L_c represent the crack band size and crush band size respectively. Parameter ε is the strain tensor at the finite element integration points. The localized softening strain in compression is defined as the strain corresponding to the maximum compressive strength after subtracting the linear portion of the stress strain curve [3]. The localized softening strain in tension is assumed 0 for plain concrete thus no hardening occurs after the first crack initiates [3]. Concrete compressive strength for the unconfined concrete used is f_c = 30 MPa, E_c = 30,000 MPa and v = 0.2. The stress strain curve of concrete was determined based on Hognestad’s parabola [8] for the ascending branch. Kent and Park [10] softening branch model was used to model the concrete cover. Modified Park and Kent [16] was used to model the confined core. The stress strain relation for concrete is shown in Fig. 2 for all cases of longitudinal and transverse reinforcement ratios.The mechanical properties used for transverse and longitudinal reinforcement are listed in Table 1.

Fig. 2

Concrete stress strain curve for different sections and transverse stirrup spacing

Table 1 Longitudinal and transverse reinforcement properties

2.2.2 Reinforcement Bond

The longitudinal reinforcement is connected to the concrete through interface springs endowed with a proper bond stress-strain relationship to allow the slip between the bar and concrete. In this manner pullout or splitting behavior of the longitudinal reinforcement can be considered. The bond stress-slip relationship was determined based on fib Model Code 2010 [13] assuming good bond conditions. Splitting failure is estimated to occur along the shear span of the column. For the case of 100 mm stirrup spacing the maximum bond strength was used as shown in Eq. 5. However, in the case of 200 mm stirrup spacing the unconfined maximum bond stress was used as shown in Eq. 6, as the effective confining pressure in this case is, K_e < 0.3. Equation 7, 8, 9 and 10 are used to calculate the milestone points of the bond-slip relationship. Bond strength T_max is calculated as 2.5 $\sqrt {{\text{f}}_{{\text{c}}} }$. The residual strength for the 100 mm stirrup spacing case is 0.4*T_max. A value of 0 is assumed for the residual strength for lower confinement according to the code. The bond stress-slip relation used for the column models are shown in Fig. 3.

Fig. 3

Bond stress-slip relationship

$${\text{T}}_{{\text{u,split}}}^{1} = \, 8.\left( {{\text{f}}_{{\text{c}}} /25} \right)^{0.25}$$

(5)

$${\text{T}}_{{\text{u,split}}}^{2} = \, 7.\left( {{\text{f}}_{{\text{c}}} /25} \right)^{0.25}$$

(6)

For slip calculations:

$${\text{T}}_{{\text{o}}} = {\text{ T}}_{{{\text{max}}}} .\left( {{\text{s}}/{\text{s}}_{1} } \right)^{0.4} ,\;{\text{for}}\;0 < = {\text{s}} < = {\text{s}}_{1}$$

(7)

$${\text{T}}_{{\text{o}}} = {\text{ T}}_{{{\text{max}}}} ,\;{\text{for}}\;{\text{s}}_{1} < = {\text{s}} < = {\text{s}}_{2}$$

(8)

$${\text{T}}_{{\text{o}}} = {\text{T}}_{{{\text{max}}}} \left( {{\text{T}}_{{{\text{max}}}} - {\text{T}}_{{\text{f}}} } \right).\left( {{\text{s}} - {\text{s}}_{2} } \right)/\left( {{\text{s}}_{3} - {\text{s}}_{2} } \right),\;{\text{for}}\;{\text{s}}_{2} < = {\text{s}}_{3}$$

(9)

$${\text{T}}_{{\text{o}}} = {\text{T}}_{{\text{f}}} ,\;{\text{for}}\;{\text{s}}_{3} < {\text{s}},\;{\text{where}},\;{\text{T}}_{{{\text{max}}}} = 2.5.{\text{fc}}^{0.5}$$

(10)

3 Results and Analysis

3.1 Deformation and Strength Capacities and Modes of Failure for Cantilever Models

Table 2 displays the deformation capacities and strengths for each case shown in Fig. 1a. The strength is plotted against the drift ratios (which is defined as the displacement divided by the length of the shear span (1.5 m)) as shown in Fig. 4. To make a fair comparison of the different resistance curves in order to reveal the influence of the parameters studied, the apparent loss of lateral load resistance owing to P-∆ effect was eliminated in the plots by adding the product of the axial load multiplied by the drift ratio at each point. Figure 5 depicts the failure of the columns under ultimate loading capacity.

Table 2 Deformation and strength capacities for 5 different columns

Fig. 4

Load-displacement for all cases of longitudinal reinforcement. Here, the drift ratio is defined as the displacement of the column divided by the column’s shear span length

Fig. 5

Column failure modes at near collapse state. Strain XX are the longitudinal strains in reinforcement and strain ZZ are the strains in concrete

4 ASCE/SEI 14/17 and Eurocode 8-Part III 2005 Deformation Capacity Correlation

The deformation capacities at different performance stages using the ASCE/SEI 14/17[1] and Eurocode 8 III 2005 were calculated. The performance levels are determined in the codes at Near Collapse (NC), Significant damage (SD) and damage limitation (DL) performance limit states in the Eurocode 8 [6] which corre spond to Collapse prevention (CP), Life safety (LS), Immediate Occupancy (IO) respectively in the ASCE/SEI 14/17. Figures 6a, b and c plots the 3 limit stages according with the ASCE/SEI 14/17 plotted against the model analytical data. Similarly, Fig. 6d, e and f illustrates the performance levels in the Eurocode 8 III [6] plotted against the model data.

Fig. 6

Code estimated rotation capacities vs. values from F.E. Models a–c Limit states from ASCE/SEI 41-17; d–f Limit states from Eurocode 8-III

Figure 5 depicts the mode failures of the columns. Where Fig. 5a is for set 1, b is for set 2, c is for set 4, and e is for set 5.

5 Discussion and Conclusions

This study investigated the deformation capacity and its correlation to the assessment codes at three performance levels (acceptance criteria). Columns with well confined stirrups provided larger deformation capacities. The column’s shear strength started to vary at high axial loads, indicating stirrup yielding and shear failure. Moreover, it wa s found that the effective yielding increased with the increase of the axial load. A decrease in the effective yielding at an axial load of 50% was found due to the occurrence of concrete crushing failure. The hinge fabrication at higher axial loads was excluded from this behaviour as reinforcement slips were present at lower axial loading. Moreover, column yielding was delayed at low axial loads in the case of lap spliced column. This is due to the delay in the longitudinal reinforcement yielding due to the increase of slip. It is also concluded that the column’s deformation capacities decrease with an increase in the axial loads. Columns with low axial load failed due to splitting. The crushing of concrete is also present at 50% crushing load, limiting longitudinal bar yielding especially in the cases of low confinement. The effect of the longitudinal reinforcement detailing present in the lap splice and short anchored reinforcement was attenuated with the increase of the compressive loading. Thus, the reinforcement slip from the foundation was mitigated when lateral displacements were applied. At lower axial load stronger deterioration of ultimate deformation capacities occurred as these columns were dominated by the pullout slip of the reinforcement. It should also be noted that the pullout demand in these cases for columns lightly reinforced are less affected. This was observed due to the prevalence of flexural yielding. For deep cross sections, the larger internal lever arm also increased the shear demand. Shear failure could occur at lower shear strengths with lower deformation capacities.

The ASCE/SEI 41-17 procedures showed better correlation to the rotation at yielding when compared to deformation capacities from models as opposed to the Eurocode 8 III [6]. This is because shear force demands are considered in calculating the rotation capacities in the ASCE/SEI 41-17. For the second set (the lapped splice), the finite element models show a decrease in the yield rotation compared to columns in the first set (columns with continuous reinforcement). This increase is attributed to the increase of slip rotations along the reinforcement bar. However, this is not reflected in the assessment codes properly. The codes demand a decrease in the slip rotation in a lapped splice column at yielding compared to columns with full anchorage. Generally, the assessment codes show a decrease in the life safety/significant damage state with the increase of the axial load for all column cases.  However, from the finite element models it was found that this trend was not applicable to columns that fail ultimately due to reinforcement slips, which are columns in set 2 and 3 (lapped splice and short anchorage length) that have high reinforcement ratios. This is because the compression forces of the axial load mitigate the pullout forces when applying lateral load. In the case of 50% the ultimate rotations decrease, as the column were controlled by concrete crushing rather than pure slip. At Near Collapse or Collapse Limit state the scatter increased when the column was controlled by crushing of concrete according with the Eurocode at higher axial loads. However, according with the ASCE SEI 14/17, values successfully matched the calculated ones in computing the deformation capacities when concrete crushing failure controlled. The Eurocode 8 III [6] showed lesser correlation to the computational results for columns failing ultimately in buckling/shear.