Seismic Capacity Reduction Factors for RC Beams and Columns

P.M. Barlek Mendoza, D.M. Scotta & E.E. GalíndezInstituto de Estructuras “Ing. Arturo M. Guzmán”, Universidad Nacional de Tucumán, Argentina

ABSTRACT:

Many building structures can be damaged and even collapse during a severe earthquake. For this reason it is important to take immediate decisions about the safety of damaged structures in order to avoid possible human losses in case of aftershocks. Therefore, a quantitative damage assessment should be made to estimate the residual seismic capacity. RC Frames are one of the most common earthquake resistant elements used in Argentina. Consequently, it was considered necessary to study the residual seismic capacity of the basic components of this structural type. Different studies were made in order to establish the reduction factors for different damage classes. These studies include an experimental test carried out on an RC beam and numerical models of both beams and columns. The results were compared with the values suggested by the revised JBDPA Guideline for Post-Earthquake Damage Evaluation and Rehabilitation (2014) and a satisfactory agreement was found between them.

Keywords: Residual Seismic Capacity, RC Beams, RC Columns, Crack Width.

1. INTRODUCTION

A well planned reconstruction strategy is vital to restore an earthquake damaged community as quickly as possible. In this sense, damage inspections are needed to identify safe buildings from those that are not. Consequently, it is important to develop a technical guide to help engineers perform this task with a unified criterion.

Many evaluation methodologies were proposed in different countries over the years. The Japanese post-earthquake damage assessment methodology is one of the most widespread. Seismic capacity reduction factors are fundamental in this methodology. These factors are obtained from experimental tests on structural members. In this paper seismic capacity reduction factors are calculated for RC beams and columns. The results are compared with values recommended by the Japanese guideline.

2. POST-EARTHQUAKE DAMAGE ASSESSMENT BASED ON RESIDUAL SEISMIC CAPACITY

In Japan, the Guideline for Post-Earthquake Damage Evaluation and Rehabilitation (JBDPA, 1991) was originally developed in 1991 and revised in 2001 and 2014. The main objective of the guideline is to provide rational criteria in order to identify and rate building damage quantitatively. Residual seismic capacity of reinforced concrete (RC) buildings is evaluated through the R-index. This index is defined as the ratio of post-earthquake seismic capacity to the structure’s original capacity. Mathematically,

R [ % ]=I SD

❑

I S⋅100 (1)

Where:

I SD❑ : Seismic Performance Index of the structure after earthquake damage.

I S: Seismic Performance Index of the structure before earthquake damage.

Experience in post-earthquake damage assessment has shown that R=95 %, the limit between slight and light damage, is a limit state of serviceability. Thus, structures with slight damage do not need to be repaired and can continue to operate after the seismic event. Additionally, R=60 %, the limit between moderate and severe damage, is a limit state of reparability. Therefore, most of the buildings found with severe damage were demolished and rebuilt. Maeda, Matsukawa and Ito (2014).

Seismic performance index I S is widely applied in the assessment of existing RC buildings in Japan. Its definition can be found in the Standard for Seismic Evaluation of Existing Reinforced Concrete Buildings (1977). This index takes into account such factors as strength, ductility, irregularities in structural configuration and the ageing process of materials.

Post-earthquake performance index I SD❑ can be evaluated using seismic capacity reduction factors (η).

These factors account for the loss of lateral strength and ductility corresponding to the damage state of each structural member.

The Damage Evaluation Guideline (1991) recognizes five different damage classes for structural members (Table 1). These categories are classified according to the maximum residual crack width (max W 0) measured on the member.

Table 1. Damage Classes of Structural Members.

Damage Class Observed Damage

I Some cracks are found. max W 0≤ 0.20 mm

II 0.20 mm ≤ max W 0 ≤1.00 mm

IIIHeavy cracks are found. Some spalling of concrete is observed.

1.00 mm ≤ maxW 0≤ 2.00 mm

IVMany heavy cracks are found. Reinforcing bars exposed due to spalling of the

covering concrete. max W 0>2.00mm

VBuckling of reinforcement. Crushing of concrete and vertical deformation of

columns and/or shear walls. Side-sway, subsidence of upper floors and/or fracture of reinforcing bars. Member collapse.

Figure 1. Concept of the seismic capacity reduction factor η

Figure 1 shows a diagram that illustrates a hypothetical lateral force – displacement curve. During an earthquake when a member reaches the peak displacement (δ p), a residual displacement (δ 0) occurs. The areas corresponding to Ed and Er are the dissipated energy during the earthquake and the residual

energy dissipation capacity of the structural member after the event, respectively. The η factor can be defined as the ratio of residual energy dissipation capacity (Er) to the member’s original energy dissipation capacity (ET=Ed+Er). Hence,

η=E r

ET=

E r

Ed+E r(2)

The 2014 revision of the Guideline is based on the information and knowledge gathered from the 2011 East Japan Earthquake. The two main aspects of this revision include the re-evaluation of the η factors and the proposal of a methodology to calculate the R-index for structures with total collapse mechanism.

The new values for η factors suggested by the JBDPA were enhanced by more experimental data on beams and a new category for quasi-ductile columns was added. These values can be seen on Table 2.

Table 2. Seismic Capacity Reduction Factors η Suggested by JBDPA in 2014 Revision of Guideline.

Damage ClassBeams Columns

Ductile Brittle Ductile Quasi-Ductile BrittleI 0.95 0.95 0.95 0.95 0.95II 0.75 0.70 0.75 0.70 0.60III 0.50 0.40 0.50 0.40 0.30IV 0.20 0.10 0.20 0.10 0.00V 0.00 0.00 0.00 0.00 0.00

The previous versions of the Guideline only considered the story collapse mechanism on RC frames. In this type of mechanism the damage is prominent on vertical members, i.e. columns. Seismic codes try to avoid the story collapse mechanism and recommend the total collapse mechanism instead. Ductile failure patterns such as beam yielding are observed in the total collapse mechanism. This has led to the development of a new evaluation method for structures that exhibit this kind of behaviour. As a result, there is a renewed interest in the seismic reduction factors for beams.

3. RC BEAM

3.1. Experimental Study

A pin supported RC beam with a central stub was tested by Scotta, Galíndez and Pavoni (2012). The objective was to determinate the seismic capacity reduction factors for a ductile beam. The detail of the specimen is showed in Figure 2. Symmetrical longitudinal reinforcement was adopted. Sufficient transverse reinforcement was provided to ensure adequate deformation capacity in the hinge region. Material parameters are presented in Table 3.

Figure 2. RC beam tested by Scotta, Galíndez and Pavoni (2012). Dimensions in mm.

The beam was subjected to a series of controlled pseudo-static cyclic vertical displacements. The load was applied on the central stub. The vertical load and displacement history imposed consisted of two

cycles of ductility ratio (μ) ranging from 1 to 5. Test configuration did not allow displacements higher than 52.5mm. However, at this displacement value the specimen still showed capacity to resist loads. Load – displacement hysteresis loops were obtained from this test.

Flexural crack widths were measured in the hinge area, Figure 3(a). The cracks were measured using a 10mm total length ruler with 0.1mm precision. Measurements were carried out at the peak of each cycle and when the vertical force was unloaded. The relationship between maximum residual crack widths and peak displacements is shown in Figure 3(b).

Table 3. Material Properties of RC Beam (see section 3) and RC Columns (see section 4).

Concrete

Scotta et al. (2012)

RC beam[MPa]

U1 – Tanaka (1990)RC column

[MPa]

U8 – Zahn (1986)RC column

[MPa]

Concrete Compressive Strength (f ´ c) 42.1 25.6 40.1

Concrete Young Module (Ec) 29761 25298 31662

Concrete Tensile Strength (f t=0.1 f ´c) 4.21 2.56 4.01

Steel Strength at yield point (f y) 563 474 440

Steel Strength at ultimate point (f su) 668 721 674

Steel Young Module (E s) 215879 200000 200000

Post-Yielding Steel Young Module (E sh) 5184 1800 2000

Figure 3. RC Beam. (a) Crack Width Measurement; (b) Peak Displacement – Maximum Crack Width Relationship.

3.2. Analytical Models

A nonlinear model of the beam presented in section 3.1 was made to reproduce test results, Scotta and Galíndez (2012). A fiber element model implemented in OpenSees (Open System for Earthquake Engineering Simulation) was used. Bar slip due to strain penetration of the longitudinal reinforcement in the central stub was taken into account. In addition, this model was also used to study seismic capacity reduction factors η for beams with different ductility.

Figure 4 shows the discretization and main characteristics of the numerical model. Only half of the beam was simulated due to symmetrical geometry. Five elements were employed. Element 1 was defined with null length to apply the bar slip model. From element 2 to 5, five Gauss Points were

defined. Popovics (1973) uniaxial model was considered for concrete and a modified Chang and Mander (1994) model was used for steel. The material parameters were the same as the ones specified on Table 3.

Figure 5 presents a comparison between the analytical and experimental load-displacement hysteresis loops. Ultimate displacement was assumed as the displacement when concrete reaches a compressive strain of 0.003 in fiber F4. Good agreement between both curves was observed.

Figure 4. OpenSees Numerical Model of RC Beam tested by Scotta. Galíndez and Pavoni (2012).

Figure 5. Load – Displacement hysteresis loops for RC beam Scotta, Galíndez and Pavoni (2012).

The tested beam presented an ultimate ductility ratio of 6.00. To obtain beams with less ductility, longitudinal reinforcement in the analytical model was increased. 3.26, 3.39 and 4.52 cm2 were adopted to obtain ductility ratios of 5.00, 4.00 and 3.00, respectively.

The relationship between maximum residual crack width (max W 0), and the seismic capacity reduction factor (η), was determined by a simple analytical model, Scotta, Galíndez and Pavoni (2012). A ductile beam was assumed to only have bending deformation. If the beam is idealized as a rigid body, the flexural deformation can be represented by the rotation of the rigid body, Figure 6. This assumption gives an estimation of flexural deformation (R0 f ), due to total flexural residual crack widths (ΣW 0 f ).

R0 f =ΣW 0 f

h=

δ 0

L(3)

Rearranging Eq.3 leads to Eq.4:

max W 0=δ 0⋅h

L⋅ nf(4)

Where:

n f=Σ W 0 f

max W 0 f(5)

From the experimental results, n f=2.

Figure 6. Idealization of Residual Flexural Deformation on RC beam.

4. RC COLUMNS

4.1. Experimental Studies

In addition to the tested beam, studies were carried out on two columns tested by other authors, i.e. Unit 1 (U1) by Tanaka (1990) and Unit 8 (U8) by Zahn (1986). The former had low axial load ratio P/(Ag ⋅ f c

' )=0.20 and exhibited ductile behaviour, while the latter had high axial load ratio P/( Ag ⋅ f c

' )=0.39 and exhibited a more fragile behaviour. The main mechanical properties of these specimens can also be found on Table 3. Geometrical properties and reinforcement details are shown on Figure 7.

Figure 7. RC columns tested by other authors: (a) U1 – Tanaka (1990); (b) U8 – Zahn (1986). Dimensions in mm.

Unit 1 Tanaka (1990) had an axial load P=819,00 kN that remained constant during the test. This unit was subjected to displacement cycles that represented the following ductility ratios: 0.75, 2.00, 4.00, 6.00 and 8.00. The length of the plastic hinge formed near the stub was 400mm. Flexural failure occurred at a peak displacement of 120mm.

Unit 8 Zahn (1986) had a much higher axial load P=2502,00 kN . The displacement history imposed over this column included cycles with ductility ratios 0.75, 2.00, 4.00 and 6.00. In this case, the length of the plastic hinge was 310mm. Collapse was observed at a maximum displacement of 50mm when the longitudinal reinforcement started to exhibit prominent buckling. This specimen also exhibited flexural failure.

Crack widths were not measured in any of the column tests. As a consequence, in order to determine the relationship between maximum residual crack widths and residual displacements the expression by Choi, Nakano and Takahashi (2006) was adopted. In both cases shear deformation can be neglected

because the two specimens exhibited flexural failure. Therefore, the expression can be written as follows.

δ 0=max W 0 f ⋅ nf

( hc−x )⋅ L (6)

Where:

δ 0: Total residual displacement.

n f=∑W 0 f

max W 0 f≅ 2.00

hc=400mm: Column height, measured perpendicularly to the flexural axis.

x: Distance between the most compressed concrete fiber and the neutral axis.

L=1600mm: Column Equivalent Length.

It is important to notice that if x is approximately equal to zero, then Eq. 6 for columns becomes Eq. 4 for beams. That is to say that in the case of beams, the neutral axis is close to the most compressed concrete fiber and most of the section is tensioned.

4.2. Analytical Models

Two numerical models with fiber discretization, one for each column, were made using PERFORM-3D nonlinear analysis software, Barlek Mendoza, Galíndez and Pavoni (2014). Three different materials were considered for these models: cover concrete, confined concrete and rebar steel. Each model consisted on two frame elements. The first element extended over the plastic hinge region. The cross section was subdivided in fibers. The second element remained elastic and extended over the remaining length of the column, see Figure 8(a).

U1 – Tanaka (1990) cross section was subdivided in 57 fibers. Additionally, 48 fibers were used to discretize U8 – Zahn (1986) cross section. Figure 8(b) shows the meshing considered for each column.

Figure 8. (a) Tested RC columns vs. Analytical models; (b) Fiber Discretization used for each column.

Material properties were in accordance with Table 3.Tensile stress collaborations were neglected for both cover concrete and confined concrete. For U1 – Tanaka (1990), confined concrete stress – strain relationship was established using a Modified Kent and Park model (1971). In contrast, Mander model (1988) for confined concrete was used for U8 – Zahn (1986). Trilineal stress – strain relationships

with no strength loss were assumed for reinforcing steel. Cyclic degradation was also considered using energy degradation factors.

Load – displacement hysteresis loops for U1 – Tanaka (1990) and U8 – Zahn (1986) are shown on Figure 9(a) and Figure 9(b), respectively. Good adjustment between test results and analytical models was observed in both cases. The numerical model of U1 – Tanaka (1990) was able to reproduce higher displacement cycles with great accuracy. Contrarily, U8 – Zahn (1986) analytical model was better than U1 model for lower ductility ratio cycles.

If the maximum residual crack width is known, Eq. 6 yields the residual displacement of the member. Using analytical models it is possible to reach a peak displacement. When the member is unloaded it has a permanent deformation that equals the residual displacement previously determined. Therefore, a relationship between maximum residual crack widths and peak displacement can be established. These relationships require an analytical model and were used to calculate the seismic capacity reduction factors for the different damage classes.

Figure 9. Load – Displacement hysteresis loops for RC columns. (a) U1 – Tanaka (1990); (b) U8 – Zahn (1986).

5. RESULTS AND DISCUSSION

Seismic capacity reduction factors were evaluated using Eq. 2. Total absorbable energy ( Et) of each member was calculated with the corresponding load – displacement envelope curve. Residual energy dissipation capacity (Er) was determined as the difference between dissipated energy and Et.Values obtained for RC beams and RC columns are summarised in Table 4. Additionally, in Figure 10 these factors are compared to the values specified in the 2014 revision of the guideline.

Table 4. Seismic Capacity Reduction Factors η.

Damage ClassRC Beams RC Columns

μ=6.00 μ=5.00 μ=4 .00 μ=3.00 U1 – Tanaka (1990)

U8 – Zahn (1986)

I 0.95 0.94 0.92 0.87 0.93 0.87II 0.79 0.72 0.64 0.50 0.81 0.43III 0.56 0.42 0.26 0.00 0.67 0.33IV 0.10 0.00 0.00 0.00 0.06 0.00V 0.00 0.00 0.00 0.00 0.00 0.00

5.1. RC Beam

The η factors obtained for RC beams are shown in Figure 10(a). These results include experimental and numerical analysis. Black and cyan dashed curves indicate the values specified in the guideline for ductile and brittle beams, respectively.

Red and blue curves correspond to the results obtained experimentally for top and bottom cracks. Notice how these curves are approximately matched by the magenta straight line that represents the reduction factors for the analytical model with an ultimate ductility ratio of 6.00. The values suggested in the 2014 revision for ductile beams are lower than the ones determined in this study. In consequence, guideline suggested values are conservative with respected to the tested RC beam.

Because of the relationship assumed between residual displacements and maximum residual crack widths (Eq. 4), the curves determined for different ductility ratios were straight lines. In addition, it can be seen that the greater the ductility ratio, the less steep the slope of these lines is.

Analytical models with ductility ratios lower than 4.00 were closer to the values recommended by the JBDPA for brittle beams. Beams with ductility ratio of 3.00 tend to be below the suggested limits . On the other hand, the curve corresponding to a ductility ratio of 5.00 has higher reduction factors, that can be compared to the suggested values for ductile beams.

Figure 10. Seismic Capacity Reduction Factors η. (a) RC beams; (b) RC columns.

5.2. RC Columns

Figure 10(b) shows the seismic capacity reduction factors for the columns analysed in this paper. In addition, the values recommended in the guideline for different damage classes are also plotted for both ductile and brittle columns.

Curves corresponding to test results and analytical models were very similar, especially for U8 – Zahn (1986). This can be attributed to the fact that numerical models hysteresis loops closely matched the ones measured experimentally. For U1 – Tanaka (1990), the analytical model tended to overestimate the η factors. This tendency is notorious for high values of maximum residual crack widths. In contrast, the numerical model of U8 – Zahn (1986) underestimated reduction factors.

It is important to notice that in the case of U1 – Tanaka (1990) the reduction factors η are greater than the ones suggested for ductile columns by the JBDPA. In consequence, it can be stated that for this column the values recommended in the guideline are conservative. This last statement is true for both experimental and analytical results.

For U8 – Zahn (1986), crack widths were much narrower because of the high axial load. In this sense, there were no crack widths corresponding to Damage Class IV. This indicates that the column collapsed before reaching this damage class. Additionally, it can be seen that there are values of η below the cyan dashed curve with guideline recommendations for brittle columns. In consequence, for this column guideline specifications were not as favourable as in the ductile member. This means that

the element rapidly lost energy dissipation capacity as maximum crack widths grew.

6. CONCLUSIONS

In this study, seismic capacity reduction factors η for RC beams and columns were determined. Test results and numerical models were used to this end. The results were compared with the reference values specified in the 2014 revision of the JBDPA Guideline.

From the results obtained for ductile elements, both beam and columns, it can be inferred that the values recommended in the guideline are conservative, at least for the cases analysed in this paper. However, in the case of brittle members, the reduction factors suggested by the code were sometimes greater than the ones calculated.

Numerical studies such as the one carried out on RC beams with different ductility ratios could be used in order to define when an element can be considered ductile, quasi-ductile or brittle. In order to perform this type of analysis, it is essential to count with expressions that relate maximum residual crack widths with residual displacements.

It is of capital importance to continue studying the different type of failures that structural members can exhibit and their relationship with seismic capacity reduction factors. The beam and columns considered in this paper all exhibited flexural failure. The next step in the research project is to consider shear and flexure interaction, especially on columns with moderate to high axial load ratio.

REFERENCES

The Japan Building Disaster Prevention Association (JBDPA) (1991, revised in 2001 and 2014) Guideline for Post-earthquake Damage Evaluation and Rehabilitation (in Japanese).

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