Proc. 13th International Symposium on the Interaction of the Effects of Munitions with Structures (ISIEMS 13), Bruehl, Germany, 11-15 May 2009.

EFFECT OF SHORT-DURATION VARIABLE AXIAL AND TRANSVERSE LOADS

ON REINFORCED CONCRETE COLUMN

T. Krauthammer, Astarlioglu, S., and Tran, T.

Center for Infrastructure Protection and Physical Security (CIPPS), 2114 NE Waldo Road, PO Box 116580,

University of Florida, Gainesville, FL 32611-6580 Abstract Previous studies were conducted on the deformations of reinforced concrete columns induced by blast load that combined both axial and transverse loading components. Most of those studies assumed that the response of the mass supported by the column in its axial direction developed much slower compared to that in the lateral movement. Thus, the load transferred from the supported mass to the column in its axial direction could be treated as a static load. Moreover, it was assumed that the vertical displacement of the column was small compared to the lateral displacement, and therefore, was negligible (Biggs, 1964). Consequently, the failure of the column was assumed to be governed by the flexure caused by transverse loads. This may not be quite true since for a column that is subject to severe combinations of both transverse and axial loads, the effect of the axial load may be an important factor in determining the failure of the column. Thus, the above simplified assumption should be re-examined to determine the actual effect of variable loads in the axial direction on a column. Introduction Progressive collapse of a building is normally caused by an abrupt failure of one or more structural bearing members such as beams or columns. Therefore, the endurance of these members under short duration but highly impulsive loads is crucial for the survivability of the building. While beams are normally subject to transverse loads, columns are always exposed to both transverse and axial loads. In practice, it is assumed that failure of a column is normally caused by transverse rather than axial loads. This may not be accurate, particularly in the case where a structure is subjected to short duration but highly impulsive loads such as blast loads. While the failure of the column will most likely be induced by the transverse loads, the effect of variable axial loads should also be considered as a contributing factor. The column resistance may be reduced due to the axial load under the same material and physical properties, and the column may fail sooner. On the other hand, the alterations in directions and the eccentricity of the variable axial loads over the time period may act as an enhancement factor to the strength of the column, thus preventing it from failing in the early stage. The discussion on eccentricity is, however, not in the scope of this work and therefore is not included this study.

The objective of this paper is to determine the actual effect of the variable axial loads on the column, allowing them to be properly accounted for during the design stage of structure subjected to blast loads. Background

Numerous steps were taken in order to confirm the objective. Firstly, validations for the ABAQUS Version 6.8-1 (Dassault Systèmes, 2008) and the Dynamic Structure Analysis Suite (DSAS) Version 2.0 (CIPPS, 2008) were required to ensure that they could produce reliable results. This was completed by comparing the results obtaining from these two applications against known data of a series of experiment on reinforced concrete beams. Secondly, the validation was completed on a reinforced concrete column. However, since there was no experimental data available for the column, these two software applications were validated analytically using a standard size column with the same material properties of the experimental beam. Lastly, a series of columns of the same physical and material properties but different reinforcements configurations was arbitrarily picked from the Concrete Reinforcing Steel Institute Design Handbook (CRSI, 2002) for further analysis in the parametric study to determine the effect of axial loads on the Pressure-Impulse (P-I) relationships. This was followed by the analysis on the effects of variable axial loads on the columns. The computer code DSAS was modified to address the effects of axial force on RC columns, and it was used to derive the corresponding P-I curves. Pressure-Impulse (P-I) Diagrams P-I diagrams are graphical tools used to determine the potential damage of a structure caused by dynamic loads. Detailed descriptions on P-I diagrams can be found in numerous references (Krauthammer, 2008). There are three distinguished regions on the P-I curve, as shown in Figure 1. These are the Impulsive Loading Region, the Quasi-Static Loading Region and the Dynamic Loading Region. In addition, there are two asymptotes. The Impulse Asymptote is tangent to the Impulsive Loading Region and the Pressure Asymptote is tangent to the Quasi-Static Loading Region. In the Impulsive Loading Region, the response time of the structure is much longer than the duration of the loading. Hence, before the structure can experience any permanent deformation, the load is already dissipated. In the Dynamic Loading Region, the duration for both loading and natural period is approximately the same. The response of the structure in this region depends on the loading history. In the Quasi-Static Loading Region, the loading duration is much longer than the natural period. Therefore, the structure experiences maximum deformation before the load completely dissolves. (Smith and Hetherington, 1994)

Impulse, psi-msec

Pres

sure

, psi

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20-100

-50

0

50

100

150

200

250

300

350

400

450

500

Figure 1 – Typical Pressure-Impulse Diagram

Description of DSAS DSAS (CIPPS 2008) is a comprehensive software suite developed for the analysis and assessment of structural members subject to severe dynamic loads such as blast and impact. The primary analysis engine in DSAS is based on an advanced single degree of freedom (SDOF) formulation and is capable of developing fully non-linear resistance functions for reinforced concrete, steel, masonry and other members with diverse end conditions using force or displacement-controlled solution procedures (Krauthammer et al, 1990 and 2003, DSAS User Manual, 2008). The moment curvature relationships for RC, derived by DSAS, are based on layered section analysis with fully nonlinear material models for steel and confined and/or unconfined concrete. The resistance function is based on a displacement controlled solution approach, and the Direct Shear function uses the Hawkins model. The present study enabled the development of an enhanced version of DSAS that allows for constant gravity loads to be specified and modifications can be made to account for dynamic variations in axial force. Moreover, DSAS is also capable to provide Physics-based P-I analysis and thus P-I diagrams. DSAS Validation Beam Subject to Concentrated Load

Data of five beams from the experiment conducted by Feldman and Siess (1958) were used as part of the validation. All five beams were modeled with ABAQUS (Dassault Systèmes, 2008) using cubicle grids of 1 in. The concrete cap hardening was used as the material model for concrete. With regards to the reinforcements, two different material models were used. Beam elements were used to model the compression reinforcements, and surface elements were used for modeling the tension and transverse reinforcements. Detailed layouts of steel reinforcements for all beams are shown in Figure 2. The main difference in the physical configuration between beam 1-C and the other four beams was the transverse reinforcements. Beam 1-C used opened stirrups and the other two beams used closed stirrups. Other properties of the beams are described in Tables 1 and 2 and Figure 2. Beam 1-C is used here to illustrate the validation, and the associated load function is shown in Figure 3.

Impulsive Loading Region

Quasi-Static Loading Region

Pressure Asymptote

Impulse Asymptote

Dynamic Loading Region

Figure 2 – Beam 1-C Layout (Feldman and Siess, 1958)

Table 1 – Concrete Material Properties

b = 6 in, h = 12 in, d = 10 in, d’ = 2.0 in, span =106 in Beam Compressive Strength, f’c,

ksi Modulus of Elasticity,

Ec, ksi Rupture Strength, fr,

ksi 1-C 5.67 4292.1 0.9 1-G 6.21 4491.8 1.0 1-H 6.15 4450 0.935 1-I 6.50 4730 0.85 1-J 6.09 3890 0.935

Table 2 – Steel Reinforcements Material Properties

Tension reinforcement = 2 #7 bars, compression reinforcement = 2 #6 bars, stirrups = 16 #3 bars at 7 in on center Beam Compression Reinforcements Tension Reinforcements

f’y, Ksi E’s, Ksi ε'y, in/in ε'sh, in/in fy, Ksi Es, Ksi εy, in/in εsh, in/in 1-C 46.70 --- --- --- 46.08 29520 0.0016 0.0144 1-G 48.30 --- --- --- 47.75 --- --- --- 1-H 47.61 32280 0.0015 0.015 47.17 34900 0.0014 0.0125 1-I 47.95 --- --- --- 47.00 32600 0.0014 0.015 1-J 48.86 29560 0.0016 0.012 47.42 --- --- ---

Time, msec

Loa

d, K

ips

0 10 20 30 40 50 60 70 80 90 100 110 120-5

0

5

10

15

20

25

30

35

Figure 3 – Load Function for Beam 1-C (Feldman et al., 1958)

Validation result for mid-span displacement of beam 1-C from ABAQUS (Dassault Systèmes, 2008) and DSAS (CIPPS, UF) are shown in Figure 4. It should be noted that the unloading portion in the ABAQUS result did not match up well compared to those of experimental and DSAS results. This was because hysteresis damping was not captured by the concrete material model in ABAQUS when modeling beam 1-C. Table 3 provides the summary of the validation results for all five beams.

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.1 0.12-1

0

1

2

3

4

ExperimentDSASABAQUS

Figure 4 – Comparison of Mid-Span Displacement Time History for Beam 1-C Table 3 – Validation Resutls for the Experimental Beams

Beam Midspan Disp, in

(Experiment)

Midspan Disp, in

(ABAQUS)

% Difference

Midspan Disp, in

(DSAS)

% Difference

1-C 3.01 2.9 3.65% 3.07 1.99% 1-G 4.14 4.23 2.17% 4.02 2.90% 1-H 8.86 8.77 1.02% 8.2 7.45% 1-I 10.57 9.94 5.96% 9.55 9.65%

1-J-1st Set* 0.95 1.21 27.37% 0.84 11.58% 1-J-2nd Set* 10.17 10.6 4.23% 8.54 16.03%

* Beam 1-J was loaded accidently during test set up, and the 2nd blow was delivered to a previously damaged beam (Feldman and Siess, 1958)

Validation with Beam Subject to Uniform Load Since data for beam under uniform load was not available, beam 1-C was also used for further validation of the beam case under uniform loading. In this case, Conventional Weapons Effects (CONWEP, US Army Engineer Waterways Experiment Station, 1992) was used to derive the loading function for a blast load of 500 pounds of TNT at a distance of 20 feet. This loading function, as shown in Figure 5, was used throughout the study in this paper. The results

computed from ABAQUS and DSAS for beam 1-C under uniform pressure load is shown in Figure 6. A difference of 0.023 in or 2.3% for the peak displacement exists between the two outcomes, which is acceptable. As noted earlier, a significant difference for the residual displacement exists due to the inability of the ABAQUS material model to capture the hysteretic behavior.

Time (msec)

Pres

sure

(psi

)

2 4 6 8 10 12 140

200

400

600

800

1000

1200

Figure 5 – Loading Function for 500 lb TNT @ 20 ft

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.1 0.12-0.25

0

0.25

0.5

0.75

1

1.25

1.5ABAQUSDSAS

Figure 6 – Displacement Time History of Beam 1-C under Blast Load

Validation for Columns

A 16 in x 16 in x 144 in reinforced concrete column was generated in ABAQUS (Dassault Systèmes, 2008) and DSAS (CIPPS, 2008). The column is subject to the above-mentioned loading function. Similarly to the beam models, a cubical grid of 1 in was found to be the most effective and economical size for the modeling of the column. Material properties were also taken to be the same as those of the beam. Spacing for transverse reinforcements was set at 8 in on center. Longitudinal reinforcements consisted of 8 No. 7 and were placed into 3 layers with a minimum concrete cover of 1.5 in all around. For ABAQUS to work properly, both ends of the column were extended by 6 inches and were used as the supports. The span length of the column was still at 12 feet. A typical layout of the column is shown in Figure 7. A summary of the results for the column subjected to transverse load only is shown in Fig 8. It should be noted that the difference between the two software applications was 0.75 in (36%) at maximum displacement (mid-span). This was due to the fact that the material model parameters in ABAQUS required some minor adjustment and DSAS does not include the shear reduction factor (SRF).

Figure 7 – Typical Column Layout

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.10

1

2

3

ABAQUSDSAS

Figure 8 – Displacement Time History – Column Subject to Blast Load

Parametric Study Columns Subject to Transverse Loads and Constant Axial Load With the outcome from the above validations, both experimentally and analytically, DSAS can now be used for the necessary computations. However, since experimental data on column was not available, one additional step was required to ensure the consistency of the result. An unconfined reinforced concrete column of 16 in x 16 in x 144 in consisted 8 No. 7 grade 60 steel reinforcements was used for this confirmation. The result obtained from DSAS was compared with the pre-defined data provided in the CRSI Design Handbook (2002). The compression strength of the concrete, f’c, was 4000 psi. The yield strength of the steel, fy, was 60000 psi. The layout of steel reinforcement was based on the recommendation outlined in Table 3-1 of the CRSI. An interaction diagram for the axial loads, P, acting on this column was generated from data computed by DSAS. The result, shown in Figure 9, indicated that the axial load and moment capacity values obtained from DSAS matched those outlined in the CRSI.

The parametric study was based on four confined reinforced concrete columns with the same dimensions stated above, but with various configurations of steel reinforcements. These columns were arbitrarily selected from the CRSI Design Handbook (2002). The sizes of the transverse reinforcements were selected based on Table 3-2 of CRSI Design Handbook (2002). Spacing of

the transverse reinforcements was at 12 in on center for all configurations. A summary of column material is shown in Table 4.

Moment (ft-kips)

Axi

al L

oad

(kip

s)

0 30 60 90 120 150 180 210 240 270-500

-250

0

250

500

750

1000

1250

1500

DSASCRSI

Figure 9 – Axial-Moment Interaction Diagram – Unconfined Concrete

Table 4 – Summary of Columns Physical and Material Properties f’c = 4000 psi, fy = 60000 psi, Es = 29000 ksi, 16” x 16” x 144” Column Bars Stirrups ρ s', in Mmax, Kips*ft Pmax, Kips

1 8 #7 #3 1.88 12 323.7 1720 2 8 #10 #3 4.88 12 450.7 2140 3 4 #14 #4 3.52 12 474.5 2050 4 12 #11 #4 7.31 12 598.8 2820

A series of axial load-moment interaction (P-M), flexure resistance, and pressure-impulse (P-I) diagrams were generated for the above columns. Observations from all these results yielded common points; hence, discussion will only be based on Column 1 listed Table 4. Figs. 10 to 13 describe three different approaches in presenting the outcomes of the study. The first two approaches used the Moment-Curvature and Flexure Resistance relationships to demonstrate column behavior from the elastic to plastic range based on the analysis of loads and deformations. The results showed the changes in column stiffness for various magnitudes of constant axial loads, P. Within the elastic range, the stiffness of the column decreased slowly as P increased up to the balance load, Pbal. Once Pbal was surpassed, the stiffness of the column decreased at a much faster rate. In the elasto-plastic range for the reinforcements, the stiffness of

the column increased as P approached the balance point; after which, the stiffness of the column began to decrease sharply. In the plastic range of the reinforcements, the lower P were, the larger the displacements the column could undertake. This was probably due to the formation of the hinge at the column mid-span that allowed the column to behave this way. As P increased, the plastic range of the column became much shorter and failure took place sharply. Again, the time between the formations of the hinges at the column mid-span ant at the two support ends occurred much faster, and therefore caused the column failure. In addition, below Pbal, P actually enhanced the moment capacity and strengthened the column. Once the balance point was surpassed, P became the contributing factor to the column failure. The P-I diagrams, on the other hand, predicted the points of failure of the column based on each magnitude of P and impulse, I, that the column experienced over the time period. The Impulsive Loading region for P < Pbal had higher tolerance than that of P > Pbal. As P increased, the Dynamic Loading region became shorter and the columns exhibited longer Quasi-Static Loading Regions. Hence, the columns achieved failure.

Figure 14 shows the time-displacement history for the values of P as indicated in Figure 10. The results were obtained from DSAS and ABAQUS. In the case of DSAS, as mentioned above, since the Shear Reduction Factor (SRF) was not included in the calculation, Column 1 actually failed at Pbal. Conversely, the column did not fail in ABAQUS as it included the SRF. Regardless, it was noted that for 0 < P ≤ Pbal, the deflections were actually reduced compared to that of P=0 kips. However, as P became greater than Pbal, the deflection turned out to be higher than that of P=0 kips. As indicated in Fig 14, as P surpassed Pbal, the deflection increased until P reached Pmax, 1750 kips, where Column 1 failed.

Moment (ft-kips)

Axi

al F

orce

(kip

s)

0 50 100 150 200 250 300 350-600

-400

-200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

P4 = 1000 kips > Pb

P3 = Pb = 560 kips

P2 = 250 kips < Pb

P1 = 0

Figure 10 – Axial Load-Moment Interaction Diagram – Column 1

Curvature (1/in)

Mom

ent (

ft-ki

ps)

0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.0240

50

100

150

200

250

300

350

P1= 0

P2 = 250 kips < Pb

P3 = Pb = 560 kips

P4 = 1000 kips > Pb

Figure 11 – Moment-Curvature – Column 1

Displacement (in)

Pres

sure

(psi

)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

P1 = 0

P2 = 250 kips < Pb

P3 = Pb = 560 kips

P4 = 1000 kips > Pb

Figure 12 – Flexure-Resistance Curve – Column 1

Impulse (psi-sec)

Pres

sure

(psi

)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900

1000

Direct Shear

P = 0

P = 250 kips < Pb

P = Pb = 560 kips

P = 1000 kips > Pb

Figure 13 – PI curves – Column 1

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.10

1

2

3

P=0 - DSASP=250 - DSAS

P=560 - DSAS

P=1000 - DSAS

P=0 ABAQUS

P=250 ABAQUS P=560 - ABAQUS

P=1000 - ABAQUS

Figure 14 – Time-Displacement History – DSAS and ABAQUS

Columns Subject to Transverse Loads, Constant and Variable Axial Loads Having gone through the above process, the columns were then subject to the variable axial loads, Pvar, as shown in Figure 15, in addition to the transverse loads and the constant axial loads, P. The dynamic reactions obtained from applying the blast load to beam 1-C were used as the variable axial loads, Pvar, on the column. The result of the deflection at the mid-span of Column 1 is summarized in Figs. 16 to 18 and Table 5. It was noted that Pvar actually produced some significant effect on the deflection of the columns. For a given axial load, P, the deflection at mid-span of Column 1 was actually increased when applied with the additional variable axial loads, Pvar. Once Pbal was reached, the deflection at mid-span caused by additional Pvar doubled in magnitude. As P surpassed Pbal, the deflection at mid-span amplified significantly. In this case, at P = 1000 kips, the additional Pvar caused the deflection at mid-span to jump almost 10 times and the columns failed as shown in Figure 19.

Time, sec

Dyn

amic

Rea

ctio

n, K

ips

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-100

0

100

200

300

400

Figure 15 – Variable Axial Load Profile

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.10

1

2

3

P=0P=250P=250+Pvar

Figure 16 – Effects of Variable Axial Loads on Deflection P < Pbal

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

P=0P=560P=560+Pvar

Figure 17 – Effects of Variable Axial Loads on Deflection P = Pbal = 560 kips

Time (sec)

Dis

plac

emen

t (in

)

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

25

Disp, in P=0Disp, in P=1000Disp, in P=1000+Pvar

Figure 18 – Effects of Variable Axial Loads on Deflection P > Pbal

Figure 19 – Column Failure - P =1000 kips plus Variable Axial and Transverse Loads

Table 5 – Summary of Effects by Variable Axial and Transverse Loads

Load P=0 P=250 kips

P=250 kip +Pvar

P= 560 kips

P=560 kips + Pvar

P=1000 kips

P=1000 kips + Pvar

Midspan Disp, in 2.84 2.40 2.62 2.28 4.90 2.66 24.5

Summary and Conclusions

Three approaches were used to determine the effects of variable axial loads on a column. First, only transverse loads and constant axial loads were applied to the column. Axial load-moment interaction diagram for each column was generated to obtain the maximum load, Pmax, and the balance load, Pbal. Since the maximum moment capacity occurred at the Pbal, it was used as a benchmark for determining the behavior of the column as the constant axial loads, P, varied. The first two approaches used moment–curvature and load–deflection relationships to determine the columns behaviors in the elastic, elasto-plastic, and fully plastic ranges. Pressure– impulse relationship was used in the third approach. Rather than using the above parameters, P-I curves

for each loading case were generated and compared to that for Pbal. Time-displacement histories in all cases were plotted and comparisons were made using Pbal as the bench mark to determine the effects of the constant axial loads. Once the above process was completed, variable axial loads were applied to the columns. Displacements resulted from the variable axial loads were compared to those from the above-mentioned process. Hence, the effects of variable axial loads on the columns were determined.

In all three analyses, the P-I diagrams were the quickest way to determine the state of the columns. With regards to the effects of variable axial loads, Pvar, in this particular case, it was found that they did produce some minor effects on the column behavior. The introduction of Pvar allowed the column to behave better i.e. less deflection. However, as P approached to Pmax, the additions of Pvar pushed the bearing capacity of the columns over the limit that resulted in the column failure. The above analyses show that variable axial loads do create some effects on reinforced concrete columns. These effects are not significant to be considered in the designs of the above-mentioned columns, as long as the constant axial loads generated by the supported mass are well within the maximum allowable design limits and less than the balance load. However, if the remaining capacity between the actual loads and the maximum allowable design limit are insufficient, then there exists a possibility that the variable axial loads will contribute to the failure of the column. In addition, the effects of variable axial loads also vary depending on the load-time history and the column configurations. Hence, for buildings that are susceptible to blast loads, their designs should include the effects of variable axial loads. Acknowledgement The authors acknowledge the support from the US Army Research and Development Center (ERDC) and from the Canadian Forces. References ABAQUS Manual, 2008, Dassault Systèmes Simulia Corp., Providence, RI, USA. Astarlioglu, S., Krauthammer, T., “A Comprehensive Software Suite for Rapid Analysis of

Structural Response to Weapons Effects.” Tech. Rep. Center for Infrastructure Protection and Physical Security (CIPPS), University of Florida, Gainesville, FL, 32611

Biggs, J.M., 1964. “Introduction to Structural Dynamics” McGraw-Hill Book Company: New

York, NY. Concrete Reinforcing Steel Design HandBook, 9th Ed, 2002. Concrete Reinforcing Steel

Institute, North Plum Grove Road, Schaumburg, Illinois 30173-4758 CONWEP, US Army Engineer Waterways Experiment Station, 3909 Halls Ferry Road,

Vicksburg, MS.

Dynamic Structure Analysis Suite (DSAS), Version 2, Center for Infrastructure and Physical

Security (CIPPS), University of Florida Feldman, A., Siess, C., “An Investigation of Resistance and Behavior of Reinforced Concrete

Members Subjected to Dynamic Loading – Part II.” Tech. Rep. SRS Report No. 165, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, 1958.

Krauthammer, T., 2008. “Modern Protective Structures.” CRC Press, Taylor & Franis Group,

Boca Raton, FL, USA. Krauthammer, T., 2007. “Modern Protective Structures” Bazeos, N., Holmquist, T.J., 1986

“Modified SDOF Analysis of R. C. Box-Type Structures.” Journal of Structural Engineering, Vol. 112, No. 4, pgs 726-744

Krauthammer, T., Shahriar, S., Shanaa, H., “Response of RC Elements to Severe Impulsive

Loads.” Journal of Structural Engineering, ASCE 116, No. 4, pgs 1061-1079 Krauthammer, T., Frye, M.., Schoedel, R., Seltzer, M., Astarlioglu, S., “Single-Degree-of-

Freedom (SDOF) Computer Code Development for The Analysis of Structural Elements Subjected to Short-Duration Dynamic Loading.” Tech. Rep. PTC-TR-002-2003, Protective Technology Center, Pennsylvania State University, University Park (August 2003)

Smith, P.D. and Hetherington, J.G., 1994. “Blast and Ballistic Loading of Structures.”

Butterworth-Heinemann Ltd: London, UK.