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Publication Number: FHWA-HRT-05-063
Date: May 2007

Evaluation of LS-DYNA Concrete Material Model 159

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Appendix B. Developer Support of the Texas T4 Bridge Rail Analyses

Introduction

This appendix discusses the calculations performed by the developer in support of the evaluation of the Texas T4 bridge rail conducted by the user. These calculations examine bridge rail response and deflection as a tradeoff between the concrete material properties and the spring material properties. They were conducted after the user explored the use of reduced fracture energies for concrete, but before recalibration of the spring model and adjustment of the anchor bolt model. They should be considered preliminary calculations, not the final result. They are included here to show how the calculations evolved from being overly stiff to reasonable, and to show the thought process involved in the effort.

Pertinent conclusions drawn from these calculations are:

  • Approximately 85 to 90 percent of the available kinetic energy is absorbed by the pendulum spring and honeycomb. Therefore, a 10 percent change in pendulum energy absorption results in a 100 percent change in concrete energy absorption. This suggests that accurate models of the pendulum spring and honeycomb are needed to get accurate delivery of energy to the bridge rail.
  • Data-to-data comparisons indicate that the honeycomb in tests P3 through P7 is slightly stiffer than that in the calibration tests; therefore recalibration of the spring model was warranted. The developer's approximate technique was to scale up the stiffness of the existing spring model, as discussed in this appendix, to correlate measured and computed spring/honeycomb compressions. The user's more thorough technique was to develop a new spring model through correlations with test P5, as discussed in chapter 8. The developers data-to-data comparisons for the pendulum stiffness are not included in this report (but are available upon request), because the focus of this report is the concrete model, and not the spring/honeycomb model.
  • With use of the developer's scaled spring model, calculated deflections correlate well with test P4 with 51 mm (2 inches) of rail deflection if default concrete material properties are used (repow = 1.0 and Gfs / Gft = 1.0). Calculated deflections correlate best with test P5 with 117 mm (4.6 inches) of rail deflection if slightly reduced fracture energies are used (repow = 0.5 and Gfs / Gft = 0.5).

Unless otherwise specified, these calculations were performed with:

  • The original SBP spring model before recalibration with test P5. Scaling of the original SBP model is used to stiffen the spring in numerous calculations.
  • The original anchor bolt model with nodes merged to the base plate.
  • Concrete fracture energies that are approximately 80 percent of the preliminary properties originally tabulated by the user. These 80 percent values are the default fracture energies currently implemented in ls-dyna (at the time this report was written). They are referred to as the developer's baseline properties.

Bridge Rail Data

The user conducted five pendulum impact tests on the Texas Department of Transportation T4 Bridge Rail.(12) The pendulum contains a crushable nose with 10 stages of replaceable aluminum honeycomb cartridges. Two bridge rail designs were tested: one four-bolt design and one three-bolt design. The pendulum is 838 kg (1,847 lbs) and impacts the rail at a nominal velocity of 9,835 mm/sec (22 mi/h). The four-bolt rail experienced punching shear failure in two tests (designated P3 and P4). The three-bolt rail did not fail, although hairline cracks were visible in one of two tests (P5 and P7). A third three-bolt rail test was conducted using the previously tested, but undamaged rail (P6). Again, this rail did not fail, but hairline cracks were visible. Rail deflections and pendulum crush are listed in Table 14 for all five tests.

Table 14. Rail deflection and pendulum crush in the Texas T4 bridge rail tests.
Test No. Design Rail Deflection Honeycomb Crush Damage
P3 four-bolt 117 mm (4.6 inch) 417 mm (16.4 inch) Punching shear failure
P4 four-bolt 51 mm (2.0 inch) 414 mm (16.3 inch) Punching shear failure
P5 three-bolt 15 mm (0.6 inch) 439 mm (17.3 inch) None
P6 three-bolt 33 mm (1.3 inch) 295 mm (11.6 inch) Cracks
P7 three-bolt 30 mm (1.2 inch) 439 mm (17.3 inch) Cracks

The honeycomb crush listed in Table 14 is for nine stages of the honeycomb, as reported in Buth.(12) It does not include the first stage, which is the 76-mm (3-inch) stage that impacts the bridge rail. In all tests, the first stage crushes nearly completely; thus, the developer suggests adding an additional 70 to 76 mm (2.8 to 3 inches) to the crush listed in Table 1 to get the total crush of all 10 honeycomb stages.

The measured deflections vary by a factor of 2.3 for the four-bolt design. The average rail deflection measured in tests P3 and P4 is 81.5 mm (3.21 inches), and the standard deviation is 43.1 mm (1.7 inches). Assuming a normal distribution, approximately 68 percent of all test results are expected to lie within 1 standard deviation of the mean, 95 percent within 2 standard deviations, and 99 percent within 3 standard deviations. Therefore 68 percent of all test results are expected to lie between 38 and 125 mm (1.5 and 5 inches), 95 percent of all test results are expected to lie between 0 and 168 mm (0 and 6.1 inches) , and 99 percent of all test results are expected to lie between 0 and 211 mm (0 and 8.31 inches).

Parametric Studies

At the beginning of the computational effort discussed in this appendix, initial calculations conducted by the user correlated well with the measured deflections and damage modes of the three-bolt design, but not those of the four-bolt design. The four-bolt design calculations were stiff (less deflection and damage) relative to the test data. Therefore, the user reduced the mesh size (fewer elements) in order to improve the run time for parametric studies. This task was accomplished by removing the entire deck and fixing the nodes at the bottom of the parapet.

The developer performed a series of parametric studies to help understand and improve the correlations for the four-bolt design. Studies were conducted using both the original mesh with the full deck and the reduced mesh with the fixed parapet. The baseline impact velocity used was that measured in test P3, which is 9,555.6 mm/sec (376 inches/sec), 34.4 km/h (21.375 mi/h). The three-bolt design was not analyzed by the developer.

Parametric studies were conducted to examine three possible sources of excess stiffness:

  • The concrete material properties.
  • The steel reinforcement material properties.
  • The pendulum spring material properties.

A spring element with a nonlinear force-deflection curve is used to represent nine crushable stages of the honeycomb. Hex elements and a honeycomb material model are used to represent the first stage of honeycomb that impacts the rail. The reduced finite element mesh is shown in Figure 194.

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Figure 194. Finite element model of the pendulum, rail, and fixed end of parapet.

The studies indicate that adjustments in the steel reinforcement properties have little effect on the computed response. On the other hand, adjustments in both the concrete model properties and pendulum spring properties have significant effects on the computed response. The tradeoffs between adjustments in the properties of the concrete versus pendulum spring are discussed by the developer in the following sections.

Fixed Parapet Computational Results

Damage from the baseline property calculation is shown in Figure 195. Baseline properties are approximately those for 30.44 MPa (4,415 lbf/inch2) concrete with 24 mm (0.94 inches) maximum aggregate size. Most parametric studies examine response for variations in fracture energy; thus, fracture energy values are reported throughout this chapter. The baseline fracture energies are Gft = 0.074 N-mm with Gfs = Gf t and Gfc = 100 Gf t. Although the damage mode is similar to that observed in the tests, the damage is less severe. Rail deflection is approximately 10 mm (0.4 inches), which is substantially less than the 51 mm (2 inches) and 117 mm (4.6 inches) measured in the tests.

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Figure 195. Damage fringes calculated with baseline properties for a fixed end parapet.

Modified Concrete Property Studies . The first parametric studies were conducted on the concrete material properties. These calculations indicate that an overall reduction in concrete properties, for example, those of concrete in the 22 to 24 MPa (3,191 to 3,481 lbf/inch2) range, would produce the measured deflection. However, 22 to 24 MPa (3,191 to 3,481 lbf/inch2) is a much lower strength concrete than the 30.4 MPa (4,415 lbf/inch2) used in the test. It is doubtful that the concrete used in the test structure could somehow be as low as 22 to 24 MPa (3,191 to 3,481 lbf/inch2) when the reported value is 30.4 MPa (4,415 lbf/inch2).

For example, as a lower bound case, the four-bolt calculation was run with approximate properties for 20 MPa (2,901 lbf/inch2) concrete with 8 mm (0.3 inches) aggregate (Gft = Gfs = 0.041 N-mm and Gfc = 2.03). Recall that fracture energy and stiffness decrease with a reduction in concrete compressive strength. Fracture energy also decreases with a reduction in aggregate size. With these weak and brittle properties, the rail breaks through the concrete, with no rebound of the pendulum. Additional calculations were performed with 26 MPa (3,771 lbf/inch2) concrete with fracture energies for 8 mm (0.3 inches) (Gft = 0.049 N-mm) and 24 mm (0.94 inches) (Gft = 0.066 N-mm) aggregate. Although damage was more extensive than calculated with the baseline properties, the calculated deflections of 27 mm (1.06 inches) and 14 mm (0.6 inches) are still less than measured. Also note that a 35 percent increase in tensile fracture energy due to aggregate size results in a computed decrease in deflection of nearly 50 percent (from 27 mm (1.06 inches) to 14 mm (0.6 inches)).

Energy Absorption Studies . The next parametric studies examined the amount of energy needed to deflect the rail 51 to 117 mm (2 to 4.6 inches) using 30.4 MPa (4,415 lbf/inch2) concrete. The mass and impact velocity of the pendulum set the maximum amount of energy available to be absorbed by the pendulum and target rail. Crushing of the pendulum honeycomb and spring absorbs energy. As more energy is absorbed during pendulum crush, less energy is imparted to the rail target. Less rail energy results in less rail deflection.

The objective of these studies was to determine if it were possible to achieve the measured deflections using 30.44 MPa (4,415 lbf/inch2) concrete if no energy were absorbed by the pendulum crush. These bounding calculations maximize the energy imparted to the rail by eliminating the energy absorbed by pendulum crush. These calculations allowed the author to determine the relationship between the energy absorbed by the rail and its subsequent damage and deflection. The boundary calculation also provided a rough estimate of the energy needed to computationally damage and deflect the structure in a manner consistent with that observed in the tests.

All energy absorption studies were conducted using a noncrushable pendulum model and the fixed end parapet model. The original finite element model of the pendulum was simplified by removing the moveable crushable nose of the pendulum (spring and honeycomb) and replacing it with elastic steel. The simplified pendulum has the dimensions of the first stage of honeycomb. The mass of the simplified pendulum is equal to that of the original pendulum (838 kg (1,847.5 lb)). Little or no internal energy is absorbed by the simplified pendulum model.

Computed rail deflections as a function of kinetic energy level are given in Table 15. Six calculations were performed at six different pendulum impact velocities, which correspond to six different pendulum initial kinetic energy levels. The kinetic energy levels are listed as a percentage of the maximum kinetic energy available from impact with an 838 kg (1,847.5 lb) pendulum at a velocity of 9,555.6 mm/sec.

Table 15. Rail deflection as a function of maximum kinetic energy available.
Kinetic Energy (%) Rail Deflection
10 26 mm (1.02 inches)
15 42 mm (1.7 inches)
20 60 mm (2.4 inches)
25 95 mm (3.7 inches)
30 >163 mm (6.42 inches)
100 No rebound

Approximately 20 to 25 percent of the available kinetic energy is needed to deflect the rail between 51 and 117 mm (2 and 4.61 inches) (the measured values). At 30 percent kinetic energy, no rebound occurred by 100 msec (the duration of the calculation). Therefore, the deflection is probably infinite rather than 163 mm (6.42 inches). The deformed configuration of the structure for the 25 percent kinetic energy case is shown in Figure 196.

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Figure 196. The deformed configuration and damage of the Texas T4 bridge rail (four-bolt design) from impact with the simplified pendulum model (without a crushable nose) at 25 percent maximum available kinetic energy is similar to that observed during the tests.

The significance of these calculations is that the majority of available kinetic energy is absorbed by crush of the pendulum and not by internal energy of the structure. Small changes in the amount of energy absorbed by pendulum crush result in large changes in the energy absorbed by the structure. For example, an 11 percent change in energy absorbed by the pendulum (from 90 to 80 percent) results in a 100 percent change in energy absorbed by the structure (from 10 to 20 percent). This level of change can make the difference between the structure appearing to be too stiff and the structure deflecting in the appropriate range. Therefore, accurate modeling of the crush and energy absorption of the pendulum spring and honeycomb is needed in order to get accurate delivery of energy to the target structure.

Review of the theoretical honeycomb cartridge configurations discussed in reference 12 confirms that most of the available kinetic energy is expected to be absorbed by crush of the pendulum, as is being calculated. The available kinetic energy is approximately 38,200 kN-mm, based on the pendulum mass and impact velocity. If the theoretical potential energy of crush is calculated, based on the total crush force of each honeycomb cartridge times the thickness of each cartridge, a total of 49,500 kN-mm is estimated. However, honeycomb typically crushes to about 75 percent strain, then stiffens dramatically, so the estimated crush energy is about equal to the available kinetic energy.

Fracture Energy Studies. Additional concrete material property studies were conducted with the noncrushable pendulum model. Unlike the previously discussed studies, which examined different strengths of concrete, these studies examine only 30.4 MPa (4,415 lbf/inch2) concrete with possible modifications in baseline fracture energies.

Four calculations were conducted with 10 percent of available impact energy. In the first calculation, the fracture energy was reduced from the baseline value of Gft = 0.072 N-mm to a low end value of Gft = 0.040 N-mm. Rail deflection increased from 26 mm (1.02 inches) to about 35 mm (1.38 inches). In the other three calculations, the rate effect on the fracture energy was adjusted via the repow input parameter. This parameter scales the fracture energy in proportion to the theoretical increase in strength with strain rate, as discussed in the companion to this report Users Manual. The additional three calculations were conducted with the following parameters:

  • repow = 0 and Gft = 0.072 N-mm.
  • repow = 0 and Gft = 0.040 N-mm.
  • repow = 0 and Gft = 0.072 N-mm and Gfs = 0.5Gft.

By default, repow = 1. With this value, if the rate effects formulation increases the strength by a factor of 3, the rate effects formulation will also increase the fracture energy by a factor of 3. If repow = 0, no increase in fracture energy is calculated; the input value is used independent of the strain rate.

Calculated deflections are 36 mm (1.42 inches), 68 mm (2.68 inches), and 40 mm (1.57 inches), respectively. The first additional calculation indicates that removing the rate effect from the baseline fracture energy moderately increases deflection from 26 mm (1.02 inches) to 36 mm (1.42 inches), similar to the effect of reducing Gft from 0.072 to 0.040 N-mm. Reducing both repow and Gft has a combined effect and moves the rail deflection into the desired range. The third addition calculation indicates that reducing Gfs from one to one-half of Gft also has a small but noticeable effect on rail deflection.

These studies indicate that it is possible to achieve the measured deflection of 51 mm (2 inches) to 117 mm (4.61 inches) at 10 percent of available impact energy if the tensile fracture energies—static and dynamic—are reduced below the current default baseline values. However, the values needed are not necessarily realistic. For example, the fracture energy Gft = 0.072 N-mm is the approximate default fracture energy for 30.44 MPa (4,415 lbf/inch2) concrete with 24-mm (0.94-inch) aggregate. The fracture energy Gft = 0.040 N-mm is the approximate default for 20 MPa (2,901 lbf/inch2) concrete with 8-mm (0.31-inch) aggregate. The default value for 30.44 MPa (4,415 lbf/inch2) concrete with 8-mm (0.31-inch) aggregate is Gft = 0.054 N-mm. These values agree with CEB tabulated values. Hence, use of Gft = 0.040 N-mm for 30.44 MPa (4,415 lbf/inch2) concrete is unusually brittle and outside the range of expected values.

In general, the developer recommends a rate effect on fracture energy equivalent to repow = 1, as do many experienced analysts. However, some analysts suggest no rate effect. Thus, it may be reasonable to allow repow to drop below a value of 1. In addition, although the fracture energy in tension is well documented, the fracture energy in shear is not well documented. It therefore seems reasonable to allow the fracture energy in shear to drop below the tensile baseline value as well.

Modified Spring Stiffness . It was previously concluded that the energy absorbed by crush of the spring and honeycomb models has a significant effect on computed response. Therefore, a series of parametric studies was conducted with an increase in spring stiffness. The spring is defined by a tabulated force versus deflection curve. The increase in stiffness is accomplished by scaling the tabulated deflection listing by a scale factor. The smaller the deflection scale factor, the larger the spring stiffness. For example, a scale factor on deflection of 80 percent results in an increase in spring stiffness of 125 percent (the reciprocal of 0.8).

Results from the baseline calculation, plus 10 additional calculations, are given in Table 16. Results are reported as a function of the scale factor applied to the spring deflection, as well as the input values used for repow and the ratio Gfs/Gft (with Gft = 0.074 N-mm). Results reported are approximate crush energy (absorbed by spring and honeycomb), rail deflection, and pendulum crush (of the spring plus honeycomb). The majority of the studies were conducted with the 60 percent deflection scale factor because, at the time, it was not known that only 9 of the 10 honeycomb stages were included in the documented crush measurements. Therefore, most calculations were conducted to achieve a computed crush of about 417 mm (16.4 inches), the documented value.

Table 16. Rail deflection, pendulum crush, and crush energy as a function of spring stiffness and concrete properties for the fixed end parapet model.
Spring Deflection Factor (%) Stiffness Factor Repow Gfs / Gft Crush Energy (%) Rail Deflection (mm) Pendulum Crush (mm)
50 2.00 1 - 79 65 363
60 1.00 1 1 87 41 427
60 1.67 1 ½ 82 65 414
60 1.67 ½ 1 84 57 418
60 1.67 ½ ½ 76 129 411
60 1.67 0 1 76 ¥ 424
60 1.67 0 ½ 68 ¥ 374
70 1.43 ½ ½ 83 76 451
80 1.25 ½ ½ 89 47 513
100 1.00 0 ½ 95 39 568
100 1.00 ½ ½ 97 18 586
100 1.00 1 1 98 10 592

mm = 0.039 inch

One significant point is that as the spring stiffness increases (the applied deflection scale factor decreases), the energy absorbed by the pendulum spring and honeycomb decreases. This is the desired trend because it increases the amount of energy delivered to the structure, which increases rail deflection. In general, pendulum crush energy absorption in the 79 to 89 percent range produces computed rail deflections in agreement with measured deflections.

Stiffening of the spring also affects the calculated acceleration history. Calculated acceleration histories are given Figure 197. The calculated histories are plotted for the baseline calculation, as well as three stiffened spring calculations. The calculated trend observed is that as spring stiffness increases, the maximum deceleration increases, and the time to maximum deceleration decreases. In addition, as rail deflection increases, the "tail" on the initial triangular-shaped pulse also increases. In other words, the deceleration does not drop down to 0 as rapidly as it does with minimal rail deflection.

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mm/sec = 0.039 inch/sec

Figure 197. Stiffening the spring increases the maximum deceleration and decreases the time at which the maximum deceleration occurs.

As an additional sensitivity study, one calculation was conducted at 9,835 m/sec (32,267 ft/sec) (35.4 km/h) (22 mi/h, the nominal impact velocity) rather than 9,556 m/sec (31,352 ft/sec) (34.45 km/h) (21.4 mi/h), the measured impact velocity). The calculation was conducted with a 60 percent scale factor on spring deflection (1.67 on spring stiffness), using baseline properties. The calculated deflection is 52 mm (2.05 inches) at 35.41 km/h (22 mi/h) compared with 41 mm (1.7 inches) at 34.45 km/h (21.4 mi/h). This is an increase in rail deflection of 27 percent for an increase in velocity of 2.9 percent.

The best correlation between the measured and calculated spring compression occurs for the 80 percent deflection scale factor case (1.25 on spring stiffness). Recall that the measured pendulum crush (all 10 stages) is just below 500 mm (19.7 inches). The computed compression of 513 mm (20.2 inches) is in reasonable agreement with this value. In addition, the 47 mm (1.9 inches) of deflection is also in reasonable agreement with the 51 mm (2 inches) of measured deflection reported for test P4. Figure 198 shows that the computed acceleration history is in good agreement with the P3 and P4 test histories throughout most of the measured response. This calculation was conducted with reduced properties (Gfs/Gft = 0.5 and repow = 0.5).

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mm = 0.039 inch

Figure 198. Good comparison of measured and calculated acceleration histories for the reduced fixed parapet mesh (spring stiffened using 80 percent deflection scale factor).

Full Deck Computational Results

The bridge rail tested is mounted on top of a parapet, which is attached to a deck. All previously discussed calculations were conducted with the parapet fixed at the bottom edge, without inclusion of the deck. This was done to increase the runtime of the calculations so that parametric studies could be performed with reasonable turnaround time (overnight). Calculations discussed in this section were conducted with the deck modeled.

Comparisons With and Without Deck . To start, one calculation was conducted with the full deck mesh using baseline properties and original spring stiffness. The calculated rail deflection is 28 mm (1.1 inches) and the calculated spring compression is 578 mm (25.8 inches). This deflection is 280 percent larger than the 10 mm (0.4 inches) calculated with the fixed end condition previously reported in Table 16 using baseline properties and the original spring stiffness. Both models (full or reduced meshes) underpredict the measured deflections of 51 mm (2 inches) and 117 mm (4.6 inches).

The acceleration history for a second calculation conducted with the deck explicitly modeled is shown in Figure 199. The computed acceleration history is compared with both the P3 and P4 test data and with the corresponding calculation conducted with the fixed parapet end condition. The calculations were conducted with reduced properties (Gfs/Gft = 0.5 and repow = 0.5) and with the spring stiffened with the 80 percent scale factor.

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mm = 0.039 inch

Figure 199. Addition of the deck model improves correlations with the test data (spring stiffened using 80 percent deflection scale factor).

With the deck modeled, the calculated rail deflection is 87 mm (3.4 inches) and the calculated pendulum crush is 484 mm (19.1 inches). Without the deck modeled, the calculated rail deflection is 47 mm (1.9 inches) and the calculated pendulum crush is 513 mm (20.2 inches). Both calculations compare well with the measured P3 and P4 test results (51- and 117-mm (2- and 4.61-inch) deflection and 500-mm (19.7-inch) crush). However, the calculated accelerometer history with the deck modeled is in better agreement with the late time accelerometer measurements than that calculated with the fixed end condition. Hence, inclusion of the deck adds flexibility to the rail structure and improves correlation with the measured accelerometer records.

Note that the stiffness of the adjusted spring model is based on scaling the stiffness of the original nonlinear spring model. Additional adjustments to the spring might improve correlations if the spring is made more stiff at low deflection (below about 200-mm (7.87-inch) compression) and made less stiff at high deflection (above about 400-mm (15.7-inch) compression). Refer back to Figure 197 for the original spring stiffness modeled. This type of adjustment is discussed in subsequent sections.

Spring Stiffness Parameter Studies . Eleven calculations were run with the full deck mesh, as listed in Table 17. All were conducted at 9,555.6 mm/sec (21.375 mi/h). Three calculations were conducted with original spring stiffness (100 percent spring deflection scale factor), as provided by the user. The remainder of the calculations was calculated with a stiffened spring, using a 60 to 90 percent scale factor on the spring deflection.

Table 17. Rail deflection, pendulum crush, and crush energy as a function of spring stiffness and concrete properties for the full deck model.
Spring Deflection Factor (%) Stiffness Factor Repow Gfs / Gft Crush Energy (%) Rail Deflection (mm) Pendulum Crush (mm)
60 1.67 1 1 76 75 413
65 1.54 ½ ½ 74 ¥ 423
70 1.43 1 1 83 56 461
75 1.33 ½ ½ 79 ¥ 464
80 1.25 ½ ½ 89 87 484
80 1.25 1 1 86 44 499
90 1.11 ½ ½ 85 57 530
90 1.11 1 1 90 33 541
100 1.00 0 ½ 83 ¥ 550
100 1.00 ½ ½ 88 45 568
100 1.00 1 1 91 28 578

mm = 0.039 inch

Calculations conducted with a spring deflection scale factor of 80 percent are in best agreement with the measured pendulum crush of approximately 500 mm (19.7 inches). One of the two calculations was run with original baseline properties and the 80 percent scale factor on the spring deflection. The calculated rail deflection is 44 mm (1.73 inches) and the calculated spring compression is 499 mm (19.6 inches). These computed values compare well with measured values of 51 mm (2 inches) of rail deflection and 500 mm (19.7 inches) of pendulum crush for test P4. The other calculation was run with reduced properties (Gfs/Gft = 0.5 and repow = 0.5) and the 80 percent scale factor on the spring deflection. The calculated rail deflection is 87 mm (3.43 inches) and the calculated spring compression is 484 mm (19.1 inches). These computed values compare well with the average measured deflection and the 500 mm (19.7 inches) of pendulum crush for test P3. Hence, the baseline properties model the stiffer response associated with test P4, while the reduced properties (Gfs/Gft = 0.5 and repow = 0.5) model the more flexible response associated with the average of test P3 and test P4.

Calculations conducted with original spring stiffness do not correlate as well with the test data as do the calculations with increased spring stiffness (80 percent scale factor on deflection). Using baseline properties, the computed rail deflection of 28 mm (1.1 inches) is lower than those measured in tests P3 and P4 (51 mm (2 inches) and 111 mm (4.4 inches), respectively). The computed pendulum crush is higher than measured (500 mm (19.7 inches) in both tests). Using reduced properties, the calculated rail deflection varies from 45 mm (1.8 inches) to infinity. The 45 mm (1.78 inches) of rail deflection is in agreement with the stiffer measured value (51 mm (2 inches)) of test P4. Property adjustments also could be made to obtain agreement with test P3 (112 mm (4.41 inches) of rail deflection). However, the calculated pendulum crush is still larger than the measured value. Hence, use of reduced concrete properties with original spring stiffness does not correlate well with all test measurements.

Results from Table 17 are plotted in Figure 200, along with the range in measured data. By examining the computed results in Table 17 and Figure 200, it is apparent that only one calculated rail deflection lies outside the three standard deviations range. This is the calculation conducted with repow reduced to 0 (which means that fracture energy is independent of rate effects) and Gfs / Gft= 0.5. Although not shown in Table 17, the observed trends indicate that all calculations, regardless of spring deflection scale factor, would produce infinite deflection with repow = 0 and Gfs / Gft= 0.5. This suggests that repow = 0 with Gfs / Gft = 0.5 are not appropriate input parameter values. The response modeled is too brittle, at least for this example problem.

Only one calculated rail deflection lies outside the one standard deviation range, although it lies within the two standard deviations range. This is the calculation conducted with baseline properties and original spring stiffness. The calculated deflection of 28 mm (1.1 inches) is 26 percent lower than the one standard deviation value of 38 mm (1.5 inches). This suggests that baseline properties are on the stiff side if original spring stiffness is assumed, but they are adequate if reduced spring stiffness is assumed.

Damage fringes for two of the stiffened spring calculations (80 percent scale factor on deflection) are shown in Figures 201-204. The damage calculated with reduced properties (Gfs/Gft= 0.5 and repow = 0.5) is more severe than that calculated with baseline properties. Both results are similar to the post-test damage previously shown in chapter 8 for tests P3 and P4.

Following this computational effort, data-to-data comparisons were made between the accelerometer records of bridge rail tests P3 through P7 with those of the rigid pole calibration tests. Comparisons were made for acceleration versus time, force versus deflection, energy absorbed versus deflection, and derived spring stiffness versus deflection. These comparisons indicated that the honeycomb in the calibration tests was less stiff than that in the bridge rail tests; therefore, stiffening of the spring was warranted. Ultimately, a new stiffened spring model was setup by the user.

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mm = 0.039 inch

Figure 200. All calculations run with baseline or slightly reduced properties (repow = 0.5 and Gfs / Gft = 0.5) produce rail deflections within two standard deviations of the measured results.

Click to view alternative textFigure 201. Deflection and damage calculated with baseline properties using a stiffened spring (80 percent displacement scale factor).

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Figure 202. Closeup of damage calculated with baseline properties using a stiffened spring (80 percent displacement scale factor).

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Figure 203. Deflection and damage calculated with slightly reduced properties using a stiffened spring (80 percent displacement scale factor).

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Figure 204. Closeup of damage calculated with slightly reduced properties using a stiffened spring (80 percent displacement scale factor).

Impact Velocity Parameter Study . Additional full deck mesh calculations were conducted at 9,835 mm/sec (35.41 km/h) (22 mi/h), the nominal impact velocity) rather than 9,555.6 mm/sec (34.4 km/h (21.4 mi/h), the measured impact velocity), as shown in Table 18 and plotted in Figure 205. This is an increase of roughly 1 km/h (0.62 mi/h). In general, the increase in velocity has a much stronger effect on the deflections computed with reduced properties than on the deflections produced with baseline properties. For the calculation with baseline spring and concrete properties, the deflection is 33 mm (1.3 inches) at 35.4 km/h (22 mi/h) 9,835 mm/sec compared with 28 mm (1.1 inches) at 9,555.6 mm/sec. This is an increase in rail deflection of 18 percent for an increase in velocity of 2.9 percent and an increase in kinetic energy of 5.9 percent. The 33-mm (1.3-inch) computed rail deflection is close to the 38-mm (1.5-inch) deflection associated with 1 standard deviation.

Table 18. Rail deflection, pendulum crush, and crush energy for calculations conducted with increased impact velocity (9,835 mm/sec) for the full deck model.
Spring Deflection Factor (%) Stiffness Factor Repow Gfs / Gft Crush Energy (%) Rail Deflection (mm) Pendulum Crush (mm)
60 1.67 1 1 78 95 412
70 1.43 ½ ½ 78 ¥ 451
80 1.25 1 1 89 54 510
85 1.18 ½ ½ 83 107 510
90 1.11 ½ ½ 86 216 544
100 1.00 ½ ½ 91 62 574
100 1.00 1 1 96 33 589

mm = 0.039 inch

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mm = 0.039 inch

Figure 205. Larger rail deflections are calculated at 9,835 mm/sec (large dots) compared with 9,556 mm/sec (solid and large dashed lines).

Deck to Parapet Connection

One final study examined how the joint is modeled between the deck and parapet. In the test structure, the concrete in the deck is poured separately from the concrete in the parapet. Thus, a joint exists between the deck and parapet. In all calculations previously discussed, this joint is not modeled. Instead, a tied interface firmly connects the parapet to the deck (contact_tied_nodes_to_surface). The developer's experience is that this type of interface is more suitable for a deck and parapet that are poured at one time (without a joint), but it is much too strong for a joint. Therefore, two additional calculations were conducted with the joint explicitly modeled. This was accomplished by using an interface that allows concrete separation but not penetration (contact_nodes_to_surface). The parapet will not disconnect from the deck because it is secured by steel reinforcement.

The two calculations were conducted with baseline and reduced properties, using the spring stiffened with the 80 percent scale factor. With baseline properties, the effect of modeling the joint is to increase the calculated deflection 9 percent from 44 mm (1.7 inches) to 48 mm (1.9 inches). The calculated pendulum crush is 502 mm (19.8 inches). Damage fringes are shown in Figure 206. The effect of including the joint is to eliminate damage at the base of the parapet where it connects to the deck. With reduced properties, inclusion of the joint increases the calculated rail deflection from 87 mm (3.4 inches) to infinity. The rail breaks completely through the concrete. Maximum separation between the deck and parapet is approximately 0.36 mm (0.01 inches), which is not visible to the eye.

The developer suggests that the calculations performed with the joint modeled are a better representation of the test structure than the calculations performed without the joint modeled. Both are bounding calculations. With a tied interface, the effect is to model no joint at all, as if the deck and parapet were one piece of continuously poured concrete. This calculation provides an upper bound on joint stiffness. With a separable interface, the effect is to model a joint with no tensile strength, as if there were no bonding of the paste from the parapet with that from the deck. This provides a lower bound on joint stiffness. In reality, slight bonding and tensile strength probably exist (but are much less than those of solid concrete). The developer's recommendation is to model joints with an interface that allows for separation but not penetration. More sophisticated approaches might include adding tensile strength to the contact surface or predamage to the concrete.

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Figure 206. The tensile damage at the base of the parapet is eliminated if a flexible joint is modeled (80 percent displacement scale factor with baseline properties).

Summary of Bridge Rail Calculations

All calculations conducted with the full deck mesh produce rail deflections that lie within two standard deviations of the measured rail deflections using either baseline (repow = 1.0 and Gfs / Gft = 1.0) or slightly reduced (repow = 0.5 and Gfs / Gft = 0.5) material properties for concrete. This is true regardless of whether the pendulum model is run with original spring stiffness or increased spring stiffness. However, the developer suggests that these calculations are overly stiff because the joint between the deck and parapet was not modeled.

Data-to-data comparisons (not shown) indicate that the pendulum crushing stiffness in the bridge rail tests is greater than that in the original calibration tests. Therefore, use of a stiffened spring model is warranted. The spring stiffened with an 80 percent deflection scale factor (1.25 stiffness factor) produces spring/honeycomb compressions in agreement with those in the tests. Therefore, the spring with 80 percent deflection scale was chosen as the appropriate spring stiffness. All full deck mesh calculations conducted with the stiffened spring (80 percent scale factor on displacement) produce rail deflections that lie within one standard deviation of the measured rail deflections using either baseline or slightly reduced material properties for concrete, whether or not the joint is modeled.

Without modeling the joint, the baseline properties tend to produce a response on the stiff side of the measured deflections, whereas the slightly reduced properties tend to produce a response closer to the mean measured deflection. Extremely reduced properties (repow = 0 and Gfs / Gft = 0.5) produce infinite deflections, and therefore are not recommended for use.

With the joint modeled, more flexibility is added to the structure. The baseline properties produce a response close to that of test P4. The slightly reduced properties allow the bridge rail to break through the concrete parapet. The developer's recommendation is to leave the baseline properties modeled as currently implemented, but to make each user aware that slightly reduced properties (repow = 0.5 and Gfs / Gft = 0.5) are an option that could be examined parametrically.

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United States Department of Transportation - Federal Highway Administration