Structural performance of approach slab and its effect on vehicle induced bridge dynamic response

rn triggers higher modes in the bridge dynamic response. Among these higher modes, the torsion modes contribute more to the 139 dynamic response of exterior girders, which results in larger increase in IMs (impact factors) for exterior girders than that of interior girders. The higher modes not only affect the IMs along transverse direction but also affect the IMs along the longitudinal direction. The higher bending modes of bridges excited by the vehicle bumping, may cause larger IMs at the quarter span than that at the mid-span, and this needs to be noticed in design and evaluation of prestressed concrete girders at sections with harped strands where the section strength could be more critical than at the mid span. AASHTO specifications may also underestimate the impact factors for these slab-on-girder bridges with larger faulting conditions at the bridge ends. The bridges with vehicles moving across them under larger faulting conditions have more uniform LDFs (load distribution factors) than under static loads, and they are consistently lower than those in AASHTO specifications. 8.3 Suppression of Vehicle-Induced Bridge Vibration TMD is investigated with the purpose to suppress the vehicle-induced vibration of bridges by a finite element approach. A model for the vehicle-induced bridge vibration controlled with the TMD system takes into account the road surface conditions. The damping provided by TMDs does not result in an appreciable reduction of the maximum dynamic displacement during the period of forced vibration (i.e. when the vehicle is on the bridge). However, it is evident from the analysis results that TMD is effective in reducing the free vibration. On the other hand, for all the bridges investigated in this study, the reduction of acceleration is more significant than that of the displacement. Generally speaking, it can be concluded that for the same TMD installed in the same bridge, it is more effective for cases that trucks pass the bridge in row than for cases having only one truck. Such a study is helpful in evaluating the control performance before real control devices are designed in practice. The TMD has more effect on short bridges than on longer bridges. This is due to the fact that the vibration of short bridge is more active than longer bridge for its relatively higher natural frequencies and multi-axle load frequencies. In summary, for a given condition the most effective way to reduce bridge response may or may not be to install a TMD. The analytical result will be useful in carrying out further studies of control strategies for suppressing the vehicle-induced bridge vibration. 8.4 Recommendations for Further Research From current analysis and results, recommendations of future related researches are as follows: ¾ Regardless of the efforts made to improve the structural rigidity and long-term performance of the approach slab, the magnitude of the bump will be a function of the total settlement. A more rigid approach slab will reduce the change of the slope angle (θ1 in Fig 1-1), but may also increase the local soil pressure beneath the contact area (sleeper slab), thereby may increase the faulting deflection (Δ1 in Fig. 1). Therefore, a balanced/optimal approach slab design is desirable and should be addressed. This requires collaboration between structural and geotechnical engineers to implement the developed procedure. 140 ¾ The research on static structural performance of approach slab is based on a given differential settlement. Therefore, developing a more accurate settlement prediction procedure based on field data is necessary. Field instrumentation will help improve the prediction accuracy in terms of settlements and soil stress. Without a known settlement, the developed procedure in this study cannot be fully implemented, though the approach slab can be conservatively designed as a simple beam. ¾ Since the “bump” is a subjective description, a further study may focus on establishing an acceptance guideline for “bump”, i.e., the criteria for an acceptable slope change and faulting of approach spans. Without this information, the approach design can only be based on a strength requirement as the present study does, though deformation has been predicted. A dynamic analysis simulating the truck system and driver response will help develop such a guideline. ¾ According to the results in Chapter 7, the TMD-based system may or may not be the most appropriate countermeasure for suppressing vehicle-induced vibration in bridges. It is worthwhile to investigate other vibration reduction approaches or even to improve vehicular technologies to solve this problem more adequately. 141 APPENDIX A: EFFECT OF EMBANKMENT SETTLEMENT ON BRIDGE APPROACH SLAB DESIGN – A FEW CONCERNED ISSUES A.1 Introduction The excessive differential settlement between a bridge and the adjacent pavement causes “bumps” or uneven joints at the bridge ends. When vehicles, especially heavy trucks, enter and leave the bridge, the bumps cause large impact loads to the bridge and the pavement. To provide a smooth transition between the bridge deck and the roadway pavement, a reinforced concrete approach slab that connects the bridge deck and the adjacent roadway is commonly used. When the approach slab is initially built on the embankment soil, it has full contact with the embankment fill. However, the long-term embankment soil settlement (due to the embankment soil consolidation and erosion) will form a gap between the slab and the soil and will cause the approach slab to lose its contacts and supports from the soil (Fig. 1-1). When the soil settlement occurs, the slabs will bend in a concave manner that causes a sudden change in slope grades of the approach slab (Fig. 1-1). The loads and weight of the slab will also be redistributed to the ends of the slab and faulting may occur, which in turn will cause a secondary deformation in the approach slab. Field observations indicated that a large deformation (either faulting or a sudden change in slope grades) of approach slabs still causes this “bump”, even though the approach slab is used to alleviate the bump problem. Several comprehensive studies on the performance of bridge approach slabs have been sponsored over the years by various state DOTs. The majority of the previous researches can be categorized as (1) syntheses of practice (Ha et al. 2003; Mahmood 1990; Stewart 1985), (2) identification of the sources of differential settlement (Chini et al. 1992; Kramer and Sajer 1991; Zaman et al. 1991), and (3) soil improvement (Briaud et al. 1997). Although the bump-related problems have been commonly recognized and the causes are clearly identified, no unified engineering solutions have emerged, primarily because of the number and complexity of the factors involved. In order to solve the bump problem, it is necessary to treat it as a stand-alone design issue. Since its deformation and damage due to embankment soil settlement still causes the bump problem, the approach slab must be provided with enough stiffness and strength for such a settlement. Field observations revealed some broken approach slabs in Louisiana due to excessive soil settlement. Engineering calculations of the conventional or standard approach slab are typically not conducted since the information for the interaction of the approach slab and the embankment settlement is unknown for a routine office design. There are no guidelines in the AASHTO code specifications (AASHTO 2002, AASHTO 2004) regarding the structural design of approach slabs considering the effects of embankment settlements. Similarly, the LADOTD design manual (LADOTD 2002) specifies only the minimum reinforcement requirement, but it does not specify how to conduct the structural design of the approach slabs. The Louisiana Department of Transportation and Development has launched a major effort to alleviate this problem by changing the design of approach slabs where differential settlement is expected (LQI 2002). The objective is to find a feasible solution that allows approach slabs that are strong enough to lose a portion of their contact supports without detrimental deflection, 142 perhaps by increasing the flexural rigidity (EI) of the approach slabs. To help design engineers develop such a solution, correlations between the approach slab’s deflection and the approach embankment settlement are required. A.2 Objective and Research Approach In the point view of approach slab design, there are two extreme cases. One extreme case assumes that the slab has a full contact with the embankment soil and that the performance of the slab is the same as that of the concrete floors on the ground. This assumption is not realistic in many cases due to the embankment settlement discussed above, and it may result in an unconservative design. In the other extreme case, an approach slab can be designed as a simple beam spanning the bridge end and the pavement end, assuming that no soil supports the beam between the two ends. This assumption, while conservative, will definitely result in an uneconomical design. In the majority of these cases, the approach slab is both partially separating from and partially contacting the soil. The supports provided to the concrete slab by the embankment soil will reduce the internal force in the slab. The extent of this support and reduction depends on the slab and soil interaction for a given embankment soil settlement. As the embankment soil settlement results in the separation of the slab from the soil, the slab must be designed to provide enough strength and stiffness for such a settlement. In the study of bridge approach slab performance under different embankment settlement, which is one of the components of the LQI program, Cai et al. (2005) developed design aids which considered the effect of embankment settlement on the internal forces and deformation of approach slabs. In this method a correlation among the slab parameters, deflection of the slab, internal moment of the slab, and the differential settlement was developed by analyzing the results using the finite element analysis. The results were normalized with respect to the traditional simply-supported beams (with pin and roller supports). For a given embankment differential settlement, the predicted maximum internal moments and deflections due to the total load and dead load only were normalized and were expressed by an exponential function as follows: TM L h T T Ke M M =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −= ×− )(108.1 0 4 2 7 78.0955.0 δ (A-1) DM L h D D Ke M M =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −= ×− )(103.2 0 4 2 7 8.095.0 δ A-2) Td L h T T K L he d d =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ×−= ×− 3.0)(106.1 0 )()58.205.3( 4 2 7 δ (A-3) Dd L h D D K L he d d =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ×−= ×− 3.0)(100.2 0 )()63.201.3( 4 2 7 δ (A-4) where MT and MD are the maximum moment of the approach slab due to the total load and the dead load respectively; δ = the differential settlement (ft), shown in Fig. 1-1; h = the thickness 143 of approach slab (ft); L = the length of the approach slab (ft); KDM and KTM are the moment coefficients that are self-evidenced in the equations; MT0 and MD0 are the maximum moment of simply-supported beam due to the total load and the dead load respectively; dT and dD are the maximum deflection of approach slab due to the total load and the dead load respectively; KDd and KTd are the deflection coefficients that are self-evidenced in the equations; dT0 and dD0 are the maximum deflection of simply-supported beam due to the total load and the dead load respectively. The maximum internal moment and deflection in the approach slab due to the live load is then calculated as follows: 00 DDMTTMDTL MKMKMMM −=−= (A-5) 00 DDdTTdDTL dKdKddd −=−= (A-6) Eqs. (A-1) to (A-6) take into account the effect of different soil settlement by considering the approach slab as partially supported by and partially separated from the soil. They also provide engineers with a convenient method which can be used to obtain the slab response by multiplying the slab response of the simply-supported beam with a computed coefficient. The information (deformation and internal force) can be used for the structural evaluation and design of approach slabs without conducting a complicated finite element analysis. For example, the predicted internal moments can be used to design the slab reinforcement for a given settlement (δ). Engineers can also control the excessive settlement by either improving embankment fills or foundations or by selecting a stiffer approach slab based on the predicted deformation. However, there may be some limits in their applications because these equations are based on HS20-44 truck loads and right angle slabs. Whether they are applicable to the AASHTO LRFD HL93 truck load and to skewed approach slab needs to be confirmed. In this paper, the applicability of the previous design aids to approach slabs under HL93 truck load and to skewed approach slabs was investigated. The effect of embankment settlement was also considered. Moreover, the capacity of the approach slab to some special truck loads is rated in order to evaluate the approach slab designed by using the design aids. These results will eventually be used to systematically evaluate the effectiveness of approach slabs and develop guidelines for their structural design. This information will also help decide when settlement controls are necessary in order to have an economical design of approach slabs. In the present study, a linear settlement of embankment is assumed, as shown in Fig. 1-1. The embankment settlement, a known parameter in the present finite element analysis, will be determined in another on-going research project supported by the LADOTD LQI program. Since one end of the slab sits on the relatively stiffer abutment while the other end on the relatively weaker soil or sleeper slab, a differential movement occurs between the two ends of the slab, which results in a gap between the slab and the embankment soil (Fig. 1-1). In this study, a typical approach slab, shown in Fig. A-1, was used. The dimensions of the approach slab and the properties of soil used as embankment fill in Louisiana are listed in Table A-1 and Table A-2, respectively. A 3D finite element model was established, as shown in Fig. A-1(b), where eight-node hexahedron elements, Solid 45 (ANSYS, Canonsburg, PA,), were used to form the finite element mesh. A contact and target pair surface element available in the ANSYS element library was used to simulate the interaction between the soil and the slab. This surface 144 element is compressive only and can thus model the contacting and separating process between the slab and the soil. In addition to the dead load of the slab, two lanes of HL93 truck loads were applied on the slab. The present research will provide essential information needed for the structural design of the approach slab considering embankment settlement. A.3 Applicability of Design Aids A.3.1 Analysis of Approach Alab Aubjected to HL93 Highway Load As discussed earlier, Eqs. (A-1) to (A-6) were derived (Cai et al. 2005) to simplify the calculation of internal forces and deformations based on the HS20-44 truckload. In this study, the investigation of the applicability of the equations to the HL 93 highway load was conducted by using the finite element method. The geometries and the material conditions of the FE model are shown in Fig. A-1. The HL93 highway load, consisting of the lane load and the HS20-44 truckload, is applied on the approach slab. A parametric study was conducted by changing the slab thickness, span length, and soil settlement to investigate whether the previous equations are applicable to the HL93 highway load. The slab parameters, i.e., length (L) and thickness (h), were investigated for the following cases: (1) h was varied from 1 to 1.5 ft for the fixed L = 40 ft; and (2) h was varied from 1.5 to 2.25 ft for the fixed L = 60 ft. For each case the settlement was varied from 0.5 to 2, to 6 inches. The results of the FE analyses for approach slabs subjected to HL 93 truck loads under different settlements are shown in Fig. A-2. Meanwhile, the results obtained by using equations (1) to (6) are also plotted to compare them with the FE analyses. The M0 and d0 used in the equations were calculated by applying HL 93 loads to a simply-supported slab. For the approach slab with a span length of 40 ft and thickness of 1.5 ft, the internal moment of the slab calculated by using the equations is almost the same as those from the FE analyses. Table A-1 Dimension of approach slab, sleeper slab, abutment, embankment and natural Soil Approach slab Sleeper slab Abutment Embankment Soil Natural Soil L1 (ft) S4 L2 (ft) H3 (ft) L3 (ft) L4 (ft) W1 (ft) H1 (ft) H2 (ft) S1 S2 L5 (ft) W2 (ft) H4 (ft) H5 (ft) S3 L6 (ft) 40, 60 2% 4 2 2 4 45 4 5 6 4 40 15 5 50 2 10 Table A-2 Parameters of soil Elastic modulus E (psi) Poisson’s ratio μ Cohesion c (psi) Friction angle φ(o) Density γ ( pcf) Embankment Soil 37700 0.3 11.6 30 127.4 Natural Soil 4360 0.3 7.25 30 95.6 145 L6 S1 1 L4 L3 L1 L2/2 L5 L2 H 3 S4 Embankment soil Approach slabBridge Natural soil H 4 S2 1 S1 1 W2 H 2 H 1 H 5Natural soil Embankment soil A A A-A W1 Abutment Sleeper slab (a) Sketch of bridge abutment (b) Typical finite element mesh with 8 node cubic element Fig. A-1 Approach slab and abutment model 146 This figure shows that for different approach slabs with different dimensions under different embankment settlements the moment and deformation obtained from the equations are close to those from FE analyses. Based on the investigation of different cases, we can conclude that the equations are applicable to approach slabs subjected to the HL93 highway load. While the derived equations are applicable for both HS20-44 and HL93 highway loads, the internal forces and the load factors in the design method are different. The results of the reinforcement design for the approach slab subjected to HS20-44 and HL93 highway loads are listed in Table A-3. It is observed that when the settlement is zero, the required reinforcement at the bottom of the slab (L=40’, h=12”) is 0.65 in.2/ft. (ρ=0.0063), and it increases to 1.57 in.2/ft. (ρ=0.0145) when the settlement increases to 0.6 in. This indicates that the current design (LADOTD 2002), 0.88 in.2/ft. (ρ=0.00815), is good only for the case of zero settlement and is not adequate for a settlement larger than 0.6 in. When the embankment settlement increases, more reinforcement is required and the required reinforcement ratio, ρ, will exceed the allowed maximum reinforcement ratio, ρmax, namely 75% of the balanced reinforcement ratio (AASHTO 2002). In this case, either the slab thickness should be increased or the soil should be improved to control the settlement within the allowable limit. A.3.2 Performance of Skewed Approach Slab It is not unusual for bridges to end with large skews to pavements. In order to confirm the applicability of the previously derived equations to the skewed approach slab, skewed approach slabs with a skew angle of 45o for a few different span lengths under different differential embankment settlements were analyzed using the FE method. The geometry, material properties, and load conditions of the FE model (Fig. A-3) are the same as those used in the normal (right) approach slab analysis, except for the skew angle. Approach slabs with different span lengths, 40 ft and 60 ft, and different thicknesses, 1 ft and 1.5 ft for 40 ft long slabs, and 1.5 ft and 2.25 ft for 60 ft long slabs, were investigated. Two AASHTO HS20-44 design truckloads were applied on the slab under different embankment settlements. Application of the uniform lane loads is not necessary for this applicability study since it has been proven that the HS20-44 is equivalent to the HL93 in terms of the approach slab performance. Fig. A-4 shows the stress distribution in a skewed slab under different embankment settlements. When the settlement is small, the slab is partially supported by the soil near the sleeper slab end and separates from the embankment soil near the abutment end. The performance of the slab under this condition is more like that of a triangular slab, as shown in Fig. A-4 (a). Although the maximum moment of the total section is in the rectangular part, the maximum stress is located in the triangular part. Therefore, using the moment per unit width to describe the internal force of the slab is more reasonable for design purposes. In the following study of skewed approach slabs, the moment per unit width is thus used instead of the total moment of the section. 147 Table A-3 Reinforcement ratio of slab under different settlement (HL93 and HS20-44) (f’c = 4000 psi and fy = 60,000 psi) 40-ft Slab 60-ft Slab Differential settlement (in) ρ for thickness of 12 in ρ for thickness of 18 in ρ for thickness of 24 in ρ for thickness of 21 in ρ for thickness of 27 in ρ for thickness of 36 in 0 0.0063 (0.0061)(*) 0.0025 (0.0024) 0.0014 (0.0014) 0.0035 (0.0034) 0.0022 (0.0022) 0.0014 (0.0014) 0.6 0.0145 (0.0137) 0.0081 (0.0077) 0.0058 (0.0056) 0.0060 (0.0056) 0.0046 (0.0043) 0.0036 (0.0034) 1.2 0.0213 (0.0207) 0.0114 (0.0110) 0.0074 (0.0072) 0.0083 (0.0077) 0.0065 (0.0061) 0.0052 (0.0049) 2.4 NA(**) 0.0143 (0.0138) 0.0080 (0.0079) 0.0121 (0.0112) 0.0093 (0.0087) 0.0068 (0.0066) 3.6 NA 0.0151 (0.0147) 0.0081 (0.0079) 0.0151 (0.0140) 0.0110 (0.0104) 0.0074 (0.0072) 4.8 NA 0.0153 (0.0149) 0.0081 (0.0079) 0.0174 (0.0162) 0.0120 (0.0115) 0.0077 (0.0075) 6 NA 0.0154 (0.0149) 0.0081 (0.0079) 0.0191 (0.0178) 0.0126 (0.0121) 0.0077 (0.0075) 7.2 NA 0.0154 (0.0150) 0.0081 (0.0079) NA (0.0191) 0.0130 (0.0124) 0.0078 (0.0076) Note: (*) The numbers in brackets are the results of approach slab due to HS20-44 truck load. (**) The required reinforcement ratio ρ exceeds the allowed maximum reinforcement of flexure, i.e., ρ > ρmax = 0.75ρb, meaning that section dimension needs to be increased. 148 0 500 1000 1500 2000 2500 3000 0 1 2 3 4 5 6 7 Settlement of Soil (in) M om en t o f S la b (k ip s- ft ) DL+HL93: L=40' H=1' (FEA) DL+HL93: L=40' H=1' (EQUATION) DL+HL93: L=40' H=1.5' (FEA) DL+HL93: L=40' H=1.5' (EQUATION) 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 3 4 5 6 7 Settlement of Soil (in) M om en t o f S la b (k ip s- ft ) DL+HL93: L=60' H=1.5' (FEA) DL+HL93: L=60' H=1.5' (EQUATION) DL+HL93: L=60' H=2.25' (FEA) DL+HL93: L=60' H=2.25' (EQUATION) (a) Moment of slab with span of 40 ft (b) Moment of slab with span of 60 ft 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 7 Settlement of Soil (in) D is pl ac em en t o f S la b (in ) DL+HL93: L=40' H=1' (FEA) DL+HL93: L=40' H=1' (EQUATION) DL+HL93: L=40' H=1.5' (FEA) DL+HL93: L=40' H=1.5' (EQUATION) 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 Settlement of Soil (in) D is pl ac em en t o f S la b (in ) DL+HL93: L=60' H=1.5' (FEA) DL+HL93: L=60' H=1.5' (EQUATION) DL+HL93: L=60' H=2.25' (FEA) DL+HL93: L=60' H=2.25' (EQUATION) (c) Displacement of slab with span of 40 ft (d) Displacement of slab with span of 60 ft Fig. A-2 Moment and displacement of approach slab versus soil settlement Fig. A-3 FE model of skewed approach slab 149 (a) Settlement = 0.5 inches (b) Settlement = 6 inches Fig. A-4 Stress distribution of skewed approach slab 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) M om en t i n Sl ab (k ip s- ft /ft ) DL: L=40' H=1' (FEM, SKEWED SLAB) DL: L=40' H=1' (EQUATION,SKEWED SLAB) DL: L=40' H=1' (EQUATION, RIGHT ANGLE SLAB) DL: L=40' H=1.5' (FEM,SKEWED SLAB) DL: L=40' H=1.5' (EQUATION, SKEWED SLAB) DL: L=40' H=1.5' (EQUATION, RIGHT ANGLE SLAB) -20 0 20 40 60 80 100 120 140 160 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) M om en t i n Sl ab (k ip s- ft /ft ) DL: L=60' H=1.5' (FEM, SKEWED SLAB) DL: L=60' H=1.5' (EQUATION,SKEWED SLAB) DL: L=60' H=1.5' (EQUATION, RIGHT ANGLE SLAB) DL: L=60' H=2.25' (FEM,SKEWED SLAB) DL: L=60' H=2.25' (EQUATION, SKEWED SLAB) DL: L=60' H=2.25' (EQUATION, RIGHT ANGLE SLAB) (a) Skewed approach slab with span of 40 ft (b) Skewed approach slab with span of 60 ft due to self weight due to self weight 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) M om en t i n Sl ab (k ip s- ft /ft ) DL+LL: L=40' H=1' (FEM, SKEWED SLAB) DL+LL: L=40' H=1' (EQUATION,SKEWED SLAB) DL+LL: L=40' H=1' (EQUATION, RIGHT ANGLE SLAB) DL+LL: L=40' H=1.5' (FEM,SKEWED SLAB) DL+LL: L=40' H=1.5' (EQUATION, SKEWED SLAB) DL+LL: L=40' H=1.5' (EQUATION, RIGHT ANGLE SLAB) -20 30 80 130 180 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) M om en t i n Sl ab (k ip s- ft /ft ) DL+LL: L=60' H=1.5' (FEM, SKEWED SLAB) DL+LL: L=60' H=1.5' (EQUATION,SKEWED SLAB) DL+LL: L=60' H=1.5' (EQUATION, RIGHT ANGLE SLAB) DL+LL: L=60' H=2.25' (FEM,SKEWED SLAB) DL+LL: L=60' H=2.25' (EQUATION, SKEWED SLAB) DL+LL: L=60' H=2.25' (EQUATION, RIGHT ANGLE SLAB) (c) Skewed approach slab with span of 40 ft (d) Skewed approach slab with span of 60 ft due to Total Load due to Total Load Fig. A-5 Moment of skewed approach slab versus embankment settlement 150 Fig. A-5 shows the moment per unit width of a skewed approach slab with a span length of 40 ft and 60 ft under different settlements of embankment soil due to the dead load and total load, respectively. The span length L for the skewed slab used here represents the length of the slab along the mid-width line. It is recalled that in the previous derived Eqs. (A-1) to (A-6), the internal force and displacement of the approach slab considering the settlement effects are calculated by using a coefficient to multiply the corresponding value of a simply-supported beam. Therefore, in these figures, in addition to the moment predicted directly from the FE analysis that is labeled “FEM, SKEWED SLAB”, two more calculations based on the derived equations were conducted. In the first one, the moment per unit width was obtained by using previously derived equations, but the M0 (moment of a simply-supported beam) was based on a simply-supported skewed slab using a finite element modeling since a direct calculation of M0 for skewed slabs is not available. This calculation is labeled “EQUATION, SKEWED SLAB”. In the second calculation, the unit width moment was calculated by using the derived equations, but the M0 is based on an equivalent simply-supported normal (right) slab where its span length is taken to be the same as the length along the mid-width line of the skewed slab. This calculation is labeled “EQUATION, RIGHT ANGEL SLAB”. The displacement of the skewed approach slab due to the dead load and total load are shown in Fig. A-6. Similarly, the displacement was calculated by using the equations with d0 based on a simply-supported skewed slab and a simply-supported normal slab. 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) D is pl ac em en t i n Sl ab (i n) DL: L=40' H=1' (FEM, SKEWED SLAB) DL: L=40' H=1' (EQUATION,SKEWED SLAB) DL: L=40' H=1' (EQUATION, RIGHT ANGLE SLAB) DL: L=40' H=1.5' (FEM,SKEWED SLAB) DL: L=40' H=1.5' (EQUATION, SKEWED SLAB) DL: L=40' H=1.5' (EQUATION, RIGHT ANGLE SLAB) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) D is pl ac em en t i n Sl ab (k ip s- ft /ft ) DL: L=60' H=1.5' (FEM, SKEWED SLAB) DL: L=60' H=1.5' (EQUATION,SKEWED SLAB) DL: L=60' H=1.5' (EQUATION, RIGHT ANGLE SLAB) DL: L=60' H=2.25' (FEM,SKEWED SLAB) DL: L=60' H=2.25' (EQUATION, SKEWED SLAB) DL: L=60' H=2.25' (EQUATION, RIGHT ANGLE SLAB) (a) Skewed approach slab with span of 40 ft (b) Skewed approach slab with span of 60 ft due to self weight due to self weight 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) D is pl ac em en t i n Sl ab (i n) DL+LL: L=40' H=1' (FEM, SKEWED SLAB) DL+LL: L=40' H=1' (EQUATION,SKEWED SLAB) DL+LL: L=40' H=1' (EQUATION, RIGHT ANGLE SLAB) DL+LL: L=40' H=1.5' (FEM,SKEWED SLAB) DL+LL: L=40' H=1.5' (EQUATION, SKEWED SLAB) DL+LL: L=40' H=1.5' (EQUATION, RIGHT ANGLE SLAB) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 Settlement of Embankment Soil (in) D is pl ac em en t i n Sl ab (k ip s- ft /ft ) DL+LL: L=60' H=1.5' (FEM, SKEWED SLAB) DL+LL: L=60' H=1.5' (EQUATION,SKEWED SLAB) DL+LL: L=60' H=1.5' (EQUATION, RIGHT ANGLE SLAB) DL+LL: L=60' H=2.25' (FEM,SKEWED SLAB) DL+LL: L=60' H=2.25' (EQUATION, SKEWED SLAB) DL+LL: L=60' H=2.25' (EQUATION, RIGHT ANGLE SLAB) (c) Skewed approach slab with span of 40 ft (d) Skewed approach slab with span of 60 ft due to Total Load due to Total Load Fig. A-6 Displacement of skewed approach slab versus embankment settlement 151 From Fig. A-5, it is obvious that the moment of skewed approach slabs based on FEM is close to that obtained from equations based on the simply-supported skewed slab, which indicates that the equations derived for the normal approach slab can be used to calculate the internal forces of the skewed approach slab. However, the simply-supported skewed slab is complicated and a hand calculation for the internal force analysis is not available because of its irregular shape. Thus a FE analysis is usually necessary. The moment obtained from the equations based on the equivalent simply-supported normal slab is larger than that of the skewed slab from the FE analysis, as shown in Fig. A-5. Therefore, it is conservative and more convenient to use the moment calculated from the equations based on an equivalent normal slab in the skewed approach slab design. The displacements of skewed slabs obtained from the FE analysis and equations based on a simply-supported skewed slab are close to each other, as shown in Fig. A-6, which means if the displacement of a simply-supported skewed slab is known, the equations derived for a normal approach slab can also be used to analyze the displacement of a skewed approach slab. However, the displacement obtained from equations based on an equivalent simply-supported normal slab is much smaller than that from the FE analysis of the skewed slab, which is caused by the large displacement of the long side of the skewed approach slab. Therefore, to use the developed equations, a longer nominal span length than that used for moment is needed. The comparison of results from the FE analysis and the results calculated by using the previous derived equations indicates that the internal force and deflection of skewed approach slabs can be obtained by using the equations derived for normal approach slabs, where M0 and d0 in the equations are internal forces and deformation of the simply-supported skewed approach slab respectively. Since the calculation of a simply-supported skewed approach slab is more complicated than a simply-supported normal slab, the FE method is also needed to analyze the simply-supported skewed slab, which makes the equation inconvenient for design purposes. Therefore, the internal forces of skewed approach slabs are compared to those of simply-supported normal slabs with the same nominal span. Results show that the internal forces of a skewed slab are less than that obtained from equations using a simply-supported normal slab, which indicates that using the equations with M0 of an equivalent normal slab to calculate a skewed approach slab internal force is conservative. A.4 Capacity Rating of Special Trucks The objective of the approach slab rating is to determine (1) the safe load-carrying capacity of the slab designed by using the design aids, and (2) whether a specific overweight vehicle may cause damage to the slab. In this study, approach slabs were rated by using trucks that may be more critical to the approach slab design. Three special trucks provided by LADOTD were used, as shown in Fig. A-7. For approach slabs, the same FE model (Fig. A-1) is used to analyze the internal force of the approach slabs subjected to different rating truck loads. The internal moments of different approach slabs under embankment settlement of 6.0 in. are listed in Table A-4. 152 Load rating was performed in accordance with the procedures given in the AASHTO Manual for Condition Evaluation of Bridges (AASHTO, 1994). The following strength condition equation was used to determine the load rating of the structure: )1( .. IM MM FR LL DDn + ∑−= γ γφ (A-7) where R.F. is the rating factor, Φ is the strength reduction factor, γD, and γL are the dead load and live load factors, respectively, Mn is the nominal moment capacity, MD and ML are the moment due to the dead load and the live load, respectively. The coefficients γD, and γL may have different values depending on the type of loading rating (inventory or operating). Load rating of the approach slabs, with reinforcement designed for HS20-44 and HL93 highway loads (Table A-3), was also conducted based on the standard AASHTO specifications and AASHTO LRFR (AASHTO 2003). The AASHTO LRFR specifications adopt three levels of rating methodology. They are: design load rating, legal load rating, and permit load rating. While the provided trucks should fit in either the legal or permit truck, all three levels of rating were conducted. The rating factors for different cases shown in Table A-5 are larger than one, which indicates that the available live load capacity of approach slabs is larger than that produced by the loads being investigated. Table A-4 Internal force of approach slab subjected to rating truck Moment of slab with L=40’ h=18” (kips-ft) Moment of slab with L=40’ h=24” (kips-ft) Moment of slab with L=60’ h=21” (kips-ft) Moment of slab with L=60’ h=27” (kips-ft) Truck Type DL+LL (DL: M=1697.9) DL+LL (DL: M=2293.1) DL+LL (DL: M=4429.1) DL+LL (DL: M=5724.5) Rating truck 1 2508.4 3098.6 5636.9 7519.4 Rating truck 2 3236.6 3831.9 6222.3 8611.6 Rating truck 3 2645.9 3236.3 5764.1 7746.5 A.5 Conclusions The approach slabs are supposed to prevent “bump”, but the large deformation of approach slabs designed according to conventional methods still causes this “bump”. The current approach slab design is still more an art than a science. There are no AASHTO guidelines for designing approach slabs with embankment settlements (due to embankment soil long-term consolidation and erosion). An appropriate approach slab design will directly affect the safety and economy of the transportation infrastructure. It will be a trend to assign the responsibility of this design issue to an engineer. A rational design is necessary not only for the serviceability requirement of the transition approach slab, but also for the life-expectancy of the whole highway system, including bridges and pavements. 153 154 9’ -9 ” 26.4 kips 26.4 kips 26.4 kips 26.4 kips 26.4 kips 26.4 kips 26.4 kips 26.4 kips 5’ -5” 5’ -5” 9’ -2” 5’ -5” 5’ -5” 7’ -5” 5’ -5” (a) Rating vehicle 1 6’ 12’ -9” 4’ -10” 9’ -10” 4’ -2” 4’ -2” ’ -2” 4’ -2” 4’ -2” 4’ -2”30 9’ 4’ -2” 4’ -2” 4’ -2” 16.0 kips 22.0 kips 22.0 kips 20.0 kips 20.0 kips 20.0 kips 40.0 kips 40.0 kips 40.0 kips 40.0 kips 10 ’ 16.0 kips 29.5 kips 24.6 kips 19’ -4” 5’ 17’ -3” 4’ -11” 40.0 kips 40.0 kips 40.0 kips 6’ (b) Rating vehicle 2 4’ -11” 4’ -11” 4’ -11” 4’ -11” 4’ -11” 4’ -11” 4’ -11” 29.5 kips 24.6 kips 24.6 kips 24.6 kips 24.6 kips 24.6 kips 24.6 kips 24.6 kips24.6 kips (c) Rating vehicle 3 Fig. A-7 Rating vehicle (plan view) with axle loads marked 5 Table A-5 Rating result of approach slab AASHTO Standard Rating (*) AASHTO LRFR Design Load Rating (**) AASHTO LRFR Legal Load Rating (***) AASHTO LRFR Permit Load Rating(****) L=40’ h=18” L=40’ h=24” L=60’ h=21” L=60’ h=27” L=40’ h=18” L=40’ h=24” L=60’ h=21” L=60’ h=27” L=40’ h=18” L=40’ h=24” L=60’ h=21” L=60’ h=27” L=40’ h=18” L=40’ h=24” L=60’ h=21” L=60’ h=27” (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) (2) (2) (2) (2) (2) (2) (2) (2) 2.68 v1 2.18 3.63 2.19 3.66 2.10 3.50 1.59 2.65 2.68 3.48 2.72 3.52 2.62 3.40 1.99 2.58 2.61 2.64 2.55 1.94 3.61 3.66 3.53 1.67 v2 1.15 1.91 1.15 1.91 1.41 2.36 0.99 1.65 1.41 1.83 1.42 1.84 1.77 2.29 1.24 1.61 1.37 1.38 1.72 1.20 1.90 1.91 2.38 2.38 15 v3 1.86 3.11 1.87 3.12 1.90 3.17 1.41 2.35 2.29 2.97 2.32 3.01 2.37 3.08 1.77 2.29 2.23 2.26 2.31 1.72 3.09 3.12 3.19 Note: (*) Standard rating: (1) Inventory rating: ;3.1=Lγ (**)LRFD design load: (1) Inventory rating: ;25.1=Dγ ;75.1=Lγ (2) Operating rating: ;25.1=Dγ ;35.1=Lγ ;3.1=Dγ;17.267.13.1;3.1=Dγ =×=Lγ (2) Operating rating: (****)LRFD-Permit Load: (2) Operating rating: ;25.1=Dγ ;3.1=Lγ (***)LRFD legal load: (2) Operating rating: ;25.1=Dγ ;8.1=Lγ In the present study a 3-D finite element analysis has been conducted to consider the effect of embankment settlement on the approach slab performance. The parametric study has led to the confirmation of a set of equations for the prediction of internal forces and deformation of the slab for a given settlement. Based on a parametric study, we can conclude that the equations are applicable to approach slabs subjected to the HL93 highway load and also applicable for skewed slabs. The internal force and deflection of the skewed approach slabs can be obtained by using the equations, where M0 and d0 are the internal forces and deformation of the simply-supported skewed approach slab respectively. However, the calculation of a simply-supported skewed approach slab is complicated. Results show that using the equations with M0 , the moment of an equivalent normal slab, to calculate the skewed approach slab internal force is conservative. Furthermore, the results of capacity rating of the approach slab indicate that the designed slab has sufficient capacity for the three special vehicles. By using the design aids confirmed in this study, engineers, without using finite element analysis in their routine design, can conveniently design the approach slabs. This performance-based design will eventually lead to a more reliable practice in using approach slabs. These results can also be used to systematically evaluate the effectiveness of the approach slabs and develop guidelines for their structural design. This information will help decide when settlement controls are necessary in order to have an economical design of the approach slabs. A.6 References AASHTO. Standard Specifications for Highway Bridges. American Association of State Highway and Transportation Officials, Washington, D.C., 2002 AASHTO. LRFD Bridge Design Specifications. American Association of State Highway and Transportation Officials, Washington, DC., 2004 AASHTO. Manual for Condition Evaluation of Bridges. American Association of State Highway and Transportation Officials, Washington, D.C., 1994 AASHTO. Guide Manual for Condition Evaluation and Load and Resistance Factor Rating (LRFR) of Highway Bridges. American Association of State Highway and Transportation Officials, Washington, D.C., 2003 Briaud, J. L., James, R. W., and Hoffman, S. B. Settlement of Bridge Approaches (the Bump at the End of the Bridge). NCHRP Synthesis 234, Transportation Research Board, National Research Council, Washington, D.C., 1997 Cai, C. S., Shi, X. M., Voyiadjis, G. Z. and Zhang, Z. J. Structural Performance of Bridge Approach Slab under Given Embankment Settlement. Journal of Bridge Engineering, ASCE, Vol. 10, No. 4, 2005, pp 482-489 Chini, S. A., Wolde-Tinsae, A. M., and Aggour, M. S. Drainage and Backfill Provisions for Approaches to Bridges. University of Maryland, College Park, 1992 156 Kramer, S.L., & Sajer, P. Bridge Approach Slab Effectiveness. Washington State Transportation Center, Seattle., 1991 LADOTD. Bridge Design Metric Manual. Louisiana Department of Transportation and Development, Baton Rouge, LA., 2002 LQI. Louisiana Quality Initiative: Preservation of Bridge Approach Ride Ability. Louisiana Department of transportation and Development, Baton Rouge, LA., 2002 Mahmood, I. U. Evaluation of Causes of Bridge Approach Settlement and Development of Settlement Prediction Models. Ph.D. Thesis, University of Oklahoma, Norman., 1990 Ha, H., Seo, J. B., Briaud, J. L. Investigation of Settlement at Bridge Approach Slab Expansion Joint: Survey and Site investigations. Report No. 4147-2 to the Texas Department of Transportation, published by the Texas Transportation Institute, Texas A&M University System., 2003 Stewart, C. F. Highway Structure Approaches. California Department of Transportation, Sacramento., 1985 Zaman, M., Gopalasingam, A. and Laguros, J. G. Consolidation settlement of bridge approach foundation. Journal of Geotechnical Engineering, Vol. 117, No.2, 1991, pp 219-239. 157 APPENDIX B: PERMISSIONS AMERICAN SOCIETY OF CIVIL ENGINEERS LICENSE TERMS AND CONDITIONS Oct 25, 2006 This is a License Agreement between xiaomin shi ("You") and American Society of Civil Engineers ("American Society of Civil Engineers"). The license consists of your order details, the terms and conditions provided by American Society of Civil Engineers, and the payment terms and conditions. License Number 1575730268289 License date Oct 25, 2006 Licensed content title Structural Performance of Bridge Approach Slabs under Given Embankment Settlement Licensed content author C. S. Cai; X. M. Shi; G. Z. Voyiadjis; Z. J. Zhang Licensed content publication Journal of Bridge Engineering Licensed content publisher American Society of Civil Engineers Type of Use Doctoral Thesis Portion used Excerpt Institution Louisiana State University Title of your work Structural performance of approach slab and its effect on vehicle indeced bridge dynamic response Publisher of your work UMI Company Publication date of your work 12/30/2006 Website Usage electronic 158 2nd November 2006 To: Xiaomin Shi Louisiana State University Dear Ms. Shi: The Transportation Research Board grants permission to use in your doctoral dissertation your paper “Design of Ribbed Concrete Approach Slab Based on Interaction with the Embankment,” coauthored with C. S. Cai, G. Voyiadjis, and Z. Zhang, as identified in your e-mail of October 31, 2006. Please note the following conditions: 1. Please credit as follows: From Transportation Research Record: Journal of the Transportation Research Board, No. 1936, Transportation Research Board of the National Academies, Washington, D.C., 2005, pp. 181-191. Reprinted with permission. 2. None of this material may be presented to imply endorsement by TRB of a product, method, practice, or policy. Every success with your dissertation. Please keep the Record in mind for future submissions. Sincerely, Javy Awan Director of Publications Transportation Research Board Email: PBARBER@nas.edu -----Original Message----- From: Xiaomin Shi [mailto:xshi1@lsu.edu] Sent: Tuesday, October 31, 2006 11:01 AM To: Barber, Phyllis Subject: Permission request Dear Phyllis Barber: This is Xiaomin Shi from Louisiana State University writing you this letter. I am completing a doctoral dissertation at entitled “Structural Performance of Approach Slab and Its Effect on Vehicle Induced Bridge Dynamic Response”. I would like your permission to reprint in my dissertation the following article published in "Journal of the Transportation Research Board": Shi, X. M., Cai, C. S., Voyiadjis, G. Z., and Zhang, Z. J. (2005) “Design of Ribbed Concrete Approach Slab Based on Its Interaction with Embankment” Transportation Research Record, J. of the Transportation Research Board, National Research Council, 1936, 181-191. I am the author of the above paper and it will be reproduced as chapter 3 in my dissertation. The requested permission extends to any future revisions and editions of my dissertation including non-exclusive world rights in all languages, and to the prospective publication of my dissertation by UMI Company. These rights will in no way restrict republication of the material in any other form by you or by others authorized by you. Your permission of this request will also confirm that you own (or your company owns) the copyright to the above-described material. If these arrangements meet with your approval, please response this email. Thank you very much. Sincerely, Xiaomin Shi 159 VITA Ms. Xiaomin Shi was born in 1976 in Jiangsu Province, P.R.China. She received her Master of Science degree from the Department of Civil Engineering at Tsinghua University, P.R.China, in 2002, her Bachelor of Science degree from the Department of Civil Engineering at Tongji University in 1999. Ms. Shi has worked as a Graduate Research Assistant at Louisiana State University since January 2003. Ms. Shi has been involved in research in several areas, such as Interaction between Approach Slab and Embankment Soil, Vehicle-Bridge Coupled System, Bridge Vibration Control, and Bridge Test. She has 15 publications, which are listed as follows: ¾ Cai, C. S., Shi, X. M., Araujo, M., and Chen, S. R. (2006) “Influence of approach span condition on vehicle-induced dynamic response of slab-on-girder-bridges: field and numerical simulations.” J. of the Transportation Research Board, (submitted). ¾ Shi, X. M., Cai, C. S., Voyiadjis, G. Z., and Zhang, Z. J. (2005) “Effect of embankment settlement on bridge approach slab design: a few concerned issues” J. of the Transportation Research Board, (submitted). ¾ Shi, X. M., Cai, C. S., Chen, S. R. (2006) “Vehicle induced dynamic behavior of short span slab bridges considering effect of approach span condition.” Journal of Bridge Engineering, ASCE, (accepted). ¾ Cai, C. S., Nie, J. G., and Shi, X. M.(2006) “Interface Slip Effect on Bonded Plate Repairs of Concrete Beams.” Engineering Structures, (accepted and in press) ¾ Zhang, Y., Cai, C. S., Shi, X. M. and Wang, C. (2006) “Vehicle Induced Dynamic Performance of a FRP Versus Concrete Slab Bridge” J. of Bridge Engineering, ASCE, 11(4), 410-419. ¾ Cai, C.S., Wu, W.J. and Shi, X. M. (2006) “Cable Vibration reduction with a Hung-on TMD System: I. Theoretical Study.” J. of Vibration and Control, 12(7), 801-814. ¾ Shi, X. M., Cai, C. S., Voyiadjis, G. Z., and Zhang, Z. J. (2005) “Design of Ribbed Concrete Approach Slab Based on Its Interaction with Embankment” Transportation Research Record, J. of the Transportation Research Board, National Research Council, 1936, 181-191. ¾ Cai, C. S., Shi, X. M., Voyiadjis, G. Z. and Zhang, Z. J. (2005) “Structural Performance of Bridge Approach Slab under Given Embankment Settlement.” Journal of Bridge Engineering, ASCE, 10(4), 482-489. ¾ Wang, Y.Q., Shi, X. M., and Chen, H. (2003) “Design and Analysis of Steel Portal Frame in Hydropower Plant Building”, Journal of Water Power, v29, n3, 63-66. ¾ Shi, X. M., Wang, Y. Q. and Zhang, Y. (2003) “Application of Large-Span Portal Frame with Prestressed Cable-Strut”, Journal of Industrial Construction, v33, n2, 68-71. ¾ Wang, Y. Q., Wu, Y. M., Wang, X. Z., Shi, X. M. (2002) “3-D stresses in a flat slab with a crack in tension and the effect on brittle fracture.” Journal of Tsinghua University (Science and Technology), v42, n6, 832-834+842. ¾ Zhang, Y., Cai, C. S., Shi, X. M. (2006) “Vehicle Load-Induced Dynamic Performance of FRP Slab Bridges” 2006 Structures Congress, ASCE, May 18-21, St. Louis, Missouri, USA ¾ Shi, X. M., Cai, C. S., and Wu, W.J. (2005) “Effect of Approach Slab on Bridge-Vehicle 160 Coupled Vibration: Numerical Analysis.” The joint /ASME/ASCE/SES Engineering Mechanics and Materials Conference, June 2005, Baton Rouge, Louisiana. (also presented) ¾ Shi, X. M., Cai, C. S., Voyiadjis, G. Z. and Zhang Z. J. (2004) “3-D Finite Element Analysis of Interaction of Concrete Approach Slab and Soil Embankment”, Geo-Trans 2004, the Geo-Institute of the American Society of Civil Engineers, Los Angeles, CA, July 27-31, 2004, ASCE Geotechnical Special Publication No. 126, 393-402. (also presented) ¾ Cai, C. S., Voyiadjis, G. Z., and Shi, X. M. (2005) “Determination of Interaction between Bridge Concrete Approach Slab and Embankment Settlement.” Final Report, LTRC Project No. [03-4GT, State Project No. [736-99-1149], submitted to Louisiana Transportation Research Center. The degree of Doctor of Philosophy will be awarded to Ms. Shi at the December 2006 commencement. 161