Flexural load rating of concrete bridge girder with deteriorated deck

Abstract

Load rating analysis of bridge live load capacity is the main technique in evaluating existing bridges in the United States. Load ratings are performed on a bridge following a detailed visual inspection mainly based on engineering experience. With the absence of an analytical evaluation procedure to determine the ultimate flexural strength of bridge girders with deteriorated concrete deck, the load rating procedure is highly dependent on the engineer’s estimation. This paper presents a Finite Element Analysis (FEA) model and an analytical model to predict the ultimate flexural strength of deteriorated Reinforced Concrete (RC) bridge girders that can be adopted in bridge evaluation analysis. Both models take into account the damaged material properties and geometry. The proposed methodology can be adopted in computing the nominal flexural strength in the procedure of load rating analysis. The authors concluded that deterioration of concrete decks leads up to 40% reduction in rating factors. This may result in posting or closing of the bridge.

Keywords

Concrete bridge girder compression reinforcement flexural strength corrosion FEA

NOTATION

A_s

area of tensile steel reinforcement

A_s (corr)

area of corroded tensile steel reinforcement

A’ _s

area of compressive steel reinforcement

A’ _s (corr)

area of corroded compressive steel reinforcement

width of cross-section

b_f

width of flange of T cross-section

distance from extreme compression fiber to the neutral axis

Structural capacity

C_c

total compressive force of concrete at critical section

C_s

total compressive force of compressive steel at critical section

distance from extreme compression fiber to centroid of tensile reinforcement

d₀

diameter of reinforcing bar

d_0 (corr)

diameter of corroded reinforcing bar

dead load effect due to structural components and attachments

dead load effect due to wearing surfaces and utilities

E_c

modulus of elasticity of concrete

E_s

modulus of elasticity of steel

f _b

critical buckling stress

f_c

concrete compressive strength at strain ɛ

f’ _c

specified compressive strength of concrete

f_y

yield strength of tensile steel

f_s

steel strength of tensile steel at strain ɛ_s

f_y (corr)

yield strength of corroded tensile steel

f_yc

yield strength of compressive steel

f_yc (corr)

yield strength of corroded compressive steel

dynamic load allowance

effective length factor

span of beam

L_corr

length of beam over which reinforcement is corroded

live load effect

L_ub

length of beam over which reinforcement is unbonded

M _n (corr)

nominal flexural bending strength of RC beams with corroded compressive steel

M _n

nominal flexural bending strength of structurally sound RC beams

permanent loads other than dead loads

Q_corr

corrosion degree, which is calculated as the mass loss (%)

radius of gyration about the axis of buckling

R _n

nominal member resistance

Rating factor

total tensile force of tensile steel at critical section

SHV

Specialized hauling vehicles

t_f

thickness of flange of T cross-section

weight of the rating vehicle

distance from extreme compression fiber to centroid of concrete in compression

concrete strain corresponding to concrete compressive strength f_c

ɛ _cu

ultimate compressive strain of concrete

ɛ _s

strain of tensile perfectly bonded steel reinforcement

ɛ _sc

strain of compressive steel reinforcement

ɛ _y

yield strain of tensile steel reinforcement

ɛ _yc

yield strain of compressive steel reinforcement

ɛ ₀

concrete strain corresponding to concrete compressive strength, taken as ɛ₀ = 1.8f’_c/E_c

γ _DC

LRFD load factor for structural components and attachments

γ _DW

LRFD load factor for wearing surfaces and utilities

γ _LL

evaluation live load factor

γ _P

LRFD load factor for permanent loads other than dead loads

midspan deflection of RC beam

LRFD resistance factor

φ _c

condition factor

φ _s

system factor

λ _c

column slenderness parameter

tensile steel reinforcement ratio

ρ’

compressive steel reinforcement ratio

1 Introduction

Deterioration of bridge decks is a major problem for US Departments of Transportation, especially with roughly 4 billion vehicles cross bridges in the United States every day. Corrosion of steel rebars is the leading cause for this problem [1]. In areas where de-icing salt is used in winter, corrosion of steel rebars is considered the main deterioration mechanism for bridge concrete decks [2]. Traffic impact also amplifies the risk of concrete spalling in cracked decks especially in bridges with high Average Daily Truck Traffic (ADTT). Of all bridge components, concrete deck is the most subject to corrosion and maintaining it is the most costly and frequent activity in bridge maintenance. This is due to the exposure to daily traffic, severe weather conditions, and the use of de-icing sault. Rehabilitation of bridge concrete decks accounts for over 50% of bridge rehabilitation expenses [3].

Corrosion of steel rebars in bridge decks leads to spalling of concrete on the compression side of bridge girders. The concrete deck forms the flange of the T-cross-section of the concrete bridge girder, which is mainly the compression zone of the cross-section. Therefore, deterioration of concrete decks could lead to a serious decrease in ultimate flexural strength of bridge girders. In addition, stirrups have smaller diameter and concrete cover when compared to longitudinal reinforcement. Therefore, they are susceptible to corrosion before compression bars. While spalling of concrete cover deteriorates the compression bars confinement, corrosion of stirrups decrease the buckling stress of the compression rebars, as it decreases the unbraced length. Despite the slight contribution of the compression reinforcing bars to the ultimate flexural strength, spalling of concrete cover and buckling of compression rebas may lead to a significant reduction in the ultimate flexural strength. Since the bridge deck is mainly the compression zone of the bridge girder, concrete spalling of the deck surface could lead to a major flexural strength reduction in bridge girder (Fig. 1).

Fig.1

Bridge deck with corroded steel rebars that lead to spalling of concrete cover.

Moreover, corrosion deteriorates the bond between the compression rebars and the adjacent concrete. This means that the strain in steel is no longer equal to the strain in the surrounding concrete, and thus, the compatibility condition is invalid. Therefore, the AASHTO equations to calculate the ultimate flexural strength of concrete girders are inapplicable.

This research presents a nonlinear FEA model to simulate the behavior of RC beams with corroded compression reinforcement and estimate the stress in the compression rebars at ultimate. After the FEA model was verified against the available experimental data, an analytical model was developed. Both models showed good agreement with the experimental data.

2 Research significance

The load rating procedure of existing bridge girders requires the computation of the cross-sectional strength. This procedure is based on the assumptions that the construction and materials of the bridge are of a high quality and there is no loss in material design strength, or the current strength is established by testing. However, the AASHTO Manual for Bridge Evaluation (MBE) suggests that, when computing the load rating, any reductions in cross-sectional area due to deterioration must be taken into account [4]. With the absence of an analytical procedure to compute the flexural strength of concrete girders with deteriorated decks, the computation of the deteriorated cross-sectional capacity is highly based on the engineer’s experience and expertise. This paper presents a FEA and analytical methods to evaluate the flexural strength of girders with a deteriorated concrete deck that can be adopted in current bridge load rating practices.

3 Load rating and bridge posting analysis

Bridge load rating delivers a foundation for estimating the safe load carrying capacity of an existing bridge. The above procedure involves the engineering judgement in estimating a rating value that is appropriate to sustain a level of safety of the bridge [4]. The Manual for Bridge Evaluation requires the following equations to be used in computing the load rating factor for each bridge component. $RF = \frac{C - (γ_{DC}) (DC) - (γ_{DW}) (DW) \pm (γ_{P}) (P)}{(γ_{LL}) (LL + IM)}$ (1) $C = φ_{c} φ_{s} φ R_{n}$ (2) where: RF = rating factor; C = structural capacity; γ_DC= LRFD load factor for structural components and attachments; DC = dead load effect due to structural components and attachments; γ_DW= LRFD load factor for wearing surfaces and utilities; DW = dead load effect due to wearing surfaces and utilities; γ_P= LRFD load factor for permanent loads other than dead loads = 1; P = permanent loads other than dead loads; γ_LL= evaluation live load factor; LL = live load effect; IM = dynamic load allowance; φ_c= condition factor; φ_s= system factor; φ= LRFD resistance factor; and R_n= nominal member resistance.

The condition factor presents a decrease in the cross-sectional strength to account for the escalation in the uncertainty in the deteriorated members’ strength, and the possibility of future deterioration during the time span between inspection cycles [4]. The above factor is based on the condition rating scale suggested by National Bridge Inventory (NBI). Condition ratings are employed to evaluate the existing in-place bridge in comparison with the as-built bridge condition, and are assigned by engineers, based on the visual observations, following a 10-point scale system, where code 0 indicates a failed condition, and code 9 indicates excellent, as new condition [5]. The Manual for Bridge Evaluation suggests a condition factor value of 1 if the bridge member inspected is in a good/satisfactory condition (condition rating is larger than 5), 0.95 for a fair condition member (condition rating is equal to 5), and 0.85 for a poor condition member (condition rating that is smaller than 5). For instance, if the concrete deck is subject to any deterioration, the condition rating ranges from 0 to 4, and therefore, the condition factor is assumed to be 0.85. This is a fixed value regardless of the degree or location of the deterioration. Thus, the condition factors may not reflect the actual reduction in the cross-sectional strength. However, The AASHTO MBE requires the bridge investigated to undergo design and legal load rating, and if the RF calculated for any legal vehicle type is between 0.3 and 1.0, Equation 3 is used to establish the safe posting load for the vehicle type. Moreover, if the RF for any legal truck (AASHTO vehicle) is less than 0.3, this specific vehicle type should not be allowed on the bridge. $LRFR Posting Load = \frac{W}{0.7} [(RF) - 0.3]$ (3) where W is the weight of the rating vehicle. Furthermore, AASHTO MBE requires any bridge that is unable to carry a minimum gross live load weight of 3 tons to be closed [4].

4 Effect of corrosion on material properties

While corrosion decreases the cross-sectional area and the strength of the steel reinforcement, it does not affect its stiffness. Therefore, the slope of the stress-strain diagram of corroded reinforcement is similar to that of non-corroded. However, the yield strength and the cross-sectional area of corroded bars are computed based on the empirical Equations 4–6 [6] and [7]. $A_{s (corr)} = (1 - 0.01 \cdot Q_{corr}) \cdot A_{s}$ (4) $f_{y (corr)} = (1 - 0.005 \cdot Q_{corr}) \cdot f_{y}$ (5) $Q_{corr} = [1 - {(\frac{d_{0 (corr)}}{d_{0}})}^{2}] \times 100 %$ (6)

5 FEA model

The authors developed a Finite Element Model to simulate the behavior of bridge girders with corroded compressive reinforcement. The model was created using the commercial software ANSYS. The model accounts for material properties and geometry.

5.1 Element types

The authors employed 3-D SOLID65 element to model concrete elements. The SOLID65 element permits the modeling of nonlinear material properties. Eight nodes define the SOLID65 element, each of these nodes experience three degrees of freedom; translations in the nodal x, y, and z directions. In addition, SOLID65 element has the capability of cracking in three orthogonal directions at every integration point and the ability of crushing, undergoing plastic deformation and creep [8].

The uniaxial spar, LINK180, simulated the behavior of steel reinforcing bars. The above element is capable of carrying tension and compression. Two nodes with three degrees of freedom at each node define the element. These degrees of freedom are translations in the nodal x, y, and z directions. The X-axis of the element is oriented along the length of the element from node I to node J. The element does not allow bending. In addition, the element accounts for plasticity, creep, rotation, large deflection, and large strain capabilities [8].

LINK180 also simulated the behavior of corroded steel rebars. However, stress-strain diagram of the corroded bars described the behavior of the corroded bars.

Spring elements, COMBIN14, modeled the loss of bond between reinforcing steel and surrounding concrete. This element has longitudinal or torsional capability in 1-D, 2-D, or 3-D applications. COMBIN14 behaves as a uniaxial tension-compression element when the user activates the longitudinal spring-damper option. The element connects two nodes and experiences three degrees of freedom at each node: translations in the nodal x, y, and z directions. The element has no mass and the spring or the damper capability can be deactivated [8].

5.2 Material properties and real constants

The authors used Von Mises failure criterion along with William and Warnke’s [9] constitutive model to define concrete failure. The modified Hognestad stress-strain relationship defined the multilinear isotropic concrete stress-strain diagram as shown in Fig. 2. The linear branch of the diagram that satisfies Hook’s law connects the origin to the first point (0.30f’ _c ) [10] and [11]. The curve between 0.30f’ _c and ɛ₀ is drawn based on the equation describing the non-linear branch of the modified Hognestad stress-strain relationship (Fig. 2). The linear branch towards the end of the curve is assumed horizontal as the most recent edition of ANSYS does not accept negative slopes in stress-strain plots.

Fig.2

Modified Hognestad Stress-strain diagram for concrete.

The modulus of elasticity of concrete is 4,750√f’ _c , modulus of rupture (uniaxial cracking stress) is 0.085√f’ _c , and Poisson’s ratio is 0.2. The shear transfer coefficient for a closed crack is 1, and the shear transfer coefficient for an open crack is 0.3 [10 –12]. The concrete uniaxial cracking stress and uniaxial crushing stress represented the concrete modulus of rupture and compressive strength respectively. The tensile crack factor, hydrostatic pressure, biaxial crushing stress, hydro uniax crush stress, and hydro biax crush stress are set equal to their default values determined by ANSYS, which is zero. It is important to mention that the authors performed a preliminary analysis to validate the values of the above coefficients to the best agreement with the available experimental data.

The steel stress-strain curve is assumed to be elastic perfectly plastic as shown in Fig. 3. The steel yield stress varied based on each case studies. The modulus of elasticity of steel is 200,000 MPa (29,000 ksi), and Poisson’s ratio is 0.3. The cross-sectional area of the steel reinforcement also varied based on each experiment.

Fig.3

Elastic perfectly plastic stress-strain diagram for steel.

The COMBIN14 spring is set to behave as linear longitudinal spring with a vertical degree of freedom UY by setting KEYOPT(1) and KEYOPT(2) to zero and two respectively. A large value of spring stiffness (100,000 N/mm (571 kip/in.)) is selected, whereas the damping coefficients and initial force were set equal to zero.

5.3 Building of the model

The FEA model was created by creating the nodes first. A cuboid SOLID65 element connected 8 nodes. This element was copied along the X, Y, and Z axes. Longitudinal reinforcing bars and stirrups were modeled by connecting LINK180 elements to the concrete element nodes at locations where reinforcement exists. In regions of corrosion/unbond, identical nodes were created at the same location of the concrete nodes and LINK180 were connected to the latest nodes. However, the nodes of the concrete elements were connected to the nodes of the steel elements with vertical springs, i.e. COMBIN14, as shown in Fig. 4.

Fig.4

Modeling of loss of bond.

Due to symmetry, the authors modeled only one-half of each beam. The plane of symmetry is a vertical YZ plane that splits the beam in half. The authors defined the above plane by constraining the nodes defined by this plane along the X axis. A line of nodes along the Z axis was constrained in the Y and Z directions (UY = UZ = 0). This line of nodes represents the beam supports. The authors applied the loads gradually with sub-steps as high as 100 at regions where concrete cracks or rebars start to yield. The authors applied the load on a line of nodes along the Z axis, with the total load divided by the number of nodes (Fig. 4).

5.4 Validation of the FEA model

The authors verified the FEA model by comparing the output of the model to 30 experimental beams ([13 –15], and [16]). The experimental data comprised of two structurally sound beams, two exposed to different corrosion levels on the compression side, 12 exposed to different corrosion levels on the tension side, and 14 subjected to unbond between steel rebars and adjacent concrete. Table 1 shows that the FEA model shows a good agreement with the available experimental data. Thus, the model can be used to increase the database of beams with corroded reinforcement on the compression side of the cross-section.

Table 1
A comparison between the FEA model and the available experimental data

Specimen reference Deg. of Corrosion (% mass loss) L, mm b, mm h, mm d, mm L_ub, mm # of bars d₀, mm ρ% M_nExp ., kN·m M_nFEA, kN·m M_nExp ./M_nFEA

Du et al. [13] T682 8.8 1800 150 200 180 600 2 15.92 1.6 29.08 29.23 0.99

T322 2.4 1800 150 200 180 600 2 31.61 6.2 65.02 65.42 0.99

T162 3.4 1800 150 200 180 600 4 15.92 3.1 57.26 56.46 1.01

T163 6.6 1800 150 200 180 600 4 15.92 3.1 51.38 50.00 1.03

T164 10.3 1800 150 200 180 600 4 15.92 3.1 49.25 49.01 1.00

T282 11.1 1800 150 200 180 600 2 12.01 0.87 17.79 18.42 0.97

T122 10.2 1800 150 200 180 600 2 12.01 0.87 16.67 17.00 0.98

R122 10.5 1800 150 200 180 600 2 12.01 0.87 13.29 15.72 0.85

C124^* 13.6 1800 150 200 180 600 2 31.61 6.2 63.75 63.75 1.00

C162^* 11.4 1800 150 200 180 600 4 15.92 3.1 50.625 50.84 1.00

Sharaf and Soudki [14] C1 0 1500 100 150 120 0 1 16 1.7 10.43 10.48 0.99

D1 0 1500 100 150 120 750 1 16 1.7 10.1 10.04 0.98

D2 0 1500 100 150 120 1050 1 16 1.7 9.7 9.61 0.98

D3 0 1500 100 150 120 1200 1 16 1.7 9.2 9.08 0.98

D4 0 1500 100 150 120 1350 1 16 1.7 6.8 6.89 0.98

El Maaddawy et al. [15] C 0 3000 152 254 213 0 2 16 1.24 36.43 36.09 1.01

CN-50 8.9 3000 152 254 213 1400 2 16 1.24 33.97 32.36 1.05

CN-110 14.2 3000 152 254 213 1400 2 16 1.24 32.24 29.63 1.09

CN-210 22.2 3000 152 254 213 1400 2 16 1.24 28.87 27.18 1.06

CN-310 31.6 3000 152 254 213 1400 2 16 1.24 25.46 24.87 1.02

Cairns and Zhao [16] S2 0 2700 225 405 372 2500 2 20 0.75 96.5 93.9 1.03

S3 0 2700 225 400 380 1700 2 20 0.73 113 110.94 1.02

S4 0 2700 230 290 253 2520 2 20 1.08 61.4 63.03 0.97

S4B 0 2700 221 270 225 2520 2 20 1.27 46.1 49.26 0.94

S5 0 2700 230 230 195 2540 2 20 1.4 28.9 31.52 0.92

S7 0 2700 228 410 358 2320 3 20 0.77 135.5 132.16 1.03

S8 0 2700 150 405 340 2320 2 25 1.93 92.1 84.28 1.09

S9 0 2700 230 400 350 2560 2 16 0.5 69.2 68.85 1.01

S10 0 2700 230 230 200 2550 3 12 0.74 27.4 27.55 0.99

S11 0 2700 230 230 200 1620 3 12 0.74 30 32.34 0.93

Mean 0.996

Standard deviation 0.049

	Specimen reference	Deg. of Corrosion (% mass loss)	L, mm	b, mm	h, mm	d, mm	L_ub, mm	# of bars	d₀, mm	ρ%	M_nExp ., kN·m	M_nFEA, kN·m	M_nExp ./M_nFEA
Du et al. [13]	T682	8.8	1800	150	200	180	600	2	15.92	1.6	29.08	29.23	0.99
	T322	2.4	1800	150	200	180	600	2	31.61	6.2	65.02	65.42	0.99
	T162	3.4	1800	150	200	180	600	4	15.92	3.1	57.26	56.46	1.01
	T163	6.6	1800	150	200	180	600	4	15.92	3.1	51.38	50.00	1.03
	T164	10.3	1800	150	200	180	600	4	15.92	3.1	49.25	49.01	1.00
	T282	11.1	1800	150	200	180	600	2	12.01	0.87	17.79	18.42	0.97
	T122	10.2	1800	150	200	180	600	2	12.01	0.87	16.67	17.00	0.98
	R122	10.5	1800	150	200	180	600	2	12.01	0.87	13.29	15.72	0.85
	C124^*	13.6	1800	150	200	180	600	2	31.61	6.2	63.75	63.75	1.00
	C162^*	11.4	1800	150	200	180	600	4	15.92	3.1	50.625	50.84	1.00
Sharaf and Soudki [14]	C1	0	1500	100	150	120	0	1	16	1.7	10.43	10.48	0.99
	D1	0	1500	100	150	120	750	1	16	1.7	10.1	10.04	0.98
	D2	0	1500	100	150	120	1050	1	16	1.7	9.7	9.61	0.98
	D3	0	1500	100	150	120	1200	1	16	1.7	9.2	9.08	0.98
	D4	0	1500	100	150	120	1350	1	16	1.7	6.8	6.89	0.98
El Maaddawy et al. [15]	C	0	3000	152	254	213	0	2	16	1.24	36.43	36.09	1.01
	CN-50	8.9	3000	152	254	213	1400	2	16	1.24	33.97	32.36	1.05
	CN-110	14.2	3000	152	254	213	1400	2	16	1.24	32.24	29.63	1.09
	CN-210	22.2	3000	152	254	213	1400	2	16	1.24	28.87	27.18	1.06
	CN-310	31.6	3000	152	254	213	1400	2	16	1.24	25.46	24.87	1.02
Cairns and Zhao [16]	S2	0	2700	225	405	372	2500	2	20	0.75	96.5	93.9	1.03
	S3	0	2700	225	400	380	1700	2	20	0.73	113	110.94	1.02
	S4	0	2700	230	290	253	2520	2	20	1.08	61.4	63.03	0.97
	S4B	0	2700	221	270	225	2520	2	20	1.27	46.1	49.26	0.94
	S5	0	2700	230	230	195	2540	2	20	1.4	28.9	31.52	0.92
	S7	0	2700	228	410	358	2320	3	20	0.77	135.5	132.16	1.03
	S8	0	2700	150	405	340	2320	2	25	1.93	92.1	84.28	1.09
	S9	0	2700	230	400	350	2560	2	16	0.5	69.2	68.85	1.01
	S10	0	2700	230	230	200	2550	3	12	0.74	27.4	27.55	0.99
	S11	0	2700	230	230	200	1620	3	12	0.74	30	32.34	0.93
												Mean	0.996
												Standard deviation	0.049

^*corrosion in compression reinforcement. Notes: 1 mm = 0.0394 in.; 1 kN·m = 8.85 kip·in.

6 Analytical procedure

As mentioned above, corrosion deteriorates the bond between steel rebars and surrounding concrete. With the absence of bond, the strain in steel reinforcement is unequal to that in the adjacent concrete and the compatibility condition is no longer valid. Because AASHTO equations for calculating the ultimate flexural capacity of RC girder are based on the compatibility conditions, these equations are inapplicable in the presence of corrosion.

6.1 Stress in unbonded reinforcement

As soon as the concrete cover spalls and the bond between the reinforcement and concrete is lost, compression reinforcing bars are very likely to buckle under relatively small stresses. In other words, once the stress in the compression rebars exceeds the buckling stress, the rebars will buckle before reaching their yield capacity. The authors adopted Equations 7 and 8 to calculate the critical buckling stress in reinforcing bars.

However, in order to calculate the actual stress in the unbonded compression reinforcement, the authors employed the FEA model to investigate different cases of RC beams with unbond between the steel reinforcement bars and the adjacent concrete in the compression zone. Thirty-six beams subjected to different unbonded lengths between compression steel reinforcement and adjacent concrete were investigated. The 36 beams studied shared the same cross-section and span length. However, three different compression reinforcement ratios were investigated as follows, ρ’ = 0.25ρ, ρ’ = 0.40ρ, and ρ’ = 0.56ρ. The unbonded length over the span length varied from 1.3% to 100% (Fig. 5).

Fig.5

Dimensions of the FEA model beam.

The authors employed Equations 7 and 8 in order to compute the buckling stress of steel reinforcement bars in the compression zone. The above two equations were presented by the Structural Stability Research Council (SSRC), in its third edition of the Guide, and account for the critical buckling stress of solid circular columns [17]. The authors assumed that the bars are pinned at both ends [18]. After running the model, the authors obtained the stress in the compression reinforcing bars of all of the cases studied. As expected, the strain in the compression reinforcement in all of the cases studied exceeded the buckling strain obtained from buckling stress provided by Equation 8. This means that in the absence of the concrete cover in the compression zone, the compression reinforcement will buckle at ultimate flexural strength. In other words, the buckling stress of the compression reinforcement may be used in place of the yielding stress when computing the ultimate flexural strength of girders with corroded compression reinforcement.

6.2 Ultimate flexural strength

Based on the above outcomes, the authors present the following procedure to calculate the flexural strength of a T girder with a deteriorated deck:

Step 1: Calculate the corroded compressive steel area and yield strength from Equations 1 and 2

Step 2: Calculate the slenderness ratio as follows $λ_{c} = \frac{K \cdot L}{r \cdot π} \cdot \sqrt{\frac{f_{y}}{E_{s}}}$ (7) Step 3: Compute the buckling stress using Equation 8 $\frac{f_{b}}{f_{yc}} = {\begin{matrix} 1 (Yieldlevel) & (0 \leq λ_{c} \leq 0.15) \\ 1.035 - 0.202 λ_{c} - 0.222 λ_{c}^{2} & (0.15 \leq λ_{c} \leq 1.0) \\ - 0.111 + 0.636 λ_{c}^{- 1} + 0.087 λ_{c}^{- 2} & (1.0 \leq λ_{c} \leq 2.0) \\ 0.009 + 0.877 λ_{c}^{- 2} & (2.0 \leq λ_{c} \leq 3.6) \\ λ_{c}^{- 2} (Eulerbuckling) & (λ_{c} > 3.6) \end{matrix}$ (8)

Step 4: Calculate the conncrete modulos of elasticity and the concrete strain corresponding to concrete compressive strength as follows $E_{c} = 4750 \cdot \sqrt{f_{c}^{'}}$ (9) $ɛ_{0} = 1.8 \cdot \frac{f_{c}^{'}}{E_{c}}$ (10)

Step 5: Assuming that the depth of the neutral axis (c) is smaller than the thickness of the flange (t_f) (Fig. 6), Equations 11 and 12 can be used to calculate the compressive force in the concrete (C_c) and the distance from the centroid of this force to the neutral axis (y) as functions of the unknown depth of the neutral axis (c)

Fig.6

Stress distribution of a T cross-section where the neutral axis falls within the flange.

$C_{c} = \int_{0}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} b_{f} \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c} b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ$ (11)

$y = \frac{\int_{0}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} ɛ \cdot b_{f} \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c} ɛ \cdot b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ}{\int_{0}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} b_{f} \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c} b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ}$ (12)

Step 6: Equations 13 and 14 are used to calculate the forces in the compressive and tensile reinforcement

$C_{s} = A_{s}^{'} \cdot f_{b}$ (13) $T = A_{s} \cdot f_{y}$ (14)

Step 7: The depth of the neutral axis (c) can then be computed from the following equilibrium equation $T = C_{s} + C_{c}$ (15)

Step 8: If the actual depth of the neutral axis is smaller than the thickness of the flange c < tf, the above assumption (Step 5) is correct.

Step 9: Calculate the strain in the tensile steel $ɛ_{s} = ɛ_{cu} \cdot \frac{d - c}{c}$ (16)

Step 10: If ɛ_s > = ɛ_y compute the flexural strength of the cross-section $M_{n (corr)} = C_{c} \cdot (d - c + y) + C_{s} \cdot d$ (17)

Step 11: If ɛ_s < ɛ_y, the tensile steel did not reach its yield strength, calculate the stress and the force in the tensile reinforcement from Equations 18 and 19 respectively, then compute the flexural strength of the cross-section using Equation 17 $f_{s} = E_{s} \cdot ɛ_{s}$ (18) $T = A_{s} \cdot f_{s}$ (19)

Fig.7

Stress distribution of a T cross-section where the neutral axis falls within the web.

Step 12: If c > t_f, Fig. 7, assume that the flange lies within the linear branch of the concrete stress-strain diagram, the compressive force in concrete (C_c) and the distance from the centroid of this force to the neutral axis (y) as functions of the unknown depth of the neutral axis (c) could be calculated using the following equations

$\begin{matrix} C_{c} = \int_{0}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} b \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ \\ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c - t_{f}} b \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \\ + \int_{c - t_{f}}^{c} b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \end{matrix}$ (20)

$y = \frac{\begin{matrix} \int_{0}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} ɛ \cdot b \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c - t_{f}} ɛ \cdot b \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \\ + \int_{c - t_{f}}^{c} ɛ \cdot b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \end{matrix}}{\begin{matrix} \int_{0}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} b \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c - t_{f}} b \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \\ + \int_{c - t_{f}}^{c} b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \end{matrix}}$ (21)

Step 13: Calculate the depth of the neutral axis using Equation 15

Step 14: If $t_{f} < [c - ɛ_{0} \cdot \frac{c}{ɛ_{cu}}]$ , the above assumption is correct and the flange lies within the linear branch of the concrete stress-strain diagram. Repeat steps 9–11

Step 15: If $t_{f} > [c - ɛ_{0} \cdot \frac{c}{ɛ_{cu}}]$ , the flange extends to the nonlinear branch of the concrete stress-strain diagram. In this case, the compressive force in concrete (C_c) and the distance from the centroid of this force to the neutral axis (y) as functions of the unknown depth of the neutral axis (c) could be calculated using the following equations

$\begin{matrix} C_{c} & = & \int_{0}^{c - t_{f}} b \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{c - t_{f}}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} b_{f} \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ \\ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c} b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \end{matrix}$ (22)

$y = \frac{\begin{matrix} \int_{0}^{c - t_{f}} ɛ \cdot b \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{c - t_{f}}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} ɛ \cdot b_{f} \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ \\ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c} ɛ \cdot b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \end{matrix}}{\begin{matrix} \int_{0}^{c - t_{f}} b \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ + \int_{c - t_{f}}^{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}} b_{f} \cdot f_{c}^{'} \cdot [2 . \frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}} - {(\frac{ɛ_{cu} \cdot \frac{ɛ}{c}}{ɛ_{0}})}^{2}] d ɛ \\ + \int_{ɛ_{0} \cdot \frac{c}{ɛ_{cu}}}^{c} b_{f} \cdot f_{c}^{'} \cdot [1 - 0.15 \cdot \frac{ɛ_{cu} \cdot \frac{ɛ}{c} - ɛ_{0}}{ɛ_{cu} - ɛ_{0}}] d ɛ \end{matrix}}$ (23)

Step 16: Calculate the depth of the neutral axis using Equation 15

Step 17: Repeat steps 9–11 to compute the compute the cross-sectional flexural strength

6.3 Effect of deterioration on RF and load posting

Corrosion of steel rebars and spalling of bridge deck concrete will cause a decrease in rating factors and load posting. The authors investigated the effect of deck deterioration on load rating factors and load posting. The authors examined the decrease in flexural load rating at different stages of strength reduction due to corrosion of compression bars and loss of concrete cover. The authors assumed 5, 10, 15, and 20% reduction in the cross-sectional capacity. For each of the above cases, DC varied from 15 to 40% of the flexural capacity of the cross-section. The load rating factors of the structurally sound girder are set equal to one. This was performed by computing the live load that yields a rating factor of 1 under the above considerations. The reductions in load rating factors are calculated as a percentage of the load rating factors of the structurally sound girders as shown in Fig. 8.

Fig.8

Decrease in rating factors due to corrosion.

One can note from Fig. 8 that reduction of cross-sectional capacity due to corrosion of compression bars can lead to a serious decrease in load rating factors. This decrease increases in long span bridges where the own weight of the bridge accounts for the majority of the carried load. For instance, if the decrease in cross-sectional capacity due to corrosion is 15%, and the dead load is 20% of the cross-sectional capacity, the reduction in rating factor is 20%. Moreover, the decrease in RF is associated with a decrease in the posting load, and can potentially cause posting or closing of the bridge as shown in the following example.

7 Example

An interior concrete bridge girder with a deteriorated concrete deck as shown in Fig. 9. The bridge is built in the State of Louisiana in 1950. Corrosion rate is 25% and the corrosion exposed length is 10 ft. The bridge consists of 4 girders with a clear roadway of 24 ft, a span of 23 ft, and a 3.0 in. asphalt wearing surface. The compressive strength of concrete, f’_c, is 3 ksi, and the yield strength of steel reinforcement f_y, is 40 ksi.

Fig.9

Girder cross-section.

The authors used AASHTOW Bridge Rating analytical software to run design and legal vehicles. The flexural analysis and load ratings at the mid-span under Strength-I Limit State are summarized in Table 2. Note that the weight of the parapets is included in the analysis of DC.

Table 2

Load rating of a structurally sound girder

Limit State	Load Combination			LL (kip-ft)	DC (kip-ft)	DW (kip-ft)	d (in.)	φ _c	R_n (kip-ft)	C (kip-ft)	RF
STR-I	HL-93	Des-Inv	Tr. + Lane	196.83	64.06	16.78	18.7	1	383	344.7	0.70
STR-I	HL-93	Des-Op	Tr. + Lane	196.83	64.06	16.78	18.7	1	383	344.7	0.91
STR-I	HL-93	Des-Inv	Tan. + Lane	245.62	64.06	16.78	18.7	1	383	344.7	0.56
STR-I	HL-93	Des-Op	Tan. + Lane	245.62	64.06	16.78	18.7	1	383	344.7	0.73
STR-I	Type 3-3	Legal		121.3	64.06	16.78	18.7	1	386	344.7	1.53
STR-I	NRL	SHV		200.18	64.06	16.78	18.7	1	386	344.7	0.93
STR-I	SU4	SHV		174.65	64.06	16.78	18.7	1	386	344.7	1.06
STR-I	SU5	SHV		187.42	64.06	16.78	18.7	1	386	344.7	0.99
STR-I	SU6	SHV		187.42	64.06	16.78	18.7	1	386	344.7	0.99
STR-I	SU7	SHV		200.18	64.06	16.78	18.7	1	386	344.7	0.93

Note: AASHTO vehicles Type 3 and 3S2 are not included in the analysis. 1 in. = 25.4 mm; 1 kip·ft = 1.36 kN·m.

The girder is analyzed assuming that the bridge is in a good condition and no deterioration is recorded. The load rating factors are shown in Table 2. The following factors are adopted following the MBE [4] recommendations.

γ_DC= 1.25, γ_LL(Inv.)= 1.75, γ_LL(Ope.)= 1.35, γ_LL(Legal)= 1.3, φ_c= 1, φ_s= 1, φ= 0.9, and IM = 0.33

Even though that the HL-93 vehicles rate less than 1, there is no requirement for posting the bridge in the state of Louisiana because the legal load rating factor is higher than 1.

The girder is analyzed again assuming that the girder is in a poor condition (φ_c= 0.85), and a summary of the analysis is demonstrated in Table 3. Note that even though that the design and Specialized Hauling Vehicles (SHV) rate lower than 1, there is no necessity to post the bridge in the State of Louisiana as the legal load rating is still larger than 1.

Table 3

Load rating of a girder in a poor condition (φ_c= 0.85)

Limit State	Load Combination			LL (kip-ft)	DC (kip-ft)	DW (kip-ft)	d (in.)	φ _c	R_n (kip-ft)	C (kip-ft)	RF
STR-I	HL-93	Des-Inv	Tr. + Lane	196.83	64.06	16.78	18.7	0.85	383	293	0.54
STR-I	HL-93	Des-Op	Tr. + Lane	196.83	64.06	16.78	18.7	0.85	383	293	0.71
STR-I	HL-93	Des-Inv	Tan. + Lane	245.62	64.06	16.78	18.7	0.85	383	293	0.44
STR-I	HL-93	Des-Op	Tan. + Lane	245.62	64.06	16.78	18.7	0.85	383	293	0.57
STR-I	Type 3-3	Legal		121.3	64.06	16.78	18.7	0.85	383	293	1.19
STR-I	NRL	SHV		200.18	64.06	16.78	18.7	0.85	383	293	0.72
STR-I	SU4	SHV		174.65	64.06	16.78	18.7	0.85	383	293	0.83
STR-I	SU5	SHV		187.42	64.06	16.78	18.7	0.85	383	293	0.77
STR-I	SU6	SHV		187.42	64.06	16.78	18.7	0.85	383	293	0.77
STR-I	SU7	SHV		200.18	64.06	16.78	18.7	0.85	383	293	0.72

Note: AASHTO vehicles Type 3 and 3S2 are not included in the analysis. 1 in. = 25.4 mm; 1 kip·ft = 1.36 kN·m.

The girder is re-analyzed assuming that the condition factor is 0.85, however, the proposed procedure is used to compute the flexural strength of the girder and the reduction of cross-sectional area and buckling of compressive bars are taken into account. The results are shown in Table 4.

Table 4

Load rating of a girder using the proposed methodology

Limit State	Load Combination			LL (kip-ft)	DC (kip-ft)	DW (kip-ft)	d (in.)	φ _c	R_n (kip-ft)	C (kip-ft)	RF
STR-I	HL-93	Des-Inv	Tr. + Lane	196.83	64.06	16.78	16.7	0.85	340	260	0.45
STR-I	HL-93	Des-Op	Tr. + Lane	196.83	64.06	16.78	16.7	0.85	340	260	0.58
STR-I	HL-93	Des-Inv	Tan. + Lane	245.62	64.06	16.78	16.7	0.85	340	260	0.36
STR-I	HL-93	Des-Op	Tan. + Lane	245.62	64.06	16.78	16.7	0.85	340	260	0.47
STR-I	Type 3-3	Legal		121.3	64.06	16.78	16.7	0.85	340	260	0.98
STR-I	NRL	SHV		200.18	64.06	16.78	16.7	0.85	340	260	0.59
STR-I	SU4	SHV		174.65	64.06	16.78	16.7	0.85	340	260	0.68
STR-I	SU5	SHV		187.42	64.06	16.78	16.7	0.85	340	260	0.63
STR-I	SU6	SHV		187.42	64.06	16.78	16.7	0.85	340	260	0.63
STR-I	SU7	SHV		200.18	64.06	16.78	16.7	0.85	340	260	0.59

Note: AASHTO vehicles Type 3 and 3S2 are not included in the analysis. 1 in. = 25.4 mm; 1 kip·ft = 1.36 kN·m.

Note that the bridge experiences 36% reduction in load rating factors, which results in a rating factor of AASHTO vehicle Type 3-3 that is less than 1. Therefore, the bridge is required to be posted in the state of Louisiana, and the posting sign is 20–35.

8 Summary and conclusion

Deterioration of bridge decks is one of the most common problems in concrete bridges and corrosion of compressive rebars is the main cause of this problem. This paper presents a Finite Element and analytical strength evaluation methodologies for bridge girders with a deteriorated concrete deck. These methodologies can be adopted in computing the nominal flexural strength in the procedure of load rating analysis. In addition, the proposed method provides a better understanding of the effects of deck deterioration on the bridge performance and allow engineers to better assess the load rating capacity of deteriorated bridge girders. Moreover, deterioration of bridge decks can cause a reduction in bridge rating factors as high as 40%, which may result in posting or closing of the bridge.

References

Marks

. Detection of Steel Corrosion in Bridge Decks and Reinforced Concrete Pavements. Ames (IA): Iowa Highway Research Board, Iowa Department of Transportation; 1977, 25p. Report No.: HR-156.

, Feng

, Dekelbab

, Moses

, Cohen

, Mertz

, Thompson

. Effect of Truck Weight on Bridge Network Costs. Washington (DC): Transportation Research Board; 2003, 197p. Report No.: 495. Doi: 10.17226/21956

Lee

. Current State of Bridge Deterioration in the U.S.-Part 2. . Mat Sel Des. 2012;51(2):1–7.

American Association of State Highway and Transportation Officials (AASHTO). The Manual for Bridge Evaluation: Second Edition. Washington (DC): AASHTO Publications; 2011, 575p.

Federal Highway Administration (FHWA). Recording and coding guide for the structure inventory and appraisal of the nation_s bridges.Washington (DC): Office of Engineering, Bridge Management Branch; 1995. 124p. Report No.: FHWA-PD-96_001.

, Clark

, Chan

AHC

. Effect of corrosion on ductility of reinforcing bars. Mag Concrete Res. 2005;57(7):407–19.

, Clark

, Chan

AHC

. Residual capacity of corroded reinforcing bars. Mag Concrete Res. 2005;57(3):135–47.

ANSYS. Mechanical APDL theory reference. Release 15.0., Canonsburg, PA, ANSYS, Inc.; 2013, 988p.

Willam

, Warnke

. Constitutive Model for Triaxial Behavior of Concrete. Seminar on Concrete Structures Subjected to Triaxial Stresses, International Association of Bridge and Structural Engineering Conference; 1974 May; Bergamo (Italy). 31p.

10.

Kachlakev

, Miller

, Yim

, Chansawat

, Potisuk

. Finite Element Modeling of Reinforced Concrete Structures Strengthened With FRP Laminates. California Polytechnic State University, San Luis Obispo (CA), Oregon State University, Corvallis (OR), Oregon Department of Transportation; 2001, 99p. Report No.: FHWA-OR-RD-01-XX.

11.

Wolanski

. Flexural behaviour of reinforced and pre-stressed concrete beams using finite element analysis [thesis]. Milwaukee (WI): Marquette University; Wisconsin, 2004. 77p.

12.

Dahmani

, Khennane

, Kaci

. Crack identification in reinforced concrete beams using ansys software. Strength Mater. 2010;42(2):232–40.

13.

, Clark

, Chan

AHC

. Impact of reinforcement corrosion on ductile behavior of reinforced concrete beams. Aci Struct J. 2007;104(3):285–93.

14.

Sharaf

, Soudki

. Strength Assessment of Reinforced Concrete Beams with Unbonded Reinforcement and Confinement with CFRP Wraps. 4th Structural Specialty Conference of the Canadian Society for Civil Engineering; 2002 June 5–8; Montreal (Canada), 10p.

15.

El Maaddawy

, Soudki

, Topper

. Long-term performance of corrosion-damaged reinforced concrete beams. Aci Struct J. 2005;102(5):649–56.

16.

Cairns

, Zhao

. Behavior of concrete beams with exposed reinforcement. Proc Inst Civ Eng Struct Build. 1993;99(2):141–54.

17.

Chen

, Lui

. Structural Stability: Theory and Implementation. New Jersey: PTR Prentice Hall; 1987, 490p.

18.

Rodriguez

, Ortega

, Casal

. Corrosion of reinforcing bars and service life of reinforced concrete structures: Corrosion and bond deterioration. International Conference on Concrete Across Borders. 1994;Odense (Denmark)2:315–26.