Capacity of Reinforced/Prestressed Concrete Compression Members Strengthened/Repaired Using Ultra-High-Performance Concrete Encasement

Abstract

Currently, 36% of bridges in the United States need major repair work or replacement and 7% of them are classified as structurally deficient. Most of these bridges can be repaired or strengthened to meet the current load demands and withstand the environmental impacts for the remaining service life. Ultra-high-performance concrete (UHPC) has shown immense potential as a repair and strengthening material thanks to its exceptional characteristics such as high workability, excellent strength and durability, and remarkable energy absorption capacity. This paper presents an analytical approach to predict the capacity of reinforced or prestressed concrete compression members with UHPC encasement (jacket) under combined axial and bending effects. The approach is based on strain compatibility and uses idealized UHPC material models in tension and compression according to the new AASHTO Guide Specifications. The approach uses integration to develop accurate interaction diagrams for any section with complex geometry, which overcomes the approximations of the lamina approach. The paper also provides a comprehensive literature review on UHPC usage in repairing and strengthening concrete bridge columns and piers. A design example of a circular reinforced concrete column is presented to illustrate the proposed approach and to calculate the nominal and design capacities of the composite section. This example has shown that increasing the thickness of the UHPC jacket has a prominent effect on enhancing the axial capacity but only a slight effect on the flexural capacity of compression members.

Keywords

infrastructure deterioration bridges piers Jacketing composite sections interaction diagrams

As of 2022, around 36% of all bridges in the United States need repair or replacement work and 7% of them are considered structurally deficient. The cost of the identified repair work is estimated at $260 billion to restore their condition and maintain their service life ( 1 ). Therefore, it is important to develop new repairing and strengthening techniques and materials to optimize the use of the available funds. Recently, ultra-high-performance concrete (UHPC) has proven to be a promising material for repair work with respect to structural performance, durability, economy, and speed of construction (2 –4). UHPC shows exceptional characteristics including excellent water and chemical resistance, high compressive, tensile, and fatigue strength, and remarkable energy absorption capacity (5 –10).

The use of UHPC encasement (jacketing) to repair and/or strengthen a conventional concrete (CC) compression member, such as bridge pier or column, has been shown to be an efficient and cost-effective technique compared with the methods of jacketing ( 11 , 12 ). However, the development of interaction diagrams of composite CC-UHPC sections that account for the differences in mechanical properties of the two materials has not been adequately investigated. Some researchers have proposed simplified analytical methods to calculate the capacity of CC columns repaired with UHPC that either ignore the tensile strength of UHPC ( 13 ) or approximate the stress-strain relationships of both materials ( 14 ).

This paper proposes an analytical approach based on strain compatibility for determining the capacity of reinforced or prestressed composite CC-UHPC sections subjected to combined axial and bending effects. This approach uses integration to construct the interaction diagram for any section geometry considering the idealized UHPC material models outlined in ( 15 ) and the conventional concrete compression model as specified in ( 16 ) and ( 17 ). For more detailed information about the basic UHPC characteristics in compression and tension used by different codes and guidelines, refer to Tadros et al. ( 18 ). Four different modes of failure were considered in calculating the capacity of the composite section, namely compression failure of UHPC or of CC, crack localization of UHPC, and rupture of tension reinforcement. The paper also provides a comprehensive literature review on the use of UHPC in the strengthening and repair of conventional concrete bridge columns and piers, and a numerical design example using the proposed approach.

Literature Review

This section presents the experimental investigations conducted on composite CC-UHPC columns. Hossain et al. ( 19 ) tested three circular columns with a core diameter of 6 in. (152 mm) and height of 3.3 ft (1,000 mm) and reinforced with 3#3 bars in the longitudinal direction and #2 ties at 4 in. (102 mm) spacing as transverse reinforcement to study the effect of the thickness of UHPC jacket on the structural behavior of the bridge columns as well as the effect of the jacket thickness to core diameter ratio. Figure 1a shows the dimensions and reinforcement details of the original column that is considered a control specimen, which was tested until failure to measure the axial capacity without UHPC jacket. The other two specimens were loaded to cause damage, then repaired using UHPC jackets with a thickness of 1.5 in. (38 mm) and 2 in. (51 mm) and reinforced with 3#3 (10-mm) bars as longitudinal reinforcement and ties with a diameter of 0.16 in. (4 mm) at 4 in. (102 mm) spacing as shown in Figure 1, b and c. The specimens were then tested again under concentric axial loading until failure. The results indicated that the axial capacity of the repaired specimens increased by 2.48 and 2.86 times the original capacity for the jacket’s thickness of 1.5 in. (38 mm) and 2 in. (51 mm), respectively. However, the results also showed that increasing the ratio of UHPC thickness to core diameter results in a more brittle crushing failure and spalling of the UHPC jacket ( 19 ).

Figure 1.

Dimensions and reinforcement details of: (a) original column; (b) repaired column with 1.5-in. (38-mm) thick jacket; and (c) repaired column with 2-in. (51-mm) thick jacket ( 19 ).

Hung et al. ( 20 ) tested seven 13.78-in. (350-mm) square columns with a height of 47 in. (1,200 mm) to study the seismic behavior under different axial load levels and the effect of adding reinforcement in UHPC jacket. Two columns were control specimens without UHPC jackets, and five columns were retrofitted using 1.57-in. (40-mm) thick UHPC jackets. Two jackets were reinforced with a sheet of #3 welded wire mesh with a 3.93-in. (100-mm) grid size, two jackets were unreinforced, and one jacket was constructed using prefabricated panels as shown in Figure 2. The specimens were tested under biaxial bending at two levels of axial loads. The results indicated that increasing the level of axial loading resulted in a brittle failure for the control columns and the columns retrofitted with unreinforced UHPC jackets. However, the columns retrofitted with reinforced cast-in-place or precast UHPC jackets showed ductile behavior under high axial loads. The results also showed that using a UHPC jacket without reinforcement increased the shear capacity by only 20% and had no effect on the drift capacity as a result of the formation of the crack localization. However, adding reinforcement to cast-in-place or precast UHPC jackets increased the shear capacity by 50% as well as the drift capacity by 2% and 5% for specimens under low and high levels of axial load, respectively ( 20 ).

Figure 2.

Dimensions and reinforcement details of: (a) original column; (b) column repaired with plain UHPC jacket; (c) column repaired with reinforced UHPC jacket; and (d) column repaired with reinforced precast UHPC jacket ( 20 ).

Ronanki and Aaleti ( 21 ) conducted experimental investigations to assess the confinement effect of unreinforced UHPC shell. The experimental investigation was composed of two phases. The first phase was conducted on nine short columns with heights of 4 in. (102 mm) and 8 in. (203 mm) and diameters of 4 in. (102 mm) and 6 in. (152 mm) as shown in Figure 3a. The specimens were composed of one control specimen and eight specimens repaired with UHPC jacket. The thickness of UHPC jackets varied between 0.5 in (13 mm) and 1.5 in. (38 mm) so that the ratio between the area of UHPC shell to core area varied from 0.4 to 3. In the second phase, a total of 13 larger specimens were tested as shown in Figure 3b: four specimens were 11.5-in. (292-mm) square columns, three specimens were 12-in. (305-mm) circular columns, three specimens were unreinforced 9-in. (229-mm) square columns, and three specimens were unreinforced 8-in. (203-mm) circular columns. All specimens had a height of 26 in. (686 mm) with UHPC shell throughout the height except the top and bottom 2 in. (51 mm). Two specimens were considered control specimens with no UHPC confinement, and 11 other specimens were confined with UHPC shells with thicknesses of 1 in. (25 mm) and 2 in. (51 mm). The findings indicated that the peak compressive stress and crushing strain values of the core concrete increased by 15%–30% and 26%–46%, respectively, given the confinement effects of unreinforced UHPC shell. The effectiveness of UHPC shell in confinement was found to be higher in circular columns than square columns ( 21 ).

Figure 3.

(a) Phase 1: small specimens without reinforcement; and (b) Phase 2: large reinforced and unreinforced specimens ( 21 ).

Zhang et al. ( 13 ) tested seven columns, including two control columns and five UHPC strengthened columns under combined lateral cyclic load and constant axial load to study the effect of axial load level, fiber content, and the presence of steel reinforcement in UHPC. The results revealed that for the strengthened specimens, the effect of jacket reinforcement on the lateral load capacity was insignificant and the energy dissipation and post-peak ductility decreased with the reduction of either steel fibers or reinforcement mesh bars. This study also presented an approach for predicting the flexural capacity of strengthened columns, showing a good agreement with the test data. This approach ignores the tensile strength of UHPC in the tension zone and the confining effect of transverse reinforcement and UHPC jacket ( 13 ). Additionally, the study employs idealized stress-strain relationships for UHPC, and conventional concrete as outlined in ( 22 ) and ( 23 ), respectively.

Farzad et al. ( 14 ) developed a method for predicting the flexural capacity of circular columns repaired with UHPC considering the tensile strength of UHPC but based on a simplified assumption of the tension model. Also, the method assumed that all the steel reinforcements are merged into an equivalent steel ring as suggested in Cosenza et al. ( 24 ). Additionally, the reinforcement was assumed to be rigid—ideally plastic material—and the stress distribution to be constant at any height in a section in accordance with ( 25 ). This method was assessed by comparing with the results of the moment-curvature approach using the constitutive models proposed by Mander et al. ( 26 ) for both confined and unconfined concrete. The results show that the method demonstrates superior performance when applied to UHPC with smaller thicknesses and lower axial load levels (around 10%) ( 14 ).

Proposed Approach

The approach proposed for calculating the combined axial and flexural capacity of CC compression members repaired or strengthened using reinforced or unreinforced UHPC jackets is developed by formulating all the key parameters as functions of the neutral axis depth from the compression side, c. By assuming different values of c, various points on the interaction diagram are obtained. This approach assumes perfect bond between CC and UHPC, so that the section behaves as a fully composite section, and the confinement effect of the UHPC jacket is ignored. To confirm the perfect-bond assumption, interface shear between CC and UHPC has to be checked, which is not presented in this paper. The critical interface shear force is calculated for the strength limit state and compared with the interface shear resistance calculated according to the provisions in the AASHTO LRFD UHPC guide specifications ( 15 ). It is highly recommended that UHPC be placed against a clean conventional concrete substrate surface, free of laitance, with surface intentionally roughened to an amplitude of 0.25 in. Post-installed concrete anchors might be used to enhance the interface shear resistance if needed.

Material Models

Compression and tension models of UHPC are shown in Figure 4 ( 15 , 18 ), while the compression model of CC proposed by Thorenfeldt et al. ( 16 ) and Collins and Mitchell ( 17 ) is shown in Figure 5. These models were selected for their simplicity and common use in design practices.

Figure 4.

Material models of UHPC (a) compression model; (b) elastic perfectly plastic tension model; and (c) bilinear strain-hardening tension model ( 15 , 18 ).

Figure 5.

Material model of conventional concrete (CC) in compression ( 16 , 17 ).

With regard to UHPC, the idealized uniaxial stress-strain model in compression is defined by modulus of elasticity, E_uc; compressive strength, f^′_uc; and ultimate compressive strain, $ε_{cu}^{UHPC}$ , as shown in Figure 4a. Similarly, the idealized uniaxial stress-strain model in tension is defined by the same modulus of elasticity; effective cracking strength, f_t,cr; crack localization strength, f_t,loc; and crack localization strain, ε_t,loc, as shown in Figure 4, b or c, based on the values of cracking strength and crack localization strength. If f_t,loc ≥ 1.20 f_t,cr, Figure 4c will be used, otherwise, Figure 4b is used.

To illustrate labels in figures

f^′_uc = ultimate compressive strength of UHPC,

$ε_{cu}^{UHPC}$ = strain when f_uc reaches α_uf^′_uc,

ε_ucp = elastic compressive strain limit, can be determined using Equation 2,

f_t,cr = effective cracking strength of UHPC,

f_t,loc = crack localization strength of UHPC,

ε_t,loc = crack localization strain of UHPC,

E_uc = modulus of elasticity for UHPC, can be determined using Equation 1 ( 15 ),

α_u = reduction factor to account for the nonlinearity of the compressive stress-strain response; it shall not be greater than 0.85 ( 15 ), and

E_{uc} = 2, 500 K_{1} {f_{uc}^{'}}^{0.33}

(1)

where K₁ is a correction factor which shall be taken as 1.0 and $f_{uc}^{'}$ in ksi ( 15 ).

The ultimate compressive strain of UHPC, $ε_{cu}^{UHPC}$ , shall be taken as the greater of the elastic compressive strain limit, ε_ucp, or 0.0035 (Figure 4a) ( 15 ).

ε_{ucp} = \frac{α_{u} f_{uc}^{'}}{E_{uc}}

(2)

Likewise, the stress-strain curve for conventional concrete can be developed using Equation 3

f_{cc} (ε_{cc}) = (\frac{n (\frac{ε_{cc}}{ε_{cc}^{'}})}{n - 1 + {(\frac{ε_{cc}}{ε_{cc}^{'}})}^{nk}}) f_{cc}^{'}

(3)

where

$f_{cc}^{'}$ = compressive strength of conventional concrete,

$f_{cc}$ = stress at any strain $ε_{cc}$ ,

$ε_{cc}^{'}$ = strain when $f_{cc}$ reaches $f_{cc}^{'}$ ,

$E_{cc}$ = strain at different loading and assumed (0 to $ε_{ccu}$ ),

$ε_{cu}^{CC}$ = ultimate strain of conventional concrete and equal to 0.003 ( 27 ),

N = a curve-fitting factor equal to $0.8 + (\frac{f_{cc}^{'}}{2, 500})$ where $f_{cc}^{'}$ is in psi,

$k$ = a factor to control the slop of the ascending and descending branches of stress-strain curve, taken equal to 1.0 for $\frac{ε_{cc}}{ε_{cc}^{'}}$ less than 1.0 and taken $0.67 + (\frac{f_{cc}^{'} (psi)}{9, 000})$ for $\frac{ε_{cc}}{ε_{cc}^{'}}$ greater than 1.0, and

E_cc = modulus of elasticity for conventional concrete is determined using Equation 4 ( 27 )

E_{cc} = 120, 000 K_{1} γ_{cc}^{2} {f_{c c}^{'}}^{0.33} k s i

(4)

where $γ_{cc}$ is the unit weight of conventional concrete in kcf, and K₁ is a correction factor which shall be taken equal to 1 and $f_{cc}^{'}$ in ksi.

The material models of the commonly used non-prestressing steel and prestressing steel are shown in Figures 6 and 7, respectively. Other material models can be used instead when other types of reinforcement are used, such as high strength steel.

Figure 6.

Material model of non-prestressing steel ( 26 ).

Figure 7.

Material model of prestressing steel ( 28 ).

The elastic perfectly plastic model of the non-prestressing steel is given by Equation 5 according to Mattock ( 28 ).

f_{s} (ε_{s}) = {\begin{matrix} E_{s} . ε_{s} when 0 < | ε_{s} | < | ε_{y} | \\ f_{y} when | ε_{y} | < | ε_{s} | < | ε_{u} | \end{matrix}

(5)

where

f_y = yield strength,

ε_y = yield strain,

ε_u = ultimate strain, which depends on the grade and diameter of bars,

$f_{s}$ = stress at a given strain ε_s, and

E_s = modulus of elasticity, equal to 29,000 ksi ( 27 ).

Prestressing steel is modeled using a power formula shown in Equation 6 according to References ( 29 ) and ( 30 ).

f_{p} (ε_{p}) = E_{p} . ε_{p} (Q + \frac{1 - Q}{{(1 + {(\frac{E_{p} . ε_{p}}{K . f_{py}})}^{R})}^{\frac{1}{R}}}) \leq f_{pu}

(6)

where

f_py = yield strength of prestressing steel, equal to 0.9 f_pu,

f_pu = ultimate strength of the prestressing steel,

f_p = stress at a given strain, ε_p,

ε_pu = ultimate strain of prestressing steel,

$ε_{pe}$ = effective strain of prestressing steel, equal to $\frac{f_{pe}}{E_{p}}$ ,

E_p = modulus of elasticity of prestressing steel,

Q = power formula factor, equal to 0.031 for grade 270 low-relaxation strands,

K = power formula factor, equal to 1.043 for grade 270 low-relaxation strands, and

R = power formula factor, equal to 7.36 for grade 270 low-relaxation strands.

Section Properties

To determine compression member section properties, the width is expressed as a function of the distance from the compression side to allow the properties of any section to be calculated by integration. For example, a typical section of a circular CC column with radius, r, and outer UHPC shell with thickness, t_uhpc, will have a total height, h, of the composite section is equal to 2(r + t_uhpc), as shown in Figure 8. Equations 7 and 8 express the width of the outer shell and inner circular core as a function of the height, z, measured from the compression side as shown in Figure 8.

b_{o} (z) = 2 \sqrt{zh - z^{2}}

(7)

b_{cc} (z) = 2 \sqrt{r^{2} - {(z - \frac{h}{2})}^{2}}

(8)

Figure 8.

Strain distributions of a composite section for: (a) compression failure of CC and UHPC; and (b) tension failure of steel and UHPC.

The following equations are used to determine the section properties using the defined width of the section, $b (z)$ , at any level, z, as the difference between $b_{o} (z)$ and $b_{cc} (z)$ which is shown in Equation 9

b (z) = b_{o} (z) - b_{cc} (z)

(9)

Equations 10 and 11 are used to determine the section area, A_g, and its center of gravity, cg.

A_{g} = \int_{0}^{h} b (z) . dz

(10)

cg = \frac{\int_{0}^{h} b (z) . z . dz}{A_{g}}

(11)

Methodology

Modes of Failure

For CC-UHPC composite sections, the failure mode may occur by crushing UHPC in compression, crushing CC in compression, crack localization of UHPC in tension, or tension rupture of reinforcement, whichever occurs first for a given location of the neutral axis as shown in Figure 8. These modes of failure are determined by evaluating the calculated values of the lesser sectional curvature, $ψ_{n}$ , when:

the compressive strain at the extreme compression fiber of the composite section is equal to the compression strain limit, $ε_{cu}^{UHPC}$ ;

the compressive strain at the extreme compression fiber of the core is equal to the compression strain limit, $ε_{cu}^{CC}$ ;

the net tensile strain at extreme tension fiber of the composite section is equal to the UHPC crack localization strain limit, $ε_{t, loc}$ ; and

the strain in the extreme tension steel is equal to the ultimate strain of reinforcing steel, $ε_{su}$ .

Figure 8 illustrates the strain profiles for different modes of failure, and it describe the sectional curvature. The value of the sectional curvature, $ψ_{n} (c)$ , can be calculated from Equation 12:

ψ_{n} (c) = {\begin{matrix} \min of (\frac{ε_{t, loc}}{h - c}, \frac{ε_{su}}{d_{t} - c}) when c \leq c_{b} \\ min of (\frac{ε_{cu}^{UHPC}}{c}, \frac{ε_{cu}^{CC}}{c - t_{uhpc}}) when c > c_{b} \end{matrix}

(12)

where

$ε_{cu}^{UHPC}$ = strain in the extreme compression fiber of the composite section when the UHPC tensile strain limit, ε_t,loc, at extreme tension fiber is reached (in./in.),

$ε_{cu}^{CC}$ = strain in the extreme compression fiber of the core (in./in.),

c = distance from the extreme compression fiber of the composite section to the neutral axis at different values of strain,

d_t = distance from the extreme compression fiber of the section to the centroid of the extreme tension steel, equal to either the maximum depth of non-prestressing steel, d, or the maximum depth of prestressing steel, d_p (in.), and

c_b = distance from the extreme compression fiber to the neutral axis determined by the first occurrence of either the UHPC crack localization strain limit, ε_t,loc, or reinforcement yield strain limit, ε_y, simultaneous with either UHPC compression strain limit, $ε_{cu}^{UHPC}$ , or CC compression strain limit, $ε_{cu}^{CC}$ , which can be determined using Equation 13.

This equation demonstrates that the sectional curvature depends on the neutral axis depth at balanced point, c_b. If the neutral axis depth, c, is less than the neutral axis depth at the balanced point, c_b, either UHPC crack localization failure or yield of steel reinforcement occurs. Conversely, if the neutral axis depth, c, is greater, compression failure mode occurs, which might be in UHPC or CC, depending on whether the lesser sectional curvature is corresponding to UHPC crushing or CC crushing. c_b can be determined using Equation 13

c_{b} = \max of {\begin{matrix} \frac{ε_{cu}^{UHPC}}{ε_{cu}^{UHPC} + ε_{t, loc}} x h \\ \frac{ε_{cu}^{CC}}{ε_{cu}^{CC} + ε_{t, loc}} x (h - t_{uhpc}) \\ \frac{ε_{cu}^{UHPC}}{ε_{cu}^{UHPC} + ε_{y}} x d_{t} \\ \frac{ε_{cu}^{CC}}{ε_{cu}^{CC} + ε_{y}} x (d_{t} - t_{uhpc}) \end{matrix}

(13)

Nominal Axial Capacity

Figure 9 shows the resultant internal forces of the composite section which are composed of CC compressive force ( $C_{cc}$ ), UHPC compressive and tensile forces ( $C_{uc}, T_{c}$ ), and either reinforced steel tensile and compressive forces, $T_{si}$ , or prestressing steel tensile forces, $T_{pi}$ . To determine the contribution of each material to the axial capacity of a member, it is necessary to use its stress-strain relationship and integrate it over the corresponding area. It is permissible to ignore the tensile stresses in CC according to Devalupura and Tadros ( 30 ). However, it is crucial to account for the tensile stresses in UHPC because of its enhanced ductility and higher tensile strength. Static equilibrium of the section requires that the sum of internal forces be equal to the nominal axial capacity of the section. Equation 14 can be used to determine the axial capacity of a compression member expressed as a function of c.

\begin{matrix} P_{n} (c) = C_{uc} (c) + C_{cc} (c) - T_{c} (c) - \sum_{0}^{i} T_{si} (c) \\ - \sum_{0}^{i} T_{pi} (c) \end{matrix}

(14)

where

$P_{n} (c)$ = nominal axial capacity of the section as a function of the neutral axis depth, c,

$C_{uc} (c)$ = resultant compressive force in the UHPC as a function of the neutral axis depth, c,

$C_{cc} (c)$ = resultant compressive force in the CC as a function of the neutral axis depth, c,

$T_{c} (c)$ = resultant tensile force in UHPC as a function of the neutral axis depth, c,

$T_{si} (c)$ = force of the reinforcement steel in row (i) as a function of the neutral axis depth, c, and

$T_{pi} (c)$ = force of the prestressing steel in row (i) as a function of the neutral axis depth, c.

Figure 9.

Stress and strain distributions of a composite section at a given neutral axis depth.

Figure 10, a and b, shows a strip in the UHPC shell and CC core at a height, y, measured from the neutral axis, which might be above or below the neutral axis. The strip thickness is $dy$ and the width is equal to $b (c - y)$ and $b_{cc} (c - y)$ for the UHPC shell and the CC core, respectively, above the neutral axis. Below the neutral axis, the strip has the same thickness while its width is $b (c + y)$ for UHPC shell and zero for CC core since tension is neglected in CC. Equations 15 and 16 show the resultant compressive forces of CC and UHPC respectively. These equations use the integration of the stress distribution over the area at the top part with intervals beginning from the extreme top fibers of the section at $y = c$ , down to the neutral axis level at $y = 0$ . Equation 17 shows the resultant tensile strength of UHPC which is calculated using the integration of stress distribution over the area below the neutral axis with intervals beginning from a neutral axis level at $y = 0$ , down to the extreme bottom of the section at $y = h - c$ . Figure 10a shows the calculation of the resultant compression and tensile forces of UHPC shell while Figure 10b shows the calculation of the resultant compression force of CC core.

Figure 10.

Calculation of forces in (a) UHPC shell, and (b) CC core.

The resultant forces in reinforcement steel, prestressing strands, or both, are calculated as the sum of forces in each individual row (i), as shown in Equations 18 and 20, respectively. Equations 19 and 21 may be used to determine the stress in each row based on the corresponding strain. The stress calculation includes the effect of the concrete stress corresponding to the strain in the steel.

C_{cc} (c) = \int_{0}^{c} f_{c c} (ψ_{n} (c) . y) . b_{cc} (c - y) . dy

(15)

C_{uc} (c) = \int_{0}^{c} f_{u c} (ψ_{n} (c) . y) . b (c - y) . dy

(16)

T_{c} (c) = \int_{0}^{h - c} f_{t} (ψ_{n} (c) . y) . b (c + y) . dy

(17)

T_{s} (c) = \sum_{1}^{i} T_{si} (c) = \sum_{0}^{i} A_{si} . f_{si} (c)

(18)

f_{si} (c) = {\begin{matrix} f_{s} (ε_{s} {(c)}_{i}) when c < d_{i} < h \\ f_{s} (ε_{s} {(c)}_{i}) + f_{cc} (ε_{s} {(c)}_{i}) when 0 < d_{i} < c \end{matrix}

(19)

T_{ps} (c) = \sum_{1}^{i} T_{pi} (c) = \sum_{0}^{i} A_{pi} . f_{pi} (c)

(20)

f_{pi} (c) = {\begin{matrix} f_{p} (ε_{ps} {(c)}_{i}) when c < d_{pi} < h \\ f_{p} (ε_{ps} {(c)}_{i}) + f_{cc} (ε_{ps} {(c)}_{i} - ε_{pe}) when 0 < d_{pi} < c \end{matrix}

(21)

where

$f_{cc} (ψ_{n} (c) . y)$ = compression stress in CC corresponding to a strain at a level, y, measured from neutral axis level and equal to $ψ_{n} (c) . y$ ,

$b_{cc} (c - y)$ = width of CC core above neutral axis at a distance, c – y, measured from the top,

$f_{uc} (ψ_{n} (c) . y)$ = compression stress in UHPC corresponding to strain at a level, y, measured from neutral axis level and equal to $ψ_{n} (c) . y$ ,

$b (c - y)$ = width of UHPC shell above neutral axis at a distance, c – y, measured from the top,

$f_{t} (ψ_{n} (c) . y)$ = tensile stress in UHPC corresponding to strain at a level, y, measured from neutral axis level and equal to $ψ_{n} (c),$

$b (c + y)$ = width of UHPC shell below neutral axis at a distance, c + y, measured from the top,

$f_{si} (c)$ = stress in reinforcement steel in row (i) at a given c, determined using Equation 19,

$A_{si}$ = area of reinforcement steel in row (i),

$d_{i}$ = depth of row (i) of reinforcement steel,

$f_{pi} (c)$ = stress in prestressing steel in row (i) at a given c, determined using Equation 21,

$A_{pi}$ = area of prestressing steel in row (i), and

$d_{pi}$ = depth of row (i) of prestressing steel

Nominal Flexural Capacity

The moment capacity of the section (M_n) is determined by summing the moments of each force about the centerline of the section. Equation 22 shows the calculation of the moment capacity of a CC-UHPC composite section as terms in the neutral axis depth, c.

\begin{matrix} M_{n} (c) = C_{cc} (c) . (cg - Z_{cc} (c)) + C_{uc} (c) . (cg - Z_{uc} (c)) \\ + T_{c} (c) . (X (c) - cg) + \sum_{1}^{i} T_{si} (c) . (d_{i} - cg) \\ + \sum_{1}^{i} T_{pi} (c) (d_{pi} - cg) \end{matrix}

(22)

where

$Z_{cc} (c)$ = depth of resultant compression force of CC measured from extreme top fibers as a function of c,

$Z_{uc} (c)$ = depth of resultant compression force of UHPC measured from extreme top fibers as a function of c, and

$X (c)$ = depth of resultant tension force of UHPC measured from extreme top fibers as a function of c.

Figure 9 depicts the distance between the centroid of the resultant compression CC and UHPC forces and the extreme top fiber of the composite section which can be determined using Equations 23 and 24, respectively. Additionally, it shows the distance between the resultant tensile force of UHPC, and the extreme top fibers which can be determined using Equation 25. The equations are expressed as a function of neutral axis depth, c, so that different flexural and axial capacities for various locations of neutral axis could be calculated.

Z_{cc} (c) = \frac{\int_{0}^{c} f_{cc} (ψ_{n} (c) . y) . b_{cc} (c - y) . (c - y) . dy}{C_{cc} (c)}

(23)

Z_{uc} (c) = \frac{\int_{0}^{c} f_{uc} (ψ_{n} (c) . y) . b (c - y) . (c - y) . dy}{C_{uc} (c)}

(24)

X (c) = \frac{\int_{0}^{h - c} f_{t} (ψ_{n} (c) . y) . b (c + y) . (c + y) . dy}{T_{c} (c)}

(25)

Pure Axial Capacity

For the pure compression point, the axial capacity could be determined using Equation 26 ( 15 ). This capacity is the nominal axial capacity when the nominal flexural capacity is zero (no eccentricity).

P_{o} (c) = k (α_{u} . f_{uc}^{'} . A_{g} + α_{1} . f_{cc}^{'} . (A_{gc} - \sum_{0}^{i} A_{s}) + f_{y} . \sum_{0}^{i} A_{s})

(26)

where k = confinement factor taken 0.8 for ties and 0.85 for spirals, and α₁ = compressive strength factor and taken equal to 0.85.

Ultimate Design Axial and Flexural Capacities

To determine the ultimate design capacity of a compression member, Equations 27 and 28 could be used.

P_{u} = ϕ . P_{n}

(27)

M_{u} = ϕ . M_{n}

(28)

where $ϕ$ is a resistance factor which depends on the curvature ductility ratio, μ.

The resistance factor, ϕ, shall be taken to be equal to 0.9 for sections with curvature ductility ratio, μ, greater than the curvature ductility ratio limit, μ_ℓ, which is equal to 3.0, and 0.75 for compression members, tension members, members subjected to combined tension and flexure, and sections with curvature ductility ratio, μ, less than 1.0. When curvature ductility ratio, μ, is between the curvature ductility ratio limit, 3.0 and 1.0, the value of ϕ associated with the ductility ratio may be obtained by a linear interpolation from 0.75 to 0.90. as shown in Figure 11 ( 15 ) using Equation 29:

ϕ (c) = {\begin{matrix} 0.75, 1.0 > μ (c) \\ 0.75 + 0.15 \frac{(μ (c) - 1.0)}{(μ_{ℓ} - 1.0)}, 1.0 \leq μ (c) \leq μ_{ℓ} \\ 0.9, μ (c) > μ_{ℓ} \end{matrix}

(29)

Figure 11.

Relation between resistance factor, ϕ, and curvature ductility ratio, μ, in UHPC ( 15 ).

The curvature ductility ratio, μ is defined as the ratio of the sectional curvature at the nominal moment resistance over the baseline sectional curvature and can be computed using Equation 30 ( 15 ).

μ = \frac{ψ_{n}}{ψ_{s ℓ}}

(30)

ψ_{s ℓ} = \frac{ε_{s ℓ}}{d_{t} - c_{s ℓ}}

(31)

where

$ψ_{s ℓ}$ = sectional curvature of the composite section when the steel stress in the extreme tension steel is equal to the steel service stress limit, f_sℓ, calculated using Equation 31 (1/in.),

$ε_{s ℓ}$ = service strain in the tension steel when the steel stress in the extreme tension steel is equal to the steel service stress limit, f_sℓ (in./in.),

F_sℓ = stress limit in steel at service loads (ksi), and

$c_{s ℓ}$ = distance from the extreme compression fiber of the section to the neutral axis when the steel stress in the extreme tension steel is equal to the steel service stress limit, f_sℓ (in.).

Interaction Diagram

The interaction diagram could be constructed using Equations 14 and 22 by substituting different values for the neutral axis depth, c. The range of neutral axis depth begins from the depth corresponding to pure flexural failure and ends at a depth corresponding to a pure compression failure which could happen at c > h.

Design Example

A demonstration project was selected by Nebraska Department of Transportation (DOT) to repair a deteriorated column in the Adams Street bridge in Lancaster County, over Highway I-180. The plan was to remove the damaged section and concrete cover, shown in Figure 12, then cast a layer of UHPC around the column that had the same cover thickness. The proposed approach is used to construct the interaction diagram for that column after repair and show the effect of adding UHPC cover on the capacity of the column.

Figure 12.

Example of a deteriorated bridge column.

Details of the Bridge Column

The bridge column as 28 in. in diameter and is reinforced with 12 #7 bars in the longitudinal direction and with #4 ties in the transverse direction at 12-in. spacing. Section dimensions and reinforcement details are shown in Figure 13. The repair procedures began with removing the 2-in. thick conventional concrete cover of the column, cleaning the surface, forming for UHPC encasement, and finally casting UHPC. The material properties of CC, UHPC, and steel reinforcement are listed in Table 1.

Figure 13.

Cross section and reinforcement of the example column: (a) before repair; and (b) after repair.

Table 1.

CC, UHPC, and Reinforcing Steel Properties

Property	CC	UHPC
Compressive strength, ksi	5.0	17.5
Modulus of elasticity, ksi	4,291	6,249
Tensile strength, ksi	neglected	0.75
Localized strength, f_t,loc, ksi	na	0.75
Maximum strain in Compression	0.003	0.0035
Localized strain, ε_t,loc	na	0.005
Property	Reinforcing steel (A615)
Yield strength, ksi	60
Modulus of elasticity, ksi	29,000
Yield strain, ε_y	0.002
Service strain, ε_sl	0.00166
Ultimate strain, $ε_{u}$	0.09

Note: UHPC = ultra-high-performance concrete; CC = conventional concrete; ksi = kips per square inch; na = not applicable.

Section Properties

The area of the shell, A_g, is 163.38 in.² and the area of the core, A_gc, is 452.37 in.². The centroid of both is at the mid-height of the section.

Calculations for a Point on the Interaction Diagram for a Given Depth of the Neutral Axis

For a given depth of the neutral axis, c = 10 in., the axial and flexural capacities are calculated as follows: the sectional curvature is first computed to capture the mode of failure. The result shows c = 10 in. is less than balanced neutral axis depth, c_b = 15.63 in., so that the curvature at the given neutral axis depth, ψ_n (10 in.) = 0.00028 which corresponds to crack localization failure of UHPC jacket. Thus, the actual compressive strains in UHPC and CC, $ε_{c}^{UHPC}$ and $ε_{c}^{CC}$ , corresponding to the crack localization strain are calculated using strain compatibility and are equal to 0.0028 and 0.0022, respectively. The pre-existing strain and curvature in the CC column were ignored for simplicity, but they can be determined by analyzing the loads on the column when UHPC jacketing is applied. This strain impacts the ultimate strain of the CC column and needs to be subtracted accordingly. The strain in each row of the reinforcing steel is then calculated using strain compatibility. Consequently, the stresses and the capacity of the reinforcing steel are determined for each row of steel as shown in Table 2.

Table 2.

Strain, Stress, and Force at Each Reinforcement Row

Row (i)	Number of bars (n_i)	Area of bars (A_si) (in²)	Depth of bars from the top (d_i) (in)	Strain in bars* (ε_si) $ε_{si} = (d_{i} - c) . ψ_{n} (c)$	Stress in bars** (f_si) (ksi)	Force in bars* (T_i) (kip) $T_{i} = f_{si} A_{si}$	Distance between CG of bars to CG of section** (in.)
1	2.0	1.2	3.125	−0.00191	−55.38	−66.46	+10.875
2	2.0	1.2	6.0	−0.00111	−32.22	−38.67	+8.0
3	2.0	1.2	11.125	+0.00031	+9.06	+10.87	+2.875
4	2.0	1.2	16.875	+0.00191	+55.38	+66.46	−2.875
5	2.0	1.2	22.0	+0.00333	+60.0	+72.0	−8
6	2.0	1.2	24.875	+0.00413	+60.0	+72.0	−10.875
TOTAL						+116.21

Note: CG = center of gravity; ksi = kips per square inch. * Positive sign is tension and negative sign is compression; ** Positive sign is above centerline of the section and negative sign is below.

Equations 15 and 16 are used to determine the resultant compression forces in CC and UHPC, respectively, while Equation 17 is used to determine the tensile force in UHPC. Substituting for c = 10 in., internal forces are as follows: C_uc (10 in.) = 665.8 kip, C_cc (10 in.) = 422.2 kip, T_c (10 in.) = 72.8 kip. Substituting these forces in Equation 14 leads to nominal axial capacity, P_n = 902.6 kip. By substituting in Equations 23 to 25 for c = 10 in., the depths of the internal forces measured from the top are Z_uc (10 in.) = 2.86 in., Z_cc (10 in.) = 5.68 in., and X (10 in.) = 20.67 in. Consequently, the nominal flexural capacity, M_n (10 in.) in Equation 22 is equal to 1,160.82 kip.ft.

Another point on the interaction diagram, point of pure axial, is determined. The nominal axial load capacity should not exceed the pure compression capacity, P_o (10 in.) = 3,803.4 kip, given in Equation 26.

To develop the interaction diagram, the neutral axis depth, c, is assumed to range from the pure flexural value, c_eq, to about 3 h, which represents the pure compression value. The c_eq value is equal to 6.064 in. for this example. The nominal and ultimate interaction diagrams for the original and repaired sections are shown in Figure 14.

Figure 14.

Interaction diagrams of the bridge column before and after repair.

The diagram in Figure 14 illustrates that the ultimate pure axial and flexural capacities of the repaired column section increased by 82% and 20%, respectively, compared with their original column. The limited increase in the flexural capacity is a result of the controlling failure mode of UHPC in tension, which is reaching crack localization strain.

Figure 15 shows the effect of increasing the thickness of the UHPC jacket, t_uhpc, on the column interaction diagram. The figure indicates a significant increase in the column’s capacity above the balanced failure line. Conversely, below the balanced failure line, the capacity slightly increases because the UHPC crack localization strain controls the design even with an increased overall height of the column section, which could limit the strain in the reinforcement steel. For example, in comparison with a UHPC jacket with a thickness of 2.0 in., the diagram depicts a 14% increase in the ultimate pure axial capacity and a 9% increase in the pure flexural capacity for a UHPC cover thickness of 2.5 in. Furthermore, a UHPC cover thickness of 3.0 in. exhibits an increase of 28.5% and 19% in the ultimate pure axial capacity and the pure flexural capacity, respectively, compared with the 2.0-in. thickness.

Figure 15.

Ultimate capacity interaction diagrams for repaired column with different UHPC jacket thicknesses.

It should be noted that the capacity calculations shown in the study assume a short column. As the column gets slender, P-δ analysis needs to be conducted to consider the secondary moments associated with the deformations of the composite CC-UHPC column, which will result in capacity reduction depending on the slenderness ratio and end conditions.

Conclusions

This study presents an approach to predicting the capacity of reinforced or prestressed concrete compression members strengthened or repaired using UHPC encasement and subjected to combined axial and bending. This approach uses the idealized material models of UHPC in compression and tension according to the new AASHTO Guide Specifications for Structural Design with UHPC. The approach uses integration to determine the internal forces for any section geometry without the approximation associated with the lamina method. The approach assumes a perfect bond between conventional concrete and UHPC and with internal reinforcing bars, strands, or both. The strength reduction factor is determined according to the curvature ductility approach to accurately represent the structural behavior of the compression members. Four different modes of failures were considered in calculating the capacity of the composite section, namely compression failure of UHPC or CC, crack localization of UHPC, and rupture of tension reinforcement.

Applying the proposed method to the design of a circular reinforced concrete column repaired using UHPC jacket has shown a significant increase of 82% in the pure axial capacity, and a slight increase of 20% in the pure flexural capacity compared with the original column, as shown in Figure 14. Studying the effect of UHPC jacket thickness on both flexural and axial capacities has shown a more prominent increase in the axial compression capacity compared with a slight increase in the flexural capacity when the thickness of UHPC jacket increases, see Figure 15.

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: Mohammed H Hedia and George Morcous; data collection: Mohammed H Hedia and George Morcous; analysis and interpretation of results: Mohammed H Hedia and George Morcous; draft manuscript preparation: Mohammed H Hedia and George Morcous. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research work here was supported by the Nebraska Department of Transportation (DOT), which is gratefully acknowledged. The research project name is “Repair and Strengthening of Bridge Girders Using Ultra-High-Performance Concrete (UHPC)” with accession number of 01849003.

ORCID iDs

Mohammed H. Hedia

George Morcous

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