**Piles**

**Figure 10.** Model of Six-Column Bents

244 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**Figure 11.** Bent Element Cross-Sections

The columns were considered fixed in the cap beam. Nonlinear springs along the pile shafts were used to model the resistance provided by the surrounding soil. The L-Pile software (2002) was used to compute the P-Y curves, based on the stiff sand soil model with free water at 15 depths.

To build an exact computer model of a structure beard against underground elements-piles it is necessary to know how interaction between soil and a pile can be simulated, to get more precise result of the analysis. The p-y curves is a strait interpretation of the relation between deflection of an element and soil pressure on a particular depth. The pressure from the soil on the element is distributed within certain length which depends on the number of springs assigned to it Figure 12.

**Figure 12.** Model of laterally loaded pile

A physical definition of the soil resistance *p* is given in Figure 13. There was made an assump‐ tion that the pile has been installed without bending so the initial soil stresses at the depth *xi* are uniformly distributed as shown in Figure 13*b*. If the pile is loaded laterally so that a pile deflection *yi* occurs at the depth *xi* the soil stresses will become unbalanced as shown in Figure 13*c*. Integration of the soil stresses yielding the soil resistance *pi* with units F/L equation 1.

$$\mathbf{p}\_i \mathbf{=} \mathbf{E}\_s \mathbf{y}\_i \tag{1}$$

*u p y y y*

 j

in which ϕ<sup>y</sup> is yield curvature, ϕ<sup>p</sup> is plastic curvature, and ϕ<sup>u</sup> is summation of yield curvature

Figure 15 and Figure give a moment-curvature diagram for the column sections in the North Bound Bridge, calculated by the XTRACT. Curvature properties are section dependent and can be determined by numerical integration methods. Input data of a cross-section include nonlinear material properties of concrete and steel, and the detailed configuration of the section. For the North Bound bridge, all the columns have the identical section dimension, however, the moment-rotation relationships may not be the same because of the different axial

The plastic hinge length for piles depends on whether the hinge is located at the pile/deck interface or is an in-ground hinge. For prestressed piles where the solid pile is embedded in

the deck, the plastic hinge length at the pile/deck interface can be taken as (PIANC):

<sup>+</sup> = = (2)

Depth 24 in Depth 26 in Depth 48 in Depth 52.8 in Depth 57.6 in Depth 62.4 in Depth 67.2 in Depth 79.2 in Depth 96 in Depth 120 in Depth 168 in Depth 240 in Depth 288 in

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j j

**Displacement (in)**

j

m

**Figure 14.** Bilinearized Force-Displacement of SSI at Different Depths

loads.

**Force (kips)**

**Hinge length**

j

j

and plastic curvature that presents the ultimate curvature capacity of a section.

where,

Es – a parameter with the units F/L2 , relating pile deflection *y* and soil reaction *p*.

**Figure 13.** Definition of p and y as Related to Response of a Pile to Lateral Loading

Once the p-y curves at various depths of the pile have been obtained, a force-displacement relationship can be calculated by multiplying p with the tributary length of the pile between springs. Figure 14 shows a bilinearization of the force-displacement relationship at different depth based on the data retrieved from LPILE single pile analysis. These results were used to define multi-linear elastic links (springs) in SAP2000 in order to represent the SSI of the piles. The piles of all bents were assumed to extend 27 ft under the ground, so all bents had the same pile modeling.

### **5. Plastic hinge**

It is well known that well-confined concrete structures can deform inelastically without significant strength loss through several cycles of response. Ductility describes such ability of structures, which is often defined as the ratio of deformation at a given response level to the deformation at yield response. Commonly used ductility ratios include displacement ductility, curvature ductility and rotation ductility. In the software of XTRACT, developed by Imbsen & Associates Company (2002) with the capability of analyzing structural cross sections, curvature ductility can be calculated for a given section and are defined in Equation 2 (Paulay and Priestley, 1992).

**Displacement (in)**

**Figure 14.** Bilinearized Force-Displacement of SSI at Different Depths

$$
\mu\_{\varphi} = \frac{\varphi\_u}{\varphi\_y} = \frac{\varphi\_p + \varphi\_y}{\varphi\_y} \tag{2}
$$

in which ϕ<sup>y</sup> is yield curvature, ϕ<sup>p</sup> is plastic curvature, and ϕ<sup>u</sup> is summation of yield curvature and plastic curvature that presents the ultimate curvature capacity of a section.

Figure 15 and Figure give a moment-curvature diagram for the column sections in the North Bound Bridge, calculated by the XTRACT. Curvature properties are section dependent and can be determined by numerical integration methods. Input data of a cross-section include nonlinear material properties of concrete and steel, and the detailed configuration of the section. For the North Bound bridge, all the columns have the identical section dimension, however, the moment-rotation relationships may not be the same because of the different axial loads.

### **Hinge length**

i s i p =E y (1)

, relating pile deflection *y* and soil reaction *p*.

where,

pile modeling.

**5. Plastic hinge**

and Priestley, 1992).

Es – a parameter with the units F/L2

246 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**Figure 13.** Definition of p and y as Related to Response of a Pile to Lateral Loading

Once the p-y curves at various depths of the pile have been obtained, a force-displacement relationship can be calculated by multiplying p with the tributary length of the pile between springs. Figure 14 shows a bilinearization of the force-displacement relationship at different depth based on the data retrieved from LPILE single pile analysis. These results were used to define multi-linear elastic links (springs) in SAP2000 in order to represent the SSI of the piles. The piles of all bents were assumed to extend 27 ft under the ground, so all bents had the same

It is well known that well-confined concrete structures can deform inelastically without significant strength loss through several cycles of response. Ductility describes such ability of structures, which is often defined as the ratio of deformation at a given response level to the deformation at yield response. Commonly used ductility ratios include displacement ductility, curvature ductility and rotation ductility. In the software of XTRACT, developed by Imbsen & Associates Company (2002) with the capability of analyzing structural cross sections, curvature ductility can be calculated for a given section and are defined in Equation 2 (Paulay

The plastic hinge length for piles depends on whether the hinge is located at the pile/deck interface or is an in-ground hinge. For prestressed piles where the solid pile is embedded in the deck, the plastic hinge length at the pile/deck interface can be taken as (PIANC):

p p L =0.5 D (3)

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For in-ground hinges, the plastic hinge length depends on the relative stiffness of the pile and the foundation material. Because of the reduced moment gradient in the vicinity of the inground hinge, the plastic hinge length is significantly longer there. In this report CALTRANS interpretation of in-ground hinges for a non-cased pile shaft was used. Figure 17 describes the

In order to locate the plastic hinge locations, a separate push over analysis was run on single column. Figure 18 shows the single column element modeled in SAP2000. Top of the column is restrained against rotation to represent the rigid connection between the column and the deck. The SSI is represented by links just as discussed for general bents. The pin connection at

Figure 19 provides the moment diagram of the above column/pile under horizontal loading. The diagram has two points of maximum moment. The plastic hinge should be placed at these locations in order to represent the most conservative nonlinear behavior of the column/pile.

the bottom of the pile restricts the pile from vertical movement.

where,

Dp – diameter of a pile

**Figure 17.** In-Ground Hinge Length

**Hinge location**

calculation steps provided by CALTRANS.

**Figure 15.** Bilinearization of the Moment–Curvature Curve for Hollow Column

**Figure 16.** Bilinearization of the Moment–Curvature Curve for Filled Hollow Column

$$\mathbf{L\_{p}} = 0.5 \,\mathrm{D\_{p}}\tag{3}$$

where,

**Figure 15.** Bilinearization of the Moment–Curvature Curve for Hollow Column

248 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

**Figure 16.** Bilinearization of the Moment–Curvature Curve for Filled Hollow Column

Dp – diameter of a pile

For in-ground hinges, the plastic hinge length depends on the relative stiffness of the pile and the foundation material. Because of the reduced moment gradient in the vicinity of the inground hinge, the plastic hinge length is significantly longer there. In this report CALTRANS interpretation of in-ground hinges for a non-cased pile shaft was used. Figure 17 describes the calculation steps provided by CALTRANS.

**Figure 17.** In-Ground Hinge Length

### **Hinge location**

In order to locate the plastic hinge locations, a separate push over analysis was run on single column. Figure 18 shows the single column element modeled in SAP2000. Top of the column is restrained against rotation to represent the rigid connection between the column and the deck. The SSI is represented by links just as discussed for general bents. The pin connection at the bottom of the pile restricts the pile from vertical movement.

Figure 19 provides the moment diagram of the above column/pile under horizontal loading. The diagram has two points of maximum moment. The plastic hinge should be placed at these locations in order to represent the most conservative nonlinear behavior of the column/pile.

**Figure 18.** Single Column Finite Element Stick Model

The placement of the first hinge should be at the column/bent connection as expected before. The second hinge has to be place under the ground, but the location of maximum moment in that area changes in a pushover analysis. A parametric study was run in order to locate the worst location for an in-ground hinge. The placement of the in-ground hinge was varied for multiple pushover analysis. Figure 20 shows the results of this parametric study, where the hinge depth below ground level is compared to column top displacement capacity. The plot in Figure 20 shows that placing the hinge 20% of pile length under the ground would give a displacement capacity of 2.25 in, which is less than any other location. Figure 21 shows the placement of the plastic hinges in four column bent.

**Push Capacity of Pile** 

**Figure 20.** Single Column Parametric Study Results

**Figure 19.** Moment Diagram of Single Column under Horizontal Load

**% of Pile Length under the Ground**

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#### Pushover Analysis of Long Span Bridge Bents http://dx.doi.org/10.5772/52728 251

**Figure 19.** Moment Diagram of Single Column under Horizontal Load

**Figure 20.** Single Column Parametric Study Results

The placement of the first hinge should be at the column/bent connection as expected before. The second hinge has to be place under the ground, but the location of maximum moment in that area changes in a pushover analysis. A parametric study was run in order to locate the worst location for an in-ground hinge. The placement of the in-ground hinge was varied for multiple pushover analysis. Figure 20 shows the results of this parametric study, where the hinge depth below ground level is compared to column top displacement capacity. The plot in Figure 20 shows that placing the hinge 20% of pile length under the ground would give a displacement capacity of 2.25 in, which is less than any other location. Figure 21 shows the

placement of the plastic hinges in four column bent.

**Figure 18.** Single Column Finite Element Stick Model

250 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

a generalized force-deformation relation model shown in Figure 22 for the nonlinear static analysis procedure, which is the defaulted model in SAP for the Axial-Moment hinge.

Three parameters, *a*, *b* and *c* are defined numerically in FEMA-365, and are permitted to be determined directly by analytical procedures. The moment and rotation are normalized by

In Table 6-8 of FEMA 356, modeling parameters and numerical acceptance criteria are given for reinforced concrete columns in various categories. Columns investigated are all primary structural elements. A conforming transverse reinforcement is defined by hoops spaced in the

(*Vs*) being greater than three-fourths of the design shear. Thus, the category of the column is decided in Table 6-8 of FEMA 356, and values and relationship of the performance levels can

In SAP, an absolute rotation value can overwrite the default value in defining a hinge property. The plastic rotation capacity angle, *a*, calculated with Equation 4-12 for a given column is at point C. The ultimate rotation angle, which is inputted as *b* in SAP, is taken as 1.5 times the plastic angle. It is indicated at point E, which defines a local failure at a plastic hinge. A larger

The three discrete structural performance levels are Immediate Occupancy (IO), Life Safety

The ultimate plastic hinge angle calculated by the XTRACT was taken as the Collapse Pre‐ vention level. Its value was indicated as "*a*" in Figure 1. The permissible deformation for the Life Safety performance level is taken as three quarters of the plastic rotation capacity "*a*".

value could be used to allow the structure to form a global failure due to instability.

the yield forces and the yield rotation based on reinforcement and section provided.

*My*

 and *<sup>θ</sup> θy*

. By default SAP will calculate

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<sup>3</sup> , and the strength provided by the hoops

**Figure 22.** Generalized Force-Deformation Relations for Concrete Elements (FEMA-356)

yield moment and yield rotation respectively, i.e., *<sup>M</sup>*

flexural plastic hinge region less than or equal to *<sup>d</sup>*

(LS) and Collapse Prevention (CP) shown in Figure 23.

be utilized.

**Figure 21.** Location of the Plastic Hinge

#### **Plastic hinge property**

The Manual of SAP2000 recommends a distributed plastic hinge model assuming 0.1 of element length as the plastic hinge length, but information on how to define distributed plastic hinge properties is not provided. In this research, a concentrated plastic hinge model is used with the assumption that plastic rotation will occur and concentrate at mid-height of a plastic hinge. Input hinge properties consist of the section yield surface, plastic rotation capacity, and acceptance criteria.

A plastic rotation, *θp*, can be calculated by the plastic curvature given the equivalent plastic hinge length *Lp* as shown in Equation 4.

$$\mathcal{O}\_p = \mathcal{O}\_p \mathcal{L}\_p = \mathcal{L}\_p(\mathcal{O}\_u - \mathcal{O}\_y) \tag{4}$$

The plastic rotation is an important indicator of the capacity of a section to sustain inelastic deformation and is used in SAP to define column plastic hinge properties. FEMA 356 provides a generalized force-deformation relation model shown in Figure 22 for the nonlinear static analysis procedure, which is the defaulted model in SAP for the Axial-Moment hinge.

**Figure 22.** Generalized Force-Deformation Relations for Concrete Elements (FEMA-356)

**Figure 21.** Location of the Plastic Hinge

252 Engineering Seismology, Geotechnical and Structural Earthquake Engineering

hinge length *Lp* as shown in Equation 4.

The Manual of SAP2000 recommends a distributed plastic hinge model assuming 0.1 of element length as the plastic hinge length, but information on how to define distributed plastic hinge properties is not provided. In this research, a concentrated plastic hinge model is used with the assumption that plastic rotation will occur and concentrate at mid-height of a plastic hinge. Input hinge properties consist of the section yield surface, plastic rotation capacity, and

A plastic rotation, *θp*, can be calculated by the plastic curvature given the equivalent plastic

 jj

The plastic rotation is an important indicator of the capacity of a section to sustain inelastic deformation and is used in SAP to define column plastic hinge properties. FEMA 356 provides

= = - *L L* (4)

( ) *p pp p u y*

qj

**Plastic hinge property**

acceptance criteria.

Three parameters, *a*, *b* and *c* are defined numerically in FEMA-365, and are permitted to be determined directly by analytical procedures. The moment and rotation are normalized by yield moment and yield rotation respectively, i.e., *<sup>M</sup> My* and *<sup>θ</sup> θy* . By default SAP will calculate the yield forces and the yield rotation based on reinforcement and section provided.

In Table 6-8 of FEMA 356, modeling parameters and numerical acceptance criteria are given for reinforced concrete columns in various categories. Columns investigated are all primary structural elements. A conforming transverse reinforcement is defined by hoops spaced in the

flexural plastic hinge region less than or equal to *<sup>d</sup>* <sup>3</sup> , and the strength provided by the hoops

(*Vs*) being greater than three-fourths of the design shear. Thus, the category of the column is decided in Table 6-8 of FEMA 356, and values and relationship of the performance levels can be utilized.

In SAP, an absolute rotation value can overwrite the default value in defining a hinge property. The plastic rotation capacity angle, *a*, calculated with Equation 4-12 for a given column is at point C. The ultimate rotation angle, which is inputted as *b* in SAP, is taken as 1.5 times the plastic angle. It is indicated at point E, which defines a local failure at a plastic hinge. A larger value could be used to allow the structure to form a global failure due to instability.

The three discrete structural performance levels are Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP) shown in Figure 23.

The ultimate plastic hinge angle calculated by the XTRACT was taken as the Collapse Pre‐ vention level. Its value was indicated as "*a*" in Figure 1. The permissible deformation for the Life Safety performance level is taken as three quarters of the plastic rotation capacity "*a*".

A concrete interaction surface was obtained from XTRACT for the frame hinges under combined bending and axial load. A generated interaction surface is shown in Figure 25.

**Moment (kin)**

The results of the pushover analysis are organized in the following pages. Each bent has two pages of results. The first page shows the general characteristics of the bent and the push over curve. The second page shows the results of the step by step push over analysis and the development of hinges. The first and the last bent on the plan are abutments, so they are not

Maximum displacement capacity exhibited is 3.9 inches by the fifth bent. The lowest bent capacity is 2.08 inches by the second bent. The reason for this low displacement capacity in

Using two hinges per column creates a much lower capacity than using only one hinge at the top of the column. The shear force capacity stays almost the same, but the ductility reduces for

bent number two is because of its shorter columns compared to other bents.

**Figure 25.** Axial Load-Moment Interaction Curve (Compression force is negative)

In-Ground Hinge Top Column Hinge

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**Axial Force (Kip)**

**6. Results**

all bents.

included in this report.

**Figure 23.** Performance Level on Generalized Force-Deformation Relations for Concrete Elements (FEMA-356)

The increase of moment strength at point C is taken as the over strength factor computed by XTRACT, ignoring the strength softening effect. The actual moment strength at point C is the product of the factor and the yielding moment. FEMA 356 defines a 0.2 residual strength ratio before plastic hinge eventually fails. Figure 24 presents moment-rotation curves for one of the columns.

**Figure 24.** Moment-Rotation Relationship of the Columns

A concrete interaction surface was obtained from XTRACT for the frame hinges under combined bending and axial load. A generated interaction surface is shown in Figure 25.

**Figure 25.** Axial Load-Moment Interaction Curve (Compression force is negative)
