INTRODUCTION

A skew bridge is the one for which the span of the bridge is not orthogonal to its supports. Skew

bridges are to be considered under a different category from the normal orthogonal bridges

because of the complicated behavior of such bridges. The analysis, design and detailing has to

be separately dealt for skew cases. Because of these apprehensions it is always tried to avoid

skew spans but at times they are necessitated due to site considerations such as alignment

constraints, land acquisition problems, etc. Hence skew bridges are pretty common throughout

United States and around the globe and a lot….

Characteristics of Skew Deck

The some of the characteristic differences in behavioral aspects of a skew deck when compared

to right deck are as follows:

High reactions at the obtuse corners.

Possible uplift at acute corners, especially at very high skew angles.

High shear and torsion on obtuse corners.

Sagging moments are orthogonal to the abutments in the central region.

(More to be added)

Theory for analyzing and designing skew slabs

The behavior of skew slabs is found to be not just dependent on the skew angle but other

factors such as the span and the width of the slab. The introduction of aspect ratio- ratio of

skew span to right width, accounts for the effect of span length and width in skew behavior.

Combining the two parameters, namely skew angle and aspect ratio, skew behavior can be

assessed with a single parameter namely, S/L ratio, where S is the span length if the skew

corners are not considered, See Fig. 1. (below).

The bridges considered in the project have a skew span of 40’ and right width of 30’. There are

in total 10 skew slab bridges designed for skew angles going from 20 degrees to 35 degrees. The

S/L ratio for these bridges varies from 0.73 to 0.47 as the skew angle goes from 20 degrees to

35 degrees respectively. From figure 1 (below) it can be seen that for S/L ratio around 0.71 the

skew effect is predominant on edges and as it goes to below 0.45 the skew effect is expected

to spread all over the span of the slab. So, all the bridges considered in the project have skew

effect to some extent and for 30 and 35 degrees the skew effect is all over the slab.

The next important thing to look at it is the trajectories of principal stresses in the skew slabs.

The figure 2 gives a rough idea of these trajectories for varying S/L ratios. The figure tells that

when the skew is predominant the principal stresses run from obtuse corner to obtuse corner

rather than running along the skew length. This has certain implications on how to model the

slab bridges for analysis which is covered in the next section.

Analysis Methodology- Grillage

The skew slab bridges have very different behavior than right slab bridges and it is vital to pay

great attention to the analysis methodology used for such slabs. It is generally a good idea to

use modeling software to capture the behavior of these slabs as accurately as possible. The

grillage to be used greatly influence the true representation of principal stresses. Generally the

grillage should be chosen in the direction of principal trajectories to best represent the

behavior of skew slab.

Fig. 3. Grillages for skew decks: (a) skew mesh (b) mesh orthogonal to span and (c) mesh

orthogonal to support.

The grillage in figure 3 (b) works best for higher skew angles and L/B ratio greater than 1 and

the grillage in figure 3(c) works best for higher skew angles but L/B ratio less than 1. The grillage

shown in figure 3 (a) is the easiest one to construct and analyze but it works good only for small

skew angles as at higher skew the principal stresses does not run along the grillage elements

defined.

(Put in figure from the book) figure 3

Problem Statement

1. To figure out a critical skew angle beyond which it becomes unavoidable to use

orthogonal reinforcement layout in preference over skew lay out.

2. To study the contribution of well integrated barrier in reducing the principal stresses in

the slab.

MODEL

The 10 skew bridge cases were modeled using Finite Element Software SAP 2000’s new bridge

wizard. The bridges were designed using nodes, lines, and 3D meshes to design the concrete

slab. The slab was segmented into 12”x12” squares for design convenience. The slab was

modeled as a simply supported slab with pins at the left and rollers at the right ends of the

bridge.

Geometry

Fig. 4. General

All the 10 bridges considered in the project have same skew span length, right width, and depth

and it is just the skew angle which is varying from 20 degrees to 35 degrees. The dimensions of

the slab bridge are given below.

Skew Span Length, L = 40 feet

Right Width, B = 30 feet

Slab Thickness, T = 2 feet

The barrier deployed for the study is Texas Constant Slope Barrier and the dimensions of the

barrier are given below in the figure 5.

Fig. 5. Texas Constant Slope Barrier dimensions.

Grillage Used

The grillage orthogonal to span- as shown in figure 3 (b), is deployed for all the skew cases as

L/B ratio is greater than 1. A uniform grillage irrespective of skew angle gives uniformity in

analysis and is important from point of view of a parametric study where everything other than

the parameter studied must be kept uniform. The grillage is one way supported and it does not

have stiffened free edges so the load distribution for the grillage is expected to follow a

behavior as per the figure 6 below.

Fig. 6. Qualitative load distribution diagram.

Load Application

The loads considered for the analysis and design of Slab Bridge are as follows:

Dead Load

The load was not applied externally to the grillage rather the geometry of the slab was

defined in the model and the unit weight of concrete was fed in as 0.150 ksi.

Dead Load Barrier

The weight of the barrier was calculated manually using the dimensions of the Texas

Constant Slope Barrier and the unit weight of concrete as 0.150 ksi. The calculated load

came out to be 0.636 kips per linear foot for a 2 feet wide barrier. Consequently an area

load of 0.363 kips/ft2 was applied over a 2 feet strip on the edges.

Live Load

The live load was taken into account by applying HL 93 loading, which consists of the

two cases mentioned below.

- Lane Load and Design Truck Load

- Lane Load and Design Tandem Load

The numbers of design lanes to be considered are given by the equation below.

NL = S/12

= 26/12 ≈ 2 lanes

Where,

NL = Number of design lanes

S = Clear roadway width excluding the barriers

The lane load is a 0.64 kip/ft2 area load applicable for a design lane of 12 feet. The design

truck is HS 20 and design tandem consists of two axles of 25 kips each separated by a

distance of 4 feet.

The model runs the HL 93 loading along the span for one lane and two lanes loaded and try to

capture the maximum forces and moments for the governing case. Apart from the barrier load

all the other loads are applied internally by the modeler itself.

Multiplication Factors

The dead load for the slab is

Critical Points of Analysis

A set of points were considered on the slab to represent the positions of design bending

moments in a skew deck (Rajagopalan N. 2006). These points have been determined by Hubert

Rusch based on an experimental study and they represent various types of critical bending

moments. The figure 7 below gives a visual idea of where the points lie on the slab and the

table 1 is to be used in conjunction with figure 7 to decide for what all points to be considered

for a particular skew angle.

Fig. 7. Position for design bending moments in a skew deck

Table 1. Distance lc along skew span from obtuse corner for various skew angles and aspect

ratios.

The chosen critical points are supposed to cover up the different regions on the slab which

show a change in behavior of the slab due to redistribution of stresses at different boundary

conditions. The qualitative diagram of the redistribution of loads is given in the figure 8 below.

Fig. 8. Qualitative load distribution diagram

It can be seen from Fig. 8. that the principal stresses try to run from obtuse corner to obtuse

corner and the triangular skew regions left out on acute corners outside the right region try to

redistribute their loads onto the nearest boundary condition which can be the abutment or the

right span in the middle depending upon the location of the load point.

The dead load of the acute triangular regions and any live load coming onto it causes a high

torsion on the obtuse corners and this effect gets severe with increasing skew angle. For

example, the moments at point E, which lies in the obtuse corner region, are expected to

capture the design moments for torsion.

The point C is taken into account only for high skew cases of 30 and 35 degrees. From table 1

the value of lc comes out to be 0.26 l for 30 degrees skew for b/ l ratio of 0.6 and it can be

taken as 0.3 l for 35 degrees skew by interpolation between its values for 30 and 45 degrees.

The moments were obtained for all the critical points for the skew cases of 20, 25, 30 and 35

degrees for strength limit state I and it was observed that only A, B and E were the governing

points in all the cases, which conforms to the reference points A, B, C and E for design moments

as per table 1 for b/ l ratio of 0.6. Eventually the points A, B and E were considered for design

bending moments for strength limit state I and service limit state I for all the bridges.

Calculation of Principal Moments

The SAP results did not give the principal design moments directly for the combined load

effects. It rather gives separate values of M11, M22 and M12 and that too the absolute

maximum or minimum at any point rather than giving all the values for a given load case. This

way it is not possible to find the resultant principal moments using the individual values of M11,

M22 and M12. Moreover, the model did not reveal the angle of these moment vectors for the

combined load cases. An assumption was made to overcome this problem by suggesting that

the angle poised by the principal moments for the combined load case is going to be the same

as for the case of just the dead loads. Using this assumption the local axis of the area elements

is rotated so as to align with the direction of the principal moments for the dead load. Now the

M11 and M22 represents the principal longitudinal and transverse moments and the torsion or

the M12 values drops drastically as expected and can be neglected for design as it has already

been taken into account be the increased M11 and M22 moments.

A uniform distributed load of .636 kips was placed along the free sides of the bridge to model

the Barrier. The FEA dead and moving load analyses were used to determine the required

design moments used for the design of the steel reinforcement. As the skew is introduced into

the slab, there is a distribution of stresses in the center area between the two obtuse angles in

the slab, also torsional moments are induced at the free edges of the slab due to loading on

skew spans. See figure(………) (Rajagoplan 2006) Due to this abnormal distribution of stresses

and loading; SAP 2000 was key to getting our design moments. The calculation of these

moments by hand may not be very accurate and could lead to an over estimation if not done

with a finite element software.

According to the text “Bridge Superstructures” by N.Rajagoplan based on experimental study

critical points were determined …..