EXAMPLE 3.1.1. SOLID SHAFT

1. Determine the torque, T, that causes a maximum shearing stress of 100 MPa in the steel cylindrical shaft shown.

2. For the cylindrical shaft shown, determine the maximum shearing stress caused by a torque of magnitude T=1kN-m.

1. Required: Torque, T:

The general formula to be used is as follows, where all are given except the torque, T, and the polar moment of inertia, J so compute for J first.
                    

For polar moment of inertia, use the solid shaft:


                          where c will be taken as the radius to have the maximum shearing stress.

So,


                  

Thus,

               

             

2. Required: Shearing stress:

              

           
             
         
                       

LOADING ANALYSIS FOR A REINFORCED CONCRETE DECK SLAB- EXAMPLE

Determine the positive moment to be used in the design of the deck slab of an ordinary simple span bridge as shown. 

Materials used are the following:
Slab:                   Normal weight concrete, f'c = 35 MPa
Wearing surface:   Asphalt concrete
Thickness of future wearing surface:         60mm
Thickness of top integral wearing surface: 12mm

Given as an ordinary bridge, the load modifier, = 1.0. 

Assuming that we are going to use #36 bars:

     Top cover           = 40mm

     Bottom cover      = 25mm

Using Normal weight concrete (f'c = 35 MPa), 

                                              ' or taking the unit weight,

                                        

Future wearing surface uses Asphalt concrete:
                                    or using the unit weight: 

                                               


So let's start computing for DW and DC. In order to solve for the dead loads, the dimensions of the bridge should be identified first.


1. For the overhang width, using 50% of S:

                  

2. For the thickness of the slab (overall depth) and overhang: (simple span)

Bridge Design and Evaluation LRFD and LRFR

So considering the design depth for the concrete slab:

Design depth =  overall depth - sacrifical surface= 250mm -12mm=238mm

With bridge dimensions complete, we can compute for DW and DC (using 1m strip).

1. Solve for DW:

It should be noted that since the future wearing surface is much thicker than the integrated thickness, DW should account for the FWS.

                                


                               

2. Solve for DC of parapet:

                             


                            

So, computing for DC of parapet:


                              

                             

3. Solve for DC of slab. There are different parameters which should be considered for the DW of slab such as:

• Weight of the slab
• Miscellaneous (railings, lighting, etc) 
• Stay in place formworks

a. For DC of slab:

                           

                          
                       
                         

b. For miscellaneous DC, let's take 5% of the slab weight:

                                              
                     

c. For Stay-in-place formworks, the problem has not identified girder specifications.


So the total DC for the slab:


                   



Now that you already have determine the dead loads, you can convert such loads to moment using any method you are familiar with for indeterminate structures (moment distribution, three moment equation, slope deflection method)

For the uniformly distributed load:

For the parapet DC:

By using moment distribution, the resulting moment for DC total:

The moment for DW:

The moment for the parapet: (Note that at the location of max. moment for the distributed loads, the moment shown for the parapet = 0.92 kN/m.

The moment due to live load is computed from the table:

By interpolation, compute for the positive moment of S=2650.

Moment due to LL = 27670 N

Finally, we can compute for the positive moment of the bridge from the formula:

COMPUTATION OF LOADS FOR REINFORCED CONCRETE DECK SLABS

The design of the superstructure includes the deck system. Gongkang Fu shows a flowchart on the procedure of designing the deck system as follows:

Bridge Design and Evaluation LRFD and LRFR

Specifications included in the flowchart are articles from AASHTO Bridge.

Similar to how the design of any structure starts, the design loads should be identified to determine the dimensions required.

PRIMARY REINFORCEMENT REQUIREMENTS

The traditional design method involves the different steps in the design of the primary deck reinforcement:

1. Obtain design criteria.
2. Determine the minimum slab and overhang thicknesses.
3. Select slab and overhang thickness.
4. Compute dead load and live load effects.
5. Compute factored positive and negative design moments.
6. Design for positive flexure in deck.
7. Check for positive flexure cracking under service limit state.
8. Design for negative flexure in deck.
9. Check for negative flexure cracking under service limit state.
10. Design for flexure in deck overhang.
11. Check for cracking in overhang under service limit state.
12. Compute overhang cut-off length requirement.
13. Compute overhang development length.
14. Design bottom longitudinal distribution reinforcement.
15. Design top longitudinal distribution reinforcement.
16. Design longitudinal reinforcement over piers.
17. Draw schematic of final concrete deck design.

The basic design criteria include:

• Girder spacing
• Number of girders
• Deck top cover
• Deck bottom cover
• Deck unit weight
• Deck concrete strength, f'c
• Reinforcement strength, fy
• Future wearing surface thickness
• Future wearing surface unit weight

LOAD ANALYSIS FOR DESIGN

The main goal for the load analysis is to come up with the moments needed to design the reinforcement. The formula for the moment is as follows:




The   are taken as load factors which can be determined from the AASHTO tables for load factors of permanent loads and all other loads.

AASHTO 2007 Edition SI units

AASHTO 2007 Edition SI units

The moments result from using any method to determine moments for indeterminate structures such as:

• Moment distribution
• Three-moment equation
• Slope deflection method, etc.

SOLVING FOR THE DEAD LOAD

The DL can be solved simply by using the equation:

Dead load = unit weight (cross-sectional area)

For unit weight of materials, AASHTO provides the following table:

AASHTO 2007 Edition SI units

 

DIMENSIONING THE BRIDGE

A bridges length and width will be determined through site inspection. The length of which can be measured from end to end of the valley or water form which requires a passage. The bridge's width adopts the road width.

The number of girders will be upon the bridge designer's discretion; guided by the conditions that:

• The minimum number of girders for a bridge is 4
• 5 girders are efficient to carry loads all throughout the bridge

THE OVERHANG WIDTH

As a rule of thumb, a bridge's overhang is taken within 30% to 50% of the girder spacing. 

SLAB AND OVERHANG THICKNESS

The minimum depth and cover as stated in the AASHTO is 175mm (which excludes a sacrificial wearing surface). The thickness will have to account for the:

•  wearing surface thickness, 
• the cover, and 
• the main bar diameter.    

Table 5.12.3-1 lists the cover for unprotected main reinforcing steel.

AASHTO 2007 Edition SI

The slab thickness can be determined from the table:

AASHTO 2007 Edition SI

HEIGHT OF TRAFFIC PARAPET OR RAILING (13.7.3.2.)

The minimum height for a concrete parapet with a vertical face shall be 685mm. The typical height considered in most designs is 1000mm.

SOLVING FOR THE LIVE LOAD

Most bridge designers compute for the live load effects using the tenth rule whereby the moving loads (and influence lines) would have to be analyzed in 10 divisions.  

Table A4-1 from AASHTO 2007 lists maximum live load moments.

AASHTO 2007 Edition SI

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