Page 163 - Bridge and Highway Structure Rehabilitation and Repair

138 SECTION 2 STRENGTHENING AND REPAIR WORK

2
Since (dw/dx) is small, (dw/dx) is negligible
2
2
I/R 3 (d w/dx ) (4.3b)
as given by Newton’s law of curvature
Slope 3 EI dw /dx

where w is the vertical deflection due to bending
2
Bending Moment M 3 EI d w/dx 2 (4.4a)
3
Shear Force V 3 dM/dx 3 EI d w/dx 3 (4.4b)
Load intensity q 3 dV/dx 3 EI d w/dx 4 (4.4c)
4
Beam torsion: in a similar format to equation 4.2, the general torsion equation is given by:
/r 3 C 0/L 3 T/J (4.5)
where 3 shear stress, r 3 radius of a point on x-section of beam, C 3 shear modulus,
0 3 angle of twist w.r.t. center line of x-sec., L 3 span, T 3 torsional moment.
This equation expresses shear stress resulting from a torsional moment to torsional rigidity
and twist of sections. Generally, torsion is accompanied by bending moment due to self weight.

They simultaneously cause deflection, twist, and strain in a member.
Analysis used as basis of design: For equilibrium, compressive force C 3 tensile force T.
The distance between the point of application of the two opposite forces (commonly known
as lever arm) when applied by compressive or tensile force will be equal to applied moment.
Bending moment Bending stress : Allowable material bending stress
Shear force Shear stress : Allowable material shear stress
Reaction Bearing stress : Allowable material bearing stress
Compressive force Compressive stress : Allowable material compressive stress
Buckling Compressive stress 9 Allowable material compressive stress
Tensile force Tensile stress : Allowable material tensile stress
Axial force Axial stress and buckling : Allowable axial and buckling stress.
Torsion Shear stress : Allowable shear stress.
Allowable stress for a given material is usually obtained from laboratory tests on
specimens. Stresses acting in the same plane are combined.
Physical parameters: Analytical results are used for selecting member sizes:

1. Stress criteria based on the magnitude of flexure and shear forces.
2. Deformations based on serviceability and limiting live load defl ections.
Every physical parameter should be included in any analysis, namely:
1. Geometry and curvature.
2. Deck width, number of lanes, and overhang.
3. Girder spacing.
4. Materials—reinforced concrete, steel, aluminum, prestressed concrete, timber, or masonry.
5. Plan aspect ratio of deck slab and thickness.
6. Skew angle.

4.4 ANALYSIS OF SLAB BEAM BRIDGES
4.4.1 Analytical Approach to Composite Bridge Decks
1. Drastic changes in distribution coefficients: The two theories based on idealization of com-

posite deck and girder action are:
• The simplified beam theory, using transverse distribution factor—The bridge deck is

idealized as a series of T-beams (line girders composite with deck slab) in the longitudinal