GIRDER DESIGN

Any cross-section of a plate girder is normally subjected to a combination of shear force and bending moment. The primary function of the top and bottom flange plates of the girder is to resist the axial compressive and tensile forces arising from the applied bending moment. The primary function of the web plate is to resist the applied shear force. Under static loading, bending and shear strength requirements will normally govern most plate girder design, with serviceability requirements such as deflection or vibration being less critical.

The first step in the design of plate girder section is to select the value of the web depth. For railway bridges, the girder depth will usually be in the range L^Ro^R/12 to L^Ro^R/8, where L^Ro^R is the length between points of zero moment.

However, for plate girder roadway bridges the range may be extended to approximately L^Ro^R/20 for non-composite plate girders and to L^Ro^R/25 for composite plate girders.

Having selected the web plate depth, the effective flange area to resist the applied moment can be computed from the relation, see Fig. 5.4(b):

M = F^Re^R A^Re^R h^Re^R ………..… (5.1)

Fig. 5.4 Proportioning of Plate Girder Flanges

Flange Stress: According to ECP 2001, girders with laterally supported compression flanges can attain their full elastic strength under load, i.e., F^Rb^R = 0.64*F^Ry^R for compact sections and F^Rb^R = 0.58 * F^Ry^R for non-compact sections. If the compression flange is not supported laterally, then appropriate reduction in the allowable bending stresses shall be applied to account for lateral torsional buckling as set in the Code.

The equivalent flange area A^Re^R is made up of the actual area of one flange, plus the part of the web area that contributes in resisting the applied moment.

The moment resistance M^Rw^R of the web can be defined by; Fig. 5.4 (c):

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M^Rw^R = (0.5 F^Rw^R) (0.5 A^Rw^R) (2h^Rw^R/3) = F^Rw^R h^Rw^R A^Rw^R/6 ………(5.2) where A^Rw^R = area of web and F^Rw^R = maximum bending stress for web. From the above equation it can be seen that one sixth of the total web area can be considered as effective in resisting moment M^R_w^R with lever arm h^R_w^R and stress F^R_w^R. Consequently, the area required for each flange will be:

A^Rf ^R = A^Re^R - A^Rw^R / 6 ... (5.3) Substituting for A^Re^R from Eqn. 5.1 gives:

A^Rf^R = ( M / F^Rb^R d ) - A^Rw^R / 6... (5.4) 5.2.2 OPTIMUM GIRDER DEPTH

An optimum value of the plate girder depth d which results in a minimum weight girder can be obtained as follows:

Express the total girder area as: A^R_g^R = d t^R_w^R + 2 A^R_f^R ... (5.5) The moment resistance of the girder can be expressed as

M = F^R_b^R Z^R_x^R ... (5.6) Where Z^Rx^R is the section modulus of the girder. Substituting from Eqn. 5.6 into Eqn. 5.4 gives:

A^Rf^R = Z^Rx ^R/ d - A^Rw^R / 6 ... (5.7) Substituting from Eqn 5.7 into Eqn. 5.5 gives:

A^Rg^R = 2 Z^Rx ^R/ d + 2 A^Rw^R / 3 = 2 Z^Rx ^R/ d + 2 d t^Rw^R / 3 ... (5.8) By introducing a web slenderness ratio parameter, β = d/t^Rw^R, Eqn 5.8 can be expressed as

A^R_g^R = 2 Z^R_x ^R/ d + 2 d^P2P / 3 β ... (5.9) A^Rg^R is minimum when ∂ A^Rg^R / ∂ d =0 which gives:

d^P3P = 1.5 β Z^Rx^R ... (5.10)

Substituting Z^Rx^R = M / F^Rb^R, Eqn 5.10 gives:

d = ³ 1.5β M / F ... (5.11) The value of β will normally lie in the range 100 to 150. With M expressed in meter-ton units and F in t/cm^P2P units, the above equation gives the optimum girder depth in meters as:

d = (0.25 ~ 0.3)³ M / F ... (5.12) For steel St. 52 with F^Rb^R = 0.58 F^Ry^R this equation gives:

d = (0.2 ~ 0.24)³ M ... (5.13) Design Considerations:

For efficient design it is usual to choose a relatively deep girder, thus minimizing the required area of flanges for a given applied moment. This obviously results in a deep web whose thickness t^Rw^R is chosen equal to the minimum required to carry the applied shear. Such a web may be quite slender, i.e. has a high d/t^Rw^R ratio, and may be subjected to buckling which reduces the section strength. A similar conflict may exist for the flange plate proportions.

The desire to increase weak axis inertia encourages wide, thin flanges, i.e.

flange with a high b/t^Rf^R ratio. Such flanges may also be subjected to local buckling.

Design of plate girders therefore differs from that of rolled sections because the latter generally have thicker web and flange plates and thus are not subjected to buckling effects. In contrast, the freedom afforded in material selection in plate girder design makes buckling a controlling design criterion.

Thus, in designing a plate girder it is necessary to evaluate the buckling resistance of flange plates in compression and of web plates in shear and bending. In most cases various forms of buckling must be taken into account.

Figure 5.5 lists the different buckling problems associated with plate girder design. A brief description of each form is given below:

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Fig. 5.5 Plate Buckling Problems Associated with Plate Girders

a-

b-

c-

d-

e-

f-a) Shear Buckling of the Web Plate

If the web width-to-thickness ratio d/t^Rw^R exceeds a limiting value, the web will buckle in shear before it reaches its full shear capacity. Diagonal buckles, of the type shown in Fig.5.5a, resulting from the diagonal compression associated with the web shear will form. This local buckling reduces the girder shear strength.

b) Lateral Torsional Buckling of girder

If the compression flange is not supported laterally the girder is subjected to lateral torsional buckling which reduces the allowable bending stresses, see Fig. 5.5b.

c) Local Buckling of the Compression Flange

If the compression flange width-to-thickness ratio exceeds a limiting value, it will buckle before it reaches its full compressive strength as shown in Fig.

5.5c. This local buckling will reduce the girder’s load carrying resistance.

d) Compression Buckling of the Web Plate

If the web width-to-thickness ratio d/t^Rw^R exceeds a limiting value, the upper part of the web will buckle due to bending compression as shown in Fig. 5.5d.

Consequently, the moment resistance of the cross section is reduced.

e) Flange Induced Buckling of the Web Plate

If particularly slender webs are used, the compression flange may not receive enough support to prevent it from buckling vertically rather like an isolated strut buckling about its minor axis as shown in Fig. 5.5e. This possibility may be eliminated by placing a suitable limit on d/t^Rw^R.

f) Local Buckling of the Web Plate

Vertical loads may cause buckling of the web in the region directly under the load as shown in Fig. 5.5f. This buckling form is known as web crippling.

The level of loading that may safely be carried before this happens will depend upon the exact way in which the load is transmitted to the web and the web proportions.

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Detailed considerations of these buckling problems will be presented in the following sections.

5.3 INFLUENCE OF BUCKLING ON PLATE GIRDERS DESIGN