A grillage member in bending behaves according to the well-known flexure formula:
(5.55)
where M is the moment, I the second moment of area, E the modulus of elasticity and R the radius of curvature. By substituting the curvature 1/R with κand rearranging, the moment per unit breadth, m is found:
(5.56)
Equation (5.34) gives an expression for the moment per unit breadth in the X direction in the slab. For an isotropic slab, there is only one value for E and ν. Substituting E for Exand ν
for νxand νyin that equation gives:
As Poisson’s ratio, v, is relatively small in bridge slabs (approximately 0.2 for concrete), it is common practice to ignore the second term in this equation, giving:
A further simplification is made by equating the term below the line to unity. This can be justified by the fact that Poisson’s ratio is small. Further, if this approximation is applied to both mxand my, they are both affected by the same amount. As it is the relative values of
stiffness that affect the calculated bending moments and shear forces, such an adjustment has very little effect on the final results. The moment/curvature relationship then becomes:
(5.57)
To achieve similitude of moments between a slab and the corresponding grillage, the stiffness terms of equations (5.56) and (5.57) must be equated. This can clearly be achieved by
adopting the same elastic modulus and second moment of area per unit breadth in the grillage as that of the slab.
A grillage member in torsion behaves according to the well known equation:
(5.58)
where is the angle of twist, T is the torque, l is the length of the beam, G is the shear modulus and J is the torsion constant (St. Venant constant). Figure 5.15shows a portion of a beam of length δx in torsion. The displacement in the Z direction is given by w and the angle of twist over the length δx is given by:
Hence:
Fig. 5.15 Segment of beam subjected to torsion
Substituting equation (5.18) into equation (5.59) gives:
(5.60)
Applying equation (5.58) to the beam of Fig. 5.15gives:
(5.61)
Substituting equation (5.60) into equation (5.61) gives:
This can be rewritten in terms of torque per unit breadth, t:
(5.63)
where jgrilis the torsion constant per unit breadth in the grillage member.
Equation (5.42) gives an expression for the twisting moment per unit breadth in the bridge slab:
(5.64)
To achieve similitude of moments, mxy, in the slab and torques, t, in the grillage members, the
stiffness terms of equations (5.63) and (5.64) must be equated. This can clearly be achieved by adopting the same shear modulus and torsion constant in the grillage member as is in the slab.
Equation (5.43) gives an expression for the torsion constant of the slab. Equating this to jgril gives:
(5.65)
where d is the slab depth. Equation (5.65) ensures that the grillage members in both directions will have the same torsional constant per unit breadth. However, they will not necessarily have the same total torsional constant as they may represent different breadths of slab if the grillage member spacing in the longitudinal and transverse directions differ. The torsion constant for the grillage member can alternatively be expressed in terms of the slab second moment of area:
(5.66)
Although equations (5.65) and (5.66) are based on the grillage member having the same shear modulus as the slab, it will not generally be necessary to specify Gxyfor the grillage model.
The behaviour of a grillage member is essentially one dimensional and consequently its shear modulus can be derived from the elastic modulus and Poisson’s ratio directly using the well- known relationship:
(5.67)
Typically, this is carried out automatically by the grillage program.
The preceding derivation of grillage member torsional properties is applicable to thin plates of rectangular cross-section where equation (5.43) for the torsional constant is valid. Torsion in beams is complicated by torsional warping (in all but circular sections) and formulas have been developed to determine an equivalent torsional constant for non-rectangular sections such that equation (5.58) can be applied.
For rectangular beams with depth d and a breadth of greater than 10d, the torsional constant may be approximated with:
(5.68)
It can be seen that equation (5.68) predicts a torsion constant for the beam which is twice that predicted by equation (5.66) for isotropic slabs. The reason for this lies in the definition of torsion in a beam and of moment mxyin a slab.Figure 5.16shows a portion of a beam of
breadth b and depth d in torsion. The shear stresses set up in the beam are shown, in both the horizontal and vertical directions. The torque in the beam results from both of these shear stresses and is given by:
(5.69)
In the slab, equation (5.36) shows that the moment mxyis arrived at by summing only the
shear stresses in the horizontal direction (i.e. only). Consequently the torsion constant for a grillage member representing a portion of an isotropic slab is only half that of a regular beam (or a grillage member representing a regular beam). In the slab, the shear stresses in the vertical direction are accounted for by the shear force per unit breadth, qxas illustrated in
Fig. 5.17. It has been recommended that the edge grillage members be placed at 0.3 times the slab depth from the edge so as to coincide with the resultant of the shear stresses. The vertical shear stresses are accounted for in the grillage in the same manner by the shear forces qyin
the transverse beams.
5.3.3 Grillage member properties—geometrically orthotropic slabs