Substituting equations (5.19)–(5.21) into equations (5.23)–(5.25) respectively gives expressions for the stresses in terms of curvature:
(5.26)
(5.27)
(5.28)
5.2.4 Moments in materially orthotropic thin plates
Figure 5.7 shows a small cube taken from a thin plate with the associated normal stresses σx, σy and σzand shear stresses. It is well established that, to satisfy
equilibrium, pairs of shear stresses must be equal as follows:
(5.29)
Considering the normal stresses first, Fig. 5.8(a)shows a vertical line of cubes (such as that of
Fig. 5.7) through the depth of the plate in the X−Z plane. Each of these cubes is subjected to a normal stress in the X direction as indicated in the figure. When there are no in-plane forces in a bridge deck, the sum of the forces in these cubes is zero. As each cube is of the same surface area, it follows that:
However, there is a bending moment caused by these stresses. The term mxis used to
represent the moment per unit breadth due to the σx stresses, summed through the depth of the
deck. Figure 5.8(b)shows the depths of the cubes δz and their distances from the origin, z1, z2,
z3, etc. Each cube has a width perpendicular to the page of δy (not shown in the figure). The
forces F1, F2, F3, etc., due to each of the stresses are also shown. The ith cube contributes a
component of hogging bending moment of magnitude (σxiδzδy)zi. Taking sagging moment as
positive and summing over the depth of the plate gives:
(5.30)
Substituting equation (5.26) into equation (5.30) gives:
which gives:
Fig. 5.8 Vertical line of elemental cubes through the depth of a plate: (a) stresses on each cube; (b) forces on the cubes and distances from the origin
Applying a similar method it can be shown that the stress σy causes a moment per unit breadth mywhich is given by:
(5.32)
The second moment of area per unit breadth of the plate, i is defined by:
(5.33)
Therefore equations (5.31) and (5.32) can be rewritten in terms of the second moment of area as follows:
(5.34)
(5.35) It is important to remember that mxis the moment per unit breadth on a face perpendicular to
the X axis and not about the X axis, i.e. in a reinforced concrete deck it is the moment which would be resisted by reinforcement parallel to the X axis. Likewise, myis the moment per unit
breadth on a face perpendicular to the Y axis.
Referring to Fig. 5.7, it can be seen that the shear stresses result in forces parallel to the Y axis which will also cause a moment. The moment per unit breadth due to is termed mxy.
Figure 5.9 shows a number of cubes through the depth of the plate in the Y−Z plane. The shear force on the face of each cube is given by:
and the moment per unit breadth due to this force is given by:
Taking anti-clockwise as positive on the +X face, the total moment per unit breadth due to is given by:
(5.36)
Fig. 5.9 Stack of elemental cubes in the Y−Z plane showing shear stresses
which gives:
(5.37)
Similarly the moment per unit length, myx, caused by (on the Y face) can be shown to be:
(5.38)
(5.39) However, as indicated in equation (5.29), equilibrium requires and to be equal and comparison of equations (5.36) and (5.38) yields:
(5.40)
It follows from the definition of curvature (equation (5.18)) that the two twisting curvatures are the same:
(5.41)
so there is no contradiction between equations (5.37) and (5.39). These equations can be rewritten as:
Fig. 5.10 Bending and twisting moments in a plate: (a) segment of plate and directions of moments; (b) associated distortions
where j is known as the torsional constant and is given by:
(5.43)
The moment mxy(=myx) is often referred to as a twisting moment and is distinct from the
normal moments mxand my.Figure 5.10(a)shows the direction in which each of these
moments acts while Fig. 5.10(b)shows the type of deformation associated with each of them. 5.2.5 Shear in thin plates
Vertical shear forces occur in bridge decks due to the shear stresses, and
illustrated inFig. 5.7. Unlike beams, there are two shear forces at each point, one for each direction (X and Y). Defining qxand qyas the downward shear forces per unit breadth on the
positive X and Y faces respectively then gives:
(5.44)
and:
(5.45)
It was assumed earlier (equations (5.8) and (5.9)) that shear deformations in the plate were negligible. This is a reasonable assumption as shear deformation is generally small in bridge slabs relative to bending deformation. However, shear stresses, while numerically small, can be significant, particularly in concrete slabs which are quite weak in shear. In the simple flexural theory of beams, the same phenomenon exists and an expression is found from equilibrium of forces on a segment.Figure 5.11shows a segment of a beam of length dx in bending. The moment and shear force at the left end are M and Q respectively and at the right end are M+dM and Q+dQ respectively. Taking moments about the left hand end gives:
Rearranging and ignoring the term, dQdx which is relatively small, gives an expression for the shear force Q:
(5.46)
i.e. the shear force is the derivative of the moment. In thin-plate theory, a similar expression can be derived.
Fig. 5.11 Equilibrium of small segment of beam
A small element from the plate of base dimensions dx×dy is shown inFig. 5.12, with varying bending moment and shear force. The terms qxand qyrefer to shear forces per unit breadth
while mx, myand mxy refer to moments per unit breadth. This is different from the beam
example above where Q and M referred to total shear force and total moment. Taking moments about the line a–b (Fig. 5.12) gives:
where Fzis the body force acting on the segment of slab (for example, gravity). Dividing
across by dx dy gives:
where fzis the body force per unit area. The second and third terms of this equation represent
very small quantities and can be ignored giving:
(5.47)
By taking moments about the line b–c (Fig. 5.12), an equation for qy can be derived in a
similar manner:
(5.48)
It can be seen that the expressions for the shear forces per unit breadth (equations (5.47) and (5.48)) are of a similar form to that for a beam (equation (5.46)) except for the addition of the last term involving the derivative of mxyor myx.
Fig. 5.12 Equilibrium of small segment of slab