## Reinforced Foamed Plastic

Frequently cases arise in which ribs are used to reinforce plastic (RP) plates such as in tanks, boat hulls, bridges, floors, towers, buildings, and so on. The design of ribbed plates such as these is somewhat analogous to reinforced concrete and T-beam design. In view (a) of Fig. 4.8, for example, a construction is shown consisting of a plate composed of balanced fabric RP 0.15 in. thick and mat RP 0.05 in. thick, combined with a rib, making a structure whose overall depth is 1.500 in. The rib is formed of a cellular material such as foamed plasdc, plus a cluster of resin-bonded parallel fibers such as roving, at the bottom. The mat is carried around the rib and serves to de the rib and plate together.

The construction of a plate and rib form a T-beam. The principal design problem is to determine how much of the plate can be considered to be acting as a flange with the rib, that is, the magnitude of b'm Fig. 4.8 (a). For purposes of illustration, b is taken as 5 in. If the T-beam is loaded in bending so as to induce compression at the top and tension at the bottom, the balanced fabric and the mat will be in compression at the top, and the roving and mat will be in tension at the bottom. Because the roving is much stronger than the mat, it is evident that the mat adjacent to it will break before the roving reaches its maximum stress. That is if the roving were stressed 50,000 psi the adjacent mat would be stressed 10,000 psi, which is double its strength. Consequendy, in finding the neutral axis and compudng the strength of the cross-secdon, the mat is neglected on the tension side. Above the neutral axis the mat is in compression, but in order to simplify the computadons, only the mat in the flange of the T is considered.

The foamed plasdc has such a low modulus of elasticity and such low strength that it contributes little to either the stiffness or the bending strength of the T-beam. It must, however, be stiff enough to prevent buckling or wrinkling of the mat or the roving.

The acdve elements of the T-beam are therefore as shown in Fig. 4.8 (b). The flange consists of balanced fabric 5.000 in. wide and 0.150 in. thick plus mat 4.200 in. wide and 0.050 in. thick. The web consists of the bundle of roving 0.800 in. wide and 0.200 in. thick. By the application of equation 4.32, the neutral axis is found to be 1.105 in. from the bottom, or 1.055 in. above the lower edge of the roving.

The basic assumptions discussed in the introduction imply that when this beam is bent, strains at any point in both tension and compression are proportional to the distance from the neutral axis, and that stress is equal to strain multiplied by modulus of elasticity. As an example, the stress in the lower-most roving fiber is 50,000 psi, the stress in the topmost fiber of the flange is 50,000 (3 x 106/5) (0.395/1.055) or 11,250 psi. Similarly, the stresses in the upper edge of the bundle of rovings, at the lower edge of the balanced fabric, and at the upper and lower edges of the balanced fabric, and at the upper and lower edges of the mat in the flange are as shown in Fig. 4.8 (c). These are all less than the corresponding values of <j], o2, and o3 listed.

The internal resisting moment, or resistance to outside bending forces, can be found by computing the total resultant compression Q in the balanced fabric, total resultant compression C2 in the mat, finding the distances ot] and a2 between the lines of action of these two resultants and the line of action of the total resultant tension T in the roving, computing the values Qc*! and C2a2, and adding.

Resultant Q acts at the centroid of the trapezoidal stress area 1 a, resultant C2 at the centroid of area 2a, and resultant T at the centroid of area 3a. Solving for these centroids, the distance CC\ is found to be 1.285 in. and distance a2 is 1.179 in. These are the internal moment arms of the two resultant compressive forces C\ and C2.

C, = [(11250 + 7000J/2][5.000 x 0.150] = 6,840 lb (4-3)

Cross-section of a rib applied to a plate o.ir

balanced fabfic Mol o.ir

balanced fabfic Mol

0.03

distribution

£j = I « 10* pti A3 = 0.2 » 6.8 = 0.16 in.1 E 3 = 5 » 10* pw

0.03

distribution

£j = I « 10* pti A3 = 0.2 » 6.8 = 0.16 in.1 E 3 = 5 » 10* pw

A check on the accuracy of the computations is afforded by the fact that Tmust equal C.

T= [(40,600 + 50,000)/2][0.800 x 0.200] = 7,250 lb (4-4)

The initial resisting moment is C\CC\ + C2(X2

8790 in-lb 520

Evidently, if the mat in the flange had been left out of the computation (C2CX2), the error in the calculated result would have been approximately 5%. Omitting the mat in the rib between the flange and the neutral axis affected the result much less.

If a shear force is imposed on the rib, two critical planes of internal shear stress occur, one at the neutral axis and one at the plane between the mat and the fabric in the flange [Fig. 4.8 (a)]. Shear stresses are computed by equation,

For example, suppose the T-beam is 30 in. long (L) and carries a uniformly distributed load W. Then M = wL/8 =9310 in-lb and W= 2500 lb. The maximum shear Fis half the total load or 1250 lb.

At the neutral axis the stadsdcal moment Q is the weighted moment of the flange or of the roving about the neutral axis; El is the sdffness factor of flange plus roving; and b is the total thickness of the mat at the neutral axis plus the effective thickness of the cellular filler. This effective thickness may be computed in accordance with the principles set forth on combined action. If, for example, the shear modulus of the cellular core is 1/15 the shear modulus of the mat-reinforced material, the effective width of the core is 0.8/15 or 0.053 in. The total effective width of the mat and the core at the neutral axis is therefore 0.05 + 0.05 + 0.053 = 0. 153 in.

The computed value of Q' is 0.725 x 106 lb-in, and the value of El is 0.968 x 106 psi. The shear stress intensity in the mat at the neutral axis is:

1250x0.725

= 6100 psi

The shear stress intensity in the cellular core is

If either r„, or r,. is excessive it is necessary to increase the rib thickness at the neutral axis, probably by increasing the thickness of the mat.

Properties of the cellular core may not be known well enough or may be too low to warrant inclusion with the mat in calculating shear. If the core is not included, the thickness at the neutral axis is that of the mat alone, or 0.10 in. The shear stress r„, then becomes 9400 psi instead of 6100 psi.

At the interface between mat and fabric in the flange of the T-beam, the value of Q' is that of the fabric alone. This is found to be 0.720 x 106 psi. The width b is 4.2 in. (neglecting the width of the cellular core). Therefore:

1250x0.720 4.2 x 0.968

220 psi

In all probability the shear stress intensity is actually higher adjacent to the rib, and lower near the outer ends of the flange, but in any event it is not likely to be excessive.

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