Midspan

A polyester matrix is reinforced with continuous glass fibres. A 15 mm wide beam made from this material is to be simply supported over a 300 mm length and have a point load at midspan. For a fixed beam weight of 90 g/m investigate how the stiffness of the beam changes with the volume fraction of glass and state the optimum volume fraction, (p/ = 2560 kg/m. p , = 1210 kg/m Ef = 76 GN/m = 3 GN/m ). 

Uniformly distributed Both simply supported Midspan 5/384 1/8... 

Concentrated at midspan Both simply supported Midspan 1/48 1/4... 

Thickness variations from the intrados to the extrados and along the length of the bend shall be gradual. The thickness requirements apply at the midspan of the bend, y/2, at the intrados, extrados, and bend centerline radius. The minimum thickness at the end tangents shall not be less than the requirements of para. IP-3.2 for straight pipe (see Fig. IP-3.3.1). 

M is the ultimate moment capacity at midspan M psis the ultimate moment capacity at support. 

Because the roof diaphragm is a deep and relatively short beam, the stiffness must include shear deformations. Compute the total midspan deflection for an arbitrary load of 1,000 Ib/in. 

Flexibility (% at midspan) 3-5 Expected to meet Expected to meet ... 

FIG. 25.3 Simple beam loaded at (A) midspan (B) one-quarter points (C) cantilever beam loaded at free end (reproduced from Schmitz and Brown (1965-1969), Vol. 2, p. 323). 

The building code requires that the maximum deflection of deck boards used as stair treads under concentrated load of 300 lb placed at midspan shall be 1/8 in. (3.2 nun) or l/180th of the span (AC 174, Section 4.1.1 2000 International Building Code, Section 1607.1). For 16-in. span, the allowed deflection is either 0.125 in. or 16"/180 =... 

ASTM D 790-03, which we will consider below, refers to maximum surface stress, whereas the earlier version, ASTM D 790-97, refers to maximum fiber stress and says that the maximum stress in the outer fibers occurs at the midspan. Because maximum fiber stress is often used in strain and stress literature, the term outer fibers needs an explanation. It has nothing to do with the actual fiber in the material. The term outer fibers refers to the material near the specimen snrfaces, where the maximum strains occur when the specimen is loaded at, say, midspan, as described in ASTM 790. 

The most severe stress for the given load is at 3-pt loading. Maximum moment at center (at midspan) is... 

The ASTM procedure describes procedures for determining flexural properties of materials in the form of rectangular bars. The ASTM emphasizes that it is applicable to only those materials that break or fail in the outer surface within the 5% strain limit at midspan. The method employs a 3-pt loading system (concentrated load) applied at midspan of a simply supported beam (Fig. 7.4). 

The stress in the outer surface occurring at the midspan is calculated using the formula (see eq. (7.11) above)... 

The stress in the onter snrface occnrring at the midspan is calcnlated nsing the formula (see Eq. (7.12))... 

The same as in ASTM D 790, the specimen should fail (ruptured) within 5% maximum outer surface strain. As it was described above (see ASTM D 6109), the maximum strain (at midspan) in the outer surface at 1/3 span loading is equal to (see Eq. (7.24))... 

Eirst, let us consider how typical composite deck boards would satisfy the ASTM D 790 criterion on the 5% strain limit at midspan in the outer surface of boards (Table 7.2). 

TABLE 7.2 Limiting values for deflection at midspan for 3-pt load for GeoDeck boards (1.25-in. depth) ... 

TABLE 7.10 Limiting values for deflection at midspan for 4-pt load (third-point load) for commercial composite deck board and actual data... 

Fig. 7.6) is about 27% lower compared to the strain for center-point load. Hence, third-point load is closer to a uniformly distribnted load than a load at midspan. Tables 7.9 and 7.10 show the actual values of maximum strain in the outer surface (i.e., strain at break). Despite those differences, maximum strain in the outer surface (i.e., strain at break) was in all the cases well below 3%. 

A minimum concentrated load of 750 lb (apphed over a 2 X 2 in. area at midspan),... 

A minimum concentrated load of 300 lb (average of 15 specimens) at midspan at the deflection at l/180th of the test span,... 

The maximum deflection of 0.125" at 300-lb -concentrated load at midspan, plus adjustments for end use as stated in Section 5.4 of ASTM D 7032 (Creep-Recovery Test). 

Comparison of Eq. (7.47) with Eq. (7.46) shows that at the same support span and the load, the same profile (with the same moment of inertia) should deflect by 18% less under third (4-pt)-point load compared to concentrated, center-point load at midspan. 

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