This paper proposes a method to optimize reinforcement layout of three-dimensional members under a state of complex stress and multiple load cases (MLCs). To simulate three-dimensional members, the spatial truss-like material model is adopted. Three families of truss-like members along orthotropic directions are embedded continuously in concrete. The optimal reinforcement layout design is obtained by optimizing the member densities and orientations. The optimal design of three-dimensional member is carried out by solving the problem of minimum volume of reinforcing bars with stress constraints. Firstly, the optimized reinforcement layout under each single load case (SLC) is obtained as per the fully stressed criterion. Second, on the basis of the previous results, an equivalent multi-case optimization is proposed by introducing the idea of stiffness envelope. Finally, according to the characteristics of the truss-like material, a closed and symmetrical surface is adopted to fit the maximum directional stiffness under all SLCs. It can be proved that the densities and orientations of truss-like members are the eigenvalues and eigenvectors of the surface coefficient matrix, respectively. Several three-dimensional members are used as examples to demonstrate the capability of the proposed method in finding the best reinforcement layout design of each reinforced concrete (RC) member and to verify its efficiency in application to real design problems.

An unusual 6–8 cm layer of prismatic cartilage and matrix containing some 8,800 teeth, coprolites, incomplete

occipital spines, and denticles of Orthacanthus platypternus (Cope, 1883) occurs in the lower Permian

(Artinskian) Craddock Bonebed in Texas, USA. It is the only species of shark present in the Clear Fork

Group except for three worn Xenacanthus Beyrich, 1848 occipital spine fragments and two teeth of ?Lissodus

(Polyacrodus) zideki (Johnson, 1981) (Hybodontoidei), both being the first occurrences in this unit. Analysis

of measurements of teeth with complete bases randomly selected from 3,050 initially available teeth failed to

reveal the presence of sexual dimorphism or the discrete presence of juveniles as expected, based on an independent

study which identified the presence of Orthacanthus juvenile occipital spines. A few highly symmetrical

small teeth are present, which had not been previously observed in the Texas lower Permian. They may be

symphyseals and restricted only to juveniles. Other unusual teeth include germinal teeth and deformed teeth,

both of which occur in the Clear Fork and underlying Wichita groups. One tooth displays an apparent example

of the equivalent of an “enamel pearl” on one of its cusps. The most unusual teeth are those that appear to have

undergone various stages of resorption. Only the lingual margin of the base is affected in which the apical button

is resorbed to varying degrees until only the labial margin with the basal tubercle and the three cusps are

all that remain. If the teeth were undergoing resorption, then the perplexing problem is why the apical button

is resorbed and not the superjacent basal tubercle. Other vertebrate remains include palaeoniscoid scales and

teeth and unidentified tetrapod bone fragments, jaw fragments, and teeth. Rare fragments of bones (scales?) bear

a “comb edge” which have not been previously observed in the Texas lower Permian.