A simple expression using a critical fillet hoop stress (CFHS) model was derived to predict the capacity of a simply supported wood beam with a notch on the tension face between the supports. The derivation used the hypothesis that cracking initiates when the hoop stress tangent to the free surface of a round-cornered notch exceeds a critical value. This critical value is characteristic to the material. Finite element modeling was used to explore the effects of a broad range of notch geometries, notch locations, beam sizes, loading configurations, and material elastic properties on fillet hoop stress. The analyses assumed homogeneous, orthotropic, linear elastic behavior, and used a hybrid element to provide accurate results in the region of high stress gradients. Simplified, closed form expressions to predict maximum hoop stress were developed from the numerical results. Notched beam tests included nine wood materials, encompassing hardwoods and softwoods in both green and kiln-dried conditions. A broad array of notch geometries was tested. A theoretical framework related the experimental failure loads with the calculated maximum fillet hoop stress values. The dependence of failure loads on notch geometry, location, and loading condition was described well by the predictive expression derived from the finite element modeling. The CFHS model can be applied to sharp-cornered notches when an appropriate effective fillet radius is substituted into the strength equation. Preliminary test results showed the effective fillet radius to be material dependent; theoretical analysis suggested a beam depth dependence as well. The notched beam strength equation utilizes a single material constant which can be experimentally determined from tests of beams with a single notch geometry. The notched beam strength parameter, κ, was found to be strongly related to specific gravity and cross-grain tensile strength. The regression equation from this work can be used to estimate κ for solid wood materials outside of this study. CFHS results compared favorably with those of earlier models shown to be accurate over a more limited set of cases. In addition to its broad applicability, the CFHS method benefits from its reliance on only one, easily determined, material parameter and avoids the need for directional fracture toughness and elastic parameter data which are very difficult to obtain. / Ph. D.

Cubes of coal have been tested in compression in the past, and it has been found. that the following formula, relating the compressive strength to the size of the cube can be applied: P = k .a- n Where P is the compressive strength in pounds per square inch. a is the edge dimension os specimens tested. n is a constant. k is a constant. The value of n has been found by a majority of people working on coal to be 0.5, however, lower values have also been found. In this study limestone shale and Plaster of Paris cubes, varying in size between one and three inches, and one and five inches in the case of shale, were tested in compression. The results were converted to logarithmic form, and the value of n determined for each material. It was found for the limestone and the Plaster of Paris that the value of n was close to zero over the range of sizes tested, indicating that the strength is independent of the size of the spec1men over the range one inch to three inch cubes. A value of 0.20 was found for the shale over the range one inch to five inches. The Griffith crack theory of failure gives the following result: P = k.c -0.5 Where P is the compressive strength in pounds per square inch. 2c is the length of cracks in the material. k is a constant. Thus, depending upon the relationship between c and a, the Griffith theory predicts that the value of n should be 0.5 for cracked materials where the length of crack is directly proportional to the edge dimension of the specimen, and zero where the length of crack is independent of the edge dimension of the specimen, in relatively uncracked materials. The Griffith theory is supported both by the results of compression tests, and by the results of the tests in this study and those previously conducted on coal. / Master of Science