Orthotropic

[Cameron Fluet]

Inmaterial science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength can be quantified with Hankinson's equation.

A familiar example of an orthotropic material is wood. In wood, one can define three mutually perpendicular directions at each point in which the properties are different. It is most stiff (and strong) along the grain (axial direction), because most cellulose fibrils are aligned that way. It is usually least stiff in the radial direction (between the growth rings), and is intermediate in the circumferential direction. This anisotropy was provided by evolution, as it best enables the tree to remain upright.

If orthotropic properties vary between points inside an object, it possesses both orthotropy and inhomogeneity. This suggests that orthotropy is the property of a point within an object rather than for the object as a whole (unless the object is homogeneous). The associated planes of symmetry are also defined for a small region around a point and do not necessarily have to be identical to the planes of symmetry of the whole object.

Orthotropic materials are a subset of anisotropic materials; their properties depend on the direction in which they are measured. Orthotropic materials have three planes/axes of symmetry. An isotropic material, in contrast, has the same properties in every direction. It can be proved that a material having two planes of symmetry must have a third one. Isotropic materials have an infinite number of planes of symmetry.

Transversely isotropic materials are special orthotropic materials that have one axis of symmetry (any other pair of axes that are perpendicular to the main one and orthogonal among themselves are also axes of symmetry). One common example of transversely isotropic material with one axis of symmetry is a polymer reinforced by parallel glass or graphite fibers. The strength and stiffness of such a composite material will usually be greater in a direction parallel to the fibers than in the transverse direction, and the thickness direction usually has properties similar to the transverse direction. Another example would be a biological membrane, in which the properties in the plane of the membrane will be different from those in the perpendicular direction. Orthotropic material properties have been shown to provide a more accurate representation of bone's elastic symmetry and can also give information about the three-dimensional directionality of bone's tissue-level material properties.[1]

It is important to keep in mind that a material which is anisotropic on one length scale may be isotropic on another (usually larger) length scale. For instance, most metals are polycrystalline with very small grains. Each of the individual grains may be anisotropic, but if the material as a whole comprises many randomly oriented grains, then its measured mechanical properties will be an average of the properties over all possible orientations of the individual grains.

Material behavior is represented in physical theories by constitutive relations. A large class of physical behaviors can be represented by linear material models that take the form of a second-order tensor. The material tensor provides a relation between two vectors and can be written as

where d , f \displaystyle \mathbf d ,\mathbf f are two vectors representing physical quantities and K \displaystyle \boldsymbol K is the second-order material tensor. If we express the above equation in terms of components with respect to an orthonormal coordinate system, we can write

An orthotropic material has three orthogonal symmetry planes. If we choose an orthonormal coordinate system such that the axes coincide with the normals to the three symmetry planes, the transformation matrices are

It can be shown that if the matrix K _ _ \displaystyle \underline \underline \boldsymbol K for a material is invariant under reflection about two orthogonal planes then it is also invariant under reflection about the third orthogonal plane.

In linear elasticity, the relation between stress and strain depend on the type of material under consideration. This relation is known as Hooke's law. For anisotropic materials Hooke's law can be written as[3]

where summation has been assumed over repeated indices. Since the stress and strain tensors are symmetric, and since the stress-strain relation in linear elasticity can be derived from a strain energy density function, the following symmetries hold for linear elastic materials

The stiffness matrix C _ _ \displaystyle \underline \underline \mathsf C satisfies a given symmetry condition if it does not change when subjected to the corresponding orthogonal transformation. The orthogonal transformation may represent symmetry with respect to a point, an axis, or a plane. Orthogonal transformations in linear elasticity include rotations and reflections, but not shape changing transformations and can be represented, in orthonormal coordinates, by a 3 3 \displaystyle 3\times 3 matrix A _ _ \displaystyle \underline \underline \mathbf A given by

An orthotropic elastic material has three orthogonal symmetry planes. If we choose an orthonormal coordinate system such that the axes coincide with the normals to the three symmetry planes, the transformation matrices are

We can show that if the matrix C _ _ \displaystyle \underline \underline \mathsf C for a linear elastic material is invariant under reflection about two orthogonal planes then it is also invariant under reflection about the third orthogonal plane.

No further information can be obtained because the reflection about third symmetry plane is not independent of reflections about the planes that we have already considered. Therefore, the stiffness matrix of an orthotropic linear elastic material can be written as

The compliance matrix is symmetric and must be positive definite for the strain energy density to be positive. This implies from Sylvester's criterion that all the principal minors of the matrix are positive,[6] i.e.,

In the early 1980s, San Francisco's Golden Gate Bridge was in need of a tuneup. Completed in 1937, the landmark bridge spanning the bay between San Francisco and Marin County, CA, began to show signs of deterioration in its concrete deck. Salt fog had reached the rebar, causing corrosion and concrete spalling. Engineers at the Golden Gate Bridge, Highway, and Transportation District made the decision to switch deck systems. In 1985, with assistance from construction engineers at the California Department of Transportation (Caltrans), the Golden Gate Bridge was restored using steel deck panels. The project not only restored the bridge to prime condition but also used fewer materials and reduced the deck weight by 11,160 metric tons (12,300 tons).

The unsung hero in the retrofit is orthotropic technology. Engineers define an orthotropic deck as one that consists of steel plates supported by ribs underneath, overlain by an integrated wearing (driving) surface. An orthotropic deck is a collage of steel plates welded together with a flat, solid steel deck stiffened by a grid of deck ribs welded to framing members like floor beams and girders. By integrating the structural system and the driving surface, orthotropic deck bridges are more lightweight and efficient on long-span structures.

A staple feature in transportation networks in Europe and East Asia, these bridges also are valued for their seismic performance, maneuverability (as in movable bridges), and versatility for construction in cold weather.

First used in Germany in the 1950s, orthotropic technology facilitated the cost-effective replacement of bridges destroyed during World War II. Today, Japan is home to the world's longest suspension, floating, and cable-stayed orthotropic deck bridges. In fact, major orthotropic viaducts in Tokyo are composed of more than 1,100 spans, and there are more than 250 orthotropic deck bridges of various sizes throughout the country.

Despite their popularity overseas, the percentage of orthotropic decks constructed in the United States remains low. But that may be about to change. Orthotropic structures have earned the trust of a few American bridge designers and owners who are pushing the technology forward. In fact, the Federal Highway Administration (FHWA) and many other organizations recently sponsored the world's first conference in nearly 30 years focused exclusively on orthotropic deck bridges.

"Over the last few decades, improved knowledge in regard to the design and performance of orthotropic decks has made them a popular choice for renovating older decks here in the United States," says John Fisher, professor emeritus of civil engineering at Lehigh University. "And if properly designed, orthotropic decks could offer a lifespan of more than 100 years."

When first introduced in the United States in the 1950s and 1960s, orthotropic decks represented a new and relatively unfamiliar technology for bridge designers. As a result of inadequate knowledge about the performance characteristics, particularly in regard to fatigue and traffic loading, early designers created bridges that were too light and tended to crack in the welds under repeated use by trucks. An experimental bridge built in the 1960s in Maryland, for example, only lasted a few years. The problem, according to Fisher, was that not enough experimentation had been carried out to define the details.

"Designers had bad experiences with early applications, using deck plates that were too thin," Fisher says. "In the United States, we were using thicknesses of 10 to 12 millimeters [0.39 to 0.47 inch], which is too thin to carry the wheel loads of heavy trucks. Many bridge decks failed, and that turned off owners."

Over the years, however, research in this country and abroad has helped engineers develop a more substantial base of knowledge and data on the performance of orthotropic bridges. According to Benjamin Tang, with FHWA's Office of Bridge Technology, current research on bridge performance indicates that stiffer orthotropic decks with wider ribs, along with prototype testing, could result in good performance and long bridge life.

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