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Reinforced solid : ウィキペディア英語版
Reinforced solid

In solid mechanics, a reinforced solid is a brittle material that is reinforced by ductile bars or fibres. A common application is reinforced concrete. When the concrete cracks the tensile force in a crack is not carried any more by the concrete but by the steel reinforcing bars only. The reinforced concrete will continue to carry the load provided that sufficient reinforcement is present. A typical design problem is to find the smallest amount of reinforcement that can carry the stresses on a small cube (Fig. 1). This can be formulated as an optimization problem.
==Optimization problem==

The reinforcement is directed in the x, y and z direction. The reinforcement ratio is defined in a cross-section of a reinforcing bar as the reinforcement area A_ over the total area A, which is the brittle material area plus the reinforcement area.
:\rho_ = A_ / A_
:\rho_ = A_ / A_
:\rho_ = A_ / A_
In case of reinforced concrete the reinforcement ratios are usually between 0.1% and 2%. The yield stress of the reinforcement is denoted by f_. The stress tensor of the brittle material is
:
\left(_ - \rho_ f_ & \sigma _ & \sigma _ \\
\sigma _ & \sigma _ - \rho_ f_ & \sigma _ \\
\sigma _ & \sigma _ & \sigma _ - \rho_ f_ \\
\end}\right
)
.
This can be interpreted as the stress tensor of the composite material minus the stresses carried by the reinforcement at yielding. This formulation is accurate for reinforcement ratio's smaller than 5%. It is assumed that the brittle material has no tensile strength. (In case of reinforced concrete this assumption is necessary because the concrete has small shrinkage cracks.) Therefore, the principal stresses of the brittle material need to be compression. The principal stresses of a stress tensor are its eigenvalues.
The optimization problem is formulated as follows. Minimize \rho_ + \rho_ + \rho_ subject to all eigenvalues of the brittle material stress tensor are less than or equal to zero (negative-semidefinite). Additional constraints are \rho_ ≥ 0, \rho_ ≥ 0, \rho_ ≥ 0.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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