Reinforced solid

A typical design problem is to find the smallest amount of reinforcement that can carry the stresses on a small cube (Fig.

It is assumed that the brittle material has no tensile strength.

Therefore, the principal stresses of the brittle material need to be compression.

subject to all eigenvalues of the brittle material stress tensor are less than or equal to zero (negative-semidefinite).

The solution to this problem can be presented in a form most suitable for hand calculations.

[3] It can also be presented in a form most suitable for computer implementation.

There are 12 possible reinforcement solutions to this problem, which are shown in the table below.

The second column gives conditions for which a solution is valid.

Columns 3, 4 and 5 give the formulas for calculating the reinforcement ratios.

Compute the reinforcement ratios of each possible solution that fulfills the conditions.

The principal stresses in the brittle material can be computed as the eigenvalues of the brittle material stress tensor, for example by Jacobi's method.

The formulas can be simply checked by substituting the reinforcement ratios in the brittle material stress tensor and calculating the invariants.

[3] The table below shows computed reinforcement ratios for 10 stress tensors.

Elaborate contour plots for beams, a corbel, a pile cap and a trunnion girder can be found in the dissertation of Reinaldo Chen.

[6] The solution to the optimization problem can be approximated conservatively.

For this upper bound, the characteristic polynomial of the brittle material stress tensor is

The approximation is easy to remember and can be used to check or replace computation results.

The above solution can be very useful to design reinforcement; however, it has some practical limitations.

In case of bars in one direction the stress tensor of the brittle material is computed by

Computer tools can support this by checking whether proposed reinforcement is sufficient.

{\displaystyle \left[{\begin{matrix}\sigma _{xx}-\rho _{x}f_{y}&\sigma _{xy}&\sigma _{xz}\\\sigma _{xy}&\sigma _{yy}-\rho _{y}f_{y}&\sigma _{yz}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}-\rho _{z}f_{y}\\\end{matrix}}\right]}

{\displaystyle \left[{\begin{matrix}{\frac {\sigma _{xx}}{\rho _{x}f_{y}}}&{\frac {\sigma _{xy}}{{\sqrt {\rho _{x}\rho _{y}}}f_{y}}}&{\frac {\sigma _{xz}}{{\sqrt {\rho _{x}\rho _{z}}}f_{y}}}\\{\frac {\sigma _{xy}}{{\sqrt {\rho _{x}\rho _{y}}}f_{y}}}&{\frac {\sigma _{yy}}{\rho _{y}f_{y}}}&{\frac {\sigma _{yz}}{{\sqrt {\rho _{y}\rho _{z}}}f_{y}}}\\{\frac {\sigma _{xz}}{{\sqrt {\rho _{x}\rho _{z}}}f_{y}}}&{\frac {\sigma _{yz}}{{\sqrt {\rho _{y}\rho _{z}}}f_{y}}}&{\frac {\sigma _{zz}}{\rho _{z}f_{y}}}\\\end{matrix}}\right]}

The largest eigenvalue of this tensor is the utilization (unity check), which can be displayed in a contour plot of a structure for all load combinations related to the ultimate limit state.

This shows that the bars are overloaded and 32% more reinforcement is required.

Combined compression and shear failure of the concrete can be checked with the Mohr-Coulomb criterion applied to the eigenvalues of the stress tensor of the brittle material.

is the uniaxial compressive strength (negative value) and

is a fictitious tensile strength based on compression and shear experiments.

Cracks in the concrete can be checked by replacing the yield stress

in the utilization tensor by the bar stress at which the maximum crack width occurs.

Clearly, crack widths need checking only at the surface of a structure for stress states due to load combinations related to the serviceability limit state.