Hyperoperation

File:Reinforced solids cube.jpg

Figure 1: Small cube of a material with reinforcing bars. The cube is cracked and the material above the crack is removed to show the reinforcement that crosses the crack.

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 $A_r$ over the total area $A$, which is the brittle material area plus the reinforcement area.

${\rho }_x$ = $A_{rx}$ / $A_x$

${\rho }_y$ = $A_{ry}$ / $A_y$

${\rho }_z$ = $A_{rz}$ / $A_z$

In case of reinforced concrete the reinforcement ratios usually are between 0.1% and 2%. The yield stress of the reinforcement is denoted by $f_y$. The stress tensor of the brittle material is

$\left[\begin{array}{ccc}{\sigma }_{xx}-{\rho }_x{f}_y& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{xy}& {\sigma }_{yy}-{\rho }_y{f}_y& {\sigma }_{yz}\\ {\sigma }_{xz}& {\sigma }_{yz}& {\sigma }_{zz}-{\rho }_z{f}_y\\ \end{array}\right]$.

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 ${\rho }_x$ + ${\rho }_y$ + ${\rho }_z$ subject to all eigenvalues of the brittle material stress tensor are less than or equal to zero (negative-semidefinite).

Solution

The solution to this problem can be presented in a form most suitable for hand calculations. It can be presented in graphical form. It can also be presented in a form most suitable for computer implementation. In this article the latter method is shown.

There are 12 possible reinforcement solutions to this problem, which are shown in the table below. Every row contains a possible solution. The first column contains the number of a solution. The second column gives conditions for which a solution is valid. Columns 3, 4 and 5 give the formulas for calculating the reinforcement ratios.

	Condition	${\rho }_x$ $f_y$	${\rho }_y$ $f_y$	${\rho }_z$ $f_y$
1	$I_1$ ≤ 0, $I_2$ ≥ 0, $I_3$ ≤ 0	0	0	0
2	${\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{yz}^2$ > 0 $I_1({\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{yz}^2)-I_3$ ≤ 0 $I_2({\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{yz}^2)-I_3({\sigma }_{yy}+{\sigma }_{zz})$ ≥ 0	$\frac{I_3}{{\sigma }_{yy}{\sigma }_{zz}-{\sigma }_{yz}^2}$	0	0
3	${\sigma }_{xx}{\sigma }_{zz}-{\sigma }_{xz}^2$ > 0 $I_1({\sigma }_{xx}{\sigma }_{zz}-{\sigma }_{xz}^2)-I_3$ ≤ 0 $I_2({\sigma }_{xx}{\sigma }_{zz}-{\sigma }_{xz}^2)-I_3({\sigma }_{xx}+{\sigma }_{zz})$ ≥ 0	0	$\frac{I_3}{{\sigma }_{xx}{\sigma }_{zz}-{\sigma }_{xz}^2}$	0
4	${\sigma }_{xx}{\sigma }_{yy}-{\sigma }_{xy}^2$ > 0 $I_1({\sigma }_{xx}{\sigma }_{yy}-{\sigma }_{xy}^2)-I_3$ ≤ 0 $I_2({\sigma }_{xx}{\sigma }_{yy}-{\sigma }_{xy}^2)-I_3({\sigma }_{xx}+{\sigma }_{yy})$ ≥ 0	0	0	$\frac{I_3}{{\sigma }_{xx}{\sigma }_{yy}-{\sigma }_{xy}^2}$
5	${\sigma }_{xx}<0$	0	${\sigma }_{yy}-\frac{{\sigma }_{xy}^2}{{\sigma }_{xx}}+|{\sigma }_{yz}-\frac{{\sigma }_{xz}{\sigma }_{xy}}{{\sigma }_{xx}}|$	${\sigma }_{zz}-\frac{{\sigma }_{xz}^2}{{\sigma }_{xx}}+|{\sigma }_{yz}-\frac{{\sigma }_{xz}{\sigma }_{xy}}{{\sigma }_{xx}}|$
6	${\sigma }_{yy}<0$	${\sigma }_{xx}-\frac{{\sigma }_{xy}^2}{{\sigma }_{yy}}+|{\sigma }_{xz}-\frac{{\sigma }_{yz}{\sigma }_{xy}}{{\sigma }_{yy}}|$	0	${\sigma }_{zz}-\frac{{\sigma }_{yz}^2}{{\sigma }_{yy}}+|{\sigma }_{xz}-\frac{{\sigma }_{yz}{\sigma }_{xy}}{{\sigma }_{yy}}|$
7	${\sigma }_{zz}<0$	${\sigma }_{xx}-\frac{{\sigma }_{xz}^2}{{\sigma }_{zz}}+|{\sigma }_{xy}-\frac{{\sigma }_{yz}{\sigma }_{xz}}{{\sigma }_{zz}}|$	${\sigma }_{yy}-\frac{{\sigma }_{yz}^2}{{\sigma }_{zz}}+|{\sigma }_{xy}-\frac{{\sigma }_{xz}{\sigma }_{yz}}{{\sigma }_{zz}}|$	0
8	${\sigma }_{yz}+{\sigma }_{xz}+{\sigma }_{xy}$ ≥ 0 ${\sigma }_{xz}{\sigma }_{xy}+{\sigma }_{yz}{\sigma }_{xy}+{\sigma }_{yz}{\sigma }_{xz}$ ≥ 0	${\sigma }_{xx}+{\sigma }_{xz}+{\sigma }_{xy}$	${\sigma }_{yy}+{\sigma }_{yz}+{\sigma }_{xy}$	${\sigma }_{zz}+{\sigma }_{yz}+{\sigma }_{xz}$
9	$-{\sigma }_{yz}-{\sigma }_{xz}+{\sigma }_{xy}$ ≥ 0 $-{\sigma }_{xz}{\sigma }_{xy}-{\sigma }_{yz}{\sigma }_{xy}+{\sigma }_{yz}{\sigma }_{xz}$ ≥ 0	${\sigma }_{xx}-{\sigma }_{xz}+{\sigma }_{xy}$	${\sigma }_{yy}-{\sigma }_{yz}+{\sigma }_{xy}$	${\sigma }_{zz}-{\sigma }_{yz}-{\sigma }_{xz}$
10	${\sigma }_{yz}-{\sigma }_{xz}-{\sigma }_{xy}$ ≥ 0 ${\sigma }_{xz}{\sigma }_{xy}-{\sigma }_{yz}{\sigma }_{xy}-{\sigma }_{yz}{\sigma }_{xz}$ ≥ 0	${\sigma }_{xx}-{\sigma }_{xz}-{\sigma }_{xy}$	${\sigma }_{yy}+{\sigma }_{yz}-{\sigma }_{xy}$	${\sigma }_{zz}+{\sigma }_{yz}-{\sigma }_{xz}$
11	$-{\sigma }_{yz}+{\sigma }_{xz}-{\sigma }_{xy}$ ≥ 0 $-{\sigma }_{xz}{\sigma }_{xy}+{\sigma }_{yz}{\sigma }_{xy}-{\sigma }_{yz}{\sigma }_{xz}$ ≥ 0	${\sigma }_{xx}+{\sigma }_{xz}-{\sigma }_{xy}$	${\sigma }_{yy}-{\sigma }_{yz}-{\sigma }_{xy}$	${\sigma }_{zz}-{\sigma }_{yz}+{\sigma }_{xz}$
12	${\sigma }_{xy}{\sigma }_{xz}{\sigma }_{yz}<0$	${\sigma }_{xx}-\frac{{\sigma }_{xz}{\sigma }_{xy}}{{\sigma }_{yz}}$	${\sigma }_{yy}-\frac{{\sigma }_{yz}{\sigma }_{xy}}{{\sigma }_{xz}}$	${\sigma }_{zz}-\frac{{\sigma }_{yz}{\sigma }_{xz}}{{\sigma }_{xy}}$

$I_1$, $I_2$ and $I_3$ are the stress invariants of the composite material stress tensor. Additional constraints are ${\rho }_x$ ≥ 0, ${\rho }_y$ ≥ 0, ${\rho }_z$ ≥ 0.

The algorithm for obtaining the right solution is simple. Compute the reinforcement ratios of each possible solution that fulfills the conditions. Further ignore solutions with a reinforcement ratio less than zero. Compute the values of ${\rho }_x$ + ${\rho }_y$ + ${\rho }_z$ and select the solution for which this value is smallest. 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. The first invariant needs to be less than or equal to zero. The second invariant needs to be greater than or equal to zero. These provide the conditions in column 2. For solution 2 to 12, the third invariant needs to be zero.

Examples

The table below shows computed reinforcement ratios for 10 stress tensors. The applied reinforcement yield stress is $f_y$ = 500 N/mm². In the table ${\sigma }_m$ is the computed brittle material stress.

	${\sigma }_{xx}$	${\sigma }_{yy}$	${\sigma }_{zz}$	${\sigma }_{yz}$	${\sigma }_{xz}$	${\sigma }_{xy}$		${\rho }_x$	${\rho }_y$	${\rho }_z$	${\sigma }_m$
1	1 N/mm²	2 N/mm²	3 N/mm²	-4 N/mm²	3 N/mm²	-1 N/mm²		1.00%	1.40%	2.00%	-10.65 N/mm²
2	-5	2	3	4	3	1		0.00	1.36	1.88	-10.31
3	-5	-6	3	4	3	1		0.00	0.00	1.69	-10.15
4	-5	-6	-6	4	3	1		0.00	0.00	0.00	-10.44
5	1	2	3	-4	-3	-1		0.60	1.00	2.00	-10.58
6	1	-2	3	-4	3	2		0.50	0.13	1.80	-10.17
7	1	2	3	4	2	-1		0.40	1.00	1.80	-9.36
8	2	-2	5	2	-4	6		2.40	0.40	1.40	-15.21
9	-3	-7	0	2	-4	6		0.89	0.00	0.57	-14.76
10	3	0	10	0	5	0		1.60	0.00	3.00	-10.00

Extension

The above solution can be very useful to design reinforcement, however, it has some practical limitations. The following aspects can be included too if the problem is solved using convex optimization.

• Multiple stress tensors in one point due to multiple loads on the structure instead of only one stress tensor,
• A constraint imposed to crack widths at the surface of the structure,
• Shear stress in the crack (aggregate interlock),
• Reinforcement in other directions than x, y and z,
• The whole structure instead of one small material cube in turn.

See also

• Reinforced concrete
• Solid mechanics
• Structural engineering