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In
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
, a reinforced solid is a
brittle A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Bre ...
material that is reinforced by
ductile Ductility is a mechanical property commonly described as a material's amenability to drawing (e.g. into wire). In materials science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stres ...
bars or fibres. A common application is
reinforced concrete Reinforced concrete (RC), also called reinforced cement concrete (RCC) and ferroconcrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having hig ...
. 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 Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
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 In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
of the reinforcement is denoted by f_. The stress tensor of the brittle material is : \left
right Rights are law, legal, social, or ethics, ethical principles of Liberty, freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convent ...
. 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 In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
of the brittle material need to be compression. The principal stresses of a stress tensor are its
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. 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.


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. {, class="wikitable" , - , , , 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^2_{yz} > 0
I_{1}(\sigma_{yy}\sigma_{zz} - \sigma^2_{yz}) - I_{3} ≤ 0
I_{2}(\sigma_{yy}\sigma_{zz} - \sigma^2_{yz}) - I_{3}(\sigma_{yy}+\sigma_{zz}) ≥ 0 , , \frac{I_{3{\sigma_{yy} \sigma_{zz} - \sigma^2_{yz , , 0 , , 0 , - , 3 , , \sigma_{xx}\sigma_{zz} - \sigma^2_{xz} > 0
I_{1}(\sigma_{xx}\sigma_{zz} - \sigma^2_{xz}) - I_{3} ≤ 0
I_{2}(\sigma_{xx}\sigma_{zz} - \sigma^2_{xz}) - I_{3}(\sigma_{xx}+\sigma_{zz}) ≥ 0 , , 0 , , \frac{I_{3{\sigma_{xx} \sigma_{zz} - \sigma^2_{xz , , 0 , - , 4 , , \sigma_{xx}\sigma_{yy} - \sigma^2_{xy} > 0
I_{1}(\sigma_{xx}\sigma_{yy} - \sigma^2_{xy}) - I_{3} ≤ 0
I_{2}(\sigma_{xx}\sigma_{yy} - \sigma^2_{xy}) - I_{3}(\sigma_{xx}+\sigma_{yy}) ≥ 0 , , 0 , , 0 , , \frac{I_{3{\sigma_{xx} \sigma_{yy} - \sigma^2_{xy , - , 5 , , \sigma_{xx}<0 , , 0 , , \sigma_{yy}- \frac{\sigma^2_{xy{\sigma_{xx +, \sigma_{yz}-\frac{\sigma_{xz}\sigma_{xy{\sigma_{xx, , , \sigma_{zz}-\frac{\sigma^2_{xz{\sigma_{xx+, \sigma_{yz}-\frac{\sigma_{xz}\sigma_{xy{\sigma_{xx, , - , 6 , , \sigma_{yy}<0 , , \sigma_{xx}-\frac{\sigma^2_{xy{\sigma_{yy +, \sigma_{xz}-\frac{\sigma_{yz}\sigma_{xy{\sigma_{yy, , , 0 , , \sigma_{zz}-\frac{\sigma^2_{yz{\sigma_{yy +, \sigma_{xz}-\frac{\sigma_{yz}\sigma_{xy{\sigma_{yy, , - , 7 , , \sigma_{zz}<0 , , \sigma_{xx}-\frac{\sigma^2_{xz{\sigma_{zz +, \sigma_{xy}-\frac{\sigma_{yz}\sigma_{xz{\sigma_{zz, , , \sigma_{yy} -\frac{\sigma^2_{yz{\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. 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. The
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
of the reinforcing bars is 7800 kg/m3. In the table \sigma_{m} is the computed brittle material stress. m_{r} is the optimised amount of reinforcement. {, class="wikitable" , - style="height: 30px;" , width="50pt" , , , \sigma_{xx} , , \sigma_{yy} , , \sigma_{zz} , , \sigma_{yz} , , \sigma_{xz} , , \sigma_{xy} , , , , \rho_{x} , , \rho_{y} , , \rho_{z} , , \sigma_{m} , , m_{r} , - , 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 , , 343 kg/m3 , - , 2 , , -5 , , 2 , , 3 , , 4 , , 3 , , 1 , , , , 0.00 , , 1.36 , , 1.88 , , -10.31 , , 253 , - , 3 , , -5 , , -6 , , 3 , , 4 , , 3 , , 1 , , , , 0.00 , , 0.00 , , 1.69 , , -10.15 , , 132 , - , 4 , , -5 , , -6 , , -6 , , 4 , , 3 , , 1 , , , , 0.00 , , 0.00 , , 0.00 , , -10.44 , , 0 , - , 5 , , 1 , , 2 , , 3 , , -4 , , -3 , , -1 , , , , 0.60 , , 1.00 , , 2.00 , , -10.58 , , 281 , - , 6 , , 1 , , -2 , , 3 , , -4 , , 3 , , 2 , , , , 0.50 , , 0.13 , , 1.80 , , -10.17 , , 190 , - , 7 , , 1 , , 2 , , 3 , , 4 , , 2 , , -1 , , , , 0.40 , , 1.00 , , 1.80 , , -9.36 , , 250 , - , 8 , , 2 , , -2 , , 5 , , 2 , , -4 , , 6 , , , , 2.40 , , 0.40 , , 1.40 , , -15.21 , , 328 , - , 9 , , -3 , , -7 , , 0 , , 2 , , -4 , , 6 , , , , 0.89 , , 0.00 , , 0.57 , , -14.76 , , 114 , - , 10 , , 3 , , 0 , , 10 , , 0 , , 5 , , 0 , , , , 1.60 , , 0.00 , , 3.00 , , -10.00 , , 359 , -


Safe approximation

The solution to the optimization problem can be approximated conservatively. \rho_{x} f_{y}\sigma_{xx} + , \sigma_{xy}, + , \sigma_{xz}, \rho_{y} f_{y}\sigma_{yy} + , \sigma_{xy}, + , \sigma_{yz}, \rho_{z} f_{y}\sigma_{zz} + , \sigma_{xz}, + , \sigma_{yz}, This can be proofed as follows. For this upper bound, the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
of the brittle material stress tensor is \lambda^3 + 2(, \sigma_{yz}, +, \sigma_{xz}, +, \sigma_{xy}, )\lambda^2 + 3(, \sigma_{xz}, , \sigma_{xy}, +, \sigma_{yz}, , \sigma_{xy}, +, \sigma_{yz}, , \sigma_{xz}, )\lambda + 2, \sigma_{yz}\sigma_{xz}\sigma_{xy}, - 2\sigma_{yz}\sigma_{xz}\sigma_{xy}, which does not have positive
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
, or eigenvalues. The approximation is easy to remember and can be used to check or replace computation results.


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 Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization probl ...
: *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 *Reinforcing bars that already have been placed in the reinforcement design process *The whole structure instead of one small material cube in turn *Large reinforcement ratio's *Compression reinforcement


Bars in any direction

Reinforcing bars can have other directions than the x, y and z direction. In case of bars in one direction the stress tensor of the brittle material is computed by \left \begin{matrix} \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\ \sigma _{xy} & \sigma _{yy} & \sigma _{yz} \\ \sigma _{xz} & \sigma _{yz} & \sigma _{zz} \\ \end{matrix\right - \rho f_{y} \left[{\begin{matrix} \cos^2(\alpha) & \cos(\alpha)\cos(\beta) & \cos(\alpha)\cos(\gamma) \\ \cos(\beta)\cos(\alpha) & \cos^2(\beta) & \cos(\beta)\cos(\gamma) \\ \cos(\gamma)\cos(\alpha) & \cos(\gamma)\cos(\beta) & \cos^2(\gamma) \\ \end{matrix\right] where \alpha, \beta, \gamma are the angles of the bars with the x, y and z axis. Bars in other directions can be added in the same way.


Utilization

Often, builders of reinforced concrete structures know, from experience, where to put reinforcing bars. Computer tools can support this by checking whether proposed reinforcement is sufficient. To this end the tension criterion, The
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of \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] shall be less than or equal to zero. is rewritten into, The eigenvalues of \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] shall be less than or equal to one. The latter matrix is the utilization tensor. The largest eigenvalue of this tensor is the utilization (unity check), which can be displayed in a Contour line, contour plot of a structure for all load combinations related to the ultimate limit state. For example, the stress at some location in a structure is \sigma_{xx} = 4 N/mm², \sigma_{yy} = -10 N/mm², \sigma_{zz} = 3 N/mm², \sigma_{yz} = 3 N/mm², \sigma_{xz} = -7 N/mm², \sigma_{xy} = 1 N/mm². The reinforcement yield stress is f_{y} = 500 N/mm². The proposed reinforcement is \rho_{x} = 1.4%, \rho_{y} = 0.1%, \rho_{z} = 1.9%. The eigenvalues of the utilization tensor are -20.11, -0.33 and 1.32. The utilization is 1.32. 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. \frac{\sigma_{1{f_{t + \frac{\sigma_{3{f_{c ≤ 1, where \sigma_{1} is the largest principal stress, \sigma_{3} is the smallest principal stress, f_{c} is the uniaxial compressive strength (negative value) and f_{t} is a fictitious tensile strength based on compression and shear experiments. Cracks in the concrete can be checked by replacing the yield stress f _{y} in the utilization tensor by the bar stress at which the maximum crack width occurs. (This bar stress depends also on the bar diameter, the bar spacing and the bar cover.) 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.


See also

*
Reinforced concrete Reinforced concrete (RC), also called reinforced cement concrete (RCC) and ferroconcrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having hig ...
*
Solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
*
Structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...


References

Andreasen B.S., Nielsen M.P., Armiering af beton I det tredimesionale tilfælde, Bygningsstatiske meddelelser, Vol. 5 (1985), No. 2-3, pp. 25-79 (in Danish). Foster S.J., Marti P., Mojsilovic N., Design of Reinforced Concrete Solids Using Stress Analysis, ACI Structural Journal, Nov.-Dec. 2003, pp. 758-764. Hoogenboom P.C.J., De Boer A., "Computation of reinforcement for solid concrete", Heron, Vol. 53 (2008), No. 4. pp. 247-271. Hoogenboom P.C.J., De Boer A., "Computation of optimal concrete reinforcement in three dimensions", Proceedings of EURO-C 2010, Computational Modelling of Concrete Structures, pp. 639-646, Editors Bicanic et al. Publisher CRC Press, London. Nielsen M.P., Hoang L.C., Limit Analysis and Concrete Plasticity, third edition, CRC Press, 2011. Composite materials Plasticity (physics) Reinforced concrete Structural analysis