Reinforcement that is not orthogonal or that is arranged in more than two directions occurs quite frequently in concrete slabs. In such cases, ultimate strength can be calculated from an equivalent distribution of orthogonal reinforcement.
Given n arbitrarily oriented groups of parallel bars with corresponding reinforcement ratios A(1), A(2),…, A(i),…, A(n), the equivalent reinforcement ratios Ax, Ay, Axy for the structural model axis XY can be obtained from the following equations
Where a(i) is the angle between the group of parallel bars “i” and X axis.
These values of reinforcement are then transformed into the equivalent reinforcement ratios in the principal directions p and q.
Where angle b can be obtained as
A model of shells of the slab in axis XY can provide axial forces Nx, Ny, Nxy and bending moments Mx, My, Mxy.
When reinforcement is arranged non-orthogonally or in more than two directions the design moments must be obtained in the principal directions of the reinforcement p and q, this is, we need to obtain from our FE software Np, Nq, Npq and Mp, Mq, Mpq before applying Wood and Armer rule or any similar rule.
In 1968, Wood and Armer proposed a popular design method that explicitly incorporates shell twisting moments. The Canadian code allows a simplified version of the Wood and Armer method that I assume here for the sake here of simplicity and safety.
Moment design rule can be stated as follows
All plus signs apply only to bottom reinforcement and all minus signs apply only to top reinforcement. Mpd and Mqd will be negative for tension in the top reinforcement and positive for tension in the bottom reinforcement. In the Canadian simplification, when the assumed-to-be-negative design moment is positive (adds compression) that moment is taken as zero. When the assumed-to-be-positive design moment is negative (adds compression) that moment is taken as zero.
Axial design rule can be stated as follows
It is generally assumed that tension (positive axial force) governs the reinforcement design of the slab and the plus sign generally applies. When the assumed-to-be-tension design axial force is a compression the axial force can be taken as zero.
After this step, the reinforcement Apd can be checked with the design forces Npd and Mpd; and the reinforcement Aqd can be checked with the desing forces Nqd and Mqd.
Some final clarifications:
- When the slab is very thin the effective depth difference between different groups of reinforcement may be important. In the case of simple bending this can be taken into account simply by using capacities rather than ratios, but if there is bending and axial forces the formulation becomes slightly messy. In commons slabs it is usually considered an average depth since the error is small and the Wood& Armer rule tends to overestimate the necessary reinforcement. An alternative simple and conservative approach is to consider the least effective depth.
- The rule of Wood and Armer presented colloquially here is a simplified version. Full version and its variations as implemented in well known software packages are much longer algorithms. Also, mind that Wood and Armer rule was derived for ULS reinforcement design. For SLS, cracking or fatigue analysis other rules and other methods may be more appropriate.
- The current post has been written as a revision of my former post http://notonlybridges.blogspot.com.es/2009/05/skew-or-non-orthogonal-reinforcement.html and it is not intended to be published as a peer reviewed paper. Avelino Samartin and other researchers have more rigorous procedures based on generating and rotating a tensor of resistances.
Unfortunately, I have not had time to apply those procedures in my professional practice.
- Spanish version of this post has been gently published by J. A. Agudelo in http://estructurando.net/2014/09/29/el-problema-del-diseno-de-armado-oblicuo-no-perpendicular/