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Masonry Beam Design: How to design beams with what might seem like less than the minimum reinforcement ratio
When a design seems like it might not meet minimum steel, all hope is not lost
While there is no way around satisfying the minimum steel requirements in the CSA S304, there is a clause that can be taken advantage of that is not used by MASS. By following the procedure outlined in this article, beams can be designed within the restraints of the CSA standards while also containing less than the minimum reinforcement area as prescribed by clause 188.8.131.52. This often overlooked clause can be very helpful when it comes to masonry beam design, especially those that would seem at face value to be the easiest to design due to nominal loads and short spans.
July 2020 Update: This clause has been incorporated into the release of MASS Version 4.0.
Disclaimer: This post is exclusively intended to provide insight into the approach taken by the MASS design software in interpreting a CSA S304-14 code compliant design. It is up to the professional discretion of the designer to input an appropriate layout, boundary and loading conditions, interpret the results, and determine how they should be incorporated into their designs. In the event of any descrepencies between the contents of this post and the referenced codes and standards, the codes and standards shall apply. As per the end user license agreement (and also recommended within PEO’s guidelines for using engineering software), a tool cannot be considered competent and reliance on a tool does not relieve the user of responsibility.
To jump straight to a summary further down the post, click here.
The smallest beams with only nominal loading can also be the trickiest to design. One of the reasons is the need to satisfy minimum reinforcement ratio requirements in S304-14: 11.2.3:
Currently in the MASS software, all beam designs not meeting ρmin in clause 184.108.40.206 will fail moment and deflection design, shown below in the simplified moment results tab:
While these designs are failed before attempting an incrementally larger design with more reinforcement, there is an option at the engineer’s disposal outside of MASS: invoke the mighty power of clause 220.127.116.11 which often goes overlooked.
There was some initial due diligence behind including this clause in the MASS software. It was not included as programming costs would be very high. The change would involve functionally designing two beams; the first being the actual beam used for the design, and the second beam being the theoretical beam containing one third less reinforcement which would be used to resist the factored moment. While at first glance, it might seem acceptable to instead ensure that the beam has additional moment resistance by factor of 4/3. However, while it is very close, there is not quite a directly proportional relationship between a beam’s moment resistance and its longitudinal reinforcement area so this approach would not adequately reflect the phrasing of clause 18.104.22.168.
How to use this clause
There are two different approaches that can be taken to satisfy clause 11.2.3 and design a beam that contains less than the required reinforcement ratio, or ρmin, of clause 22.214.171.124:
Compare the area of reinforcement to the area required by analysis increased by one third
Compare the factored load to the load resisted by the beam if the reinforcement area were reduced by a third
Unfortunately, there is no way around completing a separate analysis to determine “the area of reinforcement required by analysis” quoted directly from the standard. Since the minimum reinforcement is a function of loading, one cannot be solved without assuming the other. Method 1 is likely more intuitive as it returns an alternative minimum reinforcement area to clause 126.96.36.199. However, Method 2 can be more straight forward to calculate as there is no need solve a quadratic or cubic function.
Method 1: Comparing areas of reinforcement
The approach taken in method 1 is to answer the question: “How much reinforcement is required in this beam design?”. This can be expanded to “What area of steel results in factored moment being equal to moment resistance?”. For beams with compression steel or intermediate steel which may not necessarily be yielding, it is likely easiest to solve for this value using a spreadsheet and GoalSeek or Solver since a simplified expression would likely require finding the roots of a cubic function and would also rely upon an assumed strain profile if the steel is yielding. Minimum reinforcement failure messages do not tend to present themselves for beams with several layers of steel so this article will focus on the simple beam designs where these issues arise.
The area of steel required, or Asreq, can be solved for using force and moment equilibrium for any beam configuration where the beam’s moment resistance is equal to the maximum factored moment (Mr = Mf). The expression below can be used calculate this minimum area of reinforcement for beams with only tension reinforcement:
Note that this expression is only applicable to beams that exclusively contain primary tension steel to resist flexure (ie. no intermediate or compression steel). This formula also assumes that the tension steel is yielding which is a requirement in clause 11.2.2. Also, while the area of reinforcement required is typically a function of a beam’s moment resistance, it is possible that other factors such as cracking and deflection may govern the design and as such, should never be assumed to be satisfied and always checked manually (in addition to using this formula).
The units of each input are shown below as well as the derivation which can be expanded.
Method 2: Comparing Loads
Rather than ask the question, “What is the minimum area of reinforcement needed?” for a beam design, the other way to approach satisfying clause 188.8.131.52 is to assume that the area of steel present is equal to the minimum allowable and determine the largest possible factored moment that the assumption is valid for. Since this clause takes loading into account to evaluate minimum steel, rather than use a load to check the area of steel, method 2 involves using an area of steel to check the load. As mentioned earlier, this method may appear less intuitive as it is not as simple as comparing the reinforcement present to another minimum value. However, the steps used to determine the maximum allowable are the same as those used to calculate the moment resistance of any other beam design which is why method 2 may be preferable.
The table below shows a summary of the maximum allowable factored moments resisted by beams which contain less reinforcement than allowable by clause 184.108.40.206 for four possible beam geometries:
Maximum applied factored moment for a beams having less than the required reinforcement ratio in accordance with S304-14: 220.127.116.11 to satisfy S304-14: 18.104.22.168
Disclaimer: These values should not be relied upon as part of the design process. They are meant to illustrate the concept that very low areas of reinforcement may be acceptable, depending on the applied loads. The full cross section should be analyzed by the engineer by hand (or other tool) to check these requirements in a way that is applicable to the situation.
As stated under the simplified method 1 expression earlier, although the area of steel required is typically governed by moment resistance, it is possible that other factors such as cracking and deflection may govern the design and as such, should never be assumed to be satisfied and always checked manually.
Note that for designs using 2 No. 10 bars in tension (same bar area as 1 No. 15 bar), the placement is affected as the distance from the compression face of the beam is slightly further away from the vertical centroid of the bars. This change increases the moment arm separating the coupled internal forces and slightly increases the maximum allowable moment for these designs. The lower (and more conservative) values are shown in the table for simplicity however the comparison can be expanded below.
All hope is not lost when a beam design fails due to not satisfying minimum steel using MASS, which only checks against CSA S304-14: 22.214.171.124. It is possible to satisfy minimum reinforcement requirements by instead using clause 126.96.36.199 by approaching the design in one of two ways:
- Method 1: checking bar area against one third greater than the area resulting in Mf being equal to Mr
- This can be done quickly for beams with only primary tension reinforcement using the following expression:
- Method 2: checking factored moment against moment resistance for a beam with a third less steel than what is actually present within the beam.
- This can be quickly checked by comparing the factored moment to the maximum allowable moment in the table below:
Disclaimer: These values should not be relied upon as part of the design process. They are meant to illustrate the concept that very low areas of reinforcement may be acceptable, depending on the applied loads. The full cross section should be analyzed by the engineer by hand (or other tool) to check these requirements in a way that is applicable to the situation. Regardless of which method is used, it is up to the designer to ensure that in addition to satisfying the minimum reinforcement requirements in S304-14:11.3.2, all other provisions must be considered and independently verified. The guides in both methods are based on the area of reinforcement being governed by applied bending moment at any section within the beam which is not true for all cases.
As always, feel free to contact us if you have any questions at all. CMDC is the authorized service provider for the MASS software which is a joint effort of between CCMPA and CMDC.