May 4th, 2010
Analytical (Manual) Verification of a Reinforced Masonry Beam-Column Stress Calculation
Introduction
The data and equations from the Building Code Requirements and Specifications for Masonry Structures (BCRSMS), 2008 edition, cover minimum requirements for the structural design and construction of masonry elements consisting of masonry units bedded in mortar. This case explores a scenario that is not covered in the code.
The strength of concrete in tension is negligible, requiring the use of steel reinforcement. This example corresponds to steel reinforcement that is placed centrally in a beam-column made from concrete. This article demonstrates that the contribution of steel reinforcement to the total moment of inertia can be neglected when the steel is placed at the center of the cross section of the concrete beam-column.
During the calculation of the structural capacity of a reinforced masonry wall (or beam-column), it is necessary to calculate the gross (net) moment of inertia with respect to the centerline of the cross section. In the case of a centrally located reinforcing bar (rebar), it is normally known that the moment of inertia about the centerline is normally negligible for small areas. However, the area of steel, though small (a No. 5 bar is 5/8 of an inch in diameter, for instance) compared to the gross cross-sectional area of the concrete cell being reinforced, when transformed from steel to concrete area, Ast, is the product of As times the moduli ratio n, where n is the dimensional parameter and normally exceeds 15 (i.e., n = Es/Ev = Ast/As > 10). This case demonstrates that the area of rebars transformed from steel to concrete may not be neglected when rebars are centrally located, and further investigation is warranted (see Figure 1).
Figure 1: Original and transformed reinforced beam-column cross section.
Nomenclature:
A = b × t = cross-sectional area of the concrete (or masonry)
As = cross-sectional area of the reinforcement steel
Ast = n × As = steel area transformed to concrete (or masonry)
b = width of the section, 8-inch minimum
bst = width of the steel transformed to masonry
Es = elastic modulus of steel
Ec = elastic modulus of concrete
Em = elastic modulus of masonry
n = elastic moduli ratio, Es/Ec or Es/Em
t = depth of the section, 8-inch minimum
The most expedient method of analyzing a section made from concrete (or masonry) and reinforced with centrally located bars is to transform the reinforcement to the material surrounding it, such as concrete (or grout). Therefore,
Therefore:
Substitute Equation (6) into Equation (4) to get:
where:
Since 0 < λ <<1 is negligible, it is possible to approximate the transformed moment of inertia as:
It = Ig Q.E.D.
Conclusion
The results indicate that it is appropriate, convenient, practical, and conservative to use the gross (net) moment of inertia of a centrally located reinforcement bar and neglect the contribution of the rebars. The contribution of the steel reinforcement area in a section assumed not to be cracked can be neglected for centrally located reinforcement bars in a reinforced concrete (or masonry) section, as shown in Figure 1.
Reference:
ASME/ACI: Building Code Requirements (TMS 402-08/ACI 530.1-08/ASCE 5-08) and Specifications (TMS 602-08/ACI 530.1-08/ASCE 6-08) for Masonry Structures, pp. C-44–C-45. The Masonry Society.
May 7th, 2010 at 4:00 pm
I am puzzled as to why anyone designing a steel reenforced concrete beam would seek to reenforce that beam with a single rebar at the center of the beam, where it would do little in the way of reenforcing the beam.
However, my analysis for a centrally located rebar in an 8″X *8″ beam is different from the one you propose and instead would be as follows:
(1) (S steel)=((M)(D rebar/2))/I rebar
(2) (( S steel/((e steel)/L beam))/((S concrrete)/((e concrete/L beam))= (E stee)l/(E conc)
(3) (e steel)/(e concrete) = 4/8
Solution of !,2,3 will provide the maximum stress in the steel and concrete
Alan Cross
May 7th, 2010 at 5:20 pm
Dear Mr. Cross, the work is a result of my analysis of a beam-column consisting of
16″ wide CBS “wall” with a single No. 5 bar in each cell. In Florida, this is a typical
design for a garage having a 8-foot and 16-foot garage doors. The design is simple
and therefore inexpensive.
Please note that most masonry walls in residential design in Florida use a single No. 5
at a spacing not to exceed four (4) feet. Therefore, the article is appropriate for those
instances where masonry walls are reinforced with centrally placed rebars.
The work is not intended to endorse the use of centrally-located rebar but instead, it
simply provides the analyst with an assurance that the computation of a gross moment
of inertia need not be overly concerned with the added complexity of considering the
steel contributions.
Sincerely appreciative for your interest in the article,
Julio C. Banks, MSME, PE & CGC
Solutions Consultant
Port Saint Lucie, Florida
May 7th, 2010 at 8:30 pm
Mr. Banks
As an afterthought, that occurred when I attempted to evaluate the average concrete stress relative to the average stress of the reenforcing steel, when centered in the concrete I had to make a few revisions and additions as follows:
(4) (e conc.)/(e stl.) = (2)/((D stl.)/(4))
(5) (S conc.)/(e conc.)/(L beam) = E conc.
(6) ( S stl.)/ (e stl.)/ (L beam) = E stl.
Solving (4), (5), (6), for a steel stress of 20000 PSI gives the following:
(S conc.) = (8)(3)(20000)/(0.625)(30) = 25625 PSI
It is apparent that this average stress would result in a very deep crack in the concrete
Of course the average stress for the steel is an approximation. To get a more accurate value, it would be necessary to know what the value of the bending momement happens to be and check the resulting stress in the steel which is the sum of the resisting moment of the concrete and the steel, or:
(7) M total = M stl + M conc where:
(8) M stl = (A rebar)(S stl)((D rebar/(2))
(9) M conc = (A conc)(S conc)(4)
Alan Cross
April 2nd, 2011 at 4:33 pm
One again, your article is very nice.