# Load Transfer from Slab to Beams – A Comparative Analysis

In the design of reinforced concrete structures, floor loads are usually transferred from slabs to beams, and from the beams, the loads are transferred to the columns, and finally to the foundation. Load transfer from slab to beams is one of the most intriguing aspects of design, especially for beginners in the design of reinforced concrete structures. Generally slab pressure loads (kN/m2) are transferred to supporting beams as line loads (kN/m) which can be triangular, trapezoidal, or partially distributed.

In the manual design of structures, some formulas can be used to represent slab loads on beams as uniformly distributed loads. The main essence of this is to simply analysis and the formula can be obtained from Reynolds and Steedman (2005) for transfer of load from two-way slab to beams.

Two-way slab (ly/lx < 2)
Long span: p = nlx/2(1 – 1/3k2)
Short span: p = nlx/3

One-way slab (ly/lx > 2)
Long span: p = nlx/2
Short span: p = nlx/5

Where;
ly = length of long side of the slab
lx = length of short side pf the slab
k = aspect ratio = ly/lx

In this article, we are going to review load transfer from slab to beams using three approaches;

(1) Full finite element analysis of beams and slabs using Staad Pro
(3) Manual method using formula

#### CASE 1: Two way slab of dimensions (5m x 6m) simply supported by beams on all sides and subjected to a pressure load of 10 kN/m2

(a) Finite element analysis

Long span beam:
Maximum span moment = 73.063 kNm
Support moment = -2.71 kNm
End shear = 37.6 kN

Short span beam:
Maximum span moment = 54.495 kNm
Support moment =-0.814 kNm
End shear = 31.9 kN

(b) Yield line method

Long span beam:
Maximum span moment = 76.562 kNm
Support moment = -9.897 kNm
End shear = 39.4 kN

Short span beam:
Maximum span moment = 46.987 kNm
Support moment =-5.096kNm
End shear = 30.151 kN

(c) Manual analysis using formula
k = ly/lx = 6/5 = 1.2
Load on long span beam = nlx/2(1 – 1/3k2) = [(10 x 5)/2] x [1 – 1/(3 x 1.22)] = 19.212 kN/m
Maximum span moment = ql2/8 = (19.212 x 62)/8 = 86.454 kNm
End shear = ql/2 = (19.212 x 6)/2 = 57.636 kN

Load on the short span beam = nlx/3 = (10 x 5)/3 = 16.667 kN/m
Maximum span moment = ql2/8 = (16.667 x 52)/8 = 52.084 kNm
End shear = ql/2 = (16.667 x 5)/2 = 41.6675 kN

#### CASE 2: One-way slab of dimensions (2.5 m x 7 m) simply supported by beams on all sides and subjected to a pressure load of 10 kN/m2

k = ly/lx = 7/2.5 = 2.8

(a) Finite Element Analysis

Long span beam:
Maximum span moment = 60.689 kNm
Support moment = -6.337 kNm
End shear = 29.7 kN

Short span beam:
Maximum span moment = 12.091 kNm
Support moment = +2.81 kNm
End shear = 11.6 kN

(b) Yield line method

Long span beam:
Maximum span moment = 63.4 kNm
Support moment = -9.9 kNm
End shear = 35.9 kN

Short span beam:
Maximum span moment = 6.16 kNm
Support moment = -0.346 kNm
End shear = 7.81 kN

(c) Manual analysis using formula
Load on long span beam = nlx/2 = (10 x 2.5)/2 = 12.5 kN/m
Maximum span moment = ql2/8 = (12.5 x 72)/8 = 76.56 kNm
End shear = ql/2 = (12.5 x 7)/2 = 43.75 kN

Load on the short span beam = nlx/5 = (10 x 2.5)/5 = 5 kN/m
Maximum span moment = ql2/8 = (5 x 2.52)/8 = 3.906 kNm
End shear = ql/2 = (5 x 2.5)/2 = 6.25 kN

#### Discussion of results

(a) Two-way slab systems
(1) In the long span direction, finite element analysis and yield line method gave very close results for bending moment and shear forces. Manual analysis overestimated the load transferred.
(2) In the short span direction, yield line method underestimated the load transferred to the short span beams when compared with finite element analysis. Manual analysis gave close result to finite element analysis.
(3) Manual analysis using formula gave bending moment values that can be used for design purposes, but generally overestimated the shear forces. In the long span direction, the sagging moment can be redistributed to the supports by 10% (reduce the span moments by 10% and take the value taken away as support moment).

(b) One-way slab systems
(1) As with two-way slabs, finite element analysis and yield line method gave very close results for bending moment and shear forces in the long span beams. Manual analysis overestimated the load transferred.
(2) In the short span direction, yield line method underestimated the load transferred to the short span beams when compared with finite element analysis. Manual analysis using formula underestimated the load transferred.
(3) As with two-way slabs, manual analysis using formula gave bending moment values that can be used for design purposes, but overestimated the shear forces in the long span beams. The shear force and bending moment in the short span beam were underestimated when formula was used.
(4) In the long span direction, the sagging moment can be redistributed to the supports by 10% (reduce the span moments by 10% and take the value taken away as support moment).

#### Conclusion and Recommendation

(1) In a strict technical sense, there is nothing like one-way slab for a slab supported on all the edges. There is always a two way action even though it is greater in the long span.
(2) Formula should not be applied when assessing the shear force induced in beams supporting floor loads.
(3) Yield line method of load transfer from slab to beams should be used for manual design of structures, despite the more onerous computational effort.

A dynamic civil engineer with vast experience in research, design, and construction of civil engineering infrastructures. He is a member of the Nigerian Society of Engineers. Reach him at ubani@structville.com