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Saturday, January 13, 2018

Analysis of Column Loads in a Building By Considering Beam Support Reactions


Introduction
In the manual design of reinforced concrete structures, column loads are usually assessed by considering the support reactions from beams they are supporting, or by the tributary area method. The later is more popular due to its simplicity and speed, but usually fails to capture all loads that are imposed directly on the columns, while the former is more complex and time consuming, but very representative of all the possible loads that are imposed on the column.



Steps on how to load a column from beam support reaction
(1) Load the floor slab adequately, and factor the loads at ultimate limit state, considering the appropriate load combination.
(2) Transfer the loads from slab to the beam using the appropriate relationship. Based on yield lines, the loads are usually triangular or trapezoidal, but this is cumbersome to analyse manually. An equivalent UDL (reasonably accurate) can be used to transfer slab loads by employing the formulars below;

One way slabs
Long span   q = nlx/2
Short span   q = nlx/5

Two way slabs
Long span    q = nlx/2(1 - 1/3k2)
Short span     q = nlx/3

Where;
q =  Load transferred from slab to the beam
n = load at ultimate limit state
      1.4gk + 1.6qk (BS 8110)
      1.35gk + 1.5qk (Eurocode 2)
Ly = Length of long span of slab
Lx = Length of short span of the slab
k = Ly/Lx

(3) Analyse the floor beams completely using any suitable method of your choice, while also considering any additional load that may be on the beam such as wall load and finishes.

(4) Obtain the support reactions of the beam, which represents the load that is transferred from the floor to the column.



Design Example
In this post, the floor plan shown below is for a shopping complex, and it desired to obtain the column axial loads at ultimate limit state.


Design Data
Size of all columns = 230 x 230mm
Size of all beams = 450 x 230mm
Thickness of slab = 200mm
Unit weight of concrete = 25 kN/m3
Unit weight of sandcrete block = 3.47 kN/m2
fck = 25 N/mm2
fyk = 500 N/mm2

Load combination = 1.35gk + 1.5qk
Design variable load (qk) = 4 kN/m2

k = Ly/Lx = 6/5 = 1.2 (in all cases)
Only external beams are carrying block work load.

Load Analysis

Roof beam
Permanent load on roof beams gk = 6 kN/m
Variable load on roof beam  qk = 1.5 kN/m
(These values are assumed)
At ultimate limit state, n = 1.35(6) + 1.5(1.5) = 10.35 kN/m

Slab Load Analysis
Concrete own weight = 25 kN/m3  × 0.2m  = 5.0  kN/m2
Screeding and Finishes (say)    = 1.35  kN/m2
Partition allowance   = 1.5  kN/m2
Total (gk)    =  7.85 kN/m2
Variable action (qk)    = 4 kN/m2

 n = 1.35gk + 1.5qk = 1.35(7.85) + 1.5(4) = 16.6 KN/m2

Wall load on beam
Unit weight of sandcrete block = 3.47 kN/m2
Height of wall = 3.5m
Wall load on beam =  3.47 kN/m2  × 3.5m  = 12.145  kN/m

Load on Beams
External Longitudinal beams (Axis 1:A-D and Axis 3:A-D)
Self weight of beam (factored) = 1.35 × 0.23m × 0.25m × 25 = 1.94 kN/m
Load from slab = nlx/2(1 - 1/3k2) = [(16.6 × 5)/2] × (1 - 1/(3 ×1.22)) = 31.893 kN/m
Load from block work = 12.145  kN/m
Total load = 45.978 KN/m

External Transverse beams (Axis A:1-3 and Axis D:1-3)
Self weight of beam (factored) = 1.35 × 0.23m × 0.25m × 25 = 1.94 kN/m
Load from slab = nlx/3 = (16.6 × 5)/3 = 27.667 kN/m
Load from block work = 12.145  kN/m
Total load = 41.752 KN/m

Internal Longitudinal Beam (Axis 2:A-D)
Self weight of beam (factored) = 1.35 × 0.23m × 0.25m × 25 = 1.94 kN/m
Load from slab = nlx/2(1 - 1/3k2) = [(16.6 × 5)/2] × (1 - 1/(3 ×1.22)) = 2 × 31.893 kN/m = 63.786 kN/m (we multiplied by two because this beam is receiving slab load from two directions.)
Load from block work = 0 (there are no block works inside the building)
Total load = 65.726 KN/m



Internal Transverse beams (Axis B:1-3 and Axis C:1-3)
Self weight of beam (factored) = 1.35 × 0.23m × 0.25m × 25 = 1.94 kN/m
Load from slab = nlx/3 = (16.6 × 5)/3 = 2 × 27.667 kN/m = 55.334 kN/m (slab load is coming from two directions)
Load from block work = 0
Total load = 57.274 KN/m

Self weight of Columns
Self weight of column (factored) = 1.35 × 0.23m × 0.23m × 3.75m × 25 kN/m3 = 6.695 kN

Structural Analysis
A little consideration will show that the support reactions for the floor beams (equal spans) can be obtained by considering the shear force coefficients given below;

Fig 3: Shear force coefficient (2 - span beam)

Fig 4: Shear force coefficient (3 span beam)


Analysis of the column axial loads
Load Analysis of Column A1
1st Floor
Load from longitudinal roof beam (1:A-D) = 0.4 × 10.35 kN/m × 6m = 24.84 kN
Load from transverse roof beam (A:1-3) = 0.375 × 10.35 kN/m × 5m = 19.41 kN
Self weight of column = 6.695 kN
Total = 50.945 kN


Ground Floor
Load from above = 50.945 KN
Load from longitudinal floor beam (1:A-D) = 0.4 × 45.978 kN/m × 6m = 110.347 kN
Load from transverse floor beam (A:1-3) = 0.375 × 41.752 kN/m × 5m = 78.285 kN
Self weight of column = 6.695 kN
Total = 246.272 kN

Load Analysis of Column A2
1st Floor
Load from longitudinal roof beam (2:A-D) = 0.4 × 10.35 kN/m × 6m = 24.84 kN
Load from transverse roof beam (A:1-3) = 2(0.625 × 10.35 kN/m × 5m) = 64.687 kN
Self weight of column = 6.695 kN
Total = 96.222 kN

Ground Floor
Load from above = 96.222 KN
Load from longitudinal floor beam (2:A-D) = 0.4 × 65.726 kN/m × 6m = 157.742 kN
Load from transverse floor beam (A:1-3) = 2(0.375 × 41.752 kN/m × 5m) = 156.57 kN
Self weight of column = 6.695 kN
Total = 417.229 kN

Load Analysis of Column B1
1st Floor
Load from longitudinal roof beam (1:A-D) = (0.5 × 10.35 kN/m × 6m) + (0.6 × 10.35 kN/m × 6m) = 68.31 kN
Load from transverse roof beam (B:1-3) = 0.375 × 10.35 kN/m × 5m = 19.41 kN
Self weight of column = 6.695 kN
Total = 94.415 kN

Ground Floor
Load from above = 94.415 KN
Load from longitudinal floor beam (1:A-D) = (0.5 × 45.978 kN/m × 6m) + (0.6 × 45.978 kN/m × 6m)   = 303.455 kN
Load from transverse floor beam (B:1-3) = 0.375 × 57.274 kN/m × 5m = 107.388 kN
Self weight of column = 6.695 kN
Total = 511.953 kN

Load Analysis of Column B2
1st Floor
Load from longitudinal roof beam (2:A-D) = (0.5 × 10.35 kN/m × 6m) + (0.6 × 10.35 kN/m × 6m)= 68.31 kN
Load from transverse roof beam (A:1-3) = 2(0.625 × 10.35 kN/m × 5m) = 64.687 kN
Self weight of column = 6.695 kN
Total = 139.692 kN

Ground Floor
Load from above = 139.692 KN
Load from longitudinal floor beam (2:A-D) = (0.5 × 65.726 kN/m × 6m) + (0.6 × 65.726 kN/m × 6m)  = 433.7916 kN
Load from transverse floor beam (A:1-3) = 2(0.625 × 57.274 kN/m × 5m) = 357.9625 kN
Self weight of column = 6.695 kN
Total = 938.141 kN





As you can see, that was very straightforward with the use of shear force coefficients, given the fact that the beams were of equal span. When the beams are not equal, you have to carry out the analysis, and transfer the shear forces accurately. Now, let us compare load analysis of column B2, by the tributary area method.

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12 comments:

  1. This made what i have being working on easy for me. But please can this be used for steel structures too?. Thanks

    ReplyDelete
    Replies
    1. I am happy​ that you found this helpful. The answer to your question is yes, shear force from beams are transferred to columns as axial load in steel structures.

      Delete
    2. halo i see your articles as very helpful. I have checked the calculations above and checked with Staad.pro i have realized that the loads are higher in staad.pro.
      Can you give an explaination please. Thank you very much

      Delete
  2. Boss in fact you are too much! I always like your approach each subject. Thank you for your educational materials

    ReplyDelete
  3. A little observation here bro.
    The loads from each beam to the columns should be divided by 2 since adjacent columns bear half the load of the beam they support not all.
    You multiplied by 6m instead of 6/2 in the analysis of Col A1 and so on.
    Pls review them coz I think they are wrong.

    Eg.
    Load Analysis of Column A1
    1st Floor
    Load from longitudinal roof beam (1:A-D) = 0.4 × 10.35 kN/m × 6/2m = 12.42 kN
    Load from transverse roof beam (A:1-3) = 0.375 × 10.35 kN/m × 5/2m = 9.703 kN
    Self weight of column = 6.695 kN
    Total = 28.818 kN

    Likewise other columms analysis.

    ReplyDelete
    Replies
    1. No Sir. The values are the support reactions being directly transmitted to the columns. All load distribution has been done during the analysis of the beams.

      So 0.4 x q x 6m = support reaction at A. Why share it again?

      Delete
    2. I see bro. Can you explain why or how the 0.4, 0.375,0.625,x q x l
      . I thought its half loads transfered from beams. ie 0.5
      Thanks man

      Delete
    3. OK. I got your work now. I just learnt a new thing.

      How do we calculate the Shear force coefficients? In case of other beams of unequal span?
      Pls thanjst

      Delete
    4. If the spans are unequal, it is still the same thing... You analyse the beam using any method of your choice (force method, 3 moment equation, slope deflection method etc). No rocket science involved

      Delete
  4. Thanks for this. You are doing a great job for us that wanna know more. Thumbs up!

    ReplyDelete
  5. Good job as usual. Pls can you come up with an example of rotating or vibrating equipment support foundation base design. Thanks.

    ReplyDelete