DERIVATION OF A REFINED EQUATION FOR CALCULATING THE

ULTIMATE LOAD OF REINFORCED CONCRETE WALLS

MAGDY ABDEL MOHSEN1, TAMER ALAFANDY

2

ABSTRACT

The current paper aimed to studying available equations for calculation of

vertical ultimate load of reinforced concrete (RC) walls subjected to bending in one

direction according to different codes including Egyptian Code (ECP203-07) and

comparing the results obtained by this equations with those obtained from 24 loading

tests of reinforced concrete walls having different dimensions, cross sections,

reinforcement and slenderness ratios. From this comparison we found that existing

equations given by different codes do not give accurate estimation of vertical capacity

of walls and consequently a new accurate and practical equation has been developed

and give very close results compared to experimental ones.

KEYWORDS: Concrete, Experimental Tests, Load Bearing Walls, Ultimate Load .

1. INTRODUCTION

There is big discrepancy between different codes for calculation of ultimate

vertical capacity of reinforced concrete walls. For example the ACI318-08 and

ECP203-07 Codes neglects the effect of reinforcement in the wall. British and Euro

Codes design RC walls as columns taking into consideration the presence of

reinforcement. When comparing the results obtained from different codes with those

obtained from experimental it has been found that there is a big difference of values

1Assist. Prof., Al-Azhar University

2Assist. Prof, Housing and Building Research Center

Email: drmagdy_amb@yahoo.com

- 1 -

obtained by different codes. A modification for one of the equations available for

design of RC columns has been done to be used for design of RC walls and this

equations give a very accurate and close to the experimental ones.

2. CURRENT EQUATIONS USED FOR CALCULATING ULTIMATE LOADS

OF RC WALLS

2.1 American Code ACI318-08:

ACI 318–08 empirical wall formula which is for relatively short walls with small

moments, taking into consideration an eccentricity of h/6; the equation takes the

following form:

2

u32

1` 0.55 Ph

kLAf c

gc (1)

2.2 Egyptian Code ECP203-07:

2

u32

135.0 0.8 Ph

kHAf

ccu (2)

32

1Af0.28

2

ccu

h

kH

32

1A`f0.35

2

cc

h

kH

2.3 British, Euro and Hong Kong Codes:

Pu = 0.35 fcu Ac + 0.67 fy Asc (3)

As it has been proposed by BS, Euro and Hong Kong Codes that RC wall can be

designed as a column taking into account an eccentricity equal to 2 cm or 0.05 h

whichever is less, so we can suppose that RC wall can be designed by using column

equation given in ACI 318-08.

- 2 -

2.4 ACI 318-08 Tied Column Equation:

Pu= 0.8 [0.85 `cf (Ag-Ast) + fy Ast] (4)

For tied column =0.65

Pu = 0.442 `cf (Ag-Ast) + 0.52fy Ast (4-1)

Taking into account that `cf = 0.8 fcu

Equation (4) then becomes

Pu= 0.3536 fcu (Ag-Ast) + 0.52 fy Ast (4-2)

If we did not take into account accidental eccentricity equation (4) takes the two

following forms:

Pu= 0.5525 f`c (Ag-Ast) + 0.65 fy Ast (5)

Pu= 0.442 fcu (Ag-Ast) + 0.65 fy Ast (6)

As a suggestion for our new equation the factor 0.5 were used instead of 0.5525 or

0.442, so the proposed equation takes the form

Pu= 0.5 fcu (Ag-Ast) + 0.65 fy Ast (7)

Based on equation (7) the ultimate load of 24 tests of RC walls in one way in plane

action with an eccentricity h/6 and hinged conditions at top and bottom of the panel for

different parameters that influence the ultimate strength, which include aspect ratio

(h/L), slenderness ratio (h/t) and amounts of steel in vertical and horizontal direction

(v, h).

Typical arrangement of reinforcement is shown in Fig 1. Details of reinforcement and

yield strength of bars are shown Table 2. The load was uniformly distributed over the

length of the panel an applied at eccentricity equal to one six of the panel thickness.

The panel reinforcement was placed in two layers, symmetrically, on the two faces for

effective resistance to eccentric loading.

- 3 -

Calculated ultimate loads of RC panels using the different mentioned equations given

by different codes and by the new proposed equation are given in Table 3. From the

Table it is very clear that no one of the equations given by different codes match well

with the experimental results and only the new proposed equation gives an excellent

prediction of the ultimate load of RC panels. Also the proposed equation predicted

values have very strong correlation with the experimental ones as the coefficient of

correlation Rsqr = 0.953 which is almost equal to unity.

The maximum difference of the experimental values from those obtained by the

proposed equation is 18.7% and 45%, other values are practically equal and by the

ACI equation is 37% to 93% and by ACI column equation is 1% to 80% also this

difference for BS and Euro Codes are in the range of 5% to 80% and for the Egyptian

Code of Practice is 400% to 3100%.

F. 1. Arrangement of Reinforcement in Panels

L=900 mm

h=

60

0 m

m

10 mm

Sectional front elevation

2 mm bars 11 nos. on each face

3 mm bars 4 nos. on each face

Sectional plan

50m

m

Sectional side view

t=50mm

2 mm bars 11 nos.

3 mm bars 4 nos.

2 mm bars 11 nos. equally spaced 10mmcover

- 4 -

Table 1: Effective Length factors for Load -bearing walls

1- Walls braced top and bottom against translation and

(a) Restrained against rotation at one both ends (top and/or bottom) 0.80

(b) Not restrained against rotation at either end 1.0

2- For walls not braced against lateral transition 2.5

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Table 2.1: details of tested panels

Variable

Considered

Panel

designation

Size of panel

hLt

(mm)

h/L h/t fcu

(MPa)

(a) WAR Series

Aspect

ratio

WAR-1 600×900×50 0.67 12 22.33

WAR-2 600×600×50 1 12 22.33

WAR-3 600×400×50 1.5 12 22.33

WAR-4 600×300×50 2 12 22.33

(b) WSR Series

Slenderness

ratio

WSR-1 450×300×50 1.50 9 21.67

WSR-2 600×400×50 1.50 12 21.67

WSR-3 900×600×50 1.50 18 21.67

WSR-4 1350×900×50 1.50 27 21.67

(c) WSTV-I Series

Vertical

reinforcement

WSTV-1 600×900×50 0.67 12 25.17

WSTV -2 600×900×50 0.67 12 25.17

WSTV -3 600×900×50 0.67 12 25.17

WSTV -4 600×900×50 0.67 12 25.17

(d) WSTV-II Series

Vertical

reinforcement

WSTV-5 1200×800×50 1.50 24 22.83

WSTV -6 1200×800×50 1.50 24 22.83

WSTV -7 1200×800×50 1.50 24 22.83

WSTV -8 1200×800×50 1.50 24 2.83

(e) WSTH-I Series

Horizontal

reinforcement

WSTH-1 600×900×50 0.67 12 24.50

WSTH -2 600×900×50 0.67 12 24.50

WSTH -3 600×900×50 0.67 12 24.50

WSTH -4 600×900×50 0.67 12 24.50

(f) WSTH-II Series

Horizontal

reinforcement

WSTH-5 1200×800×50 1.50 24 20.17

WSTH -6 1200×800×50 1.50 24 20.17

WSTH -7 1200×800×50 1.50 24 20.17

WSTH -8 1200×800×50 1.50 24 20.17

- 6 -

Table 2.2: details of tested panels

Variable

Considered

Panel

designation

Vertical Steel Horizontal Steel

Failure

Load

(kN)

Bars

in

one

layer

fy

(MPa)

(%)

Bars

in

one

layer

fy

(MPa)

(%)

(a) WAR Series

Aspect

ratio

WAR-1 2-

11a

297 0.173 3-4 286 0.199 484.27

WAR-2 2-7 297 0.173 3-4 286 0.199 314.8

WAR-3 2-5 297 0.173 3-4 286 0.199 198.29

WAR-4 2-4 297 0.173 3-4 286 0.199 147.4

(b) WSR Series

Slenderness

ratio

WSR-1 2-4 297 0.165 3-3 286 0.199 214.18

WSR-2 2-5 297 0.165 3-4 286 0.199 254.1

WSR-3 2-7 297 0.165 3-6 286 0.199 298.92

WSR-4 2-11 297 0.165 3-9 286 0.199 373.65

(c) WSTV-I Series

Vertical

reinforcement

WSTV-1 2-11 297 0.173 3-4 286 0.199 N/A

WSTV -2 3-10 286 0.331 3-4 286 0.199 534.07

WSTV -3 4-9 581 0.528 3-4 286 0.199 583.52

WSTV -4 5-10 570 0.845 3-4 286 0.199 704.14

(d) WSTV-II Series

Vertical

reinforcement

WSTV-5 2-10 297 0.177 3-8 286 0.199 338.73

WSTV -6 3-9 286 0.335 3-8 286 0.199 398.56

WSTV -7 4-8 581 0.528 3-8 286 0.199 463.28

WSTV -8 5-9 570 0.856 3-8 286 0.199 503.1

(e) WSTH-I Series

Horizontal

reinforcement

WSTH-1 2-11 297 0.173 3-4 286 0.199 N/A

WSTH -2 2-11 297 0.173 4-4 581 0.352 538.10

WSTH -3 2-11 297 0.173 4-5 581 0.440 528.11

WSTH -4 2-11 297 0.173 5-4 570 0.507 528.11

(f) WSTH-II Series

Horizontal

reinforcement

WSTH-5 2-10 297 0.176 3-8 286 0.199 N/A

WSTH -6 2-10 297 0.176 4-8 581 0.352 348.74

WSTH -7 2-10 297 0.176 4-10 581 0.440 343.74

WSTH -8 2-10 297 0.176 5-8 570 0.507 348.74

- 7 -

Table 3.1: Analytical Failure Loads of Panels using the Available Codes(N)

ACI318-

08(1)

L Pv h k t fy fcu Panel

designation

11910.94 900 1.73E-03 600 1 50 297 22.33 WAR-1

7940.625 600 1.73E-03 600 1 50 297 22.33 WAR-2

5293.75 400 1.73E-03 600 1 50 297 22.33 WAR-3

3970.313 300 1.73E-03 600 1 50 297 22.33 WAR-4

4254.551 300 1.65E-03 450 1 50 297 21.67 WSR-1

5293.75 400 1.65E-03 600 1 50 297 21.67 WSR-2

6316.406 600 1.65E-03 900 1 50 297 21.67 WSR-3

3992.871 900 1.65E-03 1350 1 50 297 21.67 WSR-4

11910.94 900 1.73E-03 600 1 50 297 25.17 WSTV-1

11910.94 900 3.31E-03 600 1 50 286 25.17 WSTV -2

11910.94 900 5.28E-03 600 1 50 581 25.17 WSTV -3

11910.94 900 8.45E-03 600 1 50 570 25.17 WSTV -4

5390 800 1.77E-03 1200 1 50 297 22.83 WSTV-5

5390 800 3.35E-03 1200 1 50 286 22.83 WSTV -6

5390 800 5.28E-03 1200 1 50 581 22.83 WSTV -7

5390 800 8.56E-03 1200 1 50 570 22.83 WSTV -8

11910.94 900 1.73E-03 600 1 50 297 24.5 WSTH-1

11910.94 900 1.73E-03 600 1 50 297 24.5 WSTH -2

11910.94 900 1.73E-03 600 1 50 297 24.5 WSTH -3

11910.94 900 1.73E-03 600 1 50 297 24.5 WSTH -4

5390 800 1.76E-03 1200 1 50 297 20.17 WSTH-5

5390 800 1.76E-03 1200 1 50 297 20.17 WSTH -6

5390 800 1.76E-03 1200 1 50 297 20.17 WSTH -7

5390 800 1.76E-03 1200 1 50 297 20.17 WSTH -8

- 8 -

Table 3.2: Analytical Failure Loads of Panels using the Available Codes (N)

Proposed

Equation(7)

ACI 318-08 tied

column(4-1)

British, Euro and

Hong Kong(3)

ECP203-07

(2) Panel

designation

4.58E+05 366723.4 3.67E+05 109905.469 WAR-1

3.06E+05 244482.3 2.45E+05 73270.3125 WAR-2

2.04E+05 162988.2 1.63E+05 48846.875 WAR-3

1.53E+05 122241.1 1.22E+05 36635.1563 WAR-4

1.48E+05 118570.4 1.19E+05 47017.9746 WSR-1

1.98E+05 158093.9 1.58E+05 47403.125 WSR-2

2.96E+05 237140.8 2.37E+05 34841.2969 WSR-3

4.45E+05 355711.3 3.56E+05 6666.06445 WSR-4

5.15E+05 411835.3 4.12E+05 123883.594 WSTV-1

5.27E+05 421331.2 4.25E+05 123883.594 WSTV -2

5.88E+05 470174.1 4.89E+05 123883.594 WSTV -3

6.37E+05 509826.9 5.42E+05 123883.594 WSTV -4

4.17E+05 333270.3 3.34E+05 15981 WSTV-5

4.27E+05 341754.3 3.45E+05 15981 WSTV -6

4.81E+05 385010.3 4.02E+05 15981 WSTV -7

5.27E+05 421630.8 4.50E+05 15981 WSTV -8

5.01E+05 401192.7 4.01E+05 120585.938 WSTH-1

5.01E+05 401192.7 4.01E+05 120585.938 WSTH -2

5.01E+05 401192.7 4.01E+05 120585.938 WSTH -3

5.01E+05 401192.7 4.01E+05 120585.938 WSTH -4

3.70E+05 295655 2.96E+05 14119 WSTH-5

3.70E+05 295655 2.96E+05 14119 WSTH -6

3.70E+05 295655 2.96E+05 14119 WSTH -7

3.70E+05 295655 2.96E+05 14119 WSTH -8

- 9 -

Table 4: Ratios between Experimental and Analytical Failure Loads

Exp/(7) Exp/(6) Exp/(4-2) Exp/(3) Exp/(2) Exp/(1) Experimental

(kN)

Panel

designation

1.06 1.20 1.320532 1.32 4.406241 40.65759 484.27 WAR-1

1.03 1.17 1.287619 1.29 4.29642 39.64423 314.8 WAR-2

9.73E-01 1.10 1.216591 1.22 4.05942 37.45738 198.29 WAR-3

9.65E-01 1.09 1.205813 1.20 4.023458 37.12554 147.4 WAR-4

1.45 1.63 1.806353 1.80 4.555279 50.34139 214.18 WSR-1

1.29 1.45 1.607273 1.61 5.360406 48 254.1 WSR-2

1.01 1.14 1.260517 1.26 8.579474 47.32438 298.92 WSR-3

8.40E-01 9.51E-01 1.050431 1.05 56.05256 93.57928 373.65 WSR-4

N/A N/A N/A N/A N/A N/A N/A WSTV-1

1.01 1.15 1.267578 1.26 4.311063 44.83862 534.07 WSTV -2

9.93E-01 1.12 1.241072 1.19 4.710228 48.99027 583.52 WSTV -3

1.10 1.25 1.381136 1.30 5.683884 59.11709 704.14 WSTV -4

8.13E-01 9.20E-01 1.016382 1.02 21.1958 62.84416 338.73 WSTV-5

9.33E-01 1.06 1.166218 1.15 24.93962 73.94434 398.56 WSTV -6

9.63E-01 1.09 1.203292 1.15 28.98942 85.95176 463.28 WSTV -7

9.55E-01 1.08 1.193224 1.12 31.48113 93.33952 503.1 WSTV -8

N/A N/A N/A N/A N/A N/A N/A WSTH-1

1.07 1.21 1.341026 1.34 4.461631 45.16941 538.10 WSTH -2

1.05 1.19 1.31635 1.32 4.379532 44.33824 528.11 WSTH -3

1.05 1.19 1.31635 1.32 4.379532 44.33824 528.11 WSTH -4

N/A N/A N/A N/A N/A N/A N/A WSTH-5

9.44E-01 1.07 1.179551 1.18 24.70005 64.7013 348.74 WSTH -6

9.30E-01 1.05 1.162639 1.16 24.34592 63.77365 343.74 WSTH -7

9.44E-01 1.07 1.179551 1.18 24.70005 64.7013 348.74 WSTH -8

- 10 -

CONCLUSIONS

According the discussion mentioned above, it is clear that all the existing formulas for

predicting ultimate load of RC wall are not correct at all and a new much more

accurate equation is proposed which is working properly.

REFERENCES

[1] S. Madina Saheb and Prakash Desayi, “Ultimate Strength of RC Wall Panels in

One Way in Plane Action”, ASCE, JSED, Vol. 115, No. 10, October, 1989

[2] Jack C McCormac, James K Nelson, "Reinforced Concrete Design",7th

edition,

2006

[3] ACI 318-08 Metric Units

[4] Reinforced Concrete Code of Practice, Housing Dept. May 2008, Hong Kong

[5] British Code of Practice of Reinforced Concrete

[6] Euro Code of Practice of Reinforced Concrete

LIST OF SYMBOLS

Pu = Ultimate Load

= 0.7

Lc = vertical distance between supports

h = overall thickness of wall

k = effective length factor determined in accordance with the values given in

Table 1.

f`c = cylinder strength

Ag = gross sectional area of wall section

fcu = Cube strength

Ac = gross sectional area of wall cross section

H = vertical distance between supports

fy = steel yield strength

Asc = area of reinforcement steel

Ast = area of reinforcing steel

= 0.65 for tied column

- 11 -

:مخلص البحث

للحوائط الخرسانية المعرضة لألنحناء األقصىيهدف البحث إلى دراسة المعادالت الخاصة لحساب الحمل الرأسي

42واحد طبقا لألكواد المختلفة بما فيها الكود المصري ومقارنة نتائج هذه المعادالت مع القيم الناتجة من اتجاهفي

.قطاعات وتسليح ونسبة نحافة مختلفة تجربة تحميل لحوائط خرسانية بأبعاد و

للحوائط ى ومن هذه المقارنة تم الخلوص إلى أن المعادالت الحالية ال تعطي قيما دقيقة للحمل الرأسي األقص

ووجد أنها تعطي نتائج ى الخرسانية ومن ثم تم استنتاج معادلة دقيقة وعملية في نفس الوقت لحساب الحمل األقص

.العملية من قيم التجارب المعملية قريبة جداً من الناحية