RC-TOOL

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RC-TOOL Software

Summary

This program performs analysis and design of RC sections at ultimate state under bending and

axial force according to Saudi SBC and American ACI codes.

Analysis option includes drawing of moment - axial force interaction curve at ultimate state, as

well as moment - curvature - stiffness relationships under an increasing load requiring

numerically integrated material models.

Various section types with one, two (equal or unequal) or more than two steel layers can be

designed or analyzed.

Positive or negative moments and axial forces as well as slenderness effects are all considered.

Optimal (minimum steel) design is based on an original and powerful fast-converging re-analysis

algorithm.

Developed by Professor Abdelhamid Charif

King Saud University, Civil Engineering Department.

Last edited in April 2009.

The following section types can be analyzed and designed:

Rectangular, Double-rectangular (T), Triple-rectangular, Circular, Tubular, Sections combining

trapezes with possible holes

The last option allows generation of complex box sections.

Standard RC Design can be achieved in one of these ways:

1/ One or two unknown steel layers A and A'

2/ Two steel layers with a prefixed value for A'

3/ Two equal steel layers A = A'

4/ Many steel layers with prefixed ratios

5/ Special beam design with fixed bar diameter checking bar spacing and number of layers

A' is the compression steel layer in beam bending. It is the top layer for a positive bending

moment. For columns subjected to bending and axial force, A' is the top layer for positive

moments. A' is therefore either the most compressed or the least tensioned steel layer.

Combination of a bending moment (positive or negative) with an axial force (compression or

tension) is the general case. Beam bending is a particular case with zero axial force.

The first design option is the standard case where one or two layers are required. The software

determines whether the second layer is necessary or not. The second option with an imposed

value of A' may be useful if for instance the latter was determined by a design for a negative

moment or simply to account for top bars used to hold stirrups. Considering the existence of A'

will in general reduce the required value of A unless a top layer A' greater than the imposed value

is necessary. In this case the imposed value of A' will be discarded by the software.

The third design option is particularly convenient for columns where symmetric reinforcement is

frequently adopted.

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The fourth design option is an invaluable tool if many steel layers are to be used. This is the case

in most columns and in all shear walls. However, the steel layers cannot have any independent

values. They must be related by ratio or proportionality values R(i) which are defined as :

R(i) = A(i) / At where At is the total steel area.

For equal layers, use the same (any) number for all layers.

For layers using the same bar diameter, use the bar number in each layer i as its ratio R(i).

The first four design options are in fact reduced to two types since the first option is a particular

case of the second with a zero imposed value for A'. The third option is also treated as special

case of the fourth one where the ratios of the two layers are equal.

From a theoretical and numerical point of view, the first general case is much more complex than

the second. Both the total steel amount and its distribution over the two steel layers are unknown

in the first general case. In the other case, only the total steel amount is unknown.

An original fast-converging re-analysis algorithm is used to determine the minimum total steel

and the appropriate layer distribution with no a priori value for the strength reduction factor. The

latter is treated as an unknown variable, and updated depending on tension steel strain.

The fifth and last design option is specific to rectangular and T-beams with a fixed bar diameter.

Special RC beam design is performed by successive re-analysis-checks, by updating the number

of steel layers and checking bar spacing until the required capacity is reached or just exceeded.

Steel layer position is defined by its depth, which is the distance from the top concrete fiber to

the steel centroid.

The user must first choose "design" or "analysis" option, fix the section type and dimensions,

before entering the relevant reinforcement data. Concrete displaced by steel in compression may

be considered (default, recommended) or not. The code condition imposing a lower limit (0.005

for SBC and 0.004 for ACI) on steel tensile strain in beam bending problems or when the axial

force is less than (0.10 f'c.Ag) may be considered with either of ACI and SBC limits. The

condition is implemented in standard design with one or two layers.

Nominal and design P-M interaction curves are produced with the code compression limit. The

safe design zone is shaded.

Design option delivers the required optimum steel, the diagrams of strain and stress distributions

as well as the P-M interaction curve showing the loading point (Mu , Pu) on the border of the safe

design region. Detailed contributions of concrete and steel to nominal and design forces and

moments are also delivered.

The interaction curve is not limited to compressive forces but also includes axial tensile forces. It

is useful to remind that lateral loading (earthquake, wind) may cause some RC columns to be

subjected to axial tension forces.

Combination of loops over material strain and neutral axis depth is used to track all the

interaction curve points. This technique is more accurate than the method seeking the interaction

points for various levels of axial forces. The latter is particularly inappropriate in the transition

zone where variation of the strength reduction factor generates in some cases non-convex design

interaction curves, thus leading to many interaction points for the same axial force. The strain

compatibility method used in RC-TOOL with the adequate evaluation of the strength reduction

factor (expressed in terms of steel strain), captures all the nominal and design interaction points,

whatever the section anti-symmetry. Such a construction of the interaction curve including all

tension controlled cases requires the definition for steel of a fracture (ultimate) strain. This

parameter depends on steel grade and varies from some 20 millistrains for high yield steel to

over 150 millistrains for mild steel and is not defined in ACI and SBC codes. This omission

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assumes in fact an infinite value for steel strain, and has no major effect on the tension capacity

of the section. It discards however completely all failures by steel fracture which may yet happen

even in beam bending with no axial tension. It also overestimates ductility in moment-curvature

relations. European and international codes set conservative values for the steel ultimate strain

ranging from 0.010 to 0.020. An unlimited value (ACI / SBC) is assumed by default but the user

can change this and enter any positive value greater than or equal to 0.010. This allows interested

users to investigate the effect of this parameter.

With the drawn interaction diagram, the user may check any other combination of ultimate axial

force & moment (Pu-Mu). Detailed material contribution for particular points or any user-defined

point on the interaction curve may be obtained. This option combined with possibilities of

analyzing unreinforced sections / using zero-strength concrete offers investigation opportunities

in RC material contributions. Other investigation options include effect of displaced concrete,

value of steel ultimate strain, lower limit of steel strain in near bending problems and moment-

curvature.

Moment-curvature-stiffness figures can be obtained for any value of the axial force (compression

or tension), using either a simplified equivalent rectangular concrete block, or one of many

concrete ACI compatible analytical models, combined with numerical integration. The stress

distribution across the section is shown at any stress level. This last option is a powerful RC

research tool.

An independent module for shear design is also included. It designs stirrups (number of legs and

spacing) including spacing variation along the beam span.

Design results are sent to the output file: "design.out"

Analysis & check results are in output file: "inter.out"

Moment-curvature results are in file: "curvature.out"

Slenderness check results are sent to file: "slender.out"

The same filenames are used for all runs. Long term saving requires renaming of appropriate

output files after the run.

Steel ratio is in general computed for the total steel area compared to the concrete gross section.

However, in bending problems, tension steel ratio, compression steel ratio, as well as balanced

ratio and minimum ratio, are all computed.

The software offers original and unmatched features and is presented in compact form with one

single executable file.

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THEORETICAL BACKGROUND

1. INTRODUCTION Design of reinforced concrete sections subjected to bending moments and axial forces according

to the American code ACI or Saudi code SBC, is a very tedious task. Apart from very simple

cases (rectangular sections with one steel layer only), no direct design method is available. The

engineer can only make initial assumptions including a certain amount of steel reinforcement and

a starting strength reduction factor and, then analyze the section and check its safety as well as

the validity of his initial assumptions. The main reason for this difficulty lies first in the

nonlinear mechanical and geometrical properties and also in the unknown strength reduction

factor to be used, especially in the case of columns subjected to both bending and an axial force.

Sound design from safety and economic point of view can only be achieved with many cycles of

analysis trials and checks, which are beyond reach for hand calculations. Even some well

established commercial computer programs have recourse to simplified assumptions (with

prefixed values of strength reduction factors usually) to deliver the design results. It is frequent

that the latter contradict the initial assumptions and non-advanced users usually do not notice this

fact. RC-TOOL software relaxes all these limitations and complies fully with the American code

ACI and new Saudi Building Code SBC, and this undoubtedly constitutes an invaluable tool for

civil and structural engineers.

2. SCOPE OF RC-TOOL The main objective of this work is to develop a friendly computer program for the analysis and

design of various section types reinforced with one, two or many steel layers. Visual

programming techniques are used and a graphical user interface is developed.

The software produces graphical results such as axial force – bending moment interaction curves

and distribution of strains and stresses across the section.

Many types of sections are treated (Figure 1) such as:

Rectangular

Double Rectangular (T)

Triple Rectangular

Circular

Tubular

Composite section combining many trapezes with possible holes

Combination of trapezes with possible holes can generate complex sections as shown in Figures

1g and 1h. Unixaial bending requires, however, that the section has a vertical axis of symmetry.

Dissymmetric sections are analyzed in biaxial bending in RC-BIAX software. The analysis of a

section with no reinforcement or reinforced with up to 200 steel layers delivers the nominal and

design interaction curve P-M for positive and negative moments and axial forces. The user can

then check the safety of any combination of axial force and bending moment.

The design (determination of steel reinforcement) can be achieved in one of these ways:

One or two steel layers A and A’ ( A is the bottom layer and A’ is the possible top layer)

Two steel layers with an imposed value for A’

Two equal steel layers A = A’

Many steel layers (up to 200) with prefixed ratios

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An original and powerful fast-converging re-analysis algorithm is used for this purpose. Detailed

results include graphical strain and stress distributions and material (concrete and steel layers)

contributions to forces and moments. The P-M interaction diagram is also delivered showing the

loading combination just on the border of the safe zone (optimum design).

Figure 1: Types of sections analyzed and designed

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3. RC ANALYSIS AND DESIGN Modern reinforced concrete design philosophy is based on the so-called ultimate state behavior,

and on the concept of load and resistance factors [1-3]. The load factors depend mainly on the

type of loading (dead, live or accidental) and have known fixed values. For many materials,

strength reductions factors are also known in advance. However for reinforced concrete sections

subjected to bending and axial forces, this important factor is unfortunately variable because of

the nature of failure which can vary from a very ductile one (tension controlled), with a reduction

factor of 0.90, to a brittle compressive failure with a factor of 0.65. Its value is related to the

tensile steel strain which is unknown. In a design problem, equilibrium equations are therefore

not sufficient to solve for the unknown required steel reinforcement. Only analysis methods are

therefore available.

Reinforced concrete textbooks [1-2] describe details of the various steps required for a section

analysis with an assumed steel reinforcement. Guidelines are given for design purposes by using

recommended steel percentages. These percentages give steel areas depending only on the

concrete gross section and may turn out to be either exaggerated (over-design) or insufficient

(under-design). Many re-analysis cycles become necessary.

Computer programs dealing with reinforced concrete design are very limited in quantity and

quality. Some of them are included in large packages of structural analysis and design and are

impractical for direct RC design [4-5]. Others tend to use exaggerated simplifications such as

ignoring the variability of the strength reduction factor, and usually produce erroneous design

results from a rigorous code point of view.

The developer of RC-TOOL has a long experience with software development according to

various codes of practice [6-7].

4. SOFTWARE DEVELOPMENT STRATEGY Modern programming techniques (Visual Fortran language) are used to develop the software. A

friendly user graphical interface offering many viewing options and mouse driven menus is

included. The analysis and design concern positive and negative bending moments as well as

compressive and tensile axial forces. It is worthwhile pointing out that many textbooks

(especially American ones) do not cover the combination of bending with tensile forces. It is yet

well known that lateral loads such as winds or earthquakes may cause some columns of a

building to be subjected to tension (uplift) forces. The optimum design (minimum required steel)

is obtained using an original fast-converging re-analysis method with an accelerated double loop:

Distribute a given total steel area over various layers to obtain the nearest solution. Appropriate

correction factors for both total steel quantity and layer distribution are then obtained for a rapid

convergence with no a-priori condition about the value of the strength reduction factor.

The section analysis is performed by varying the neutral axis depth from plus infinity (pure

compression) to minus infinity (pure tension) and finding for each value the corresponding

nominal axial force and bending moment (see flow-chart in Section 6). Design values are

deduced according to the relevant strength reduction factor determined with the actual steel

strain.

Design is performed with an iterative procedure. Successive values of steel layers are assumed

until the point corresponding to the ultimate axial force and ultimate bending moment is just

inside (tangent to) the design interaction curve. Appropriate acceleration techniques are used to

determine the optimal (minimum) steel requirement. The software is validated by comparing

with standard examples in textbooks and other computer programs. Simple and complex

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examples of analysis and design of RC beams and columns are presented in details in references

[8] to [10]. Analysis results produce the nominal and design interaction curve allowing the user

to check the safety of any combination of axial force and bending moment. Design results

include the required steel as well as the graphical distributions of strains and stresses across the

section. Steel ratios below code minimum or greater than code maximum will be detected.

5. NOMINAL AND DESIGN STRENGTH OF AN RC SECTION A section is in a state of uniaxial bending if it is subjected to a single moment about one axis

while the second axis is an axis of symmetry (Figure 2). The neutral axis is then parallel to the

moment axis. If a dissymmetric section is subjected to a single moment, the neutral axis is not

parallel to the bending axis. In uniaxial bending, steel reinforcement can be described by

centered layers Asi with distances di from the top concrete fiber. Extreme layers are identified by

the depths mind and maxd .

When subjected to a positive or negative moment, and to compression or tension axial force, the

nominal capacity of the section will be reached when either of the two materials (concrete or

steel) reaches an ultimate state (Figure 2b). Either compressive concrete strain is equal to the

ultimate value cu = 0.003 or tension steel strain reaches its ultimate value

su . There is no

limitation on tension strain of concrete. Compression failure is controlled by concrete crushing

while tension failure is controlled by ultimate steel strain. The design strength is related to the

nominal strength through the strength reduction factor . This variable factor is defined by SBC

and ACI codes to account for the fundamental difference between ductile and brittle behaviors. It

is related to the tensile steel strain as shown in Figure 3 and cannot be known in advance even in

beam bending.

The possible mechanisms of failure when combining a positive or negative bending moment

with a compression or tension axial force are represented in Figure 4, with a neutral axis depth c

varying from infinity to minus infinity and considering that the top zone of the section is either

the most compressed or least tensioned. Four different parts are defined (Figures 4a to 4e)

according either to the value of the strength reduction factor or the type of failure. These parts

are described by variation in the neutral axis depth or material strain.

Part 1 (Fig.4a-4b): Neutral axis depth c varying from infinity to balanced conditions.

Concrete strain is equal to the ultimate value cu while steel strain varies until it reaches

the yield point. This part corresponds to the compression controlled zone (brittle concrete

crushing failure) with a strength reduction factor 0 (0.65 for tied columns and 0.70 for

spiral columns).

Part 2 (Fig.4b-4c): Tension steel strain varies from yield value y to 0.005. This

corresponds to the transition zone with a strength reduction factor varying linearly

from 0 to 0.90. Failure is again controlled by concrete crushing.

Part 3 (Fig.4c-4d): Tension steel strain varies from 0.005 to the ultimate value su . This

corresponds to the tension controlled zone ( = 0.90) with ductile failure controlled by

concrete crushing.

Part 4 (Fig.4d-4e): Concrete strain varies from the compression ultimate value until pure

tension where the neutral axis depth is minus infinity. This also corresponds to the

tension controlled zone ( = 0.90) but with failure controlled by steel breaking.

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Figure 2: Bending strains and stresses

(a): Symmetric section – (b): Strains in concrete and steel – (c): Stresses

Tied column: 0 = 0.65

Spiral column: 0 = 0.70

Figure 3: Variation of the strength reduction factor with tension steel strain s

Figure 4: Various parts and possibilities of strain distribution

(a): Pure compression – (b): Balanced point – (c): 0.005 steel strain

(d): Double failure point – (e): Pure tension

c

cu C

y

cu

cb

C

0.005

cu

c5

C cu

su

cu

C

S

c

su S

(a) (b) (c) (d) (e)

Compression control Transition zone Tension controlled zone

Concrete crushing control Steel breaking control

Part 1 Part 2 Part 3 Part 4

90.0

0

Strain s

s = 0.005 ys

Factor

Tension control Compression control Transition (Brittle)

Transiti

on

(Ductile)

Transiti

on

a

cuc

sus

c dmin

dmax

di

h

(b)

0.85'

cf

(c)

M

X

Y

v1

v2

c

(a)

Yb

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In the first three parts controlled by concrete crushing, variation of the strain and neutral axis

depth is performed by rotation about concrete pivot C whereas in the fourth, controlled by steel

breaking, rotation is performed about steel pivot S (Figure 4).

In parts 1 to 3, the rectangular concrete block is used with a stress 0.85 '

cf and a depth given by:

Compression block depth ca 1 with ha 0 (1)

Coefficient 1 is given by:

30)30,(

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05.085.0,65.0 '

1 cfMaxMax (2)

In part 4 however, concrete strain c is less than the ultimate value

cu and the stress block is

not rectangular with a value of '85.0 cf . In this case, we either integrate the parabolic equation or

use a mapped rectangular block with a stress cf85.0 where fc is the concrete stress related to

concrete strain c by a second degree parabolic equation.

Thus for strain c

cu

c

cc ff

' (3)

Exact tracking of the full nominal and design interaction curves requires using an iterative

procedure to scan these four parts as well as inverting the section to consider negative moments

or more generally cases where the top zone of the section is either the least compressed or the

most tensioned.

Parts 1 and 4 may each be subdivided into two subparts by separating the partial tensioned and

partial compression case from the case with a section either fully in compression (part 1) or fully

in tension (part 4).

Iterations can be performed either using the neutral axis depth c or the material strain. In order to

avoid useless points in part 1 and part 4 (with infinite values of c), alternative maximum and

minimum values of c are used. They are defined as follows:

In part 1, instead of plus infinity, the neutral axis depth c will vary from a maximum value

ensuring that the section is fully in the compression block and that the top steel layer has yielded

(if steel yield strain y is less than concrete ultimate strain cu , which is in general always the

case).

ycu

cudh

Maxc

max

1

max , (4)

In part 4, instead of minus infinity, the neutral axis depth c will vary to a minimum value

ensuring that the section is fully in tension and that the top steel has yielded.

ysu

sudddMinc

)(,0 minmaxmaxmin (5)

It must be noted that steel ultimate strain su is not defined in SBC, ACI and some other codes,

and this omission is in fact equivalent to considering an infinite value, which thus eliminates

steel breaking failure. Part 4 is thus reduced to the pure tension point. The software developed in

this work considers the SBC/ACI assumption by default but allows varying su from 0.01 to

infinity (any large number).

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Assuming that compression is positive, for each strain distribution, nominal values of axial force

and bending moment are given by:

i

sicn FBfP '85.0 (6)

)(85.0 1

'

i

i

sibcn dvFYBfM (7)

B is the area of the concrete compression block. Yb is the distance from the centroid of the

compression block to the centroid of the gross section (origin of centroid axes X and Y).

For a rectangular section b x h: abB and

22

ahYb (8)

v1 is the distance from the top fiber to the section centroid.

The expression of the force Fsi in steel layer i accounts for possible displaced concrete:

Layer outside compression block ( adi ): sisisi fAF (9a)

Layer inside compression block ( ad i ): '85.0 csisisi ffAF (9b)

fsi is the stress in steel layer i given by:

sissi Ef with ysiy fff (10)

The expression of the strain si in steel layer i, depends on the failure type:

For concrete crushing control ( cuc , parts 1 to 3): c

dc i

cusi

(11a)

For steel breaking control ( sus , part 4): cd

dc isusi

max

(11b)

It must be noted that in part 4, concrete strength fc given in expression (3) is used in equations

(6), (7) and (9b) instead of the ultimate strength '

cf .

6. ANALYSIS FLOW CHART The section analysis is performed by looping over the neutral axis depth c . It is varied from cmax

given by (4) to cmin given by (5) using the four parts described previously.

The following steps are then performed for each value of the neutral axis depth, measured from

the top fiber of the section.

For each value of c :

1. Compute nominal force (6)

2. Compute nominal moment (7)

3. Area of possible rectangular compression block is given by (8)

4. For each steel layer:

- Strain is given by (11)

- Stress is given (10)

- Force is given by (9)

5. Determine exact value of strength reduction factor from steel strains

6. Deduce design values of compression force and bending moment

7. Perform the same neutral axis depth loop and the same steps with the inverted section (by

measuring the neutral axis depth from the bottom fiber of the section)

8. Draw nominal and design interaction curves

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7. TYPICAL P-M INTERACTION CURVES Figures 5 and 6 show typical nominal and design P-M interaction curves for rectangular and T

sections. The safe design domain is shaded and limited by the SBC / ACI compression limit

given by:

stystgcn AfAAfP

'

0max0 85.0 (12)

with 80.0 for tied members and 85.0 for spiral members.

The maximum tension design force is given by:

styn AfP 90.0min (13)

The maximum compression and tension forces will correspond to pure compression and pure

tension if the section is symmetrical (concrete and steel symmetry) or more generally if the

reinforcement centroid coincides with that of the concrete gross section (about which moments

are computed).

We can see that while nominal curves are generally convex, for a T section (Figure 6) or a

rectangular section with dissymmetric reinforcement (Figure 5b), the design curve may have

non-convex parts corresponding to the transition zone of the strength reduction factor.

The nominal axial force decreases when tension steel strain increases from yield value to 0.005

but the design axial force may vary in the opposite direction because of the decrease in the

strength reduction in that zone. Iterative procedures based on axial force levels are therefore

inappropriate in the transition zone.

Figure 5b shows also that simple bending of beams is not always in the tension controlled zone

and with a variable strength reduction factor in the transition zone, many cycles of trial and

check would be required. More analysis examples are described in references [8] to [10].

8. RC DESIGN UNDER BENDING AND AXIAL FORCE Design equations in axial bending combined with an axial force are:

un PP and un MM (14)

These two equilibrium equations are rarely sufficient for direct design. The strength reduction

factor, the strain distribution (neutral axis or material strain) as well as steel areas, are all

unknown and cannot be directly determined with equations (14). With a finite value for ultimate

steel strain, the type of failure (concrete crushing or steel breaking) becomes also unknown.

Expression of concrete compression block, in cases of non-rectangular sections, adds extra

difficulty to the design problem. Direct design is possible only in the simplest case of bending of

rectangular beams with one steel layer assuming concrete crushing failure in tension-control. The

two remaining unknowns (steel area and strain distribution) are determined by equations (14).

However the initial assumption of tension-control must be checked and as mentioned previously

it frequently turns out to be untrue. If the section falls in the transition zone with a variable

strength reduction factor (as illustrated in Figure 5b), then many iterations will be required.

Computer re-analysis strategy is therefore the only systematic design method.

Four design options are considered in the axial bending case:

1/ One or two steel layers A and A’ including special beam design with a fixed bar diameter

2/ Two steel layers with an imposed value for A’

3/ Two equal steel layers A = A’

4/ Many steel layers (2 to 200) with prefixed ratios

12

(a): Two equal steel layers (b): One bottom steel layer

Figure 5: P-M interaction curve for a rectangular section

Figure 6: P-M interaction curve for a T-section

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A is the most tensioned or least compressed steel layer. A’ is the most compressed or least

tensioned steel layer. In beam bending, which is a mere particular case, A is the tension steel,

whereas A’ is the possible compression layer. The imposed top layer A’ in the second case may

be due to a negative moment or just to account for longitudinal bars used to hold stirrups. This

imposed value usually reduces the value of the required steel A. It will however be discarded if a

greater value is required.

The four cases can in fact be reduced to two only. The first standard case can be considered as a

particular case of the second (zero imposed value for A’). The third case is also a particular case

of the fourth where the two ratios are equal to 0.5.

The first group with two unknown layers is much more difficult as both total steel and layer

distribution are unknown. In the second group the reinforcement pattern is fixed and only total

steel is unknown. The various layers are given by their ratios as follows:

istsi RAA with i

iR 1 (15)

In the special beam design with a fixed bar diameter, bar spacing and steel layer are checked and

updated until a sufficient beam capacity is reached.

The re-analysis strategy used does not scan all the previous four parts. The flow chart, presented

in Section 6, is not executed in each iteration. The procedure is confined to a small relevant zone

of variation of the neutral axis depth, according to the values of ultimate moment and axial force.

The minimum starting value of steel area is easily determined using the axial force (tension or

compression) in equations (12) or (13) and therefore all design interaction curves generated will

cross the horizontal line corresponding to the ultimate axial force. Corresponding design

moments must then be checked using the ultimate moment.

The iterative procedure used is presented graphically. The minimum steel required will be

reached when equations (14) are satisfied otherwise the loading point is on the border of the safe

design region in an optimal way.

In the first group with two unknown layers, a double iterative procedure is carried out. Total steel

is varied and for each value, distribution over the two layers is performed by varying A from zero

to 100 % while A’ is reduced from 100 % to zero (Figure 7a). Convergence acceleration is in this

case based on detecting for each total steel value, the layer distribution giving the nearest

distance to the loading point (minimum moment difference M ). The next layer re-distribution

will not be performed by variation from zero to 100% but just around the previous optimum one.

As for the total steel area, the distance to the origin is used as an interpolation factor to shoot the

solution (Figure 7b).

The next total steel section (step k+1) is thus given by:

k

uk

st

k

stD

DAA 1

(16)

This simple acceleration procedure is efficient whether the initial solution is greater or less than

the exact solution. The distances Du and Dk (Figure 7b) are determined using reduced force and

moment:

22 mpD with

gA

Pp

gg AI

Mm (17)

Ag and Ig are the area and moment of inertia of the concrete gross section.

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Figure 7: RC design using re-analysis strategy

(a): Layer distribution for a given total steel section

(b): Total steel interpolation using distance to origin

Figure 8: Design of a beam with two steel layers

M

P

Mu

Pu

Curve i

Curve j

iM

jM

M

P

Mu

Pu

Curve k

Muk

Du

Dk

15

Figure 9: Design of a square column with three steel layers

Figure 10: Design of a square column subjected to negative moment and tensile axial force

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For the second group with no layer distribution, total steel area is corrected until design

equations (14) are satisfied that is when the loading point ( ), uu PM lies on the border of the

design interaction curve. Convergence is again accelerated with equation (16) using distances to

the origin (Figure 7b).

Steel area increments used in layer distribution and in total steel variation, are gradually reduced

and the convergence tolerance used is 0.1 mm2.

Figures 8 to 10 show typical design results for beams and columns subjected to pure bending or

combined with compression or tension force. The loading point appears on the border of the

design curve. More design examples are described in references [8] to [10].

9. MOMENT MAGNIFICATION FOR SLENDER COLUMNS ACI / SBC moment magnification factor is implemented in the software to account for

slenderness effects. The software delivers the magnified moment for both braced and sway

columns and this value of critical moment should be used in the design process or while

checking various P-M combinations

9.1/ Basis of the method

The moment magnification factor is used at the design stage, if it is ignored in the structural

analysis phase, for slender columns for which the slenderness ratio is greater than a certain

minimum limit and less or equal to a maximum limit equal to 100.0. Above this last value, the

method cannot be used and second order effects should be included in the structural analysis

step. The slenderness ratio is defined as the ratio of the effective length to the radius of gyration.

Slenderness ratio = r

kLu (18)

Lu is the unsupported height of the column while r is its radius of gyration defined as:

Radius of gyration: g

g

A

Ir (19)

The effective length factor k depends on the fixity conditions of the column as well as the type of

frame (braced or sway). At each column end, the degree of fixity is related to the ratio of the

sum of stiffnesses of the columns and beams connected to it as follows:

beams

columns

L

EI

L

EI

(20)

I is the moment of inertia of the cracked section and is defined with respect to the gross moment

of inertia Ig as follows:

Beams: gbb II 35.0 Columns: gcc II 70.0 (21)

The long term column stiffness EI is computed as: d

gEIEI

1

40.0 (22)

E is the concrete modulus of elasticity given by: '4700 cfE (23)

d is the ratio of sustained to ultimate axial force. For lateral (sway ) loads, d = 0.

17

The magnification factor depends on the ultimate axial force as well as the critical buckling load

which is defined as: 2

2

u

ckL

EIP

(24)

9.2/ Moment magnification for slender braced columns

For braced columns, the effective length factor k is computed as:

0.1

05.085.0

)(05.07.0

min

BA

Mink with 5.0k (25)

A and B are values of at top and bottom ends of the column while min is the smallest.

M1 and M2 are the smaller and larger factored moments at the column ends. They have the same

sign if the column is bent in single curvature.

The slenderness effects will be considered if the slenderness ratio is greater than or equal to the

limit:

0.40,120.34

2

1

M

MMin

r

kLu and 100r

kLu (26)

The moment coefficient cm is given by:

4.0,4.06.0

2

1

M

MMaxcm (27)

The moment magnification factor is given by:

0.1,

75.00.1

c

u

m

P

P

cMin (28)

The minimum moment to be considered is: hPM u 03.015min with h in mm. (29)

The magnified moment is then given by: min2 , MMMaxM c (30)

9.3/ Moment magnification for slender un-braced (sway) columns

For un-braced columns, the effective length factor k is computed as follows:

If both ends are restrained and 2m :

m

mk

1

)20(05.0 with 0.1k (31)

If both ends are restrained and 2m : mk 19.0 with 0.1k (32)

If the upper end is hinged: Bk 3.02 (33)

If the lower end is hinged: Ak 3.02 (34)

If both ends are hinged: 30k (35)

A and B are values of at top and bottom ends of the column while m is the average value.

M1b and M2b are the smaller and larger factored braced moments at the column ends. They have

the same sign if the column is bent in single curvature.

M1s and M2s are the smaller and larger factored sway moments at the column ends. They have the

same sign if the column is bent in single curvature.

The slenderness effects must be considered if the slenderness ratio is greater or equal to 22.0:

18

0.22r

kLu and 100r

kLu (36)

The column stiffness EI is computed as: gEIEI 40.0 (37)

The sway moment magnification factor is given by:

0.1,

75.00.1

1

c

u

s

P

PMax

(38)

where is the following floor to column ratio:

c

u

c

u

PP

P

P

(39)

If

gc

u

u

Af

Pr

l

'

0.35 then the magnified moments are:

ssbc MMM 111 ssbc MMM 222 (40)

If

gc

u

u

Af

Pr

l

'

0.35 then the braced moment magnification factor is computed as for

braced columns but with 0d and the total magnified moments are:

)( 111 ssbc MMM )( 222 ssbc MMM (41)

10. CONCLUSIONS A systematic method of analysis and design of various RC sections subjected to bending

combined with an axial force, implemented in software, is presented. An efficient technique

based on a simple and powerful re-analysis approach is used for RC design according to the

American code ACI and Saudi code SBC. The developed software was successfully used for

various types of sections including complex geometries with holes. RC-TOOL software includes

a friendly graphical user interface producing useful curves and diagrams, and constitutes an

invaluable tool for civil engineers.

19

REFERENCES

[1] J. G. MacGregor «Reinforced Concrete: Mechanics and Design, 4th edition »

Prentice Hall 2001.

[2] E.G. Nawy «Reinforced Concrete: A fundamental approach, 5th edition »

Prentice Hall 2003.

[3] ACI 318-95M «Building Code Requirements for Reinforced Concrete»

ACI Publications 1995

[4] E.L.Wilson, A.Habibullah “ SAP2000 -A program for the static & dynamic finite element

analysis of structures “, Comput. & Struct.Inc., Berkeley, 94704.

[5] E.L.Wilson, A.Habibullah “ ETABS Static & dynamic finite element analysis of

buildings “, Comput. & Struct.Inc., Berkeley, 94704.

[6] A.Charif "CBAEL logiciel d’analyse et ferraillage des structures en béton armé selon les

règlements CBA 93, BAEL 91, RPA 88."

1er

Colloque Maghrébin de Génie Civil, Biskra 16-17 Novembre 1998, pp.13-23

[7] A.Charif "CBAEL Software for static/dynamic analysis, design and optimization of

structures”,

6th Saudi Engineering Conference, Dhahran 14-17 December 2002, Vol.3, pp.79-92

[8] Fawaz Al-Sahli "RC-TOOL software for analysis and design of reinforced concrete beams

according to SBC and ACI codes”

Final year project, June 2006, Civil Engineering Department, King Saud University

[9] Raed Al-Rashed "RC-TOOL software for analysis and design of reinforced concrete

columns according to SBC and ACI codes”

Final year project, June 2006, Civil Engineering Department, King Saud University

[10] Saud Al-Dawud "Analysis and design of reinforced concrete beams and columns

according to SBC and ACI codes”

Final year project, June 2007, Civil Engineering Department, King Saud University

20

USER’S MANUAL

RC-TOOL MAIN SCREEN

The main RC-TOOL screen is shown in Figure 11.

Figure 11: RC-TOOL main screen

The screen is self-explanatory. The user has to choose either the Analysis or Design option. The

appropriate section button must be pushed in order to enter the dimensions. The Reinforcement

Data button must then be pushed in order to enter the steel information. This information differs

whether the section is to be analyzed or designed.

Figure 12 shows steel data in case of a Rectangular Section Analysis. The required data includes

the number of layers and the corresponding depths and areas. Blanks or commas must be used to

separate successive values. Figure 13 shows steel data in case of a Rectangular Section Design

with either two layers (Figure 13b), many layers (Figure 13c), or in case of special beam design

with an imposed bar diameter (Figure 13c). The required data is limited to the number of layers

and their depths. In each case, an assistant is provided for computing standard top and bottom

depths given values of concrete cover, main bar diameter and tie diameter. The design option

requires also the values of ultimate moment and axial force.

21

Figure 12: Steel data in case of section analysis

Figure 13a: Steel data in case of section design with one or two steel layers

22

Figure 13b: Steel data in case of section design with many steel layers

Figure 13c: Steel data in case of special beam design with a fixed bar diameter

23

GRAPHICAL ECHO OF THE SECTION

Figure 14 shows a typical graphical echo of the section highlighting the steel layer locations (and

values in case of analysis) with the section area and moment of inertia as well as the centroid

location.

Figure 14: graphical echo of the section (analysis of a T-section)

24

MATERIAL MODELS

Figure 15 shows the material models used for analysis and design of RC sections according to

ACI and SBC codes. The standard rectangular concrete compression block is used with an

ultimate concrete strain of 0.003. Figure 16 shows the models used for the moment curvature

relationships. In this last case, the model is numerically integrated and various models for

concrete compression and tension are available.

Figure 15: Material models for standard design and analysis

25

Figure 16: Material models for moment curvature relationships

26

P-M INTERACTION CURVE

Figure 17 shows a typical axial force – bending moment interaction curve (nominal and design

curves). Many options are offered including checking various combinations and viewing the

force and moment values at the mouse cursor position. Detailed information in any point of the

curve may be obtained. Lines separating the tension control / compression control / transition

zones can also be viewed.

Figure 17: Typical P-M interaction curve

27

MOMENT CURVATURE RELATIONSHIP

Figure 18 shows a typical moment curvature relationship. Many curves can be viewed in the

same screen for different values of the axial force. Stiffness variation may also be generated. The

concrete model used is shown and the values of the moment and curvature at the mouse position

as well as the stress distributions are all shown.

Figure 18: Typical Moment Curvature relationship

28

STANDARD RC DESIGN

Figure 19 shows a typical RC design showing the strain and stress distributions as well as the

required steel with detailed information at each steel layer. The design capacity of the section is

equal to the ultimate moment and axial force (the ultimate point is on the border of the design

interaction curve).

Figure 19: Typical RC design

29

SPECIAL BEAM DESIGN

Figure 20 shows a typical special beam design with a fixed bar diameter showing the detailed bar

and layer layouts. The resulting beam bending capacity is greater or equal to the ultimate

moment.

Figure 20: Typical special beam design

30

SLENDERNESS

Figure 21 shows a typical slenderness screen. Braced or un-braced (sway) columns may be

checked. The effective length factor may be user-defined or computed according to ACI and

SBC specifications.

Figure 21: Typical slenderness screen

31

Shear design

Figure 22 shows RC-TOOL output for shear design of a beam. The number of legs and stirrup

spacing are determined according to ACI / SBC specifications. Spacing variation is performed

along the beam span.

Figure 22: Shear design of a beam