Session T1H

0-7803-8552-7/04/$20.00 © 2004 IEEE October 20 – 23, 2004, Savannah, GA 34th ASEE/IEEE Frontiers in Education Conference

T1H-26

A Computer-Based Problem-Solving Courseware for Reinforced Concrete Design

Nirmal K. Das

Associate Professor, School of Technology, Georgia Southern University, Statesboro, GA 30460 ndas@georgiasouthern.edu

Abstract - Typically code-based structural design uses trial-and-error procedure that often requires several iterations, involving tedious, repetitive calculations. Also, only a limited number of examples can be presented in the classroom due to time constraint. To circumvent the situation, a logical option is to capitalize on the computer’s abilities to compute, display graphics, and interface with the user. Some asynchronous web-based course materials are available in the civil engineering technology area, but to the author’ knowledge, very few exist in structural design. The purpose of this paper is to present a computer-based problem-solving courseware that has been developed to complement traditional lecture-format delivery of the reinforced concrete design course in order to enhance student learning. The courseware consists of interactive, web-based modules. Commercial symbolic-manipulation software (e.g., Mathcad) is utilized for the calculations performed in different modules. Besides allowing for faster solution of a problem, the tool is useful for experimentation with parameter changes as well as graphical visualization. Index Terms - civil engineering technology, Mathcad,, problem-solving courseware, reinforced concrete design.

INTRODUCTION

The four-year ABET-accredited Civil Engineering Technology curriculum at Georgia Southern University includes a required, senior-level course in Reinforced Concrete Design. The two main objectives of the course are: (1) the students gain a thorough understanding of the fundamental principles underlying design of various structural components and the relevant stipulations in the ACI Code and (2) they correctly apply that knowledge to various practical design problems. A combination of homework and computational laboratory assignments is used for meeting the second objective. However, a significant amount of design activities is based on trial-and-error procedure that often requires several iterations, involving tedious, repetitive calculations. Also, only a limited number of examples can be presented in the classroom due to time constraint. To circumvent the situation, a logical option is to capitalize on the computer’s abilities to compute, display graphics, and interface with the user, letting students focus on substantive issues and thereby engage in additional problem-solving for varied scenarios, outside the classroom at their own pace.

Mathcad [1], an industry-standard calculation software, is used because it is as versatile and powerful as programming languages, yet it is as easy to learn as a spreadsheet. Additionally, it is linked to the Internet and other applications one uses everyday.

In Mathcad, an expression or an equation looks the same way as one would see it in a textbook, and there is no difficult syntax to learn. Aside from looking the usual way, the expressions can be evaluated or the equations can be used to solve just about any mathematics problem one can think of. Text can be placed anywhere around the equations to document one’s work. Mathcad’s two- and three-dimensional plots can be used to represent equations graphically. In addition, graphics taken from another Windows application can also be used for illustration purpose. Mathcad incorporates Microsoft’s OLE 2 object linking and embedding standard to work with other applications. Through a combination of equations, text, and graphics in a single worksheet, keeping track of the most complex calculations becomes easy. An actual record of one’s work is obtained by printing the worksheet exactly as it appears on the screen

ELEMENTS OF REINFORCED CONCRETE DESIGN

Concrete is basically an artificial or man-made stone made of a mixture of fine aggregate (sand) and coarse aggregate (crushed rock, or gravel), held together with a paste of cement and water. As with most rocks, concrete is very strong in compression, but very weak in tension. Reinforced concrete is a combination of concrete and steel wherein the steel reinforcement provides the tensile strength lacking in the concrete. Reinforced concrete is one of the most important materials available for construction.

A typical R.C. building frame consists of beams, columns and footings. These various reinforced concrete components are designed on the basis of the American Concrete Institute’s Building Code Requirements for Structural Concrete (ACI 318). The ACI Code [2] uses the strength design principle which can be stated as follows: the required strength of a component including all possibilities of overloading during its lifetime, must not exceed the component’s actual strength that accounts for all possible factors contributing to its under-capacity.

Two types of problems are usually dealt with by the students in a Reinforced Concrete Design course: analysis and design. Whereas in an analysis problem, the load carrying capacity of a given component is determined, in a design

Session T1H

0-7803-8552-7/04/$20.00 © 2004 IEEE October 20 – 23, 2004, Savannah, GA 34th ASEE/IEEE Frontiers in Education Conference

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problem, size of a component is determined to support a given or estimated loading. In this paper, only reinforced concrete beams are discussed. Three types of beams are commonly used in a reinforced concrete building: singly-reinforced rectangular beams, doubly-reinforced rectangular beams and T-beams.

Rectangular Beams (singly-reinforced)

Single-span, simply-supported and cantilever beams are considered. Since a simply-supported beam carries only positive moment (tension in the bottom), steel reinforcement needs to be placed near the bottom of the beam. On the other hand, a cantilever beam carries only negative moment (tension at the top), and so it needs to be reinforced with steel bars placed near the top of the beam. The moment strength of such a beam of given dimensions and specified material strengths is limited by a certain amount of steel reinforcement prescribed by the ACI Code.

Rectangular Beams (doubly-reinforced)

Sometimes, practical and architectural considerations may dictate and limit rectangular beam sizes whereby it becomes necessary to develop more moment strength from a given cross-section. When this situation occurs, the ACI Code permits the addition of tensile steel over and beyond the code-prescribed maximum limit, provided that compression steel is added in the compression zone of the cross-section. The result is a doubly-reinforced beam consisting of two sets of steel reinforcement bars – one carrying tensile stresses and the other compressive stresses.

T-Beams

A typical floor or roof framing system in a reinforced concrete building, called a beam-and-girder system, is composed of a slab on supporting reinforced concrete beams and girders. The beam and girder framework is, in turn, supported by columns. In such a system, the beams and girders are usually placed monolithically with the slab. For analysis or design of such floor and roof systems, a common assumption is that the monolithically placed slab and supporting beam interact as a unit in resisting positive moment. The slab essentially becomes the flange, and the supporting beam becomes the web of a T-shaped beam, hence the name T-beam.

MATHCAD PROGRAM FEATURES

The program developed consists of several modules. One set of modules calculates the required strengths (moment and shear) for different support types. The other set of modules calculates the design (actual) strengths. Overall, the program will require input data pertaining to the geometry of the problem, material property and the loading. Detailed discussions on fundamentals of design of reinforced concrete structures can be found in any standard textbook [3,4,5].

Required Strength Modules

The following information is required as input data: beam type (simply-supported, simply-supported with overhang, and

cantilever), beam length, magnitudes and lengths of distributed loads, and magnitudes and locations of concentrated loads, and load types (dead load, live load etc.). Based on the input data, calculations are carried out in the following steps for each concentrated load and each distributed load: 1. Multiply each load by the appropriate load factor. 2. Calculate the support reactions for the beam. 3. Calculate the shear at equal intervals along the length of

the beam. 4. Calculate the moment at equal intervals along the length

of the beam. Maximum values of shear and moment to be used in

design (or analysis) are obtained by superposing the results of steps 3 and 4, respectively.

Design Strength Modules (Analysis and Design)

For analysis problems, the following information is required as input data: compressive strength of concrete, yield strength of steel, beam cross-sectional dimensions and area of steel reinforcement (tension only/ both tension and compression). The design moment strength for the three types of beams is calculated using the equations given below, and compared to the required strength (obtained from the modules described above) to ascertain if a beam is adequate. For rectangular beams with tensile reinforcement only:

).7.1

.1.(.... '

c

yysn f

fdfAM

ρφφ −= (1)

For doubly-reinforced rectangular beams:

)].(.)2

.(..[. ''1 ddfAadfAM ysysn −+−= φφ (2)

For T-beams:

)2

.(... adfAM ysn −= φφ (3)

where φ = moment capacity reduction factor (= 0.90) Mn = nominal moment strength (in.-lb) As = area of tension reinforcement (in2) fy = specified yield strength of reinforcement (psi) d = effective depth of section (in.) ρ = ratio of tension reinforcement fc’ = specified compressive strength of concrete (psi) Asa = part of tension reinforcement balancing concrete compression (in.2) a = depth of equivalent rectangular stress block (in.) d’ = distance from extreme compression fiber to centroid of compression reinforcement (in.)

For design problems, wherein the beam size and/or steel reinforcement are to be determined, the design moment strength expression involving unknown beam cross-sectional dimensions and/or steel area, is to be set equal to the required moment strength, to solve for those unknowns.

Session T1H

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EXAMPLES

Some example problems developed for teaching Reinforced Concrete Design course are presented in this section. The first example is on determining the design moment strength of a beam of given rectangular cross-section. The second example is on design of a doubly-reinforced beam. The third example is on calculating the area of steel reinforcement for a T-beam.

Example 1

A rectangular beam cross-section is of 12 in. width and 24 in. overall thickness, with an effective depth of 21 in. The beam is reinforced with 4-#9 bars of Grade 60 steel . Determine the design moment strength of the beam, assuming compressive strength of concrete is 4000 psi. Design Moment Strength of Singly Reinforced Rectangular Beam Given: Compressive strength of concrete: fc' 4000 psi⋅:=

Yield strength of steel reinforcement: fy 60000psi⋅:=

Beam dimensions: Width: b 12 in⋅:= Effective depth: d 21 in⋅:= Overall thickness: h 24 in⋅:= Area of tensile steel reinforcement:

As 4.0 in2⋅:=

Determine: Design moment strength (φMn)

Steel ratio:

ρAsb d⋅

:=

ρ 0.016= Coefficient of resistance:

Rn ρ fy⋅ 10.59 ρ⋅ fy⋅

fc'−

⎛⎜⎝

⎞

⎠⋅:=

Rn 818.594psi=

Nominal moment strength:

Mn Rn b⋅ d2⋅:=

Mn 3.61 105

× lbf ft⋅=

φMn 0.9 Mn⋅:=

Design moment strength:

φMn 3.249 105× lbf ft⋅=

Example 2

A rectangular beam is limited to 15 in. width and 24 in. overall thickness, with an effective depth of 20 in. The required moment capacity is 600 ft kips. Assuming strength of concrete is 4000 psi and yield strength of steel is 60,000 psi, select the reinforcement required for the beam. Design of Doubly Reinforced Rectangular Beam Input: Required moment strength: Mu 600000ft⋅ lbf⋅:=

Compressive strength of concrete: f'c 4000 psi⋅:=

Yield strength of steel reinforcement: fy 60000psi⋅:=

Beam dimensions:

b

d h

d’

A’s

As

Width: b 15 in⋅:= Effective depth: d 20 in⋅:= Overall thickness:

Session T1H

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h 24 in⋅:= Effective cover of compression steel reinforcement: d' 4.0 in⋅:= Determine: Reinforcement required . Solution: Ultimate strain in concrete: εc 0.003:=

Moment capacity reduction factor: φ 0.9:= Modulus of elasticity of steel: Es 29000000psi⋅:=

Coefficient value: β1 0.85 f'c 4000 psi⋅≤if

0.85f'c 4000−

1000

⎛⎜⎝

⎞⎠

− f'c 4000 psi⋅>if

0.65 0.85f'c 4000 psi⋅−

1000 psi⋅

⎛⎜⎝

⎞⎠

− 0.65≤if

:=

β1 0.85=

If singly reinforced, maximum allowable tensile steel ratio:

ρmax 0.750.85 β1⋅ f'c⋅( ) εc Es⋅( )⋅

fy εc Es⋅ fy+( )⋅:=

ρmax 0.021=

Maximum allowable area of tensile reinforcement: As1 ρmaxb⋅ d⋅:=

As1 6.414in2

=

Maximum resisting moment of singly-reinforced section:

Mnmax ρmaxfy⋅ 10.59 ρmax⋅ fy⋅

f'c−

⎛⎜⎝

⎞

⎠⋅ b⋅ d2

⋅:=

Mnmax 5.2 105

× lbf ft⋅=

Required nominal moment strength:

MnMuφ

:=

Mn 6.667 105× lbf ft⋅=

Since

Mn > Mnmax, the beam must be doubly-reinforced

Nominal moment strength to be provided by compression steel: Mn2 Mn Mnmax−:=

Mn2 1.466 105

× lbf ft⋅=

Check if the compression steel reinforcement has yielded. Depth of Whitney's stress block:

aAs1 fy⋅

0.85 f'c⋅ b⋅:=

a 7.546in= Actual depth of compression stress block:

caβ1

:=

c 8.878in= Strain in compressive steel:

ε's εc 1d'c

−⎛⎜⎝

⎞⎠

⋅:=

ε's 1.648 10 3−×=

Yield strain of steel:

εyfyEs

:=

εy 2.069 10 3−

×=

Since ε's <

εy, the compression steel has not yielded. Stress in compression steel:

f'sε's fy⋅

εyε's εy<if

fy otherwise

:=

f's 4.78 104

× psi=

Determination of steel areas: Compression steel area:

A'sMn2

f's d d'−( )⋅:=

A's 2.301in2

=

Corresponding tensile steel area:

As2A's f's⋅

fyε's εy<if

A's otherwise

:=

As2 1.833in2

=

Total tensile steel area: As As1 As2+:=

As 8.247in2

=

Total compression steel area:

A's 2.301in2=

Total tensile steel area, As = 8.247 in2

Total compression steel area, A's = 2.301 in2

Session T1H

0-7803-8552-7/04/$20.00 © 2004 IEEE October 20 – 23, 2004, Savannah, GA 34th ASEE/IEEE Frontiers in Education Conference

T1H-30

Example 3

Determine the area of reinforcing steel (As) required for a T-beam shown below that is part of a floor framing system. The concrete compressive strength is 4000 psi and the steel yield strength is 60,000 psi. The slab thickness (hf) is 4 in., the effective depth (d) is 26 in., the overall thickness (h) is 30 in. and the width of web (bw) is 15in. Assume 40 ft simple span (l) and the center-to-center spacing of beam webs (s) is 10 ft. The required moment strength is 1000 ft-kips.

Design of T- Beam Input: Compressive strength of concrete: f'c 4000 psi⋅:=

Yield strength of steel reinforcement: fy 60000psi⋅:=

Beam span:: L 40 ft⋅:= c/c spacing: s 10 ft⋅:= Effective depth: d 26 in⋅:= Slab thickness: hf 4 in⋅:=

Overall thickness: h 30 in⋅:= Width of web: bw 15 in⋅:=

Required moment strength: Mu 12000000in⋅ lbf⋅:=

Determine: Area of steel reinforcement Solution: Effective flange width:

b s sL4

≤ s bw 16 hf⋅+( )≤∧⎡⎢⎣

⎤⎥⎦

if

L4

L4

s≤L4

bw 16 hf⋅+( )≤∧⎡⎢⎣

⎤⎥⎦

if

bw 16 hf⋅+ otherwise

:=

b 79in=

Area of flange: Af b hf⋅:=

Af 316in2

=

Moment capacity reduction factor: φ 0.9:= Initial value of moment arm:

z1 0.9 d⋅ 0.9 d⋅ dhf2

−⎛⎜⎝

⎞⎠

>if

dhf2

− otherwise

:=

z1 24in= Ast count 1←

z z1←

diff 0.01 in2⋅←

Asrev 0 in2⋅←

AsMuφ fy⋅ z⋅

←

AcAs fy⋅

0.85 f'c⋅←

aAcb

Ac Af≤if

hfAc Af−

bw+ otherwise

←

ya2

Ac Af≤if

hf2

a2Ac

Ac Af−( )⋅+ otherwise

←

z d y−←

diff As Asrev−←

Asrev As←

count count 1+←

break count 20>if

diff 0.001 in2⋅≥while

Asrev

:=

Required area of steel:

Ast 8.901in2=

b

d

hf

As

h

bw

Session T1H

0-7803-8552-7/04/$20.00 © 2004 IEEE October 20 – 23, 2004, Savannah, GA 34th ASEE/IEEE Frontiers in Education Conference

T1H-31

SOFTWARE INTEGRATION AND STUDENT RESPONSE

The use of Mathcad has been recently incorporated in this course. Typically, homework assignments and group projects require the use of this tool that is accessible in the computer laboratories located on campus. Through these exercises, students learn to write programs using Mathcad to find solutions to a variety of problems. Students are required to support their results with other manual calculations. If group projects are done using these tools, students are required to make presentations and share the experience with others in the class. It has been the author’s experience that such in-class presentations not only help students sharpen their communication skills, but also demonstrate the benefits of using such software tools in the learning process.

Many handouts were prepared with examples to clearly explain the use of Mathcad in solving several sample problems. It has been the author’s experience that such handouts were extremely useful to students in providing them with the necessary information on Mathcad and helping them learn the skills to develop simple programs for the assigned problems. Students were encouraged to use the capabilities of Mathcad to find solutions to problems with varying load conditions, varying beam support conditions, and varying beam cross-sections, examine and compare results, and learn more from these studies.

Unfortunately, no assessment information is available to show how Mathcad really helped retention and learning. However, in their semester-end evaluations of the course,

students anecdotally reported that they enjoyed using Mathcad to solve reinforced concrete design problems.

CONCLUSIONS

Mathcad has been used as a problem-solving tool in Reinforced Concrete Design course. It provided a viable means for students to solve structural design problems using a user-friendly, yet versatile software. Students exposed to Mathcad and its capabilities in computation and visualization were found to have developed better reading and studying skills, and a better understanding of the materials presented in the course. Overall, the incorporation of Mathcad resulted in a positive, efficient, and more effective teaching-learning environment.

REFERENCES

[1] Mathsoft, Mathcad 11 User’s Guide, Mathsoft Engineering & Education, Inc.

[2] American Concrete Institute, Building Code Requirements for Structural Concrete (ACI 318-02) and Commentary (ACI 318R-02), An ACI Standard

[3] McCormac, J. C., Design of Reinforced Concrete, 5th ed, John Wiley & Sons, Inc..

[4] Nawy, E. G., Reinforced Concrete-A Fundamental Approach, 4th ed, Prentice Hall, Inc.

[5] MacGregor, J. G., Reinforced Concrete- Mechanics and Design, 3rd ed, Prentice Hall, Inc.