A Procedure for Determining Maximum and Minimum Dimensions for

a Singly Reinforced Rectangular Concrete Beam Loaded to its Flexural

Capacity.

By T. B. Quimby

UAA Civil Engineering Department

Spring 2004

RC beam strength is primarily a function of width (b), height (h), and the amount of

reinforcing (A

s

). There is an infinite combination of these three variables that will result

in a beam that exactly satisfies that flexural strength requirement:

M

u

<

φM

n

(Eq. 1)

This equation can be expanded, for the case of a singly reinforced rectangular section (or

a section that can be treated as such) in which the steel yields to:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

−≤

bf

fA

dfAM

c

ys

ysu

)85(.2

φ

(Eq. 2)

Note that d (the effective depth of the steel) is a function of the height, h.

To obtain a particular strength of beam, it can be observed that increases in A

s

allow a

decrease in the size parameters, b and d, and vice versa. Consequently, determining the

size (b x h) of a section can be determined by substituting into equation (2) a

value/expression for A

s

that represents the steel condition that we are interested in

designing for.

There are three cases that are of particular interest:

1. The maximum A

s

which will minimize b and h

2. The minimum A

s

which will maximize b and h

3. The maximum A

s

which will minimize b and h and

for which deflections are not

likely to be a problem.

If we can write expressions for A

s

for each case, the we can substitute the expressions

into equation (2) and solve for relationship for b and d which can be extended to be a

relationship for b and h.

Minimum Size Beams

For this case we want to increase A

s

and still keep the beam “tension controlled” (ACI

318-02, 10.3.3 and 10.3.4). This will be the A

s

that results in a tensile strain of 0.005

RC Beam Sizing, 1/31/2008 page 1

when the beam is fully loaded (i.e. M

u

= φM

n

). To develop this expression, we use the

beam cross section and strain diagram shown in Figure 1.

Note that, in order to solve the equilibrium equations for the pure flexure condition, T

s

must equal C

c

. Writing expressions for both we get:

cccyss

AfCfAT

′

=

=

= 85. (eq. 3)

Figure 1

Beam Diagrams

Note that by setting the strain in the concrete to 0.003 and the strain in the steel to 0.005

this dictates that c = 0.375d. We also know that a = β

1

c = β

1

(.375d) and A

c

= ba.

Substituting these observations into equation (3) gives us an expression for A

s

in the two

unknowns, b and d, that we can substitute into equation (2):

bd

f

f

A

y

c

s

)375.0(

85.

1

β

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

= (eq. 4)

Maximum Size Beams

For this case, A

s

is controlled by ACI 318-02 10.5.1. The expression, when modified to

include the limiting state becomes:

RC Beam Sizing, 1/31/2008 page 2

bd

f

f

A

y

c

s

)3,200max(

′

= (eq. 5)

Like equation (4) this is an expression for A

s

in terms of the two size variables, b and d.

Substituting equation (5) into equation (2) will yield a relationship between b and d that

will result in the largest effective cross sectional size.

Minimum Beam Size for Which Deflections are NOT LIKELY

to be a Problem.

Increasing A

s

while maintaining a given strength, φM

n

, results in a smaller overall beam

size and a smaller moment of inertia. This results in increased deflection, or stiffness, of

the final beam. A rule of thumb for designing a section so that it is large enough so that it

is

not likely

to have deflection problems is to limit the neutral axis depth, c, to about

0.375 of that of the neutral axis location, c

b

, for the balanced condition.

For the balanced condition, the steel strain is exactly equal to its yield strain (f

y

/E

s

=

f

y

/29,000,000, units in psi). From a strain diagram and using similar triangles, this yields:

y

bdefl

f

d

cc

+

=≈

000,87

000,87

375.0375.0

(eq. 6)

Using the same procedure and logic as was used to develop A

s

for the minimum size

beam, we obtain the following expression for A

s

:

bd

ff

f

A

yy

c

s

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

=

000,87

)000,87(375.085.

1

β (eq. 7)

Like equations (4) and (5) this is an expression for A

s

in terms of the two size variables, b

and d. Substituting equation (7) into equation (2) will yield a relationship between b and

d that will result in the smallest effective cross sectional size for which deflection are not

likely to be a problem.

Putting It All Together

Making the substitutions for A

s

into equation (2) as indicated above and simplifying the

resulting expressions yields the following relationships for the size variables, b and d.

For the SMALLEST BEAM SIZE:

⎟

⎠

⎞

⎜

⎝

⎛

−′

≥

2

375.0

1)375)(.85(.

1

1

2

β

βφ

c

u

f

M

bd (eq. 8)

RC Beam Sizing, 1/31/2008 page 3

For the LARGEST EFFECTIVE BEAM SIZE:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

′

′

−

′

≥

)85(.2

)3,200max(

1)3,200max(

2

c

c

c

u

f

f

f

M

bd

φ

(eq. 9)

For the SMALLEST BEAM SIZE NOT LIKELY TO HAVE DEFLECTION

PROBLEMS:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

⋅−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

′

≥

yy

c

u

ff

f

M

bd

000,87

)000,87(375.0

2

1

1

000,87

)000,87(375.

)85(.

1

1

2

β

βφ

(eq. 10)

Now What Do We Do?

The above relationships establish a relationship that you can use to

guide

you in selecting

b and h for the beam you are designing. You first need to decide which criteria (smallest,

largest, smallest w/o deflection problems) that you want to guide your design. Next you

need to solve the appropriate equation for your condition. The next step is to use the

resulting solution to guide you in selecting b and h. One common way of doing this are to

establish an approximate relationship between b and d (say let d = 1.5b) then solve for the

two variables. Another method is to build a table with a column of b values and the

another with the corresponding d values. To determine h, add an appropriate estimated

value to d for the distance from the center of the steel group (which you have not yet

selected) and the tension face of the beam. Finally, select a b and h that is reasonably

close to your results and satisfies other limitations on beam dimensions.

All this work, and all you have is a b and h. You still need to select the steel for the

beam. This can be done by using equation (2). Estimate d, then substitute your values

for b and d into the equation and solve the quadratic for A

s

. Be sure to maintain the

inequality sign so that you will know what way to go when you select real bars to put in

the beam.

Finally select real bars and accurately place them in a scaled drawing of your cross

section, making sure that all spacing and cover requirements are met.

Then, ignoring all prior calculations, compute the actual φM

n

for the beam that you drew

and show that it is greater than M

u

. If it does not satisfy equation (1), then find someway

to adjust A

s

, b, or h so that equation (1) is satisfied.

Your final result should be an accurate scaled drawing/sketch and a calculation that

shows that all the limit states and criteria have been met.

RC Beam Sizing, 1/31/2008 page 4

RC Beam Size Selection

ACI 318-02

by T.B. Quimby

2/4/2004

Mu 253.3 ft-k f'c 3000 psi

phi 0.9 fy 60000 psi

Beta1 0.85

Use approximation equations to select a b and h

Smallest Defl control Largest

Required bd^2 4942.9 7751.9 17575.9 in^3

use b 12 14 18 in

computed d 20.30 23.53 31.25 in

cov + bar 2.88 2.88 2.88 in

est. h 23.17 26.41 34.12 in

Suggested h 24 27 36 in

A

ctual h 24 27 34 in

h/b 2.00 1.93 2.00

Compute a required area of steel

estimated d 21.13 24.13 31.13 in

Quad:a 52941.1765 45378.1513 35294.1176 Note, these are quadratic coefficients

Quad:b -1140750 -1302750 -1680750 to be used in the quadratic

Quad:c 3039600 3039600 3039600 equation

Strength Req'd As 3.11 2.56 1.88 in^2

ACI 10.5.1 Req'd As 0.85 1.13 1.87 in^2

Req'd As 3.11 2.56 1.88 in^2

Select the actual bars and compute the actual strength and other limiting criteria

A

ctual Bars:(4) #8 (2) #10 (2) #9

Actual As 3.16 2.54 2.00 in^2

Actual d 21.63 24.63 31.63 in

Actual pMn 263.5 257.1 272.9 ft-k

Mu/pMn 96% 99% 93% 99% OK

N.A. location 7.29 5.02 3.08 in

Steel Strain 0.00590 0.01171 0.02785

ACI 10.3.3 85% 43% 18% 85% OK

ACI 10.5.1 27% 45% 95% 95% OK

Approximate weight 300.0 393.8 675.0 lbs/ft

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