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Design of Beams for Flexure

Introduction

Beams are structural members carrying transverse loads that can cause bending moments, shear forces, and in some cases torsion.

This chapter deals only with shallow beams that are defined by ACI 10.7.1 as beams with depth to clear span ratio of less than 0.25.

Design of a beam starts with proportioning its sections to resist bending moments and choosing the required reinforcement. Once this is done, the chosen sections are checked and designed for shear and torsion. The last step in the design process is to check the bond between the reinforcement and concrete.

In this chapter, two problems are dealt with; analysis and design. In the first problem, the dimensions of the cross section together with the amount of reinforcement are given; it is only required to evaluate the bending capacity of the beam cross section. The second problem deals with evaluating the dimensions of the cross section and the required reinforcement, provided that either the bending moment or the loading causing it is given. In both cases, the properties of concrete and steel reinforcement need to be known.

In order to limit deflections, the depth of the cross section is chosen to fulfill the ACI Code serviceability requirements.

Design assumptions

The following assumptions provided by ACI 10.2 are helpful in deriving the basic equations that are used throughout this chapter.

§ The strain in both reinforcement and in concrete is assumed to be directly proportional to the distance from the neutral axis, even near ultimate strength.

§ Tensile strength of concrete is to be neglected in axial and flexural calculations of reinforced concrete.

§ Maximum usable strain at extreme concrete compression fiber is to be equal to 0.003.

§ Equilibrium of forces acting on the cross section and compatibility of strains between the concrete and the reinforcement are to be satisfied.

§ Stress in reinforcement below yf is taken as sE times steel strain. For strains

greater than that corresponding to yf , stress in reinforcement is considered to

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be independent of strain and equal to yf . Thus, for ys εε < , sss Ef ε= and

when ys εε ≥ , ys ff = .

§ The concrete compressive stress distribution may be assumed to be rectangular, trapezoidal, parabolic or any other shape that results in prediction of strength in substantial agreement with results of compressive tests.

Stress distribution for different stages of loading

Stage I (Un-cracked linear stage):

If a reinforced concrete beam is loaded in bending in such away that compressive stresses develop at the top fibers of the section while tensile stresses develop at the bottom fibers, the stress distribution according to the bending theory is given by

IyM

f =

(1)

where

f = normal stress

M = bending moment

y = distance from the neutral axis to the point under consideration, measured perpendicular to the beam axis.

I = moment of inertia of the concrete section in addition to that of the reinforcement.

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

Figure 1: (a) Cross section; (b) strains; (c),(d),(e) and (f) stress distribution

In this stage, shown in Figure 1.b and Figure 1.c, strains and stresses are distributed linearly and satisfy the following:

cc ff ′<

rct ff <

where

cf = concrete compressive stress

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cf ′= concrete compressive strength at 28 days

ctf = tensile stress of concrete

rf = modulus of rupture of concrete

This case is exclusively used in the design of un-cracked sections associated with water structures, when water is on the tension side of the section, using the working stress design method.

Stage II: Cracked linear stage

With higher loads applied to the beam, stresses are still distributed linearly and the tensile stress in concrete exceeds the modulus of rupture, as shown in Figure 1.d. Thus, the first of tensile cracks at the bottom surface starts to develop with the tensile and compressive strains satisfying the following:

cc ff ′<

rct ff >

The tensile strength of concrete in the area below the neutral axis is to be neglected. This stage is considered the basis for design of sections subjected to bending using the working stress design method.

Stage III: Cracked nonlinear stage

With further increase of the load, the compressive stresses in the concrete becomes nonlinear, while the strain is still proportional to the distance from the neutral axis, as shown in Figure 1.e.

cc ff ′<

rct ff >

This case is not used in design, as it is a transitory case between working stress and ultimate strength design methods.

Stage IV: Ultimate Strength Stage

With further increase in the load, the cracks push upward moving the neutral axis in that direction until failure takes place. The stress distribution is shown in 1.f. Depending on the properties of a beam, flexural mode of failure may be ductile or brittle as will be explained in the next section.

A. Tension-controlled sections:

In ACI 10.3.4, sections are called tension-controlled when the net tensile strain in the extreme tension steel tε is equal to or greater than 0.005 when the concrete in

compression reaches its crushing strain cε of 0.003, as shown in Figure 2.a. For these

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sections sufficient warning of failure with excessive deflection and cracking may be expected. Flexural members are usually tension-controlled.

B. Compression-controlled sections:

In ACI 10.3.3, sections are called compression-controlled when the net tensile strain in

the extreme tension steel tε is equal to or less than yε when the concrete in

compression reaches its crushing strain of 0.003, as shown in Figure 2.c. For these sections, a brittle failure may be expected with little warning of impending failure. Compression members are usually compression-controlled.

C. Transition sections:

In ACI 10.3.4, sections with small axial load and large bending moment, will have net tensile strain in the extreme tension steel between the above limits. These sections are in a transition region between compression and tension-controlled sections, shown in Figure 2.b. Failure of such sections is initiated by yield of the reinforcement, but at smaller curvature than a tension-controlled failure and hence giving less warning of failure than tension-controlled failure. D. Balanced sections:

In ACI 10.3.2, balanced strain condition exists at a cross section when tension

reinforcement reaches the strain corresponding to yf just as concrete in compression

reaches its assumed ultimate strain of 0.003, as shown in Figure 2.d. For these sections, a brittle failure may be expected with little warning of impending failure.

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

Figure 2: Classification of sections; (a) Tension-controlled sections; (b) Sections in transition between tension and compression; (c) compression-controlled sections; (d) balanced sections

Strength reduction factors, Φ :

According to ACI 9.3.2 strength reduction factors Φ are given as follows:

a- For tension-controlled sections ……….…………………….. Φ = 0.90

b- For compression-controlled sections,

• Members with spiral reinforcement …………….….……… Φ = 0.75

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• Other reinforced members ………………………...………. Φ = 0.65

The values of strength reduction factors in the transition region are permitted to be linearly increased from that for compression-controlled sections to 0.90 as the net tensile strain in the extreme tension steel at nominal strength increases from the

compression-controlled strain limit to 0.005, as shown in Figure 3. In this figure, c is

distance from the extreme compression fiber to neutral axis, and td is the distance from

extreme compression fiber to extreme tension steel.

ACI 10.3.5 specifies that for flexural members with factored axial compression load less than gc A'f1.0 , tε at nominal strength is not to be less than 0.004.

Figure 3: Variation of Φ with net tensile strain tε and tdc /

Stress distribution at strength design:

In the design for flexure, the three most common distributions of concrete stresses are the rectangular, the parabolic and the trapezoidal, shown in Figure 4.

Figure 4: Stress distribution at ultimate strength

Whitney’s rectangular stress distribution

Whitney proposed the use of an equivalent rectangular compressive stress distribution, shown in Figure 5, to replace the more exact parabolic stress distribution. This

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equivalent stress distribution does not represent the stress distribution in the compression zone at ultimate strength, but does provide basically the same results as those obtained from laboratory tests.

According ACI Code 10.2.7.1 through ACI Code 10.2.7.3, in the equivalent rectangular

block an average stress of cf ′85.0 is used with a rectangle of depth xa 1β= , where x

is the distance from the extreme compression fiber to the neutral axis, and:

85.01 =β for 2c

2 cm/kg280'fcm/kg170 ≤≤

and

( ) 21 /280

7028005.0

85.0 cmkgfforf

cc >′

−′−=β

But 1β shall not be taken less than 0.65.

Figure 5: Whitney stress distribution

Maximum reinforcement ratio, maxρ for singly reinforced rectangular

beams:

For tension-controlled sections, the maximum reinforcement ratio, maxρ , is defined as

dbAs max,

max =ρ (2)

Where

max,sA = the maximum amount of tension reinforcement evaluated when the net

tensile strain in the extreme reinforcement is 0.005 and the strain at the extreme compression fiber is equal to 0.003.

d = the effective depth of the section, measured from the centroid of the reinforcement to concrete extreme compression fiber.

From the strain diagram in Figure 6 and the use of similar triangles,

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005.0003.0003.0

dxmax

+= (3)

or,

d375.0xmax = (4)

The compressive force in concrete maxC is given by

bafC c maxmax 85.0 ′= (5)

The tensile force in the reinforcement maxT is given as

ymax,smax fAT = (6)

Figure 6: (a) cross section; (b) strain diagram; (c) Force diagram

From equilibrium of forces in the axial direction,

maxmax CT = (7)

Substituting Eqn. (5) into Eqn. (7),

bafT c maxmax 85.0 ′= (8)

but max1max xa β=

and bxfT c max1max 85.0 β′= (9)

Equating Eqns. (6) and (9), one gets

y

cs f

bxfA max1

max,85.0 β′

= (10)

Substituting Eqn. (4) into Eqn. (10),

y

cs f

dbfA 1

max,31875.0 β′

= (11)

Substituting Eqn. (2) into Eqn. (11),

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y

1cmax f

'f31875.0 βρ = (12)

For transition sections with tε at nominal strength of 0.004, Eqn. (12) is multiplied by a

factor of 78 to obtain Eqn. (13)

y

1cmax f

'f36428.0 βρ = (13)

Minimum reinforcement ratio, minρ for singly reinforced beams:

ACI 10.5.1 specifies that at every section of a flexural member where tensile

reinforcement is required by analysis, sA provided is not to be less than given by

dbf

fA w

y

cs

'

min,

80.0= and not less than db

f wy

14 . The provisions for a minimum

amount of reinforcement applies to flexural members which are larger in cross section than required for strength, with a very small amount of tensile reinforcement. With a very small amount of tensile reinforcement, the computed moment strength as a reinforced concrete section using cracked section analysis becomes less than that of the corresponding un-reinforced concrete section computed from its modulus of rupture. Failure in such a case can be sudden. However, ACI 10.5.3 states that the preceding minimums do not have to be satisfied if the area of the tensile reinforcement provided at every section is at least one-third greater than the area required by moment.

Concrete cover

Concrete cover is necessary for protecting the reinforcement against weather and other effects. Concrete cover is measured from the concrete surface to the outermost surface of steel reinforcement as shown in Figure 7. ACI 7.7 specifies minimum cover values for different exposure conditions shown in Table 1.

Figure 7: Concrete cover and clear spacing between bars

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Table 1: Minimum Concrete Cover

Type of Exposure Minimum concrete cover (cm)

Concrete cast against and permanently exposed to earth 7.50

Concrete exposed to earth or weather § Bars Φ 19 mm and larger § Bars smaller than Φ 19 mm

5.0 4.0

Concrete not exposed to weather or in contact with ground: Slabs, walls, joists

§ Bars Φ 43 mm through Φ 57 mm § Bars smaller than Φ 43 mm

Beams, columns: § Primary reinforcement, ties, stirrups, spirals

Shells, folded plate members: § Bars Φ 19 mm and larger § Bars smaller than Φ 19 mm

4.0 2.0

4.0

2.0 1.3

Spacing of reinforcing bars

The ACI Code specifies limits for bar spacing to permit concrete to flow smoothly into spaces between bars without honeycombing. According to ACI 7.6.1 and ACI 3.3.2, the minimum clear spacing between parallel bars in a layer is not to be less than the largest of: (shown in Figure 7)

§ bar diameter bd

§ 2.5 cm § 4/3 maximum size of coarse aggregate

When two or more layers are used, bars in the upper layers are placed directly above layers in the bottom layer with clear distance between layers not less than 2.5 cm as specified by ACI 7.6.2 and shown in Figure 7.

Strength of singly reinforced rectangular sections

The compressive force in concrete C is given by

bafC c′= 85.0 (14)

Knowing that the net tensile strain in the extreme tension reinforcement is larger than or equal to 0.004,

ys fAT = (15)

From equilibrium of forces in the axial direction,

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0=− CT (16)

Substituting Eqn. (14) and Eqn. (15) into Eqn. (16)

bffA

ac

ys

′=

85.0 (17)

The nominal Strength nM is evaluated using the equilibrium of moments acting on the

section,

−=

2adfAM ysn (18)

Substituting Eqn. (17) into Eqn. (18),

−=

b'f7.1fA

dfAMc

ysysn (19)

The design moment, dM , is given by Eqn. (20)

−==

b'f7.1fA

dfAMMc

ysysnd ΦΦ (20)

where

Φ = strength reduction factor for flexure

=b width of section

=sA cross-sectional area of the reinforcement

=h total height of section

=d effective depth of section

=T total tensile force in the reinforcement

=C total compressive force in the concrete

=yf yield stress of the reinforcement

=a depth of rectangular stress block

=′cf concrete compressive strength at 28 days