Strength Analysis of Beams

CE 4108 –Concrete Structures Design

Design Methods

• Working-Stress Design Method

• Strength Design Method

Working-Stress Design (WSD)

• Also known as Allowable Stress Design or Straight Line Design

• Used in the U.S. from 1900s to 1960s

• Working loads (or Service loads) are used to proportion elements

• Still allowed by AASHTO in bridge design; commonly used in the design of liquid containing structures and prestressedconcrete

Strength Design Methods

• Previously known as the Ultimate-Strength Design Method

• Factored loads are used

• Members are designed so that they would just fail under the factored loads

• It provides more economical designs than WSD

• Since 2002, it’s the only method permitted by ACI

Structural Safety

• Strength reduction factor, φ – Used to reduce theoretical ultimate strength (called nominal strength). Accounts for uncertainties in material strength, dimensions and workmanship.

For example: φMn ≥ Mu

Factored moment loadNominal or resisting moment

Derivations of Beam Expressions

Both should have the same area and the same centroid.

ACI 10.7.2.3:

β1 = 0.85 for f’c ≤ 4000 psi

For f’c > 4000 psi:

65.0)005.0(1000

4000'85.01

cf

steel tensileof percentage where'85.0'85.0

'85.0

:0

bd

A

f

df

bf

fAa

fAabf

F

s

c

y

c

ys

ysc

y

22

:0

adfA

adTM

M

ysn

Therefore, the usable flexural strength is:

2

adfAM ysn

Strains in Flexural Members

• ACI 10.2.2: Strains vary linearly from N.A.

• ACI 10.2.3: The maximum usable strain in the extreme compression fiber is 0.003

• Since a = β1c, then:

c = a /β1

Example 3.1

Balanced Sections, Tension-Controlled Sections and Compression-Controlled or Brittle Sections

• Balanced section = Tensile stress will yield at the same time that compression concrete attains a strain of 0.003.

• Compression-controlled or brittle section = Concrete fails in compression before steel yields. There is very little deformation. Fails without warning.

• ACI 10.3.4: Tension-controlled sections –Sections in which the tensile steel reaches a strain of 0.005 or greater at the same time the compression concrete reaches a strain of 0.003. These beams will exhibit large deformations before failure (i.e. they are ductile).

• Sections that have steel with strains between 0.002 and 0.005 are in the transition range between compression-controlled and tensioned-controlled sections.

ACI 9.3:

φ = 0.90 for tension-controlled beam and slabs

φ = 0.75 for shear and torsion beams

φ = 0.65 or 0.70 for columns

φ = 0.65 or 0.70 to 0.90 for columns supporting very small axial loads

φ = 0.65 for bearing on concrete

ACI Commentary Figure 9.3.2

• ACI 10.3.5: Members subjected to axial loads equal to or less than 10f’c Ag the tensile strain (εt) in steel is permitted to be as low as 0.004.

• When members are subjected to axial loads greater than 10f’c Ag , the tensile strain (εt) in steel is permitted to be as low as 0.002.

• It is more economical to have sections in the tension-controlled region.

Minimum percentage of steel

• To account for the possibility that the ultimate resisting moment could be less than the cracking moment.

• ACI 10.5.1:

where bw is the width of the web.

Expressing as a percentage:

y

ww

y

c

sf

dbdb

f

fA

200'3min,

yy

c

ff

f 200'3min

• ACI 10.5.3: The minimum doesn’t have to be met if the reinforcement area is at least 1/3 greater than the area required by moment.

• ACI 10.5.4: For slabs and footings of uniform thickness, the minimum area is the one specified for shrinkage and temperature specified by ACI 7.12.

Balanced steel percentage

df

c

fEfd

c

y

ysy

000,87

000,87

)000,000,29/(003.0

003.0

)/(003.0

003.0

As discussed previously:

The expressions for c are equated and solved for ρ:

c

y

f

dfac

'85.0 11

yy

cb

ff

f

000,87

000,87'85.0 1

Example 3.2

Example 3.3

Example 3.4