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Strength Analysis of Beams
CE 4108 –Concrete Structures Design
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Design Methods
• Working-Stress Design Method
• Strength Design Method
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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
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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
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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
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Derivations of Beam Expressions
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Both should have the same area and the same centroid.
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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
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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
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22
:0
adfA
adTM
M
ysn
Therefore, the usable flexural strength is:
2
adfAM ysn
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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
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Example 3.1
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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.
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• 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.
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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
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ACI Commentary Figure 9.3.2
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• 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.
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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
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• 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.
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Balanced steel percentage
df
c
fEfd
c
y
ysy
000,87
000,87
)000,000,29/(003.0
003.0
)/(003.0
003.0
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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
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Example 3.2
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Example 3.3
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Example 3.4
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