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Prestress Concrete Design Sessional
ULTIMATE STRENGTH DESIGNT BEAM DESIGN : Singly and Doubly
CE 416
Presented ByS. M. Rahat Rahman
10.01.03.044
Contents
Ultimate Strength Design
▫ USD (ultimate strength design)
▫ Assumption of USD
▫ Singly and Doubly Reinforced Beam
▫ T - Beam design
Ultimate Strength Design ( T BEAM ) || an introduction
Ultimat Strength Design
▫ Method of the determine the dimension of structure based on
▫ Ultimate load▫ Ultimate section
Ultimate Strength Design
▫ Historical BackgroundBeing used since 1957.
Ultimate Strength Design
▫ AssumptionsAssumptions simplify analysis.
Ultimate Strength Design || Assumption
Assumptions• Bars at the same level, provided that the
bond between the concrete and steel is adequate
• Is linearly proportional to the distance from the neutral axis.
• Modulus of elasticity for all grades of steel is taken as Es = 29 x 10 ^ 6 psi
Ultimate Strength Design || Assumption
Assumptions
• Plane cross sections continue to be plane after bending
• Concrete's tensile strength is about 1/10 of its compressive strength
• Cracked concrete is assumed to be not effective before cracking, the entire cross section is effective in resisting the external moments
Ultimate Strength Design || Assumption
Assumptions
• At high stresses, non-elastic behavior is assumed, which is in close agreement with the actual behavior of concrete and steel
• Maximum strain at the extreme compression fibers 0.003 by ACI code
• Compressive stress distribution may be assumed to be rectangular, parabolic or trapezoidal.
Ultimate Stress Design || Advantages
Advantages
▫ Better predicts strength
▫ Requires lesser material
▫ Easier to compute
▫ More rational approach
▫ Accounts for uncertainties in load.
Ultimate Strength Design
Beam Types
▫ Singly reinforced section
▫ Doubly reinforced section
▫ T-section
n w
Ultimate Strength Design
designing beam .. .
Ultimate Stress Design
▫ Singly Reinforced BeamA singly reinforced beam is one in which the concrete element is only reinforced near the tensile face and the reinforcement, called tension steel, is designed to resist the tension.
Ultimate Strength Design
▫ Doubly Reinforced BeamA doubly reinforced beam is one in which besides the tensile
reinforcement the concrete element is also reinforced near the compressive face to help the concrete resist compression. The latter reinforcement is called compression steel. When the compression zone of a concrete is inadequate to resist the compressive moment (positive moment), extra reinforcement has to be provided if the architect limits the dimensions of the section.
Ultimate Strength Design
▫ T-Section
T BEAM For monolithically casted slabs, a part of a slab act as a part of beam to resist longitudinal compressive force in the moment zone and form a T-Section.
Ultimate Strength Design
▫ T-SectionFrom ACI 318, Section 8.10.2
Effective Flange Width : Condition 1
For symmetrical T-Beam or having slab on both sides a) 16 hf + bw b) Span/4 c) c/c distance (smallest value should be taken)
▫ T-SectionFrom ACI 318, Section 8.10.2
Effective Flange Width : Condition 2 Beams having slabs on one side only a) bw + span/12 b) bw + 6hf c) bw + 1/2 * beam clear distance (smallest value should be taken)
▫ T-SectionFrom ACI 318, Section 8.10.2
Effective Flange Width :Condition 3
Isolated T Beam
a) beff ≤ 4 bw b) hf ≥ bw/2 (smallest value should be taken)
Ultimate Stress Design || T-section
T-Section basics
Ultimate Stress Design || T-section
T-Section behaviours
▫ T-section behaving as
▫ Rectangular section
▫ T-section
Continuous T-Beam
T- versus Rectangular Sections
When T-shaped sections are subjected to negative bending moments, the flange is located in the tension zone. Since concrete strength in tension is usually neglected in strength design, the sections are treated as rectangular sections of width w b . On the other hand, when sections are subjected to positive bending moments, the flange is located in the compression zone and the section is treated as a T-section shown in Figure 1
Continuous T-Beam
Continuous T-Beam
T - Section design
Strength Analysis :
1st case : (N.A. is with in the flange)
Analyze as a rectangular beam of width b = beff
Mn = As fy (d − a/2)
T-Section design
Case 2 : (N. A. is with in the web)
T beam may be treated as a rectangular if stress block depth a ≤ hf and as a T beam If a > hf .
Analysis of T-Beam
26
Case 1:
Equilibriumfha
s y
c eff0.85
A fT C a
f b
Analysis of T-Beam
27
Case 1:
Confirm
fha
005.0cus
1
ys
c
cd
ac
Analysis of T-Beam
28
Case 1:
Calculate Mn
fha
2ysn
adfAM
Analysis of T-Beam
29
Case 2: Assume steel yields fha
ys
wcw
fwcf
85.0
85.0
fAT
abfC
hbbfC
Analysis of T-Beam
30
Case 2: Equilibrium
Assume steel yields
fha
c w fsf
y
0.85 f b b hA
f
The flanges are considered to be equivalent compression steel.
s sf yf w
c w0.85
A A fT C C a
f b
Analysis of T-Beam
31
Case 2:
Confirm
fha
f
1
s cu 0.005
a h
ac
d c
c
Analysis of T-Beam
32
Case 2:
Calculate nominal moments
fha
n n1 n2
n1 s sf y
fn2 sf y
2
2
M M M
aM A A f d
hM A f d
Analysis of T-Beams
33
The definition of Mn1 and Mn2 for the T-Beam are given as:
Limitations on Reinforcement for Flange Beams
• Lower Limits– Positive Reinforcement
34
c
ysmin
w
y
4 larger of
1.4
f
fA
b d
f
Limitations on Reinforcement for Flange Beams
• Lower Limits– For negative reinforcement and T
sections with flanges in tension
35
c
y(min)
y
2 larger of
1.4
f
f
f
Ultimate Stress Design || T-section
T-Section design
Thank you