Transverse and Longitudinal Torsion Reinforcement in Non-prestressed Beams
Summary
TLDRThis lecture discusses the necessity and design of torsion reinforcement in non-prestressed beams. When the applied torsional moment exceeds a threshold, both longitudinal and transverse reinforcements are required. The ACI provides formulas for calculating the required transverse reinforcement, which considers yield stress, spacing, and crack inclination, and longitudinal reinforcement to intercept torsion-induced cracks. The goal is to ensure the nominal torsion strength exceeds the applied torsional moment.
Takeaways
- 🔍 Torsional reinforcement is necessary in beams when the factored applied torsional moment (T_U) exceeds the threshold torsion.
- 📏 The threshold torsion for non-prestressed members is calculated as \( v \times \sqrt{f'_c \times AC_P} \) divided by \( PC_P \), where \( AC_P \) is the area enclosed by the perimeter of the cross-section and \( PC_P \) is the perimeter of the cross-section.
- 🔩 Torsional reinforcement includes both longitudinal and transverse reinforcement to resist torsion.
- 🔵 Longitudinal reinforcement runs along the beam, while transverse reinforcement takes the form of stirrups or closed hoops spaced uniformly.
- ⚙️ ACI 318 recommends a specific amount of transverse reinforcement to achieve a nominal torsion strength, detailed in section 11.5.3.6.
- 📐 The nominal torsion strength equation includes terms for the area of transverse reinforcement, yield stress, and the spacing of the reinforcement, adjusted by the cotangent of the angle theta.
- 📏 The area \( a_0 \) is defined as 0.85 times the area enclosed by the centerline of the transverse reinforcement.
- 🔵 The area of transverse reinforcement (A_t) is the cross-sectional area of one leg of the stirrup, not the total area of all stirrups.
- 📐 For non-prestressed members, theta is recommended to be 45 degrees to account for the inclination of torsion-induced cracks.
- 🛠️ Longitudinal reinforcement is also required to intercept the cracks that form due to torsion, with its formula detailed in section 11.5.3.7.
- 🔄 The design must ensure that the factored torsion (T_U) is less than the torsional strength provided by the reinforcement (T_N), ensuring the design is adequate.
Q & A
What is the primary purpose of torsion reinforcement in beams?
-The primary purpose of torsion reinforcement in beams is to resist torsional moments and to provide strength against the formation of cracks induced by torsion.
When is torsion reinforcement required in beams?
-Torsion reinforcement is required in beams when the factored applied torsional moment (T_U) is greater than the threshold torsion.
What is the threshold torsion for non-prestressed members?
-The threshold torsion for non-prestressed members is calculated as V * sqrt(f'c * AC_p) / P_CP, where V is 0.75, f'c is the compressive strength, AC_p is the area enclosed by the perimeter of the cross-section, and P_CP is the perimeter of the cross-section.
What are the two types of reinforcement recommended by ACI for torsion?
-The two types of reinforcement recommended by ACI for torsion are longitudinal reinforcement and transverse reinforcement.
What is the role of longitudinal reinforcement in torsion?
-Longitudinal reinforcement runs along the beam and helps to intercept the cracks that form due to torsion.
What is the role of transverse reinforcement in torsion?
-Transverse reinforcement, in the form of stirrups, is used to resist the torsional moment directly and to provide confinement to the concrete core.
What is the formula for calculating the amount of transverse reinforcement according to ACI 318?
-The formula for calculating the amount of transverse reinforcement is 2 * a_nought * 80 * FY_t / s * cotangent(theta), where a_nought is 0.85 times the area enclosed by the center line of the transverse reinforcement, FY_t is the yield stress of the transverse reinforcement, s is the spacing, and theta is the inclination of the torsion-induced cracks.
What does a_nought represent in the context of torsion reinforcement?
-In the context of torsion reinforcement, a_nought represents 0.85 times the area enclosed by the center line of the transverse reinforcement.
Why is the yield stress of the transverse reinforcement (FY_t) important in the design?
-The yield stress of the transverse reinforcement (FY_t) is important because it directly affects the strength and ductility of the reinforcement, which in turn influences the torsional resistance of the beam.
What is the significance of the spacing (s) of transverse reinforcement in torsion?
-The spacing (s) of transverse reinforcement is significant because it affects the distribution of the reinforcement and its ability to resist torsional moments effectively.
What is the role of the cotangent of theta in the torsion reinforcement formula?
-The cotangent of theta is a factor that accounts for the inclination of the torsion-induced cracks in the beam. For non-prestressed members, theta is recommended as 45 degrees.
How does the longitudinal reinforcement formula differ from the transverse reinforcement formula?
-The longitudinal reinforcement formula includes terms such as the area of one leg of transverse reinforcement divided by the spacing (s), the perimeter enclosed by the transverse reinforcement (pH), and the yield stresses of both the transverse and longitudinal reinforcements. It also includes the cotangent of theta squared.
Outlines
🏗️ Torsion Reinforcement in Beams
This paragraph discusses the necessity and types of torsion reinforcement in non-prestressed beams. Torsion reinforcement is required when the factored applied torsional moment (T_U) exceeds a certain threshold. The threshold torsion is calculated using a formula involving the compressive strength of concrete, the area enclosed by the perimeter of the cross-section, and the perimeter of the cross-section. The paragraph introduces two types of reinforcement: longitudinal, which runs along the beam, and transverse, which is in the form of stirrups. It also explains the ACI recommendations for transverse reinforcement, including a formula to calculate the nominal torsion strength of a beam with a certain amount of transverse reinforcement.
🔍 Design Equations for Torsion Reinforcement
The second paragraph delves into the specifics of the design equations for torsion reinforcement. It emphasizes the need for both transverse and longitudinal reinforcement to resist torsion and intercept cracks. The design equation for transverse reinforcement is detailed, including factors such as the yield stress of the reinforcement, spacing, and the inclination of torsion-induced cracks. The paragraph also covers the longitudinal reinforcement requirement, which is necessary to intercept the cracks formed due to torsion. The formula for longitudinal reinforcement is also provided, incorporating the yield stress of both transverse and longitudinal reinforcement and the perimeter enclosed by the transverse reinforcement.
Mindmap
Keywords
💡Torsion Reinforcement
💡Beams
💡Non-Prestressed
💡Threshold Torsion
💡Longitudinal Reinforcement
💡Transverse Reinforcement
💡Stirrups
💡Factored Applied Torsional Moment (T_U)
💡ACI 318
💡Yield Stress (FY_t)
💡Spacing (s)
Highlights
Torsional reinforcement is needed in beams when the factored applied torsional moment (T_U) exceeds the threshold torsion.
Threshold torsion for non-prestressed members is calculated using a specific formula involving material properties and cross-sectional dimensions.
Two types of reinforcement are required for torsion: longitudinal and transverse.
Longitudinal reinforcement runs along the beam, while transverse reinforcement takes the form of stirrups.
ACI 318-14 provides recommendations for the amount of transverse reinforcement needed to resist torsion.
The formula for calculating the nominal torsion strength of a beam with transverse reinforcement is detailed.
A_o is defined as 0.85 times the area enclosed by the centerline of the transverse reinforcement.
The area of one leg of the stirrup is used in the calculation for transverse reinforcement, differing from shear calculations.
The yield stress of the transverse reinforcement (FY_t) and its spacing (s) are critical parameters in the torsion strength formula.
The inclination of torsion-induced cracks is accounted for by the cotangent of theta, with theta recommended as 45 degrees.
Designers must ensure that the nominal torsion strength (TN) multiplied by a safety factor (phi) exceeds the applied torsion (T_U).
Longitudinal reinforcement is necessary to intercept cracks formed during torsion.
The formula for longitudinal reinforcement includes the area of one leg of transverse reinforcement and other geometrical factors.
pH, the perimeter enclosed by the transverse reinforcement, is a key parameter in the longitudinal reinforcement formula.
The yield stress of both the transverse (FY_t) and longitudinal (FY) reinforcements are considered in the design equations.
The cotangent of theta squared is used in the longitudinal reinforcement formula, with theta maintained at 45 degrees.
These equations define the requirements for both transverse and longitudinal reinforcement to resist torsion in beams.
Transcripts
in this lecture we're going to talk
about the sign of torsion reinforcement
specifically torsion reinforcement in
beams non-prestressed means so first of
all when do we need design of torsion
reinforcement when do we need torsion
reinforcement in beams we need torsional
reinforcement in beams when T U which is
the factored applied torsional moment is
greater than the threshold torsion that
we talked about in our previous lecture
this threshold torsion for non
pre-stressed members is equal to v
square root of f prime c AC P Square
divided by P CP as you may remember AC P
is the area enclosed by the perimeter of
the cross section PC P is the perimeter
of the close of the cross section F
prime C is the compressive strength and
fee is point 75 so whenever the applied
external factored torsion is greater
than the threshold torsion we need
transverse reinforcement and
longitudinal reinforcement for to resist
torsion so this leads us to the two
types of reinforcement that a CI accepts
or recommends for torsion reinforcement
number one longitudinal reinforcement
presented here in red and as the name
suggests it runs along the beam and
transverse reinforcement which takes the
form of stirrups closed stirrups spaced
are a uniform distance s
okay so basically two types of
reinforcements that need to be provided
for torsion longitudinal and transverse
longitudinal is al and transverse is a t
let's see what ACI recommends for each
one of these for transverse
reinforcement ACI recommends in eleven
point five three six an amount of
transverse reinforcement that gives you
a nominal torsion equal to two times a
naught times eighty times FY t divided
by s all of that x cotangent of theta
from this equation let's first define a
o AO r a naught is defined as point
eighty five of a H and a naught H is the
area enclosed by the center line of the
transverse reinforcement so if this is
the cross section of a beam and this is
the transverse reinforcement a naught H
is this area in here enclosed by the
transverse reinforcement that's ain't
not H you multiply that times point
eight five and that gives you a naught
that's the value that goes in that
equation eighty going back to our figure
eighty is the area
of transverse reinforcement of one leg
of the stirrups right so it's important
to keep that in mind because that's
different than in shear where we are
looking at the total area of the
stirrups 80 is the area cross-sectional
area
of one leg of transverse reinforcement
so it would be the area of that leg
shown there as a t FY t is the yield
stress of the transverse reinforcement s
is the spacing of the transverse
reinforcement and cotangent of theta is
a factor that accounts for the
inclination of the torsion induced
cracks in the beam and for non
pre-stressed members theta is
recommended as 45 degrees basically this
formula gives you the nominal torsion
strength of the beam with a certain
amount of transverse reinforcement
spaced at a certain distance you must
remember that for a design to be
adequate fee times TN must be greater
than T U and so you need to provide
enough transverse reinforcement to
satisfy this equation right the basic
design equation now a CRI also states
that along with transverse reinforcement
you need to provide longitudinal
reinforcement as I'm showing here in red
longitudinal reinforcement to also
intercept the cracks that will form in
torsion and that formula can be found in
eleven point five point three point
seven and it includes the transfer the
area of one leg of transverse
reinforcement divided by s right so
that's this
right here it also includes this term P
H what is pH pH is the perimeter
enclosed by the transverse reinforcement
so if this is your beam and this is your
transverse reinforcement and here you
have your longitudinal reinforcement pH
is the perimeter of this perimeter of
transverse reinforcement FY t and FY are
the yield stress of the transverse
reinforcement and the longitudinal
reinforcement respectively and cotangent
of theta squared is gonna be taken again
with theta at 45 degrees and basically
these are the two equations that define
the transverse and longitudinal
reinforcement for torsion
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