Transverse and Longitudinal Torsion Reinforcement in Non-prestressed Beams

Summary

This 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

00:00

🏗️ 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.

05:05

🔍 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.

Keywords

💡Torsion Reinforcement

Torsion reinforcement refers to the additional steel reinforcement provided in beams to resist torsional forces. In the context of the video, it is crucial for ensuring the structural integrity of beams when subjected to twisting moments. The script discusses how to calculate and apply both longitudinal and transverse reinforcement to handle torsion, emphasizing its importance in beam design.

💡Beams

Beams are structural elements that are designed to primarily support loads transverse to their longitudinal axis. In the video, beams are the focus of the discussion on torsion reinforcement, as they are prone to torsional failure if not properly reinforced.

💡Non-Prestressed

Non-prestressed members are structural components that are not subjected to initial compressive stress before any external load is applied. The video script specifies that the torsion reinforcement discussion is relevant for non-prestressed beams, indicating that the design considerations differ from prestressed beams.

💡Threshold Torsion

Threshold torsion is a reference value used to determine when torsional reinforcement is necessary in beams. The script explains that if the factored applied torsional moment (T_U) exceeds this threshold, torsional reinforcement is required.

💡Longitudinal Reinforcement

Longitudinal reinforcement runs along the length of the beam and is used to resist torsional forces. The video script mentions that this type of reinforcement is necessary when the applied torsion exceeds the threshold torsion, and it plays a critical role in maintaining the beam's structural integrity.

💡Transverse Reinforcement

Transverse reinforcement is perpendicular to the beam's longitudinal axis and is typically in the form of stirrups or closed hoops. The script explains that this type of reinforcement is essential for providing resistance to torsional moments, with the ACI recommending specific amounts and spacing.

💡Stirrups

Stirrups are U-shaped or closed transverse reinforcements used to confine the concrete and provide torsional resistance. The video script describes stirrups as a form of transverse reinforcement, emphasizing their uniform spacing and their role in resisting torsion.

💡Factored Applied Torsional Moment (T_U)

T_U represents the torsional moment that a structure is expected to withstand, taking into account safety factors. The script uses T_U to determine the necessity of torsion reinforcement, showing its importance in structural design calculations.

💡ACI 318

ACI 318 is a building code reference in the United States for the design of concrete structures. The video script refers to ACI 318 for recommendations on the amount and spacing of reinforcement, indicating its authority in the field of civil engineering.

💡Yield Stress (FY_t)

Yield stress is the stress at which a material begins to deform plastically. In the context of the video, FY_t is the yield stress of the transverse reinforcement and is a key parameter in calculating the required amount of reinforcement.

💡Spacing (s)

Spacing refers to the distance between individual elements of reinforcement, such as stirrups. The script discusses the importance of uniform spacing (s) in transverse reinforcement to ensure effective distribution of forces and resistance to torsion.

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