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.