design for torsion (part i)

7/29/2019 Design for Torsion (Part I)

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5.5 Design for Torsion (Part I)

This section covers the following topics.

General Comments

Limit State of Collapse for Torsion

Design of Longitudinal Reinforcement

5.5.1 General Comments

Calculation of Torsion Demand

The restraint to torsion is provided at the ends of a beam. For beams in a building frame,

the restraint is provided by the columns. Precast beams are connected at the ends by

additional elements like angles to generate the torsional restraint. In bridges,

transverse beams at the ends provide torsional restraint to the primary longitudinalgirders. Box girders are provided with diaphragms at the ends.

For equilibrium torsion in a straight beam with distributed torque (tu), the maximum

torsional moment (Tu) is near the restraint at the support. The following figure shows a

schematic representation of the distributed torque.

tu

Tu

L

tu

Tu

L

Figure 5-6.1 Beam subjected to distributed torque

The torsional moment near the support is given by the following expression.

(5-5.1)

Here,

L = clear span of the beam

tu = distributed torque per unit length.

uu

t LT =

2


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For a straight beam with a point torque, the maximum torsional moment (Tu) is near the

closer support. If the location of the point torque is variable, Tu is calculated for the

location closest to a support. For a curved beam, Tu is calculated based on structural

analysis.

Design of Torsion ReinforcementThe design is done for the critical section. The critical section is defined in Clause 41.2

ofIS:456 - 2000. In general cases, the face of the support is considered as the critical

section. When the reaction at the support introduces compression at the end of the

beam, the critical section can be selected at a distance effective depth from the face of

the support.

To vary the amount of reinforcement along the span, other sections may be selected for

design. Usually the following scheme is selected for the stirrup spacing in beams under

uniformly distributed load.

1) Close spacing for quarter of the span adjacent to the supports.

2) Wide spacing for half of the span at the middle.

For large beams, more variation of spacing may be selected. The following sketch

shows the typical variation of spacing of stirrups. The span is represented by L.

L/ 4 L/ 4L/ 2L/ 4 L/ 4L/ 2

Figure 5-6.2 Typical variation of spacing of stirrups

First, an equivalent flexural moment Mt is calculated from Tu. Second, for the design of

primary longitudinal reinforcement, including the prestressed tendon, the total

equivalent ultimate moment (Me1) is calculated from the flexural moment (Mu) and Mt.

Third, the design of longitudinal reinforcement for other faces based on equivalent

ultimate moments Me2 and Me3 is necessary when the equivalent moment Mt is larger

than Mu. The following sketch shows the equivalent ultimate moments for design.


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Me1 Me2Me3

Me1Me1 Me2Me2Me3Me3

Figure 5-6.3 Equivalent ultimate moments

The design for Me1 is similar to the design of a prestressed section for flexure.

The design for Me2 is similar to the design of a prestressed concrete or reinforced

concrete section. The design forMe3 is similar to the design of a reinforced concrete

section. The design of stirrups including torsion is similar to the design of stirrups in

absence of torsion.

5.5.2 Limit State of Collapse for Torsion

The design for the limit state of collapse for torsion is based on the Skew Bending

Theory. For a beam subjected to simultaneous flexure and torsion, an equivalent

ultimate bending moment at a section is calculated.

The design for torsion involves the design of longitudinal reinforcement as well as the

transverse reinforcement. The longitudinal reinforcement is designed based on the

equivalent ultimate bending moment.

The transverse reinforcement is designed based on the Skew Bending Theory and a

total shear requirement. For the capacity of concrete, to consider the simultaneous

occurrence of flexural and torsional shears, an interaction between the two is

considered.

The equations in IS:1343 - 1980 are applicable for beams of the following sections.

1) Solid rectangular, with D> b.

2) Hollow rectangular, with D> band tb/4.

3) Flanged sections like T-beams and I-beams.

The sections are shown in the following sketch.


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D

b b

bw

t

Solid rectangular Hollow rectangular Flanged

D

b b

bw

t

Solid rectangular Hollow rectangular Flanged

Figure 5-6.4 Different sections for torsion design

The variables are as follows.

b = breadth of the section

= bw for flanged section

D= total depth of the section

t = thickness of the section.

The average prestress in a section at the level of CGC, is limited to 0.3 fck.

5.5.3 Design of Longitudinal Reinforcement

For the design of the longitudinal reinforcement, there are three expressions of the

equivalent ultimate bending moment for the three modes of failure (Reference: Rangan,

B. V. and Hall, A. S., Design of Prestressed Concrete Beams Subjected to Combined

Bending, Shear and Torsion, ACI Journal, American Concrete Institute, March 1975,

Vol. 72, No. 3, pp. 89 93). The modes of failure are explained in Section 5.4, Analysis

for Torsion. The figures of the failure pattern are reproduced here for explanation.


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Mode 1 Failure

Zoneunder Cu

Tu

Me1

Zoneunder Cu

Tu

Me1

Zoneunder Cu

Tu

Zoneunder Cu

Tu

Zoneunder Cu

Tu

Me1Me1

Figure 5-6.5 Idealised pattern and design moment for Mode 1 failure

The equivalent ultimate bending moment for Mode 1 failure (Me1) is given by the

following equation.

(5-5.2)

The equivalent bending moment forTu is given as follows.

(5-5.3)

In the previous expression,

Mu = applied bending moment at ultimate.

Mt = additional equivalent bending moment for torsion.

Tu = applied torsion at ultimate.

Since, the torsion generates tension in the reinforcement irrespective of the sign, the

sign ofMt is same as that ofMu.

e u tM = M + M 1

t u

DM = T +

b

21


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Mode 2 Failure

Zoneunder Cu

Tu Me3Zoneunder Cu

TuZoneunder Cu

Tu Me3Me3


The equivalent ultimate transverse bending moment for Mode 2 failure (Me3) is given as

follows.

(5-5.4)

In the previous expression

e = Tu/Vu, ratio of ultimate torsion and ultimate shear force at a section.

x1 = smaller dimension of a closed stirrup.

The larger dimension of a closed stirrup is represented as y1. the dimensions are

shown in the following sketch.

x1

y1

x1

y1

Figure 5-6.7 Dimensions of a closed stirrup

The transverse bending moment Me3 is considered when the numerical value ofMu is

less than Mt. Me3 acts about a vertical axis.

e t

b

+x DM = M + De +b

21

3

2

11

22 1


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Mode 3 Failure

Zoneunder Cu

Tu

Me2Zoneunder Cu

Tu

Zoneunder Cu

Tu

Me2Me2


The equivalent ultimate bending moment for Mode 3 failure (Me2) is given by the

following equation.

Me2 = Mt Mu (5-5.5)The expression of Mt is same as for Mode 1 failure, given before.

Mode 2 failure is checked when the numerical value of Mu is less than that ofMt. Me2

acts in the opposite sense of that ofMu.

The longitudinal reinforcement is designed for Me1 similar to the flexural reinforcement

for a prestressed beam. The design of flexural reinforcement is covered in Section 4.2,

Design of Sections for Flexure (Part I) and Section 4.3 Design of Sections for Flexure

(Part II). When Me2 is considered, longitudinal reinforcement is designed similar to a

prestressed concrete or reinforced concrete beam. When Me3 is considered,

longitudinal reinforcement is designed similar to a reinforced concrete beam. For a

singly reinforced rectangular section, the amount of longitudinal reinforcement (As) is

solved from the following equation.

(5-5.6)

In the previous equation,

d = effective depth of longitudinal reinforcement

fy = characteristic yield stress of longitudinal reinforcement

fck = characteristic compressive strength of concrete

Mu= one ofMe2 and Me3.

y s

y s u

ck

f Af A d - = M

f bd0.87 1

design for torsion (part i)

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