6.

0 TORSIONAL STRESSES IN I-SHAPED SECTIONS

6.1 PURE TORSION (SAINT VENANT TORSION) - Fibers are twisted

where Shear Modulus of Elasticity

6.2 WARPING TORSION - Extreme fibers translates laterally in opposite direction

Figure 6.2.1 Common Torsional Loading

Figure 6.2.2 Torsion of an I-Shaped Section

From figure 6.2.2, we define the displacement of a point at the flange uf to be a
function of the twist angle .

Consequently

eqn 1

Consider upper flange and recalling Beam Curvature Relation

Differentiating wrt x, we obtain

Substitute in (1)

Define Warping Torsion

Let
>> warping torsional constant

Then

The torsional moment is the sum of the rotational part and the lateral bending part

Recall
So that

Rewriting and dividing by ECw

The above equation is a non-homogenous PDE

Let and solve for the homogenous solution

Assume a solution in the form

Then

The above equation requires that

or

Thus
>> exponential form

Note that

for the particular solution since Mz is in general some function of z, the form of the
solution may be written as
 such that

where the terms on the left side must be paired with the terms on the right side. Rarely
will the function f1(z) be required to contain higher than second degree terms.

Study Example 8.5.1

6.3 DERIVATION OF PDE FOR LATERAL TORSIONAL BUCKLING (ELATIC THEORY)

I-SHAPED BEAM IN SLIGHTLY BUCKLED POSITION

The rotated primed axes is given by the transformation matrix equivalent to the matrix
of direction cosines.

The curvature equations in x, y and z axes are

Differentiating wrt to z

Substituting in the above equation the Moment -Curvature Relation

We get

Or rearranging

>> PDE for the

Angle of Twist

Note : See Salmon and Johnson for the complete solution of above PDE, page 437.
6.4 ELASTIC LATERAL TORSIONAL BUCKLING MOMENT Mcr
(LATERALLY UNSUPPORTED)

LATERAL SUPPORT
Two categories :
1. Continous lateral support by embedment of compression flange.
2. Lateral support intervals. (Laterally Braced)

STRENGTH OF I-SHAPED BEAMS UNDER UNIFORM MOMENT

Failure Modes even if Mu<Mp

1. Local buckling of flange in compression

2. Local buckling of web in flexural compression
3. Lateral torsional buckling

FOUR CATEGORIES OF BEHAVIOUR

1. Plastic moment strength Mp is achieved with large deformation (rotation

capacity)
2. Inelastic behaviour where Mp is achieved but with little rotation capacity.
3. Inelastic behaviour Mr, the moment above which residual stresses cause
inelastic behaviour to begin is reached or exceeded.
4. Elastic behaviour in which moment strength Mcr is controlled by inelastic
buckling.

ELASTIC LATERLA TORSIONAL BUCKLING MOMENT Mcr

Recall PDE for Elastic Lateral Torsional Buckling

The above equation may be written in this form

where
 The characteristic roots are

Substitute boundary conditions for simple support noting that for simple supports may
not twist but are free to warp.

>> end twist are zero

>> end torque are zero

Solving the above problem using Boundary Value Method, the final
solution becomes

>> basis for LRFD Eqns

6.5 LOAD AND RESISTANCE FACTOR DESIGN - I SHAPED BEAMS SUBJECTED TO

STRONG AXIS OF BENDING

The strength requirement according AISC-B3.3 may be stated as

where
 Whether or not plastic moment strength is reached, failure will be on one of the
following modes.

Figure 6.5.1 Beam Behaviour

Case 1 : Plastic moment is reached (Mn=Mp) along with large plastic rotation capacity
(R>=3 in Figure 6.5.1)

Note :
1. The section must be compact to prevent local buckling.
2. Lateral bracing must be provided where Lb<Lpd

Figure 6.5.2 Deformation requirements for developing plastic strengths

 Note: You may use M2 instead of Mp (AISC-App 1.7), because M2=Mp
under case 1.

Case 2. Plastic Moment is reached (Mn=Mp) but with relatively little rotation capacity (R<3
in Figure 6.5.1

Note :
1. The section must be compact to prevent local buckling.
2. Lateral bracing must be provided where Lb<Lp

Case 3. Lateral torsional buckling of compact sections may occur in the inelastic range
(Mp>Mn>=Mr)

Note :
1. The section must be compact to prevent local buckling.
2. Lateral bracing must be provided where Lp<Lb<Lr
3. The nominal strength Mn is a linear function of the lateral torsional
buckling strength, thus taking Mr=0.7Fy*Sx

where

Note: The length Lr is obtained by equating

and
 and solving for L which yields AISC F2.2b

where

Note: rts may be approximated as

Case 4. General limit state where nominal moment strength Mn occurs in the inelastic
range (Mp>Mn>=Mr)

Note :
1. The condition is very uncommon for rolled shapes.
2. Lateral bracing must be provided where Lp<Lb<Lr
3. The nominal strength Mn is a linear function of the lateral torsional
buckling strength, thus taking Mr=0.7Fy*Sx
4. The flanges are within the non compact limits (web local buckling
excluded)
5. Lateral bracing must be provided where Lp<Lb<Lr
6. The nominal strength Mn is a linear function of the lateral torsional
buckling strength, thus taking Mr=0.7Fy*Sx
 The strength is the lower value obtained from..

>> non compact case

and

>> inelastic LTB

Other parameters within the equation for inelastic LTB are the same for case 3.

Case 5. General limit state where nominal moment strength Mn equals the elastic
buckling strength Mcr. Mn<Mr, where Mr=0.7FySx

Note :
1. for flange and web (non compact range)
2. for slender elements

Using properties rts and ho defined previously

and the nominal strength is therefore

6.6 MOMENT GRADIENT Cb

The moment gradient Cb is a modification factor for non-uniform bending moment

variation for a beam segment laterally unbraced except at segment ends.

Old Formula 1961

where M1 and M2 are end moments such that M1<M2. M1/M2 is negative for
single curvature bending.

New Formula 1993 (Kirby and Nethercot) - Rm term added for mono
symmetric section AISC 2005. Note Cb has a minimum value of 1.0 and a
maximum value of 3.0

where

>> For doubly symmetric sections and for singly

symmetric sections bending in single curvature

Note : Rm is reduce when section is singly symmetric in reverse

curvature bending. AISC F-1. This formula is more accurate when
the unbraced segment has a non-linear moment variation.
6.7 PLOT OF BEAM MOMENT STRENGTH CURVE

Given a W 18x97 using ASTM A992 with Fy=50ksi

Material and Sectional Properties

LTB Properties

>> Formula 1

>> Formula 2

Iterate on Lb
Figure 6.7.1 Beam Moment Strength Curve using Formula 1
 Figure 6.7.2 Beam Moment Strength Curve using Formula 2

Problem Set No. 3

1. Determine the product of inertia and locate the shear center for the following steel
shapes.

2. Study Example 9.9.1 and 9.9.2. Discuss the procedure on how to undergo design of
beams in general.

3. Create a flow chart or a program on how to design a beam considering both

requirements on compactness and lateral support requirement and consider the effect
of moment gradient factor Cb.You may want to use the example above in checking
accuracy the results.
6.8 DETERMINATION OF MOMENT GRADIENT - MODIFICATION FACTOR Cb

EXAMPLE : Determine the moment gradient factor Cb for a simply supported beam with
a uniform load only. Lateral braces located at the ends and quarter points.

The moment equation may be expressed as a function of its distance x from the left
support.

Assume values and >> units are consistent

>> Moment equation

The plot of the Moment diagram is a parabola.

 Segment 1-2 and 4-5

1961 Formula

1993 Formula

Segment 2-3 and 3-4

1961 Formula

1993 Formula

*** Try solving a simple beam with point load at the midspan and braced laterally at
the ends and at quarter points.

*** Try solving a fixed end beam with uniform loading and braced laterally at the
ends and at quarter points.