Energy Method (Torsion)

 Energy Method (Torsion)

1. Assume a twisted shape.
2. Determine total potential
3. Minimize the potential

Get coeff.

o Strain Energy

1. Warping

where , , ,

- (1)

1. St. Venant torsion

- (2)

Assume a function depending on Boundary Condition

- (3)

Example)

Let

External work (Energy Loss)

Internal work (Strain Energy)

,

Know two types of integral

,

,

- (4)

for

- (5)

where

H.W ) Plot

1)When n=1, n=2, n=3 of (5) and

Compare with the know value given.

Chapter ⅢTorsion of closed thin walled Cross Sections.

1. St. Venant Torsion ( Pure Torsion). Timoshenko .T.E p294-304

◦ Application of membrane Analogy(Prandtl,1903)

a)Solid Circular section

The shape of membrane over solid circular X-sect

may be represented by paraboloid of revolution, i.e

,

From which ,

, For this const. Value of , the line integral along the perimeter of the section,

Recalling and considering the equilibrium of vertical direction in

membrane.

- (a)

Since the membrane force S = Const.

, , - (b)

The integration & Eq (b) gives,

b)Narrow rectangular section.

Neglecting end effect, the torsional stress function.

,

which yields

then ,

again , from Eq.(b)

c)St.Venant’s Approximation for the Torsion constant,

assumption.

Since the torsion constant is independent from the rotation of coordinates,

It can be observed from the expression for the torsion constant of a narrow Rect.sect. that appears in the form of the moment of inertia, the product of one dimension with the third power of the other which is,

, , ,

then where is a constant depends on a cross section.

Narrow rectangular section. = 36

If the properties of a solid circular section are considered

, ,

then

d) Closed, Thin-Walled Cross Section

Ref : Torsion in structures by Kollbrunner-Basler Springer-Verlaq, New York 1969 P10

Prandtl's membrane analogy, which has been successfully applied to solid x - sections, also may be used for hollow x - sect in the same form with condition that the inner boundary has to correspond to a contain line of the membrane. The membrane across the hollow space may be considered as being replaced by a horizontal, plane lid. This satisfies the requirement of zero slope of membrane over the stress-free hollow space.

Prandtl's membrane analogy applies to the whole region which is contained by the plane of x - section, the membrane and the lid even though the true membrane is only stretched across the effective area of the x - section. The gradient (slope) of the torsional stress functions is no longer a continuous vector function.

For a thin-walled x - sect, the analysis may be considerably simplified due to the following two reasons:

1. It is admissible to work on an average slope of the membrane at the center line of the wall, which implies a constant shear distribution across the wall.

Then the height of the lid from the plane of the x - sect can be expressed by

2. The average direction of the contour lines, which are identical to the shear stress trajectories, is assumed to be equal to the direction of the centerine of the wall, which indicates that the shear force per unit length, q, is tangential to the centerline of the wall.

The constant q shear flow Hydrodynamic analogy.

Open channel of constant depth, ideal incompressible fluid,

average velocity average shear flow

Reviewing Fig,

shear flow

( : Area under membrane, Centerline of wall)

: closed x - sect

open

area bounded by middle center

from Eq (2),

generally, 

for closed x - section







 

shape factor = 1.5 shape factor = 1.0

HW

For the triangular section shown, of side length “a” and t=a/20.

Evaluate KT0 & KTc open cross section, having a cut at point 1. If ,

compare moment capacities.

2. Lattice walls(See Nakai &Yoo, p77)

(1) Using the strain energy principle, the torsional analysis of cross sections consisting of different materials, may be reduced to the analysis of a homogeneous cross section with modified wall thicknesses for portions not consisting the sequence material.

The torsional moment T acting on the cross section represents the external force acting through the angle . The force acting one volume element and this moving by the amount represents the elemental work done by the internal forces

The shear stress in a thin-walled cross section is given by Eq(6) and the area of one element is t(s)ds. Thus

In this integral, both G and t may be functions of the arc length s. If one introduces a reference material for which go then the equation becomes,

If the transformed thickness t* of the wall of different material is used,

(2) A truss, a frame work or some other lattice structure may occasionally constitute one or more wall elements of a thin-walled box type cross section. Such a framework may be replaced by an equivalent transformed wall element of constant thickness, t* in the analysis by strain energy consideration.

Consider the shaft shown. The wall elements consists of three plates and one lattice truss system.

The strain energy U of the shaft of length a is

If each of the wall elements is of constant thickness ti and of width bi, then

Therefore, the contribution of the fictitious wall element to the strain energy is

The fictitious wall thickness t* will now be determined from the condition that the contribution ΔU has to be equal to the strain energy in the truss element of “a” .

The shear flow q results in a total shear force Q in the plane of truss ;

The shear force Q causes the force in the diagonal of length d. Since .

Fig b shows the upper chord separated from the adjacent wall element. The shear flow q acts along the line of separation.

It is introduced by the diagonal into the Gusset plates in the form of concentrated forces. There are concentrated forces causes axial forces in the chords which may be assumed to vary linearly within the distance a, from zero up to the maximum value of

The strain energy in a chord of length l and area of cross section A subjected to axial load P is

and for the case where the axial force various linearly within the distance from zero to ,

Using these results, the strain energy of a truss element of length “a” may be written down as follows ;

The thickness of the fictitious wall element is obtained when this sum is set equal to the expression for above.

Note ; 1. Eq (7) was also derived by R. Dabrowski in “Curved Thin-walled Girder Theory Analysis”, Cement Concrete Assoc. England 1968 Using Consistent deformation Theory.

2. Eq (7) was verified by prototype lab experiment by Joe Snyder, M.S. Theis, University of Maryland, 1974. Also Prof. Chen, Lehigh Univ, ASCE ST.

The following remarks are made with respect to the chord cross sectional areas and ; The wall S adjacent to the chord G will certainly assist in carrying the force . This may be considered by including a portion of the wall area with the chord area to give the equivalent chord area .

An estimate at the contribution of the wall S may be obtained from a consideration of the two limiting cases shown in the following Figure ;

a) b)

a)The wall element which connected to the wall S on the opposite side of the chord G has an area which is much smaller than i.e., .

b) The area of the wall element is much larger than i.e., .

If the wall S is assumed to have a rectangular cross section then the case a) the N.A. goes thru the outer most point of core and in case b) it goes thru the opposite boundary of the section. In the stress determination, the point of the wall S which could be added to the area of the chords in the first case, a) is and in the second case b) . These are two limiting cases and the true equivalent area of the chords is likely to be somewhere in between. Using the lower bound case ;

Note : However the test by Joe Snyder indicates that will yield the best correlation between test results and theoretical equivalent .

Without the detailed derivation, four additional lattice wall examples are given for later squence.

Ⅲ2. Warping Torsion of Closed Cross Section

Membrane Analogy

Assumptions

1. Section retains shape
2. Hooke’s Law holds
3. Thickness, t, function of s but not z
4. Shearing deformation due to warping neglected,

(second order)

To make open cross section, cut cross section at a point but there will be no relative movement at cut point in real section.

Eq. (2) previous handout

: warping at the starting point

: warping at any point on the cross section

By analogy to open cross section

Second term being correction term.

Solve for , obtain general Eqs. for closed cross section, etc.

Solving for gives

Solving for gives

Example)

,

- diagram

Warping constant

Path 1  2

Warping statical moment, uncorrected

- (21)

,

Statical moment, corrected

St.Venant shear effect

recall eqs (6), (7)

open cross section

which implies total warping around the closed x-sect must be equal to zero.

◦ Shear stress of closed x-section due to bending.

Shear flow

(1) - Determinate shear flow.

* Total shear stress

Total normal stress

(2) Indeterminate shear flow.

The compatibility requirement dictates that at cut.

(total shear stress)

(3) Combine

1