TEMA 6b

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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO

Prof. Emilio Sordo Zabay [email protected]

rea de Estructuras05 de noviembre de 2012

1TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
mailto:[email protected]:[email protected]

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Not radial symmetry, therefore plane cross sections of the torsion member normal to the z axis do notnecessarily remain plane after deformation, neither radii have to remain straight.

The torque T causes each cross section to rotate as a rigid body about the z axis (axis of the couple); this axis is

called the axis oftwist.

Experimental evidence indicates that the cross-sectional dimensions of the torsion member are not changed

significantly by the deformations, particularly for small displacements. In other words, deformation in the

plane of the cross section is negligible.

The rotation of a given section, relative to the plane = 0, will depend on its distance from the plane = 0. For small deformation

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St Venants Semi-inverse Method Establish a set of equations that represent the assumed"mathematical structure" of the solution, and typically include

various parameters to be determined.

Equilibrium

equations

= , Warping function

(alabeo)geometrical condition (compatibility condition)

to be satisfied for the torsion problem

0

0 independent

necessary and sufficient condition for the existence of a

stress function (,) (Prandtl stress function) such that

Thus, the torsion problem is transformed into the determination of the stress function

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BOUNDARY CONDITIONS. Lateral surface

Because the lateral surface of a torsion member is free of applied

stress, the resultant shear stress on the surface S of the cross sectionmust be directed tangent to the surface.

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BOUNDARY CONDITIONS. End surfaces

Assumed that stresses undergo a redistribution with distance from the ends of the bar until the distributions

are essentially given by Eqs. 6.7. (Saint-Venant principle)

0 = 0 = 0 =

0 =

=

=

+

=+=2

The stress function can be considered to represent a surface over the cross section of the torsionmember. This surface is in contact with the boundary of the cross section. Hence, the torque is equal to

twice the volume between the stress function and the plane of the cross section.
mailto:[email protected]:[email protected]:[email protected]:[email protected]

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LINEAR ELASTIC SOLUTION

The elasticity solution of the torsion problem for many practical crosssections requires special methods for determining the function .

An indirect method may be used to obtain solutions for certain types ofcross sections, although it is not a general method.

Let the boundary of the cross section for a given

torsion member be specified by the relation

is a solution of the torsion problem, provided

=

and (, ) = 0 on the lateral surface of the bar

+

=2 =

2 =

2 +

= 2 Poissons equation

Laplacian

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Prof. Emilio Sordo Zabay [email protected] de Estructuras

05 de noviembre de 2012


ELLIPTICAL CROSS SECTION

= 2

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EQUILATERAL TRIANGLE CROSS SECTION

MECNICA AVANZADA 1 P f E ili S d Z b @

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THE PRANDTL ELASTIC-MEMBRANE (SOAP-FILM) ANALOGY

The method is based on the similarity of the equilibrium

equation for a membrane subjected to lateral pressureand the torsion (stress function) equation.

It is useful in the visualization of the distribution of

shear-stress components in the cross section of a torsion

member

denotes the lateral (small) displacement of an elastic membrane subjectedto a lateral pressure in terms of force per unit area and an initial (large)tension in terms of force per unit length

Prandtl stress function


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where c is a constant ofproportionality

membrane displacement is proportional to the

Prandtl stress function

Stress components at a point (,) of the bar areproportional to the slopes of the membrane at the

corresponding point (,) of the membrane

Twisting moment T is proportional

to the volume enclosed by the

membrane and the (,)plane

The stiffnesses of torsion members

with same G are proportional to thevolumes between the membranes

and flat plate. = 2 For cross sections with equal area, one candeduce that a long narrow rectangular

section has the least stiffness and a circular

section has the greatest stiffness


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11TEXTO: ADVANCED MECHANICS OF MATERIALS Boresi and Schmidt Wiley 6th Edition

At the external comers A, B, C, E, and F, the membrane has zero slope and the shear stress is

zero; therefore, external comers do not constitute a design problem.

At the reentrant comer at D, the corresponding membrane would have an infinite slope, which

indicates an infinite shear stress. In practical problems, the magnitude of the shear stress at D

would be finite but very large compared to that at other points in the cross section

If the torsion member is made of a ductile material and subjected to staticloads, the material yields and the load is redistributed to adjacent material,so stress concentration at D is not particularly important.

If material is brittle or the torsion member is subjected to fatigue loading,shear stress at D limits the load-carrying capacity of the member

Solution Better Solution

( stiffness )

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TEMA 6b