tema 6b
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7/31/2019 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
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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
2TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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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7/31/2019 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
3TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO
Prof. Emilio Sordo Zabay [email protected]
rea de Estructuras05 de noviembre de 2012
4TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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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7/31/2019 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
5TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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.
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7/31/2019 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
6TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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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7/31/2019 TEMA 6b
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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO
Prof. Emilio Sordo Zabay [email protected] de Estructuras
05 de noviembre de 2012
7TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
ELLIPTICAL CROSS SECTION
= 2
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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO
Prof. Emilio Sordo Zabay [email protected] de Estructuras
05 de noviembre de 2012
8TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
EQUILATERAL TRIANGLE CROSS SECTION
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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO
Prof. Emilio Sordo Zabay [email protected] de Estructuras
05 de noviembre de 2012
9TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO
Prof. Emilio Sordo Zabay [email protected] de Estructuras
05 de noviembre de 2012
10TEXTO: ADVANCED MECHANICS OF MATERIALS, Boresi and Schmidt, Wiley, 6th Edition
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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7/31/2019 TEMA 6b
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MECNICA AVANZADA 1DIVISIN DE CIENCIAS BSICAS E INGENIERAUNIVERSIDAD AUTNOMA METROPOLITANA - AZCAPOTZALCO
Prof. Emilio Sordo Zabay [email protected] de Estructuras
05 de noviembre de 2012
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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