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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Preliminaries in Warping Torsion
Emanuel [email protected]
Graduiertenkolleg 1462Bauhaus-Universitat Weimar
2. Dezember 2009
Emanuel Bombasaro Preliminaries in Warping Torsion 1 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Part 1
Motivation and Introduction
Assumption
Deduction with the Analogy of Flange Bending (I ProfileSection)
St. Venant Torsion Deduction an a Solid section
Enhance the Approach for Warping Torsion
Examples
Elliptical SectionSpecial Cases of the Elliptical Section
Emanuel Bombasaro Preliminaries in Warping Torsion 2 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Part 2
Short Recapitulation
Solution for the Differential Equation in Analogy to theBending Theory of Second Order
Practice on a Simple Example
Design of a Structure
Important Remarks when Designing Structures due toWarping Impacts
Conclustion
Information
The slides represent only a minor and basic part of the holelecture, so to reach full comprehension the explanatory notes of thelecturer are indispensable!
Emanuel Bombasaro Preliminaries in Warping Torsion 3 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
St. Venant Torsion
Section can wrap freely, displacement in the x-direction arenot constrained
Torsional moment is not varying much along the elementlength
Section geometry is constant
Section is warping free.
This are very restrictive conditions!
Adhemar Jean Claude Barre de Saint-Venant (* 23. August 1797 in Villiers-en-Bie`re, Seine-et-Marne; 6. January
1886 in St Ouen, Loir-et-Cher) was a french Mathematician and Physicist.
Emanuel Bombasaro Preliminaries in Warping Torsion 4 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Lemma
Division of the torsional moment in two partsMT = MTp + MTs
primary Torsional MomentMTp = GIT
or St. Venant Torsional Moment. Section can freelywrap and so the displacements in x-direction are notconstrained. x = 0
secondary Torsional MomentMTs = M
results when the warping is constrained and sonormal stresses in x-direction arise. x 6= 0
Emanuel Bombasaro Preliminaries in Warping Torsion 5 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Appearance Warping Torsional Moment
St. Venant-Trosion (MT = GIT) exist always when a element is
twisted . Warping torsional moment appears additionally;if with twisting of the element warping occurs and
if the twist along the element is not constant
Emanuel Bombasaro Preliminaries in Warping Torsion 6 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Example
Element with constant,not constrained warpingand so without a warpingtorsional moment
Emanuel Bombasaro Preliminaries in Warping Torsion 7 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Warping
Warping of a section is only depending on the section geometrywhich means that their exist free warping and restrained warpingsections.Warping free sections are
Circular and Annuluses sections
Section which exists out of two sheets; angles and tees
Square pipes with constant wall thickness
Rectangular pipes only if b/t relation between web and flangeis the same
Warping Free SectionsIf a section is warping free only St. Venant torsionalmoment appears!
Emanuel Bombasaro Preliminaries in Warping Torsion 8 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Further Assumptions
Further constraints are assumed in the warping torsion approach
With the warping torsion connected secondary sheardeformations are neglected
With the warping torsion connected longitudinal deformationsalong a strait element of the section (web, flange) are linear,BERNOULLI hypothesis.
Section are constant along an element
The solution is based on the linear elasticity, which meanssmall deformations and twist angles.
Emanuel Bombasaro Preliminaries in Warping Torsion 9 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Reference Note Concerning Secondary Shear Deformations
The neglecting of secondary shear deformations leads at pointswith fixed warping deformations ( = 0) to bigger values for thesecondary torsional moment/ secondary shear stresses. Theinfluence decreases rapidly with the distance from the fixing and isprimary dominate for rectangular pipes.
Emanuel Bombasaro Preliminaries in Warping Torsion 10 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Derivation Schema for the Diff. Equation, Flange Bending
Emanuel Bombasaro Preliminaries in Warping Torsion 11 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Differential Equation for Flange Bending
Differential Equation of Flange Bendingx
(EIz
2vx2
)= Qy (x)
with the use ofT = Qy (x) =
MTh
v = (x) h2we obtainEIz,G
h2
4 (x) = MT it follows EI(x) = M
Definitions valid only in this special casewarping moment of second order I, refered to theshear center M, we can expressI = Iz,G
h2
4 Attention! [Lenght6]
The warping moment M [Force Lenght2] can ONLY
IN THIS CASE be computed by M(x) = MG (x)hwith flange moment MG (x).
Emanuel Bombasaro Preliminaries in Warping Torsion 12 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Basics ForcesThe equilibrium condition for torsional forces leads to
MT =
A
(z xy + y xz) dA (1)y and z coordinates are related to the the shearcenter.
Basic of Geometrical RelationsThe twisting axis of the section is called twist axis.We use =
Emanuel Bombasaro Preliminaries in Warping Torsion 13 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Relation to Gravity Axis
If the twisting axis (in the shear center) is parallel to the gravityaxis, the movement relative to the gravity axis a rigid body motion.Linear elasticity and small rotation angles are assumed.
Mx =
A
(zxy + yxz ) dA (2)
with help of the coordinative relations y = y yM andz = z zM we obtain
Mx = MT (3)
Emanuel Bombasaro Preliminaries in Warping Torsion 14 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Twisted Element Fiber When twisting an element due to torsionalaction a strait element fiber AB is turned into a helixAB . This leads to a deformation in w and vdirection but also in a displacement in u direction,which causes the section to be warped.If this warping is constrained this leads to normalstresses x and this case warping torsion has to beconsidered.
Small Deformations When the problem is reduced to smalldeformations and twisting angles, helixes can beconsidered to be strait lines. So the relation betweenrotation and twisting of the section can beexpressed = 4x .
Emanuel Bombasaro Preliminaries in Warping Torsion 15 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Derivation at a Solid Section Element
The derivation is performed at a general solid section, we use thepole coordinate system
y = r cos und z = r sin.The nodes P and Q are inthe twisted configuration P
and Q . The displacementsresults to
v = r cos ( + ) r cos, w = r sin ( + ) r sinafter linearization we getv = xz , w = xy
Emanuel Bombasaro Preliminaries in Warping Torsion 16 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Constitutive Relations
y =v
y, z =
w
z, yz = 2yz =
v
z+w
y(4)
it results y = 0, z = 0, yz = 0. Which shows that under puretorsion action on an element no distortion occurs. With x = 0,y = 0, z = 0 results x =
ux = 0 and so u=u(y,z).
Following approach u(y , z) = (y , z) is used. The constitutiverelations
xy = G(
y z), xz = G
(
z+ y
)(5)
2
y2+2
z2= 4 = 0 (6)
(y , z) is call warping function which fulfills the LAPLACEdifferential equation.
Emanuel Bombasaro Preliminaries in Warping Torsion 17 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Solving the Warping Function
Instead of matching to the boundary conditions, it is moreadvantageous to fit the PRANDTL torsion function (y , z). Sothe shear stresses xy and xz can be expressed
xy =
z, xz =
y(7)
What we see is that the PRANDTL torsion function is a stressfunction. With the help of the constitutive relations we obtain
z= G
(
y z),
y= G
(
z+ y
)(8)
Emanuel Bombasaro Preliminaries in Warping Torsion 18 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
PRANDTL Torsion Function
PRANDTL Torsion FunctionThe partial z derivation of the first equation andpartial y derivation of the second equation andsumming up leads to
4 = 2G (9)So we see that the PRANDTL torsion function fulfilsthe POSSION differential equation.
Boundary ConditionsThe surface of the element under torsional action isfree of stresses, so with the condition on the figureon slide 16 xzxy =
dzdy we obtain
xz dy +xy dz = y
dy +
zdz = d = 0 (10)
Emanuel Bombasaro Preliminaries in Warping Torsion 19 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Inserting
Inserting (7) in (2) results
MT =
A
[
y(y) +
z(z)
]dA + 2
AdA (11)
With the help of the GAUSS integration method on the first termof the equation we get
MT =
C (yny + znz ) dC + 2
AdA (12)
when looking at solid sections, along C is zero, it follows
MT = 2
AdA (13)
Emanuel Bombasaro Preliminaries in Warping Torsion 20 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Inserting
Inserting (5) in (2) results
MT = G
A
(y2 + z2 + y
z z
y
)dA (14)
The Integral
IT =
A
(y2 + z2 + y
z z
y
)dA (15)
which is call torsional second moment. IT is a section value. GITis called torsion stiffness. For the warping free element it results
MT = MTp = GIT (16)
Emanuel Bombasaro Preliminaries in Warping Torsion 21 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Warping
with the help of equation (7) and (8) we find
u
y= G
(
y z),u
z= G
(
z+ y
)(17)
and we find the separable approach function
u(x , y , z) = (x)M(y , z) (18)
the unit warping function M is relative to thesection shear centre
In previous slides we used u(y , z) = (y , z) as approachfunction. (In the literature mostly M is used.)
Emanuel Bombasaro Preliminaries in Warping Torsion 22 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Warping Stressis , Warping shear flow T
The warping stress results in the case of restrained warpingwith the help of eq. (18)
x = = Eu
x= EM , (19)
because the constraint x = 0 can not be fulfilled any more.
The warping shear flow T can be found
T = E
AM(s)dA = ES
(20)
by evaluating the equilibrium conditions d t ds + dT dx = 0.S is in analogy called warping first order moment.
Emanuel Bombasaro Preliminaries in Warping Torsion 23 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Warping Torsional Moment and Warping Moment
For a general section we obtain with equation (20) andM =
s0 r
Mt (s)ds
MTs =
T(x , s)r
Mt (s)ds = E
A
[M(s)
]2dA (21)
the integral is defined as warping second order moment I, so wecan express the warping moment in analogue to the resultingstresses
M =
A
MdA = E
A
[M]2
dA = EI (22)
Emanuel Bombasaro Preliminaries in Warping Torsion 24 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Synopsis
Returning to the Lemma
MT = MTp + MTs (23)
Primary Part; St. Venant Torsion
MTp = GIT (24)
Secondary Part; Warping Torsional Moment
MTs = EI (25)Differential Equation for Torsional Load Action
MT = GITEI GIT(x)EI(x) = mT (x)
(26)
Emanuel Bombasaro Preliminaries in Warping Torsion 25 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Example Elliptic Section
Shape Functiony2
a2+ z
2
b2 1 = 0
Stress Fucntion = a2b2
a2+b2
(y2
a2+ z
2
b2 1)
G
MT = 2
A dA =a3b3pia2+b2
G comination with the stress function
results = MTabpi(
y2
a2+ z
2
b2 1)
and so
xy = 2MTab3pi z und xz = 2MTa3bpiy . IT = a3b3pi
a2+b2
The warping function results to = a2b2a2+b2
yz
Emanuel Bombasaro Preliminaries in Warping Torsion 26 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Special Case; Circular Section
with a = b = R, y = r cosz = r sin we obtainxy = xz = 2MTR4pi r . IT = R
4pi2
The warping function results to = 0 and so obviously nowarping!
Emanuel Bombasaro Preliminaries in Warping Torsion 27 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Special Case; Rectangular Section
We replace b with the half of the narrow side of the section b andwe look at the limit b/a 0
so [(b)2 z2]G, obviouslyxy 2Gz , xz 0 and with L = 2a und b = 2b we obtainIT Lb33 .The warping function results to yz and obviously warpingexists and the section values has to be corrected!IT = 1Lb
3 und max = 2MTLb2
. are depending on the relation between length and hight L/b ofthe section.
Emanuel Bombasaro Preliminaries in Warping Torsion 28 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Differential Equation for Warping Torsion Function (withoutsecondary shear deformations)
EI(x) GIT(x) = mT (x) (27)
(x). . . torsion twist angle EI. . . warping stiffnesGIT . . . torsional stiffness (St. Venant) mT (x). . . uniform torsionalload
Differential Equation for Second Order Bending (withoutshear deformations)
EIw (x) N II w (x) = q(x) (28)
w(x). . . bending ordinate EI . . . bending stiffnessN II . . . longitudinal force q(x). . . uniform distributed load
Emanuel Bombasaro Preliminaries in Warping Torsion 29 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Warping Torsion Bending Theory 2nd OrderTerms
+ Twist Angle w Bending Ordinate
Twist w Angel of Bending Ordi-nate
M Warping Moment M Bending Moment
MT + Torsional Moment R Transversal Load
MTpPrimary Torsional Mo-ment (St. Venant)
N II w 2nd Order part of R
MTsSecondary Torsional Mo-ment
Q Shear Force
mT +Sectional Torsional Mo-ment
q Sectional Uniformal Load
MeT +Concentrated TorsionalMoment
P Concentrated Load
EI Warping Stiffness EI Bending Stiffness
GITTorsional Stiffness (St.Venant)
N II Longitudinal Force
Emanuel Bombasaro Preliminaries in Warping Torsion 30 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Warping Torsion Bending Theory 2nd OrderEquations
= MTp/GIT w = N II w /N II (Identitie) = M/EI w = M/EI
MT = MTp + MTs R = NII w + Q
M T = mT R = qM = MTs M
= QDifferential Equation
EI GIT = mT EIw N II w = q
Boundary ConditonsCradle BearingM = 0, = 0
M = 0, w = 0
Fixing = 0, = 0 w = 0, w
= 0
Free EndM = 0, MT = 0
M = 0, R = 0
Fixed HeadstockMT = 0,
= 0 R = 0, w = 0
Emanuel Bombasaro Preliminaries in Warping Torsion 31 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Application on Elements, Example
m = 10kNm/ml = 10.0mE = 21 000kN/cm2
= 0.3
Sections
I Profile Rectangular Pipe
h = 30cmb = 20cm
t = 1.5cms = 1.0cm
Emanuel Bombasaro Preliminaries in Warping Torsion 32 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
0 1 2 3 4 5 6 7 8 9 1020
10
0
10
20
x
MTs
[kN
m], M
[kN
m2]
Torsionsmomente
M
0 1 2 3 4 5 6 7 8 9 1040
20
0
20
40
MTp
[kN
m]
0 1 2 3 4 5 6 7 8 9 100
100
200
[]
Torsionsdrehwinkel und Verdrillung
x0 1 2 3 4 5 6 7 8 9 10
1
0
1
[]
Figure: Condition Diagrams I Profile
0 1 2 3 4 5 6 7 8 9 100
0.5
1x 105
x
MTs
[kNm
], Mt [k
Nm2 ]
Torsional Moment
Mt
0 1 2 3 4 5 6 7 8 9 1050
0
50
MTp
[kNm
]
0 1 2 3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
[!
]
Torsional Twist Angle and Twisting
x0 1 2 3 4 5 6 7 8 9 10
4
3
2
1
0
1
2
3
4x 103
v [
]
Figure: Condition Diagrams Rectangular PipeEmanuel Bombasaro Preliminaries in Warping Torsion 33 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Results
I Profile Rectangular Pipe
M = 19.81kNm2 M = 0.00kNm
2
MTs = 14.52kNm MTs = 0.00kNmMTp = 35.48kNm MTp = 50.00kNm = 111.87 = 0.43
= 0.66 = 3 103
Emanuel Bombasaro Preliminaries in Warping Torsion 34 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Figure: Result taken from Fiedrich + Lochner
Emanuel Bombasaro Preliminaries in Warping Torsion 35 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
x
MTs
[kNm
], Mt [k
Nm2 ]
Torsional Moments
Mt
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.5
1
0.5
0
MTp
[kNm
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52
1
0
[
]
Torsional Twist Angle and Twisting v
x0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.02
0.01
0
v [
]
Figure: System, Diagrams I Profile Section, Cantilever
Emanuel Bombasaro Preliminaries in Warping Torsion 36 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Forces and Stresses [kN/cm2]
Node a b 1 2 3
N [kN] 100.00 100.00 x ,N 1.13 1.13 1.13My [kNm] 10.00 10.00 x ,My 1.43 1.43 0.00Mz [kNm] 15.00 15.00 x ,Mz 5.00 0.00 0.00M[kNm
2] 0.10 1.50 x ,M 5.13 0.00 0.00MTp[kNm] 0.00 1.03 Tp 2.87 2.87 1.91MTs [kNm] 0.00 1.03 Ts 0.00 0.17 0.00 [] 0.00 1.49 [] 0.00 0.02v 13.63 5.85 3.50
x =N
A+
MyIy
z +MzIz
y +MIR ; = Tp + Ts (29)
v =2x + 3 2 (30)
Emanuel Bombasaro Preliminaries in Warping Torsion 37 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
= 1.0 takes into account secondary shear deformations
Emanuel Bombasaro Preliminaries in Warping Torsion 38 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Emanuel Bombasaro Preliminaries in Warping Torsion 39 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Conclusion
If a section is not warping free both the secondary torsionalmoment and the normal stresses due to the warping momenthas to be taken into account.
In addition the secondary shear deformations and sheardeformations may have to be taken into account.
When performing numerical computations, analyzations has tobe done with caution, because solving the differential equationcan lead to wrong results and so structural failure.
Emanuel Bombasaro Preliminaries in Warping Torsion 40 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Cradle Bearing in Steel Structures
Emanuel Bombasaro Preliminaries in Warping Torsion 41 / 42
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Introduction Assumption I Profile Section Derivation, Example Diff. Eq. Element Design Conclusion
Handbook of Continuum Mechanics: General Concepts -Thermoelasticity, J. Salencon, Springer 2001
Festigkeitslehre, 2. Auflage, H. Mang, G. Hofstetter, Springer2000
Baustatik Theorie I. und II. Ordnung, 4. Auflage, H. RubinK-J. Schneider, Werner Verlag 2002
Schub und Torsion in geraden Staben, 3. Auflage, W. Francke,H. Friedmann, Vieweg 2005
Mechanik der festen Korper, 2. Auflage, H. Parkus, Springer2005
Vorlesungen uber Stahlbau. Grundlagen, 2. Auflage, Kh. Roik,Ernst & Sohn 1983
Emanuel Bombasaro Preliminaries in Warping Torsion 42 / 42
TitleMotivation and IntroductionMotivation and Introduction
AssumpionBasics in Warping TorsionWarpingFurther Assumptions
Derivation with the Help of Flange BendingDerivation Schema for the Diff. Equation, Flange Bending
Derivation of Fundamental Relations on a Solid SectionPreliminariesDerivation at a Solid Section ElementSolving the Warping FunctionWarping MomentExample Elliptic SectionCircular SectionRectangular Section
Solution in Analogy to the Second Order Bending TheoryEquationsAnalogy
Application on Elements and DesignApplication on ElementsI Profile with Eccentric Horizontal Force, CantileverCondition Diagrams for Special ExamplesDesign Tables
ConclusionConclusionExample; Bearings in Steel StructuresReference