What assumptions are permissable for beams under normal force and bending in first iteration?
Linear elasticity
Geometric linearity (small displacements)
No torsion
—> LE I is mostly concerned with thin walled cross - sections reduced to the skeleton line
What stress components are to be expected when loaded with normel forces/ bending?
normal stresses
shear stresses
What are the basic assumptions ot the Euler-Bernoulli beam theory?
Slender beam —> length >> width, height
Straight and prismatic (Properties don´t change over beam length
Normal hypothesis (cross section remain orthogonal to neutral axis)
No thickness change
No warping —> Cross section will keep it´s shape
What is the static moment (name and formula) and what can it be used for?
It is the first moment of area
They can be used to find the enter of gravity —> S = 0
—> First cross sectional normalization
What is described by the 1st cross sectional normalization?
Reference of coordinate system to center of gravity (S = 0)
What happens to the loads if a 1st cross sectional normalization is carried out?
Normal fores and bending are uncoupled (for an arbitrary coordinate system, coupled via static moments).
However for a non duble symmetric cross section bending is still coupled between z and y —> Deviation moment
What is described by the 2nd cross sectional normalization?
The search of the principal axes
German term for principal axis
Hauptachse
What is the requirement that guides the 2nd cross-sectional normalization and how can it be achieved?
The deviation moment I_yz must equate to zero
—> By rotating the axis system around the center of gravity until Iyz = 0 —> Rotating angle: phi_0
How are Iyy and Izz characterized in the principal axis system?
They are the min. and max. possible Moment of inertia that can be achieved for the given cross section (as with Mohrs circle and the principle stresses)
(Remember: I_yz = 0)
Give 3 general rules when cross sectional normalizations have to be carried out.
1) In a double symmetric cross section, principel axis are parallel to symmetry axis —> Only first cross sectional normalization
2) If one plane of symmetry is present, one principal axis is parallel to that. Second one orthogonal —> Only first cross sectional normalization
3) If no symmetry is present, both cross sectional normalizations must be carried out
Give the constitutive law for a prismatic beam
How can we determine deflections from the constitutive law of a beam?
By fourfold integration of of the applied load. Integration constants will be given by boundary conditions
What assumption is a starting point to calculate shear stresses?
Equilibrium of the stresses in an infitesimal element of an Euler Bernouli beam (funny because Euler Bernoulli initially doesn´t include shear stresses)
What assumption is made regarding the normal force for Transverse shear calculation in an open cross section.
The normal force doesn´t change over the length of the beam —> N´ = 0
Give the formula for the shear flow in an open cross section
(If starting point sA is choosen wisely - e.g. free end T=0, it simplifies)
How does the shear stresses/ shear flow behave in respect to the loading direction and profile? (7 rules)
1) At free profile ends = 0
2) In loading direction (if z is linearly distributed) —> quadratic
3) Perpendicular to loading direction (if z is constant and not equal to zero) —> linear
4) Perpendicular to loading direction however at center of gravity (if z=0) —> constant
5) It assumes extreme valuves where the y-axis intersects with the skeleton line
6) Shear flow is symmetric around symmetry lines
7) The resultant of the shear stress/ shear flow must give the applied load
What is the issue with transverse shear calculation for a closed cross section?
The necessary boundary value Ts(sa) cannot be determined a priori —> statically indeterminate
What is the approach used to solve transverse shear forces in closed cross sections?
Introduction of a virtual opening (slit in longitudinal direction)
—> At the edge a displacement discontinuity will arise
To counter the displacement discontinuity, a circumferential unit shear flow is introduced.
By introducing compatibility at the edge, the size of the unit shear flow can be determinend
Superposition of the open shear flow and the adjusted unit shear flow will give the final shear flow
How are multicellular closed cross sections handled to calculate shear flow?
Every closed cell is virtually opened.
For every openend cell a countering unit shear flow is introduced.
The cells are coupled via shared parts of the cross section
What is the shear center and why is it of technical relevance?
The shear center is that point in a profile in which the resulting moment due to the shear flow equates to 0.
Technically speaking the shear center is the point where load should be applied to avoid an additional torsional load on the beam.
How can the shear center be determined?
When integrating the shear flow and the distance around the center of gravity, one can find a resultant moment. This moment is equal to the transverse force times the distance of the shear center to the center of gravity.
Give four general rules on how to determine the shear center of a profile.
1) In a double symmetric cross section, the shear center lies on the center of gravity
2) In a point symmetric cross section, the shear center lies on the center of gravity
3) In cross section with one axis of symmetry, the shear center lies on the axis of symmetry
4) If the cross section consists of thin walled straight segments that intersect at a common point, the shear center lies in that intersection
What beam theory allows for a certain shear deformation?
Which assumptions are made?
The Timoshenko beam theory.
It advances the Euler Bernulli beam by allowing for the rotation of the cross sections —> Normal hypothesis is discarded
HOWEVER - Hypothesis of plane cross sections is kept
What error is made in the Timoshenko beam theory by allowing for a rotation but not a deformation of the cross sections?
Shear strains occur, however, they are constant over the cross section.
—> This contradicts shear as we know, because it won´t be zero at a free edge. (and will not follow linear/ parabolic form)
How does the bending line calculation in Timoshenko beam theory compares to Euler Bernulli?
First term just added.
What remedy can be introduced to adjust for the error made by constant shear stresses?
A shear correction factor is introduced.
How is the shear correction factor determined and what is a typical value?
It is derrived from comparing the strain energy in the exact solution and the Timoshenko solution —> It adjusts for the strain energy
Typical value 5/6 (for rectangular cross section)
—> Used for most cross sections.
What geometric feature couples Normal forces and bending?
The static moments.
How do Shear flow and stress relate? (give formular for T(s) as function of tau(s) )
Why are the shear flow in the web of an I beam parabolic and the shear flow in the Flanges linear? (If loaded in a typical manner, transverse load acting in z)
Give a explanation based on the derrivation of the shear flow?
Profile rule explanation:
Shear stress/ flow are parabolic in the direction of the load and linear perpendicular to the direction of the load.
Explanation based on derrivation of shear flow:
The shear flow is based on an integral of the additional normal force flow nxx that is arising due to additional sigma_xx (at least in the case of a constant line load qz). This is constant in the flanges as it has the same z distance to the neutral line. The constant additional normal force flow nxx leads to a linear shear flow if integrated.
In the flange however, the normal stresses are of linear nature in respect to z. Integrating that linear normal force flow leads to a parabolic shear flow.
Explanation shear due to add. normal force flow
Integral, basis for T
Give a profile rule explanation
Give an explanation based on the static moments.
The Shear flow in respect to s is dependent on the static moment, the Moment of Inertias around the principal axis and the load
With
the moments of inertia being constant with respect to the principal axis and
the Load being constant for a cross section
The form of the shear flow is determined by the static moments. Keeping in mind that
It gehts obvious why for example for a load Qz, dependent on Sy the Shear flow is linear in the web and parabolic in the flanges
What is phy_y in context of the Timoshenko beam theory?
Give the formula for normal stress sigma-xx and shear stress tau-xz in a shear deformable beam - based on geometry and load
Give the formula for normal stress sigma-xx and shear stress tau-xz in a shear deformable beam - based on strains and based on deformations
Give the shear strain energy for a Stucture and an arbitrary location x
Structure
Arbitrary location x
Give an interpretation of the shear correction factor in Areas:
It reduces the area to an effective area.
Draw shear stress/ shear flow/ Sy for an I-beam
Draw shear stress/ shear flow/ Sy for a C cross section
Draw shear stress/ shear flow/ Sy for a T cross section
Draw shear stress/ shear flow/ Sy for an I-beam on its side (Qz applied in center of gravity)
Draw shear stress/ shear flow/ Sy for a C cross section on it´s side, load applied in center of gravity
Draw shear flow of opened single cell cross section (oppened in lower left corner) with Qz applied at the top and in the middle.
&
Draw applied unit shear flow
Draw combined resultant shear flow
Shear flow opened cross section
Unit shear flow applied
Resultant combined shear flow
Draw shear deformation of opened single cell cross section (oppened in lower left corner) with Qz applied at the top and in the middle.
Draw applied unit shear deformation
Draw shear flow of opened double cell cross section (oppened at the top of the middle web) with Qz applied at the top and in the middle.
Shear flow in opened cross section
Unit shear flow
Combined shear flow
Draw unit shear deformation of opened double cell cross section (oppened at the top of the middle web) with Qz applied at the top and in the middle. Also draw it for the bombined section
Give the way to calculate delta u-10 of opened cross section
Give the way to calculate delta u-11 (or general displacement continuity of unit shear flow)
Give the x-displacement u for an arbitrary point in the euler bernoulli beam
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