Basic Stress and Starin definition with according hookes law
What set of equations are used to describe the state of a body upon which a load is applied?
How many state variablesarise ina 3D body?
And what valuable info is gained From moment equilibrium regarding
In detail:
general definition of the stress vector
Derive teh equilibrium contition(equations) for an infinitesimally small piece of material.
Derive the Kinematic equations for strain with the help of the diagram:
What contidion applies when considering the infinitesimal strain tensor?
Give a quick explanation of Hookes generalized law for isotropic materials in 1D case
Relations in the generalized Hookes law.
What is:
The elasticity tensor
What properties must C have (even in full anisotropy)?
What are coupling effects regarding anisotropy and why are they caused?
What is a beam?
What is the goal of a designer when working with a beam?
what carachteristics define a beam?
What justifications are necessary when using a beam?
What is a disk?
What design descision can be taken when working with disk when the design space must not be filled?
Where does the load-bearing effect ocurr in a disk and what are ussually disks susceptible to?
What are the justifications necesssary when working with bars? (relevant for trusses)
What is a plate?
What are the load cases in a plate?
What charachterizes a plate and what justifications does it need?
what is a shell?
Part of airplane uan wing:
Name different types of stringers
What are the main structural elements in an aircraft?
What effects ocurr in an anisotropic material?
(example of full anisotropy)
How dos the stiffness matrix (and comliance matrix) of a fully anisotropic material look like?
Define monotropy:
Derice the stsiffnes matrix for a montropic (monoclinic) material:
(montorpy anlong x3 plane)
Considering a mirroring along x3 it is logically incudced:
13 Material constants.
What coupling effect remaines in a monotropic material?
What is orthogonal anisotropy (orthotropy)?
How does the stiffness (and compliance) matrix of an orthotropic material look like?
How many independent parameters are necessary to describe the behaviour of an orthotropic material?
nine, neun. nueve, devyat’
What coupling effects ocurr in orthotropy
That is no coupling between strain and normal stresses.
Explain transverse Isotropy:
What does teh Stiffness/ Compliance matrix look like under transverse Siotropy.
What is isotropy?
How does teh stiffness matrix look like for an isotropic material?
What are the engineering constant neded to describe orthotropic behaviour?
Represent the compliance matrix with engineering constants under Orthotropy:
What identities can be derived for the compliances in dependence of the engineering constants?
1.
2.
How many and which engineering consatnts remaing under full isotropy?
How does the compliance matrix look like under full isotropy?
How many material constants are necessary in each case of material symmetry?
What two main type of plane problems exist?
What simplifications may the loads of a plate or a disk undergo?
Where do these fluxes ocurr?
What assumption is made in order to calculate theese flows in a disk?
In the middle palne of the plate/ disk
The Stresses are constant in the z direction (across the thickness of the disk).
What is the difference in the normal stress and shear stress distrinution in disk and plates
What defines the plain state of strain.
What Strains get reduced to 0 in a plain state of strain with respect to the z palne
If :
then:
What are the strains remaining in a plain state of strain with regards to the z plane?
What shear stresses must become 0 in a plane state of strain with regards to the z- plane? (consider isotropy)
Derive the normal stress sigma_zz in a plane state of stress with respect to the z-plane.
With this information how can the ocurring strains be expressed?
How arethe replacement Elacticity modulus an Poissons ratio defined?
And how can be the strains summarized with these new definitions?
Why an to which equations are the equilibrium equations reduced in a plain state of strain with regards to the z- plane?
With which equations can a plain state of strain with regards to the z plane be described?
What assumptions are made in a plain state of stress with respect to the z- plane?
What equations remain for the generalized hook law (strains) under the plane state of stress? (Assume isotropy)
What differences (or similarities) arise between plane states of strain vs stress when comparing its contituent equations?
Can the solution for a plain state of stress be derived from a plain state of strain?
With which equations can the plane state of stress be described?
Are boundary contidions needed?
Derive the equations for the stress transformation (2D) along an arbitrary angle theta:
What is the hydrostatic stress state?
How can you find the principal axes of a load case (context: maximum normal stresses)?
What happens if you insert the value for the angle of the principal axees in the equation for the transformed stress?
What is shown in this equation?
Wich act in theta and theta +(pi/2)
Derive the equation to obtain the maximum shear stress.
What does the following relation entail?
What is an analogous equation for the maximum shear stress?
What is notet when the two shearstresses are maximal regarding the normal stresses
How do we obtain the defining equation for mohrs circle?
How can Mohrs circle be determined graphically?
How can the principal stresses, principa axees angle, maximum shear stress and angle for maximum shear stress determined via mohrs circle?
How does Mohrs circle look for a disk under uniaxial tensile load look like?
How does Mohrs circle look for a part under pure shear load?
How does Mohrs circle look under the hydrostatic case?
Consider an orthotropic disk ( thin along z).
Assume plain state of stress in z
1,2: principal axes
x,y,z: random axes
Deduce the formula for epsilon_zz
How can the principal stresses be expressed?
What are the reduced stiffnesses
This represents hookes law with respect to the principal axes af an orthotropic material
How can the relation between vectorsigma and vectorepsilon be obtained for arbitrary reference x,y?
What do theese Graphs represent?
What are the main components of a passenger aircraft an its main components?
What are the components of the wing?
What are the funtions/ task of the fuselage?
What are the significant loads the fuselage endures during flight?
What are the essential fuselage components?
What are the characteristics of the skin of the fuselage?
What Are the characteristics of the stringers in the aircraft fuselage?
Characteristics of the frames of the aircraft fuselage?
Name the main parts of different frame configurations
Nem the funtions of clips and cleats
What separates the cabin from the rear fuselage?
What are the loads of the primary srtucture?
What justifications does the fram need to undergo?
What assumptions need to be made so that the Euler Bernoulli bean Theory is valid?
Given the following Diagramm
How can the displacement up be espressed from the perspective u and the bending deformations in both tangential axes of the profile?
What relation can be assumed for the dicplacement in z and or y of the reference point and abitrary point P?
consider thereby bi-axial bedning and Euler-Bernoulli condtions
How can the strain at point P be calculated using the infinitesimal strain tensor? (Shear strains ar not considered). Use the relation from the previous slide.
using Hookes law derive sigma_xx
How can Normal forces and Torques be expressed in dependence of sigma_xx and the corresponding coordinates.
and how can it de epressed in relation to the strain and/ or displacements?
hint:
What geometrical parameter can be defined in:
How can thereby the equations be rewritten?
How can be these equations be called?
What complication arises from the fact that the point of reference is arbitrary?
How can the inegratl of the Area of a thing profile be expressed as the sum of simplified integrals?
Use the famous integral table to solve for an I- profile
Using the devils table clculate
fo a z-profile
What error occurs when calculating parameters for thin profiles?
What does the first ross-sectional normalization entail?
Referencing your profile parameters to the center of gravity of the profile.
What requisite is there so that the system can be transformed into the center of gravity?
How can you calculate the center of gravity coordinates from a randmo point of reference?
What is so great about referencing to the center of gravity?
How do the equations for the moments of inertia look when refrenced from a random reference to the center of gravity? (deduce the equations)
What does Steiners theorem allow us to do?
What Happens to the deviation moment once the reference point is set in the Center of gravity?
Moments of inertia for an I-profile:
Calculate cenetr of gravity coordinates as well as moments of inerta for a z-Profile?
Calculate Area, Satic moemnts, moments of inertia, deviation moment of an I-profile using the Integral Table an dthe center of gravitity as a reference.
How are is the contitutive law of the beam reduced when we relate the loads to the center of gravity (along with profile parameters)?
(secon cross section norm has not yet been done)
What is noteworthy in these equations compared to the arbitrary reference case?
What is the zero stress line of a profile?
(consider first cross section normalization has been made)
What is the aim of the second cross section normaliation?
What happens to the moments of inertia Izz and Iyy once the central axis are found?
Derive the Formula for the deviation moment of a rotated axis with reference toa set coordinate system
Using the equation for a rotated deviation moment find the angle for the principal axes of a profile:
Find the equation for the oments of inertia using:
Formula in order to find the principal moments of inertia:
What does this image show?
What is important to know regarding cross sectional normalization and symmetry?
Examples:
What is the polar moemnt of inertia?
What form doe the constitutive law of a beam take once both cross sectional normalizations are performed?
Knowing this
How can you calculate sigmaxx norma stres in x
hint: use Hooke and strain field.
What do these cuantities represent?
What does the Schear flow represent?
Consider the following free body diagram
Derive the eqution for shear flow Ts
Calculate the static moments of the following Profile:
What general rules can be applied to the shear stress flow ina profile due to trnsverse forces?
Simplified
What procedure should be followed when the loads are not parallel to the central axis?
What are the unit shear flows?
Derive the shear flow equation for closed off profiles (only one closed cell):
What problem arises when dealing with multiple cell profiles?
What is there to account for regarding compatibility when analyzing multiple cell profiles?
Calculate the y position of the shear center of the followin c- profile:
Steiner:
What is there to be said regarding to the Shear center and symmetry axis?
Derive the equation for the coordinates for the shear center ym and zm:
How can the equations for the shear center be simplified for thin walled profiles?
What general rules are aplicable to the shear center of thin walled profiles?
examples:
What assumptions are made in the timoshenko beam theory?
Considering Timishenko, What kinematics can be derived from the picture?
Considering Hooke; how can the tensions sigmaxx and gamma_xz be calculated?
Knowing this derive the constitutive equations for the timoshenko beam
Considering constitutive equations for a timoshenko beam
derive the equation for the deflection in z (that it: w)
What is the shear correction factor?
Considering these things:
Derive the equation with wich the stresses can be determined from the acting loads.
What is remarkable in the following equation regarding the comparison of timoshenkos theorem wiht Euler-Bernoullis?
What happens if the Shear stiffness gets increased in very large cuantity?
How can the correction factor be obtained via the Shear strain?
What does Aeff signify in a shear sensible beam?
How can the effective Area be obtained?
What assuptions are made within the fram of st venants torsion?
In th frame of st venants torsion how are strain gamma and angle theta related?
solve then for shear strain.
How can the distribution of the shear strees (St venant) be calculated knowing the strain distribution?
Derive the relation between torional moment and angle of deformation with:
Relation shear stress/ torque in st venants torsion?
Show that for thin profiles the shear stress due to torsion does not change along the circumference.
Make necessary assumptions.
Torsional moment of inertia for thin zilindircal profiles:
LAy the frame for arbitrary thin-walled cross sections:
Derive the relationship between Torsion torque and Shear flow for an arbitrary profile
deduce the resistance moment Wt
Derive the torsion moment of inertia for arbitrary profiles (thin):
HWat two special forms are to be considered
What modell is used for the deduction of shear stress due to torsion in open profiles?
Deduce the resistance moment for a thin open profile:
Remarkable difference between open and close cross sections
Compare open to close cross sections egarding torsion
Whats the ratio between maximum shear stress due to torsion in closed vs open profiles?
What is the framework under which warpin torsion is analyzed?
what do these different torques represent?
Derive the displacement u under warping torsion
What is this?
Relatiom between unit warping and swept area?
What procedure is similar with the one employed to determine the longitudinal displacement?
Derive the longitudinal displacement of a single cell unter torsion
How can the following expression be summarized?
Derive the normal stress for a problem with warping torsion andt bending in two axes (random reference):
What are these parameters?:
What paralelism can be observed with respect to warping profile parameters and steiners theorem?
What paralelism is there regarding warping in the principal axis System?
How does the constitutive law look for a bending+ Torsion warping problem in which both cross sectional normalizations have been performed?
Derive the diferential equation for warping torsion:
Whats the general solution for the warping torsion diff equateion?
What assumprions (RB) can be made inthese situations?
How do the normal an tangential stress ecuations look when considering warping?
Name three types of equilibrium regarding stability
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