Exam 2 - Study Guide

What is shear flow?

'q' is the shear flow and the formula for shear flow is as follows q = T / 2 * A where T is the torque being applied and A is distance across the cross-sectional area.

What is the relation between E, G, K and v.

...

Assumptions made in deriving the torsion formula for circular shafts

1) Circular cross sections remain circular after loading (no warping) 2) Shear stress is less than yielding shear stress 3) Material is isotropic 4) Applied torque lies in a plane perpendicular to the shaft

What assumptions are made in deriving the unsymmetrical bending stress formula?

1) Material is isotropic 2) Stresses are below the proportional limit 3) Plane of bending contains flexural axis 4) If (3) is not the case, then the external laods would cause torsion of the cross section in addition to bending.

List the six theories of failure.

1) sigma_max 2) tau_max 3) eta_max 4) U_max 5) Deviatoric U_max Theory 6) Von mises theory

What is a body force? Give an examples

A body force is a force that acts through the volume of a body, rather than through contact. Gravity and electro magnatism are examples of body forces.

In resisting torsion, is an open tube better suited or a closed tube? Why?

A closed tube is better suited for resisting torsion because the shear stress is uniformly distributed in a closed tube though te thickness of the wall of the closed section. An open section has a linear distribution.

What is Alclad?

A corrosion-resistant aluminum sheet formed from high-purity surface layers bonded to high strength aluminum alloy core material.

What is a stress funciton? How is it defined?

A function of x and y that relates the stress in a 2-dimensional case such that the substitution for the stresses in terms of this function automatically satisfies the equations of equilibrium no matter what form teh function may take. The stress function must also satisfy te 2-dimensional equations of compatibility, plus te appropriate boundary conditions.

What is a long column?

A long column is one that will buckle rather quickly and it's buckling stress is often less than that of its yield stress

What is the difference between a pinned end and a fixed end in a column?

A pinned joint is free to rotate but constrained from laterally moving while a fixed joint in constrained from rotation and lateral movement in any direction.

How do strain transformation equations and a roseete help engineers in a lab.

A roseete can give an engineer the stress in 3 directions for an arbitrary plane. From this, one can use the strain transformation equations to find te principle strains.

What is a short column?

A short column is one that will not buckle elastically easily, essentially the material strength is greater than the yield strength.

What is a strain gage?

A strain gage is a device used to measure the strain of an object. The strain is attached to the object by an adhesive. When object is deformed, the strain gage is also deformed, cause a change in the resistance of the gage. This change in the resistance can be related to strain through a gage factor.

What is a rosette?

A strain gage rosette is, by definition, an arrangement of two or more closely positioned gage grids, separately oriented to measure the normal strains along different directions in the underlying surface of the test part. Rosettes are designed to perform a very practical and important function in experimental stress analysis. It can be shown that for the not-uncommon case of the general biaxial stress state, with the principal directions unknown, three independent strain measurements (in different directions) are required to determine the principal strains and stresses. And even when the principal directions are known in advance, two independent strain measurements are needed to obtain the principal strains and stresses.

How is a stress at a point defined?

A stress at any point is defined by 3 direct/normal stresses, and 6 shear stresses.

Give an example of an open tube subjected to torsion in an aircraft structure.

A stringer in an example of an open tube subjected to shear in an aircraft structure. Also a spar.

What is a surface force? Give examples

A surace force is the force that ats across the internal or external surface element of a material body. The twi types of surface forces are pressure and stress. An example of a surface force is the lift of an airfoil, which is a pressure.

When is a tube thin?

A tube is thin when the ratio of the thickness to the average radius is less than 5%. t/Ra < 0.05

Give an example of a thin tube used in an airplane structure.

Aileron, torque tube, elevator, trim tab (most control surfaces)

What is elastic buckling?

An abrupt increase in the lateral deflection of a column at a critical load while the stresses acting on the column are wholly elastic.

What is the optimum shape for minimum weight for a compressive member?

An equilateral triangle gives about a 20% higher critical loading strength than the next highest yielding shape, a circular column.

What is an open tube?

An open tube is a thin circular shaft that doesn't have a completely closed cross sectional area. Example: a cut down the length of a tube such as a slot for a strut or some other structural component.

What is orthotropy?

An orthotropic material has different properties along each axis. An example would be a tree trunk.

What is anisoptropy?

Anisitropy is the opposite of isotropy, meaning that properties of a material depend on te direction and orientation of that material.

Will local buckling of a longeron make it a fail safe structural component?

As long as the structure does not yield as is the definition of local buckling, and remains an effective control surface throughout its operational lifetime, then it is still a fail-safe structure. However replacement would be advised.

What is a principle strain?

At a point in a deformable body, there are two mutually perpindicular planes on which the shear strain (gamma) is zero, and the normal to which the direct strain is maximum or minimum. These strains are reffered to as te "principal strains".

Identify all terms in Euler's buckling formula

E- Young's Modulus I - Moment of Inertia

What advantages does one gain by using hollow shafts?

Hollow shafts have comparative strength with solid shafts because the center region that is removed has only small stresses on it. The advantage is that hollow shafts do not weigh as much.

What is a principle stress?

In a plane where the axial/direct stresses are either maximum or minimum, and no shear stresses exist, the axial/direct stresses are referred to as the "principle stresses:

What is isotropy?

Isotropy means that a material is uniform in all directions and oreitnations.

What is the purpose of secant formula?

It is used to calculate both the maximum deflection and maximum stress in a body that is undergoing some kind of deformation.

What are the advantages of the stress function?

It provides a simple way to solve for stresses in an elastic body that satisfy the equations of equilibrium and compatibility.

What method is employed to determine the panel buckling of Alclad panels?

It uses the same method described in question 83, however the average of the two values is taken as well as the average of the plasticity factor in order to estimate it.

List some flight vehicle components subject to buckling.

Landing gear, longerons in fuselage, stringers in wing structures

Draw Mohr's Circle for a 2D element subjected to pure shear. What is max compressive stress?

Mohr's circle of element in pure shear is always centered at te orgin. In this case, the maximum compressive stress will be of the same magnitude of the maximum shear stress.

Does Euler's buckling formula hold good for composite material?

No, because it collapses instantly when the load increases by a small amount

Does warp exist when a circular shaft is subjected to torsion?

No, warp doesnt not exist in a circular shaft subjected to torsion.

When a column buckles elastically, has the column "failed" structurally?

No, while the column cannot deform to its original shape and thus will not be able to retain its original strength characteristics that does not imply overall structural failure as it can still support some kind of loading.

Is a state of plane stress also a state of plane strain?

Plane stress does not necessarily correspond to plane strain.

What is plane stress?

Plane stress is stress that exist in a 2-dimensional plane, assuming the stresses in the 3rd direction to be 0. An example would be the skin of an aircraft, made of a thin sheet of metal. The stresses in the thickness are negligible compared to those in the plane of te skin.

How is power related to torque?

Power or work done is torque x rpm or force x distance / time P = Tw = (2*pi*N*T)/60 HP = (2*pi*N*T)/(60*550)

What is shear center?

Shear center is where shear loads pass through in order for no twisting of the beam cross-section to occur. It is also the centroid of teh internal shear force system. It is a property that is dependent on structure.

How does one calculate the shear stress and twist for a composite open tube such as a tee or an angle?

Shear stress is defined as tau_max = T*r/J. T is the torsion applied, r is the radius of the shaft, and J is the second moment area of inertia of the object. The maximum overall deflection (theta_max) is defined as (T*L)/(G*J) where T is the torsion applied, L is the length of the shaft, G is the shear modulus, and J is the second moment area of inertia. J = sum((b*t^3)/3).

How does shear stress vary along the radius in a circular shaft? In a hollow shaft?

Shear stresses get larger as the radius of a shaft gets larger. For a solid shaft, the shear stress is 0 at the center. Hollow shafts have a shear stress on the inner walls, but that stress is mless than that on that outer walls.

What method is employed to determine the panel buckling in the inelastic range?

Sigma_cr = ((Eta) * Pi^2 * k_c * E) / ( 12 * (1 - v_e^2)) * (t / b)^2 where Sigma_cr is the critical panel stress, Eta is the plasticity correction factor, k_c is the compressive buckling coefficient, E is Young's Modulus of the material, v_e is Poisson's Ratio for the material, t is the thickness of the panel, and b is the length along which the compressive load is applied. The main difference is the plasticity correction factor that accounts for the material being in the inelastic range

Define strain.

Strain is the measurement of deformation ins a material or object. Normal strain is the change in location of a point, while shear strain is the difference in angle of two sets points that were at one point perpindicular.

Define stress.

Stress is the measure of the internal forces acting within a deformable body. These internal forces arise from the application of external forces on the body.

What is inelastic buckling?

Sudden increase of deflection or twist in a column when compressive stress reaches the elastic limit but before elastic buckling develops.

What is symmetric bending?

Symmetric bending is when the bending moment is applied in the plane of symmetry of the beam.

What is the Bredt Batho formula?

T = 2Aq Deals with shear flow and the distribution of stress across the cross sectional area of a thin wall vessel. Two assumptions are made, the section is a closed cross section and there is no change of shear stress across the thickness of the vessel.

Torsion Formula for Circular Shafts

T/J = tau/r = (G*phi)/L T - Torque J = Polar Moment of Inertia r - radius G - Shear Modulus L - Length tau - Resulting shear stress at radius r phi - angle of twist over length L

As one decreases stringer spacing, will the critical buckling stress increase or decrease?

The buckling strength of each stringer would stay the same, but decreasing the space between them would ultimately increase the buckling strength of the fuselage as the compressive forces are shared between more stringers.

As one increases fuselage bulkhead spacing, will the critical buckling stress increase or decrease?

The bulkheads are there mainly for structural support and for resisting span-wise compressive and torsional forces; the stringers themselves are what handle all of the compressive forces along the length of the fuse.

What is the compatibility equation? Why is it needed?

The compatibility equation ensures that the body being analyzed remaines continuous during the deformation, so that no voids are formed.

How is radius of gyration defined?

The distance from an axis at which the mass of the body may be assumed to be concentrated and at which the moment of inertia while be equal to the moment of inertia of the actual mass about the axis, equal to the square root of the quotient of the moment of inertia and mass. Radius of gyration = sqrt( I / m).

What is the effective length of a column?

The effective length of a column is defined as the distance between successive inflection points or points of zero moment in a system (also denoted as K*L or the effective length factor multiplied by the system length)

Do the equations of equilibrium depends on the material properties?

The equations of equilibrium do not depend on the material properties because they are only for elastic deformation.

Why do the equations of equilibrium not hold for the boundary of an element?

The equations of equilibrium do not hold for the boundary of an element because the surface forces must be taken into account.

Is von mises theory more conservative than maximum shear stress theory?

The maximum shear stress theory is more conservative than von mises theory

What is the maximum theoretical value for v for a material?

The maximum value is 0.5

What is the membrane analogy? Why is it now only of historic interest?

The membrane analogy compares the partial differential equation governing the torsion of a prismatics bar of arbitrary cross section with the partial differential equation governing the displacement of a membrane stretched over an identical cross section that is pressurized. Both equations are governed by Poisson equation. The following comparisons can be made: Torque: 1) Torque Applied 2) Shear stress at a point 3) Direction of stress at any point Membrane: 1) Twice the volume enclosed by the membrane 2) Slope of the membrane at A' 3) Tangent to the membrane at that point It is only of historic interest now because new technology make it no longer necessary.

Consider two columns of same material, same length. One is fixed at both ends, the other is pinned at both ends. Which one will have higher buckling load?

The overall buckling load will change because the effective length of a column that is pinned at both ends is longer than that of the effective length of a column that is pinned at both ends.

What is panel buckling?

The phenomenon where a panel subjected to large compressive forces in the plane of the panel, fold or "wrinkle" in failure.

What is structural idealization? Know how to idealize a structural cross section.

The process of replacing an actual structure with a simple system conducive to analysis

Define Poisson's ratio.

The ratio of the proportional decrease in a lateral measurement to the proportional increase in length in a sample of material that is elastically stretched.

Where is the locations of a shear center for a closed tube of arbitrary cross section?

The shear center is in the plane of symmetry of the beam cross section, or it lies with the paremiter of the tube.

Where is the location of the shear center for an open tube of arbitrary cross section?

The shear center les on the opposite side of the open face.

Give one application of panel buckling in aircraft structure

The skin of an aircraft can come into panel buckling mode when in compression

What are the limitations of the stress function?

The stress function is limited to 2 dimensions, complex geometry, stress constraints

How many equations of equilibrium exist for a structural element? What are they?

There are 3 equations of equilibrium that exist for a structural element. They are: (dsig_xx/dx) + (dtau_yx/dy) + (dtau_zx/dz) + X = 0 (dtau_xyx/dx) + (dsig_yy/dy) + (dtau_zy/dz) +Y = 0 (dtau_xz/dx) + (dtau_yz/dy) + (dsig_zz/dz) + Z = 0

How many components of strain can one identify in a structural element?

There are 9 components of strain in a structural element. There are 3 direct straings, and 6 shear strains. Equilibrium can cut the shear strains from 6 to 3.

How many components of stress can one identify in a structural element?

There are 9 components of stress in a structural element. There are 3 direct, and 6 shear stresses. This can be reduced to 6 if the struture is in equilibrium, cutting the shear stresses from 6 to 3.

When a structure is in uniaxial tension, will there be any shear stress in the structure?

There will be no shear stress in a structure in uniaxial tension.

What is the formula for calculating twist for arbitrary thin tubes?

Theta = (Length * Torque) / ( J * Shear Modulus (G)). Torsion for circular shafts is defined as Theta = (32 * Length * Torque) / (G * Pi * D^4). Torsion for a hollow circular shaft is defined as Theta = (32 * Length * Torque) / (G * Pi * (D^4 - d^4)).

What is unsymmetric bending?

Unsymmetric bending is when the bending moment is applied not in the plane of symmetry of the beam or a plane of symmetry may not exist.

Draw Morh's Circle for an element with tensile and compressive stress of equal magnitude but opposite sign in the x-y plane. Then find out what the maximum in plane shear stress is.

Use the principle stress equations

What is warp?

Warp is a twist or distortion in a shape or form of something. It is the change in cross sectional shape of a shaft due to bending.

What is plane strain?

When particles of a body suffer displacements in only one plane

Under what condition does a state of plane stress also represent a state of plane strain?

When sig_xx = -sig_yy

What happens when warp is constrained?

When warp is constrained (fixed on both sides), axial/direct stresses develop along with shear stesses.

What is the effect of temperature on buckling load?

With higher temperatures the body wants to expand out more, effectively making it easier to buckle the material

What are the boundary element equations?

X_bar = sig_x*l + tay_yx*m + tau_zx*n Y_bar = sig_y*m + tau_xy*l + tau_zy*n Z_bar = sig_z*n + tau_yz*m + tauxz*l

Does warp exist when a thin tube of arbitrary cross section is subjected to torsion?

Yes, warp will occur in a thin tube of arbitrary cross section subjected to torsion.

Define Young's modulus.

Young's Modulus is defined as the ratio between the tensile stress to the tensile strain of a material (i.e. sigma/epsilon)

What is the strain-stress relation for an isotropic material?

eta_x = (1/E) * (sig_x - v*sig_y) eta_y = (1/E) * (sig_y - v*sig_x) eta_z = (-v/E)*(sig_x-sig_y) gam_xy = tau_xy/G

What are the main displacement relations?

eta_x = du/dx eta_y = dv/dy eta_z = dw/dz gam_xz = dw/dx + du/dz gam_xy = dv/dx + du/dy gam_yz = dw/dy + dv/dz

How is shear flow related to shear stress?

q = tau*t Shear flow is the distribution of stress across the cross sectional area. Generally speaking, it is uniform across the area.

What equations do you use to calculate stresses on an inclined plane?

sig_n = ( sig_x * cos(theta)^2 ) + ( sig_y * sin(theta)^2 ) + ( tau_xy * sin(2*theta) ) tau = [(sig_x-sig_y)/2]*sin(2 * theta) - [tau_xy * cos(2*theta)]

What is the formula for finding a shear stress and angle of twist for an open tube?

tau_max = T*r/J Theta_max = (T*L)/(G*J) J = (b*t^3)/3)