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Structural Analysis Flashcards

Euler Bernoulli Beam Theory, Energy theorems, and etc...

1
Q

What are the E-B Beam theory assumptions?

A

1) Cross section is undeformable in its own plane
2) Cross section remains planar upon deformation
3) Cross Section remains normal to deformed axis of beam

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2
Q

Draw each of the EB beam theory assumptions, and what information do they tell us about our fundamental equations?

A
  • **SEE REF MATERIAL
    1) tells us beam displacement field has two rigid body translations u2(x1) and u3(x1)
    2) tells us axial displacement field of rigid body translation is u1(x1), and two rotations phi2(x1) and phi3(x1)
    3) implies slope of beam axis and the rotation of the cross section phi3 = (du2/dx1) and phi2 = 0(du3/dx1)
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3
Q

What is the major assumption of the E-B Beam Theory that contradicts the presence of transverse shear strains?

A

Assumption 2: Cross Section remains planar upon deformation i.e phi(x1)

*Implies shearing is so small that vanishes ~gamma12 = 0.

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4
Q

What do equilibrium conditions require on a beam with a distributed transverse load?

A

Non-vanishing shear force V2

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5
Q

How is the contradiction between E-B Beam theory and Equilibrium Condition assumptions about shear strains resolved?

A

By allowing E-B assumptions to assume infinite shear modulus (G) s.t shear strain vanishes, and recovering transverse shear force from equilibrium equations in bending moments (V2 = -dM3/dx1)

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6
Q

What is the implication of using equilibrium equations on the E-B assumption 3?

A

Can no longer assume plane sections remain normal to deformed axis of the beam

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7
Q

When are EB assumptions valid?

A

Long slender beams made of isotropic materials made of solid cross-sections

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8
Q

What do centroidal bending stiffnesses express?

A

Given constitutive law for bending behavior of beams, they express the proportionality between the bending moments and curvature
“Flexural Rigidity”

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9
Q

What does a material’s homogeneity change in stiffness expressions?

A

Removes the integral over the cross section to be independent of material, and only dependent on cross sectional area

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10
Q

What type of B.C’s are applied to L/R of a moving concentrated load?

A

Continuity conditions:

  1. Displacement across boundary must be the same on L/R
  2. Slope must be continuous
  3. Moment equilibrium
  4. Shear force equilibrium
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11
Q

How do we simplify a moving concentrated loading problem with 8 B.C’s to define the 4th order displacement field equations? What type of system must this be?

A

By defining the moment distribution due to the moving concentrated load to define the reaction forces we can fully define the system with a 2nd order displacement relationship.
Then apply continuity equations across boundary of load to define constants.

The system must be isotatic!

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12
Q

What does isostatic mean?

A

The number of equilibrium equations is the same as the number of force unknowns - OR - Statically determinant

The solution can be solved using equilibrium equations alone

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13
Q

What does hyperstatic mean?

A

The number of total unknown internal force/reactions is larger than the number of equilibrium equations. -OR- Statically indeterminate

The solution cannot be solved with equilibrium equations alone, requires use of strain-displacement and constitutive equations.

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14
Q

What are the main differences seen in internal force solutions of an isostatic versus hyperstatic problem?

A

For an isostatic problem, the internal forces can be expressed in terms of the externally applied loads alone. For a hyperstatic problem, internal forces depend on applied loads and the stiffness of the structure.

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15
Q

When comparing hyperstatic system for dual load paths, how does the load bearing relate to the material stiffness between the two?

A

If the load is split between the two paths according to their relative stiffness, the stiffer load path will carry more load than the compliant/more giving one.

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16
Q

What is the displacement method process of solving hyperstatic problems?

A
  1. Equilibrium Equations
  2. Constitutive laws to express internal forces w.r.t. member deformations
  3. Strain-displacement equations, express deformation w.r.t. displacement
  4. Derive internal force equations w.r.t. displacements by introducing step 3 into step 2
  5. Use internal forces in step 4 to satisfy equilibrium equations step 1 w.r.t displacements
  6. Solve equilibrium equations to find displacements in system
  7. Find deformations by back-substituting displacements into strain-displacement equations in step 3
  8. Find internal forces, by back-substituting deformations into constitutive laws in step 2.
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17
Q

What is the force method process of solving hyperstatic problems?

A
  1. Equilibrium equations
  2. Determine degree of redundancy (Nr)
  3. Cut system @ arbitrary Nr locations transforming hyperstatic system to isostatic
  4. Apply Nr redundant forces, each w/ relative displacement at Nr cuts. Express all internal forces w.r.t.Nr in equilibrium eqs.
  5. Use consitutive laws to express deformations w.r.t. Nr redundant forces
  6. Use strain-displacement eqs to express relative displacement at Nr cuts in terms of Nr redundant forces
  7. Impose VANISHING of relative displacements at Nr cuts, using compatibility eqs to solve for Nr redundant forces
  8. Recover system deformations from constitutive laws and displacements from strain-displacement eqs.
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18
Q

What is the neutral axis?

A

The locus of zero axial stress as a straight line through the cross section.

Also tells you the gradient to NA is the location of max axial stress.

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19
Q

What does the principal centroidal axes change in behavior of the constitutive + governing equations?

A

It fully decouples the constitutive equations ϵ, K2, K3, σ by reducing the cross terms. It decouples governing equations into 3-independent equations (Axial problem, 1st Bending (i1-i2), 2nd Bending (i1-i3))

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20
Q

How do you compute principal centroidal axis of bending?

A
  1. Compute centroid x2c, x3c
  2. Computer bending stiffness w.r.t. centroidal positions
  3. Compute orientation of principal axis of bending angle alpha (a) from
    sin⁡(2α)=H23/Δ ; cos⁡(2α)=(H33-H22)/2Δ where Δ=√((H33-H22)^2/2+ H23^2 )
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21
Q

How do you get min/max stiffness from principal centroidal axes def?

A

Def of tan(2α)=2H23/(H33-H22), where α, α+π/2
Where Δ=√((H33-H22)^2/2+ H23^2 )

Max bending (H33) α*: (H33+H22)/2+Δ
Min bending(H22) α*+π/2: (H33+H22)/2-Δ
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22
Q

What are the shear flow distribution and torque equipollency rules?

A

Shear Force Equipollence:

  • Integration of shear flow over cross-section must equal the applied shear force
  • Horizontal equivalence (V2) and Vertical equivalence (V3)

Torque Equipollence:

  • The equivalence of torque generated by distributed shear flow with externally applied torque
  • Moment about i1 at shear center must equal zero, and equals force*Vi when taken about other points.
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23
Q

What is the shear flow distribution process for a multi-cellular body?

A
  1. Make cuts for each cell in body, designate curves, Si
  2. Calculate shear flow, fo(Si), for each open section
  3. Calculate shear flow closing section, fci, for each cut
  4. Compute compatibility for shear flow distribution f(s) = fo(s)+fc
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24
Q

What is the shear center?

A

Location about which all torque generated by the shear flow distribution associated with transverse shear forces must vanish.

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25
Q

What is the shear flow distribution process of open thin walled sections?

A
  1. Compute centroid & select centriodal axis
  2. Select curvilinear coordinates Si, giving contour conditions ( fo(s=0) = 0, and end-end points must equate)
  3. Evaluate stiffness moments Q2(s), and Q3(s)
  4. Compute shear flow distribution f(s)
  • If need shear center
    5. Compute shear center through moment distribution relationship
26
Q

What assumptions must be made to use hooke’s law for stress-strain relationship?

A

Linearly elastic beam material, isotropic material.

σ = Eϵ

27
Q

What axis orientation should be chosen to reduce sectional constitutive coupling?

A

Choose origin of axis to coincide with centroid of the section

28
Q

What are the principal centroidal axes of bending?

A

A set of axes with their origin @ centroid of the section with cross-section orientation S.T. H23 = 0.

29
Q

What is Castigliano’s 1st Thrm?

A

For an elastic system, the magnitude of the applied load at a point is equal to the partial derivative of the strain energy with respect to projected load’s displacement.

Pi = dA/dΔ

30
Q

Which energy principle is Castigliano’s 1st thrm built from? What is its equation?

A

Built from principle of minimum total potential energy.
π=A+ϕ=A-ΣPiΔ→ dπ/dΔ=dA/dΔ-Pj=0
Pi = dA/dΔ

31
Q

What is Castigliano’s 2nd Thrm?

A

The linearly elastic complementary counter part to crotti-engesser become equal A=A’:

For prescribed deflection at a point is given by the partial derivative of strain energy w.r.t. driving force.

32
Q

Which energy principle is Castigliano’s 2nd thrm built from? What is its equation?

A

Built from principle of complementary energy. π’=A’+ϕ’ = A’ -ΣPiΔ

dπ'/dPj=dA'/dPj-d/dPj(ΣPiΔi) = dA'/dPj-Δj = 0
dA'/dPj = Δj = dA/dPj
33
Q

What is Castigliano’s 2nd used for?

A

Good for hyperstatic systems with driving forces. Solving for reactions using compatibility conditions with displacement = 0.
*dummy load/unit load method

34
Q

Which energy principle is Castigliano’s 2nd thrm built from? What is its equation?

A

Corollary to principle of minimum complementary energy.

π'=A'+ϕ' = A' -ΣPiΔ
dπ'/dPj=dA'/dPj-d/dPj(ΣPiΔi) = dA'/dPj-Δj = 0
dA'/dPj = Δj = dA'/dPj
35
Q

What is Crotti-Engesser Thrm?

A

An elastic structure with prescribed deflection at a point is given by the partial derivative of the complementary energy with respect to the driving force.

36
Q

What is the principle of virtual work?

A

Using Euler’s laws to equate to equilibrium for system of particles:
Principle of virtual work says a system of particles is in static equilibrium if/only if the sum of virtual work done by internal and external forces vanishes for all arbitrary virtual displacements.

37
Q

What is the principle of complementary virtual work?

A

Using only statically admissible virtual forces, we gather a statement of compatibility for real displacements:

Principle of complementary virtual work says a system undergoes compatible deformations if/only if the sum of internal and external complementary virtual work vanishes for all statically admissible forces.

38
Q

Principle of minimum total potential energy

A

A conservative system is in a stable state of equilibrium if/only if the total potential energy is a minimum with respect to changes in generalized coordinates.
π=A+ϕ → dπ=0: dπ/dqi = 0

39
Q

What are the assumptions of Principle of min potential energy

A
  1. Conservative internal and external forces - existence of strain energy and potential
  2. Provides equivalent equilibrium equations and stability (+ definite hessian)
40
Q

Principle of minimum complementary energy

A

A conservative system undergoes compatible deformations if/only if total complementary energy is minimum with respect to arbitrary changes in statically admissible forces.
π’=A’+ϕ → dπ’=0: dπ’/dpi = 0

41
Q

What are the assumptions of Principle of min complementary energy

A
  1. Conservative internal and external forces - existence of strain energy and potential
  2. Kinematically admissible forces only
42
Q

What are the assumptions of trusses/bars?

A

Bars have:

  • Pinned joints
  • Transmit only axial forces - no moments
  • Tension or Compression state only
  • Acts like rectilinear spring w/ stiffness k = EA/L
  • elongation e = F/k
43
Q

What are kinematically admissible displacements/forces in virtual work?

A

Kinematically Admissible virtual displacements/forces DO NOT violate any geometric boundary conditions. Therefore only equilibrium equations do not consider reaction forces

44
Q

What are kinematically inadmissible displacements/forces in virtual work?

A

Kinematically Inadmissible virtual displacements/forces violate geometric boundary conditions such that virtual work done by reactions do not vanish. Therefore treat reactions as external loads

45
Q

What is the principle of least work?

A

Complementary energy corollary - in absence of prescribed displacements.

When principle of minimum complementary energy is reduced to strain energy alone:
A conservative linearly elastic system undergoes compatible deformations if/only if strain energy is a minimum with respect to arbitrary changes in statically admissible forces.

dA/dR = 0

46
Q

What are conservative forces?

A

Forces dependent upon position NOT PATH.

  • Gravity
  • Elastic springs
  • Externally applied forces that can be derived from scalar potential forms
47
Q

What are non-conservative forces?

A

Forces dependent upon path - change with orientation of body:

  • Aerodynamic forces
  • Follower forces (rocket thrust
48
Q

How many numbers fully define a rigid bod? How many equations?

A

10 numbers:

  • Mass
  • 3 position values x,y,z of C.G
  • 3 principal MOI
  • 3 cross products of Inertia

6 equations of motion:

  • 3 position vector components
  • 3 rotational vector components

*3D translation and 3D rotation

49
Q

Newton’s three laws of motion?

A
  1. Every object at rest remains at rest unless external forces act to change its state (no net force on particle, it is in equilibrium)
  2. Momentum of particle = mass x velocity
    Force = ΔMomentum (conserved mass)
    External force changes velocity F = M*a
  3. For every action there is an equal and opposite reaction
    Fa=-Fb , Fa = Maaa = - Fb = -Mbab
50
Q

Euler’s 1st Law

A

The linear momentum of a body (system of particles) is the product of mass x velocity, but internal forces DO NOT contribute to change in momentum.

-combines Newton’s first and second s.t. it says system is in equilibrium if there is no net force, and the sum of internal forces of all particle interactions go to zero, then the forces x arbitrary distance must go to zero.

51
Q

Euler’s 2nd Law

A

Rate of change in Angular momentum about a point is equal to the sum of forces x distance (moments) acting on the body

-says if in equilibrium, the moments must go to zero such that the moment x arbitrary rotation must also go to zero.

52
Q

What is the difference between Newton’s and Euler’s Laws?

A

Newton’s laws represent forces, momentum, and equilibrium of a particle. Euler’s laws express equilibrium over a whole system of particles - a body, such that if in equilibrium the summation of forces and moments by an arbitrary/virtual distance and rotation must go to zero. Sets up the basis for virtual work.

53
Q

Are Euler’s laws necessary and sufficient conditions?

A

Euler’s laws are necessary but not sufficient to imply complete equilibrium.
*Euler laws are not satisfactory due to the arbitrary force definitions

54
Q

Are kinematically admissible virtual displacements necessary conditions satisfactions? Sufficient conditions?

A

The vanishing of virtual work by all kinematically admissible virtual displacements is a necessary condition.

It is not a sufficient condition because it does not guarantee equilibrium of the particle in the infeasible direction.

55
Q

What assumptions are made about stress components in the cross sectional plane for thin-walled beams in torsion?

A

They are assumed to be negligible compared to axial stress and axial components of shear.
σ2≪σ1,σ3≪σ1,τ23≪τ12,τ23≪τ13

56
Q

Define stress flow

A

Stress flow is the distribution of axial and transverse shearing stress components.

n(x1,s) = σ1(x1,s)*t(s) - axial stress flow
f(x1,s) = τ12(x1,s)*t(s) - shearing stress flow
57
Q

What simplifications are made for stress resultants in thin walled beams?

A

Stress resultants are identical to those with solid sections. One reduction occurs in the integration of the beam’s cross sectional area to an integration along a curve ‘C’. **from dA = tds

58
Q

What does local equilibrium of thin walled beam imply?

A

dn/dx1 + df/ds = 0
The local equilibrium conditions imply that any change in axial stress flow, n, along the beam axis must be equilibrated by a corresponding change in shear flow, f, along the curve C that defines the cross section.

59
Q

What is the physical interpretation of the stiffness static moments Q2 and Q3 w.r.t. shear flow in thin walled beams?

A

The stiffness static moment of any arbitrary area equals the product of Young’s modulus times the area, and the distance to the area centroid.

60
Q

Why do transverse loads need to be applied at the shear center?

A

When applied at the shear center, a beam bends without twisting. If not, then beam will both bend and twist.

61
Q

Why do we need a closing shear flow for close thin walled section analysis?

A

After placing a cut, the relative shear flow distribution creates a shear strain, in turn an infinitesimal axial displacement du1. Integrating over the cut edges, the total relative displacement must be adjusted to eliminate the axial displacement s.t. a constant shear flow would be seen over cut to close the section.

Structures/Solids (2 decks)

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