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Slender Structures Flashcards

1
Q

Extension

Kinematic Relation

A
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2
Q

Extension

Constitutive Relation

A
  • σ = Eε
  • N = int( σ dA )

= int ( E ε dA )

= int ( EAε )

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3
Q

Extension

Equilibrium

A
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4
Q

Extension

ODE

A
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5
Q

Extension

Free Body Diagram

A
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6
Q

Extension

FBD components & BC

A
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7
Q

Shear

Kinematic Relation

A
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8
Q

Shear

Constitutive Relation

A

τ = V / A

  • τ = Tau
  • V = GAγ
    • γ = Gamma (shear strain)
    • GA = k (Stiffness)
  • A = Area
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9
Q

Shear

Equilibrium

A
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10
Q

Shear

ODE

A
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11
Q

Shear

Free Body Diagram

A
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12
Q

Shear

FBD components & BC

A
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13
Q

Shear + Bending

What relation is affected?

How to account for bending in calculations?

A

The Kinematic Relations if affected:

  • Add additional term (Φ) to account for rotation due to Bending

Moment is the integration of Shear

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14
Q

Euler-Bernoulli

Assumptions?

Also known as?

A
  • Bernoulli-Navier hypothesis:
    • plane cross-sections remain planar and normal to the beam axis in a beam subjected to bending.
  • Shear strains = 0
  • small displacements
  • straight longitudinal axis
  • symmetric about y-axis
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15
Q

Euler-Bernoulli

Kinematic Relation

Assumptions?

A

Plane C-S remain planer and normal to the beam axis.

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16
Q

Euler-Bernoulli

Constitutive Relation

A
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17
Q

Euler-Bernoulli

Equilibrium

A
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18
Q

Euler-Bernoulli

ODE

A
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19
Q

Euler-Bernoulli

Free Body Diagram

A
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20
Q

Euler-Bernoulli

FBD components & BC

21
Q

Euler-Bernoulli

Prismatic beams

(All equations)

22
Q

Torsion

Kinematic Relation

23
Q

Torsion

Constitutive Relation

24
Q

Torsion

Equilibrium

25
Q

Torsion

ODE

26
Q

Torsion

Free Body Diagram

27
Q

Torsion

FBD components and BC

28
Q

Types of cables?

A
  • Type 1: Parabolic Cable
    • Load distributed along horizontal projection
    • Parabolic shape
  • Type 2: Catenary Cable
    • Load distributed along cable (gravity)
    • Hyperbolic cosine shape
29
Q

Parabolic Cable

Free Body Diagram

30
Q

Parabolic cable

Free Body Diagram components + BC

31
Q

Parabolic Cable

Remarks?

A
  1. Cables takes no bending
  2. Cable ODE describes an equilibrium in deflected state →No supperposition due to non-linear approach!
  3. H can be regarded as constant only if no horizontal loads are applied
  4. Deriviation based upon equilibrium only!
  5. Cable force can be expressed in H and z:
32
Q

Parabolic Cables

Geometrical Relation

remarks?

A

We do not approx. tan( α ) with α since the effect of loads on the overall geometry of cables cannot be neglected.

—►Superposition position NOT applicable

33
Q

Parabolic Cables

Moment Equilibrium

remarks?

A

Horizontal equilibrium:

dH = 0

34
Q

Parabolic Cables

Governing Equilibrium

remarks?

35
Q

Parabolic Cables

ODE

36
Q

Catenary Cable

Free Body Diagram

37
Q

Catenary Cable

Free Body Diagram components and BC

38
Q

Catenary Cable

Geometrical Relation

A

We do not approximate tan( α ) to α since the effect of loads on the overall geometry of cables cannot be neglected.

39
Q

Catenary Cable

Moment Equilibrium

A

Horizontal equilibrium:

dH= 0

40
Q

Catenary Cable

Governing Equilibrium

41
Q

Catenary Cable

ODE

42
Q

Timoshenko

Remarks

A
  • Only applicable for static determinate structures → Need M distribution
  • “Looks like” a serial system:
43
Q

Timoshenko

Kinematic Relation

44
Q

Timoshenko

Constitutive Relation

45
Q

Timoshenko

Equilibrium

46
Q

Timoshenko

ODE

47
Q

Type of system?

Remarks?

A

Serial System:

    • Load bearing capacity = Weakest link
      * both elements take the same load
    • Total deformation is summation of each element
    • Equivalent stiffness:
48
Q

Type of system?

Remarks?

A

Parallel system:

    • Total load carrying capacity = summation of each element
    • Both elements have same deformation
    • Equivalent stiffness:

Civil Engineering (4 decks)

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