Slender Structures

Question 1

Extension

Kinematic Relation

Answer 1

Question 2

Extension

Constitutive Relation

Answer 2

• σ = Eε
• N = int( σ dA )

= int ( E ε dA )

= int ( EAε )

Question 3

Extension

Equilibrium

Answer 3

Question 4

Extension

ODE

Answer 4

Question 5

Extension

Free Body Diagram

Answer 5

Question 6

Extension

FBD components & BC

Answer 6

Question 7

Shear

Kinematic Relation

Answer 7

Question 8

Shear

Constitutive Relation

Answer 8

τ = V / A

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

Question 9

Shear

Equilibrium

Answer 9

Question 10

Shear

ODE

Answer 10

Question 11

Shear

Free Body Diagram

Answer 11

Question 12

Shear

FBD components & BC

Answer 12

Question 13

Shear + Bending

What relation is affected?

How to account for bending in calculations?

Answer 13

The Kinematic Relations if affected:

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

Moment is the integration of Shear

Question 14

Euler-Bernoulli

Assumptions?

Also known as?

Answer 14

• 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

Question 15

Euler-Bernoulli

Kinematic Relation

Assumptions?

Answer 15

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

Question 16

Euler-Bernoulli

Constitutive Relation

Answer 16

Question 17

Euler-Bernoulli

Equilibrium

Answer 17

Question 18

Euler-Bernoulli

ODE

Answer 18

Question 19

Euler-Bernoulli

Free Body Diagram

Answer 19

Question 20

Euler-Bernoulli

FBD components & BC

Answer 20

Question 21

Euler-Bernoulli

Prismatic beams

(All equations)

Answer 21

Question 22

Torsion

Kinematic Relation

Answer 22

Question 23

Torsion

Constitutive Relation

Answer 23

Question 24

Torsion

Equilibrium

Answer 24

Question 25

Torsion ODE

Answer 25

Question 26

Torsion Free Body Diagram

Answer 26

Question 27

Torsion FBD components and BC

Answer 27

Question 28

Types of cables?

Answer 28

* Type 1: Parabolic Cable * Load distributed along horizontal projection * Parabolic shape * Type 2: Catenary Cable * Load distributed along cable (gravity) * Hyperbolic cosine shape

Question 29

Parabolic Cable Free Body Diagram

Answer 29

Question 30

Parabolic cable Free Body Diagram components + BC

Answer 30

Question 31

Parabolic Cable Remarks?

Answer 31

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:

Question 32

Parabolic Cables **Geometrical Relation** remarks?

Answer 32

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

Question 33

Parabolic Cables **Moment Equilibrium** remarks?

Answer 33

Horizontal equilibrium: dH = 0

Question 34

Parabolic Cables **Governing Equilibrium** remarks?

Answer 34

Question 35

Parabolic Cables ODE

Answer 35

Question 36

Catenary Cable Free Body Diagram

Answer 36

Question 37

Catenary Cable Free Body Diagram components and BC

Answer 37

Question 38

Catenary Cable Geometrical Relation

Answer 38

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

Question 39

Catenary Cable Moment Equilibrium

Answer 39

Horizontal equilibrium: dH= 0

Question 40

Catenary Cable Governing Equilibrium

Answer 40

Question 41

Catenary Cable ODE

Answer 41

Question 42

Timoshenko Remarks

Answer 42

* Only applicable for static determinate structures → Need M distribution * "Looks like" a serial system:

Question 43

Timoshenko Kinematic Relation

Answer 43

Question 44

Timoshenko Constitutive Relation

Answer 44

Question 45

Timoshenko Equilibrium

Answer 45

Question 46

Timoshenko ODE

Answer 46

Question 47

Type of system? Remarks?

Answer 47

Serial System: 1. - Load bearing capacity = Weakest link * both elements take the same load 2. - Total deformation is _summation_ of each element 3. - Equivalent stiffness:

Question 48

Type of system? Remarks?

Answer 48

Parallel system: 1. - Total load carrying capacity = summation of each element 2. - Both elements have same deformation 3. - Equivalent stiffness: