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4/5/2016

Introduction to quiz: Beam theory | Coursera

Introduction to quiz: Beam theory This document contains all the information you need to solve the quiz called 'Beam theory'. Please calculate your answers to the questions at the bottom of this page. You will then be prepared to start the quiz. We recommend using a calculation tool such as a spread sheet or a computer programming tool to solve the exercise. You can download this document in PDF format:

Introduction to quiz_ Beam theory _ Coursera.pdf (https://d3c33hcgiw…

Learning objectives When you have completed this exercise, you will be able to: · Calculate reaction forces for a cantilever beam · Calculate internal forces along the beam · Calculate deformation of the beam loaded by a tip force

Description In the quiz we consider a beam model which in a simpli䁀ed manner represents a blade from 10 MW reference wind turbine developed at DTU Wind Energy. The original blade length L is 86 meters. Variation of the blade 䁀apwise bending sti၀ness EI is shown in the 䁀gure below.

https://www.coursera.org/learn/windenergy/supplement/c9otx/introductiontoquizbeamtheory

1/3

4/5/2016

Introduction to quiz: Beam theory | Coursera

In this quiz, for simplicity reasons, the blade bending sti၀ness variation EI(x) along its length is approximated so that the beam bending sti၀ness is given as the following formula: L EI (x) = EIroot

49x + L

where EIroot is the beam bending sti၀ness at the root, L is the full beam length, and x is the coordinate along the beam length. You will see that the beam is loaded by a tip force. The load situation in the beam is such that it corresponds to a situation where a medium-sized car is hanging at the tip of the 10 MW wind turbine blade clamped at the root. Even though the blade bending sti၀ness is approximated in this exercise, this approximation is not very far from the actual blade sti၀ness variation, but allows for much simpler calculations. Therefore, the results you obtain in the quiz are quite realistic and one can get a feeling of how sti၀ this huge wind turbine blade is.

Data Beam length

L = 86m

Tip force

F = 9.07kN (ca. 925kg)

Beam root bending sti၀ness

E Iroot = 50 ⋅ 10

Beam bending sti၀ness

EI (x) = EIroot

9

Nm

2

L 49x+L

Questions Please use the data set given above to complete the following tasks. Write down your answers as you go. You will need them when you answer the quiz. 1. Write down the three equilibrium equations for the beam as it is done in lecture slide “Reaction forces”. Pay attention to the signs of the forces and moments, they should be in accordance with the given coordinate system. What are the reaction forces R x , R y and reaction bending moment M z at the clamped beam end? Hint: Take

care about direction of the forces! Use proper signs. 2. Follow the procedure described in the slide “Internal forces”. The only di၀erence would be the di၀erent set of forces acting at this beam. What is the internal bending moment distribution M (x) along the beam? Hint: Shear

force Q(x) = R y and bending moment M (x) = Q(x) ⋅ x − M z 3. From the slides “Strains and stresses in beam” and “Beam deformations” you can 䁀gure out the relation between the beam curvature and the bending moment. Remember the consistent units! What is the distribution of the bending curvature κ(x) along the beam? Hint: κ =

M (x) EI (x)

. Don’t forget that moment was expressed before in

kNm but here it should be in Nm! 4. The direct formula is not given in the lecture slides, but you can establish the relation between the beam rotation and beam curvature from the two 䁀rst formulae in the slide “Beam deformations”. The tip rotation angle is simply an integration of the curvature taken from the beam root to its tip. What is the beam tip rotation in radians?

Hint: θ(L) = ∫0

L

κdx = [3.4452 ⋅ 10

−8

x

3

− 4.3536 ⋅ 10

−6

x

2

− 1.5600 ⋅ 10

−5

x]

86 0

5. Similarly to the previous question the beam tip de䁀ection is an integral of the rotation taken from the beam root to its tip. Alternatively, you can use the last formula in the slide “Beam deformations” to calculate the tip de䁀ection directly from bending moment and sti၀ness doing double integration. What is the beam tip displacement? Hint: w(L) = ∫

L 0

θdx = [8.6130 ⋅ 10

−9

x

4

− 1.4512 ⋅ 10

−6

x

3

− 7.8002 ⋅ 10

−6

x

2

]

86 0

Summary https://www.coursera.org/learn/windenergy/supplement/c9otx/introductiontoquizbeamtheory

2/3

4/5/2016

Introduction to quiz: Beam theory | Coursera

The main learning points of this exercise were: · To understand the procedure for structural analysis of beams on a simple cantilever beam example · To get an impression of how sti၀ a 86 meter wind turbine blade is

https://www.coursera.org/learn/windenergy/supplement/c9otx/introductiontoquizbeamtheory

3/3

Introduction to quiz: Beam theory | Coursera

Introduction to quiz: Beam theory This document contains all the information you need to solve the quiz called 'Beam theory'. Please calculate your answers to the questions at the bottom of this page. You will then be prepared to start the quiz. We recommend using a calculation tool such as a spread sheet or a computer programming tool to solve the exercise. You can download this document in PDF format:

Introduction to quiz_ Beam theory _ Coursera.pdf (https://d3c33hcgiw…

Learning objectives When you have completed this exercise, you will be able to: · Calculate reaction forces for a cantilever beam · Calculate internal forces along the beam · Calculate deformation of the beam loaded by a tip force

Description In the quiz we consider a beam model which in a simpli䁀ed manner represents a blade from 10 MW reference wind turbine developed at DTU Wind Energy. The original blade length L is 86 meters. Variation of the blade 䁀apwise bending sti၀ness EI is shown in the 䁀gure below.

https://www.coursera.org/learn/windenergy/supplement/c9otx/introductiontoquizbeamtheory

1/3

4/5/2016

Introduction to quiz: Beam theory | Coursera

In this quiz, for simplicity reasons, the blade bending sti၀ness variation EI(x) along its length is approximated so that the beam bending sti၀ness is given as the following formula: L EI (x) = EIroot

49x + L

where EIroot is the beam bending sti၀ness at the root, L is the full beam length, and x is the coordinate along the beam length. You will see that the beam is loaded by a tip force. The load situation in the beam is such that it corresponds to a situation where a medium-sized car is hanging at the tip of the 10 MW wind turbine blade clamped at the root. Even though the blade bending sti၀ness is approximated in this exercise, this approximation is not very far from the actual blade sti၀ness variation, but allows for much simpler calculations. Therefore, the results you obtain in the quiz are quite realistic and one can get a feeling of how sti၀ this huge wind turbine blade is.

Data Beam length

L = 86m

Tip force

F = 9.07kN (ca. 925kg)

Beam root bending sti၀ness

E Iroot = 50 ⋅ 10

Beam bending sti၀ness

EI (x) = EIroot

9

Nm

2

L 49x+L

Questions Please use the data set given above to complete the following tasks. Write down your answers as you go. You will need them when you answer the quiz. 1. Write down the three equilibrium equations for the beam as it is done in lecture slide “Reaction forces”. Pay attention to the signs of the forces and moments, they should be in accordance with the given coordinate system. What are the reaction forces R x , R y and reaction bending moment M z at the clamped beam end? Hint: Take

care about direction of the forces! Use proper signs. 2. Follow the procedure described in the slide “Internal forces”. The only di၀erence would be the di၀erent set of forces acting at this beam. What is the internal bending moment distribution M (x) along the beam? Hint: Shear

force Q(x) = R y and bending moment M (x) = Q(x) ⋅ x − M z 3. From the slides “Strains and stresses in beam” and “Beam deformations” you can 䁀gure out the relation between the beam curvature and the bending moment. Remember the consistent units! What is the distribution of the bending curvature κ(x) along the beam? Hint: κ =

M (x) EI (x)

. Don’t forget that moment was expressed before in

kNm but here it should be in Nm! 4. The direct formula is not given in the lecture slides, but you can establish the relation between the beam rotation and beam curvature from the two 䁀rst formulae in the slide “Beam deformations”. The tip rotation angle is simply an integration of the curvature taken from the beam root to its tip. What is the beam tip rotation in radians?

Hint: θ(L) = ∫0

L

κdx = [3.4452 ⋅ 10

−8

x

3

− 4.3536 ⋅ 10

−6

x

2

− 1.5600 ⋅ 10

−5

x]

86 0

5. Similarly to the previous question the beam tip de䁀ection is an integral of the rotation taken from the beam root to its tip. Alternatively, you can use the last formula in the slide “Beam deformations” to calculate the tip de䁀ection directly from bending moment and sti၀ness doing double integration. What is the beam tip displacement? Hint: w(L) = ∫

L 0

θdx = [8.6130 ⋅ 10

−9

x

4

− 1.4512 ⋅ 10

−6

x

3

− 7.8002 ⋅ 10

−6

x

2

]

86 0

Summary https://www.coursera.org/learn/windenergy/supplement/c9otx/introductiontoquizbeamtheory

2/3

4/5/2016

Introduction to quiz: Beam theory | Coursera

The main learning points of this exercise were: · To understand the procedure for structural analysis of beams on a simple cantilever beam example · To get an impression of how sti၀ a 86 meter wind turbine blade is

https://www.coursera.org/learn/windenergy/supplement/c9otx/introductiontoquizbeamtheory

3/3