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1 Lecture 2: Exercise and Solution Rectangular steel cross-section, with constant thickness of 0.1 m. Numbers indicate height 1) measured in metres. Find end-deflection and compare. Analytical solution for 1) 23.126 mm 1) 1D elements with thickness 0.1 m and varying heights. 10 elements. 2) 2D elements with thickness 0.1 m. 20x4 elements. 3) 20x4 elements.

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Page 1: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

1Lecture 2: Exercise and Solution

Rectangular steel cross-section, with constant thickness of 0.1 m. Numbers indicate height

1)g

measured in metres. Find end-deflection and compare. Analytical solution for 1) 23.126 mm

1)

1D elements with thickness 0.1 m and varying heights. 10 elements.2)

2D elements with thickness 0.1 m.

20x4 elements.

3)

20x4 elements.

Page 2: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

2Lecture 2: Exercise and Solution

Problem Deformation at free end

1) 1 Beam element with constant height equal 0.75 m. 23.221 mm

1) 20 Beam element with constant height equal 0.75 m. 23.221 mm

2) 10 Beam elements with separately constant height. 17.696 mm

1 beam element with variable height going from 1 00 meter at fixed 17 176 mm1 beam element with variable height going from 1.00 meter at fixed end to 0.50 m at free end. Section: Tapered I (1.0;0.1;0.5;0.1;0.1;0.1;0.1)

17.176 mm

3) 4 node elements meshed 4x4 elements (height x length). Load applied at middle node at the free end. Deformation determined at the same node.

15.815 mm

the same node.

3) 4 node elements meshed 4x10 elements (height x length). Load applied at middle node at the free end. Deformation determined at the same node.

16.010 mm

3) 4 node elements meshed 4x20 elements (height x length). Load applied at middle node at the free end Deformation determined at

16.022 mmapplied at middle node at the free end. Deformation determined at the same node.

Page 3: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

3Contents

ه Element stiffness matrix for beam element

ه Assembling the global stiffness matrix from the element stiffness matrices

ه How to investigate the design (e.g. the placement of reinforcement) of a concrete beam

ه Exercise: Find the optimal placement of the longitudinal reinforcement in a concrete beam

Page 4: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

4Beam Element

General form:

Stiffness matrix: Displacement vector: Load vector:

Page 5: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

5Stiffness Matrix for a Plane Beam Element: Axial Deformation

E: Elasticity module [MPa] A: Section area [m2]

I: Bending moment of inertia [m4] L: Beam length [m]g g

Page 6: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

6Stiffness Matrix for a Plane Beam Element: Transversal Deformation

From: "Teknisk STÅBI" 18. udgave 1999, p. 108

Page 7: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

7Stiffness Matrix for a Plane Beam Element: Transversal Deformation

Page 8: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

8Stiffness Matrix for a Plane Beam Element: Rotational Deformation

Page 9: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

9Element Stiffness Matrix for a Plane 6DOF Beam Element

Page 10: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

10Shape Functions for Beam Element

u: Node displacement or rotation w(x): Section displacement

Page 11: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

11Shape Functions for a Beam Element

Page 12: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

12Assembling the Global System of Equations

Structural system of fixed beam divided into 3 elements

Page 13: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

13Stiffness Matrix for Beam Element

Page 14: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

14Assembling Finite Elements into a Structure

Page 15: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

15Structural Equation

Page 16: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

16Structural Stiffness Matrix

Page 17: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

17Structural Stiffness Matrix

Page 18: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

18Structural Stiffness Matrix

Page 19: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

19Different Numbering of Nodes and Degrees of Freedom

Page 20: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

20Boundary Conditions

Static boundary condition: f3 f6 f9 f12 are momentsStatic boundary condition: f3, f6, f9, f12 are moments

Kinematic boundary condition:

Note: One boundary condition specified for each DOF

Page 21: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

21Changing the Boundary Conditions

Static boundary condition: f3 f6 f9 f12 are momentsStatic boundary condition: f3, f6, f9, f12 are moments

Kinematic boundary condition:

Note: One boundary condition specified for each DOF

Page 22: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

22Different Stiffness Matrices

Local element stiffness matrix

ه Independent of element placement in global coordinate system

ه Dependent on material properties (e.g. elasticity modulus, E), section properties (e.g. area, A, moment of inertia, I) and element length, L

Global element stiffness matrix

ه Dependent on element placement in global coordinate system

F d f lti l i l l l t tiff t i ith t f tiه Found from multiplying local element stiffness matrix with transformation matrix

Global system matrixy

ه Dependent on degree-of-freedom numbering

ه Found from assembling global element stiffness matrices according to the degree-of-freedom numbering

Page 23: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

23Design of Concrete Structures

ه A concrete structure is not in an elastic state at failure

ه At failure the cross section is cracked and only a part of the section contributesه At failure, the cross section is cracked and only a part of the section contributes to the load-carrying capacity – i.e. the part in compression

ه The elastic section-force distribution gives large values over supportsThe elastic section force distribution gives large values over supports

ه Too expensive to design according to the elastic section forces, because the pressure zone of the cross section becomes very small, hence, large amount of p y greinforcement would be needed

ه Instead a plastic section-force distribution is chosen according to the lower-bound theory

ه Find a statically acceptable and safe section force distribution, then the section design is on the safe side

Page 24: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

24Example: Design of a Concrete Beam with Multiple Spans

Page 25: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

25Example: Design of a Concrete Beam with Multiple Spans

Example: 2.66 m wide concrete T-beam, preinforced in the longitudinal direction with a maximum of 4 bars at the top and bottom of the section.

Positive Failure Moment Negative Failure Moment

Number of reinforcement bars 4 7 8 2 6

Bars at the top 4 4 4 2 4

Bars at the bottom 0 3 4 0 2

z [mm] 414.3 392.9 388.6 390.8 318.9

Failure moment 229.1 380.1 429.7 108.0 264.5

Page 26: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

26Example: Design of a Concrete Beam with Multiple Spans

Firstly, an elastic analysis is made ...

Page 27: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

27Example: Design of a Concrete Beam with Multiple Spans

Load case 1

75 kN/m75 kN/m

34 kN/m

El ti t di t ib ti

451 kNm 476 kNm

Elastic moment distribution

98 kNm

198 kNm285 kNm

198 kNm299 kNm

Page 28: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

28Example: Design of a Concrete Beam with Multiple Spans

k

Load case 2

75 kNm

34 kN/m

El ti t di t ib tiElastic moment distribution

216 kNm 172 kNm 216 kNm

124 kNm 106 kNm 65 kNm

Page 29: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

29Example: Design of a Concrete Beam with Multiple Spans

Charniers are added left and right of the middle supports making the system statically determinate (fulfilling first part of the lower-bound theory)

In Staad.Pro, use the “general/spec” fan, choose “beam” then “release” and release Mz, both at the “start” and “end” location of the beam (two steps!)

Page 30: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

30Example: Design of a Concrete Beam with Multiple Spans

ه The chaniers model a failure point. Hence, the moment is known at this point. Over the supports the moment is negative (tension at the top of the beam. Be aware that Staad Pro uses a different sign convention) Hence the maximumaware that Staad.Pro uses a different sign convention). Hence, the maximum negative failure moment in this example is +264.5 kNm (not -264.5 kNm).

ه At the failure points the known moment (failure moment) is added as anه At the failure points the known moment (failure moment) is added as an external moment. In Staad.Pro this is done by choosing “Member Load”, “Concentrated Moment” “GZ” and applying a negative moment at right end of the beam (d1 = L, d2 = 0) and a positive moment at left end of the beam, i.e. e bea (d , d 0) a d a pos e o e a e e d o e bea , e(d1=0, d2 =0). The moment distribution from this load case should give the same sign of the moment at the failure points as the elastic analysis.

ه Then the total moment distribution is found as a sum of the moment distribution from the external load applied to the statically determinate system and the moment distribution from the applied external moments at the failure points.

Page 31: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

31

Load case 1Example: Design of a Concrete Beam with Multiple Spans

Moment distribution for actual loading on statically determinate system

216 kNm

Load case 1

216 kNm

486 kNm

662 kNm

467 kNm

264 5 kNm 264 5 kNm

Moment distribution for external applied failure moments

264,5 kNm 264,5 kNm

Page 32: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

32

Total moment distribution for load case 1Example: Design of a Concrete Beam with Multiple Spans

264,5 kNm 264,5 kNm98 kNm

Total moment distribution for load case 1

362 kNm 397 kNm 392 kNm397 kNm 392 kNm

Total moment distribution for load case 2

264,5 kNm 264,5 kNm 216 kNm

Total moment distribution for load case 2

, 216 kNm

108 kNm35,4 kNm 18,9 kNm

Page 33: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

33Example: Design of a Concrete Beam with Multiple Spans

Beam 1:

The moment distribution curve is moved 1/2⋅z⋅cotθ incurve is moved 1/2⋅z⋅cotθ in the unfavourable direction. On the safe side cotθ=2.5 (DS411 p. 40)(DS411 p. 40)

Page 34: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

34Example: Design of a Concrete Beam with Multiple Spans

Beam 2:

Page 35: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

35Example: Design of a Concrete Beam with Multiple Spans

Beam 2:

Page 36: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

36Example: Design of a Concrete Beam with Multiple Spans

Number of bars at the top

Number of bars at the bottom

Page 37: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

37Anchoring Lengths for Reinforcement in Concrete

Anchoring factor, ζ0,3 : smooth reinforcement bars0,8 – 0,9 : rough reinforcement bars

Page 38: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

38Output report

Modify report file: " " to " " and insert failure moments at the node pointsModify report file: . to , and insert failure moments at the node points

Page 39: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

39Excel for plotting the moment curve

Page 40: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

40Moment Curves

60

80

40

0

20

-2 0 2 4 6 8 10 12 14 16 18

-40

-20

-2 0 2 4 6 8 10 12 14 16 18

-60

40

-100

-80

-120

Page 41: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

41Today's Exercise

Office house

Beam to be designed

Page 42: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

42Today's Exercise

Page 43: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

43Today's Exercise

Positive Failure Moment Negative Failure Moment

Number of reinforcement bars 4 6 4 6Number of reinforcement bars 4 6 4 6

Bars at the top 2 2 2 2

Bars at the bottom 2 4 2 4

z [mm] 300 300 300 300z [mm] 300 300 300 300

Failure moment 50 100 50 70

1/2⋅z⋅cotθ=375 mm , fyk = 500 MPa , fck = 25 MPa, y ,Anchoring factor, ζ = 0.8 ⇒ anchoring length = 600 mm

Page 44: Lecture 2: Exercise and Solution - Wind Turbine · PDF fileLecture 2: Exercise and Solution 1 Rectangular steel cross-se ction, with constant thickness of 0.1 m. ... Find the moment

44Today's Exercise

Question

1 Fi d h l f h b di i f1. Find the placement of the bending reinforcement of the beam, i.e. make a moment diagram.

Hint:

Find the moment curves with Staad.Pro and use e.g. Microsoft Excel for drawing the curves.An Excel file with the failure moments has beenAn Excel file with the failure moments has been put on the homepage of the course.