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.
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.
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
4Beam Element
General form:
Stiffness matrix: Displacement vector: Load vector:
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
6Stiffness Matrix for a Plane Beam Element: Transversal Deformation
From: "Teknisk STÅBI" 18. udgave 1999, p. 108
7Stiffness Matrix for a Plane Beam Element: Transversal Deformation
8Stiffness Matrix for a Plane Beam Element: Rotational Deformation
9Element Stiffness Matrix for a Plane 6DOF Beam Element
10Shape Functions for Beam Element
u: Node displacement or rotation w(x): Section displacement
11Shape Functions for a Beam Element
12Assembling the Global System of Equations
Structural system of fixed beam divided into 3 elements
13Stiffness Matrix for Beam Element
14Assembling Finite Elements into a Structure
15Structural Equation
16Structural Stiffness Matrix
17Structural Stiffness Matrix
18Structural Stiffness Matrix
19Different Numbering of Nodes and Degrees of Freedom
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
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
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
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
24Example: Design of a Concrete Beam with Multiple Spans
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
26Example: Design of a Concrete Beam with Multiple Spans
Firstly, an elastic analysis is made ...
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
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
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!)
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.
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
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
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)
34Example: Design of a Concrete Beam with Multiple Spans
Beam 2:
35Example: Design of a Concrete Beam with Multiple Spans
Beam 2:
36Example: Design of a Concrete Beam with Multiple Spans
Number of bars at the top
Number of bars at the bottom
37Anchoring Lengths for Reinforcement in Concrete
Anchoring factor, ζ0,3 : smooth reinforcement bars0,8 – 0,9 : rough reinforcement bars
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
39Excel for plotting the moment curve
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
41Today's Exercise
Office house
Beam to be designed
42Today's Exercise
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
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.