STRUCTURAL ANALYSIS BFC Statically Indeterminate Beam & Frame
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1 STRUCTURA ANAYSIS BFC Statically Indeterminate Beam & Frame
2 Introduction Analysis for indeterminate structure of beam and frame: 1. Slope-deflection method 2. Moment distribution method Displacement Methods 3. Modified stiffness The main aim is to calculate reaction forces Shear force Moment 2 Beam Frame (non-sway/sway)
3 Introduction The main problem of indeterminate beam is to determinate support reactions. A static equilibrium is not enough to solve this problem. 3
4 Introduction In reinforced concrete building: 4
5 In steel building: Introduction 5
6 In frame: Introduction 6
7 In bridge: Introduction 7
8 Introduction Two method that can be used: Slope deflection method 1. Fixed end moment 2. Moment result deflection and moment deposit (support shift) 3. Slope of support 4. Moment of support Moment-distribution method 1. Stiffness member 2. Distribution factor 3. Cary over factor 4. Fixed end moment 5. Distribution process 6. Moment of support 8
9 Slope-Deflection Method For indeterminate structure, moment of member happen from: i. Fixed end moment ii. Deflection slope or rotation iii. Support shift (support settlement) To form equation of state, member must have uniform and homogeneous among two support. Redundancy create the value of unknown that related to force method, e.g. flexibility method. Deformation can also contribute to the value of unknown that related to deformation method, e.g. slope-deflection method. 9
10 Slope-Deflection Method Consider typical beam BC from continuous beam: Moment resultant at the end B and C can be identified as: 1. Fixed end moment (FEM) 10
11 Slope-Deflection Method 2. Moment of Slope (MS) 3. Moment of Support Displacement (MSD) 11
12 Slope-Deflection Method FEM: moment resultant at end to end outside tax incidence member that imposed to stated member when both supports are assumed as fixed, therefore the rotation is zero. 12
13 Slope-Deflection Method MS: Moment that occur at different direction of rotation. End gradient member would be positive if the rotation is clockwise. MS can be phased as following: - End gradient B if C s end control/fixed MS BC 4EIθ B 1 2EIθ B ; MS - End gradient C if B s end control/fixed MS CB CB MS 2 BC 4EIθ C 1 2EIθ C ; MS BC MS 2 CB 13 The results are: MS MS cer BC cer CB 4EIθ B 2EIθ C 2EI2θ B θc 4EIθ C 2EIθ B 2EI2θ C θb
14 Slope-Deflection Method MSD: Moment that occur due to deformation (displacement) at one end to another end, e.g. B s end and C constrained. - Moment of support displacement is generally calculated using: MSD BC MSD DC 6EIΔ 2 where δ Δ MSD BC MSD DC 6EIδ 14
15 Slope-Deflection Method Moment resultant for beam BC is: M BC FEM BC MS BC MSD BC M BC Pab 2 2 EI 2 2θ B θ C 6EIδ M BC Pab 2 2 2EI 2θ B θ C 3δ M CB FEM CB MS CB MSD CB M CB Pab 2 2 EI 2 2θ C θ B 6EIδ 15 M CB Pab 2 2 2EI 2θ C θ B 3δ
16 Slope-Deflection Method Principle of analysis: 16 - Equilibrium equation: - Boundary condition: Fixed end slope, =0 M M M A B C M M M AB BA CB 0 M 0 BC 0
17 Slope-Deflection Method Example 4.1: Determine the moment value and shear force to each support and draw shear force diagram (SFD) and bending moment diagram (BMD) for structure beam below. Assume EI is constant. 17
18 Slope-Deflection Method Procedure: 1. Calculate fixed end moment (FEM) 2. Determine boundary conditions at support 3. Calculate moment resultant for both ends 4. Calculate rotation based on equilibrium equation 5. Calculate end moments, M AB and M BA 6. Calculate reaction forces 7. Calculate shear forces 8. Draw shear forces diagram 9. Calculate moment from end moments and shear forces diagram 10. Draw bending moment diagram 18
19 E Slope-Deflection Method E 19
20 E Slope-Deflection Method E 20
21 E Slope-Deflection Method E 21
22 E Slope-Deflection Method 22
23 Slope-Deflection Method Example 4.2: Determine the moment value and shear force to each support and draw shear force diagram (SFD) and bending moment diagram (BMD) for structure beam below. Assume EI is constant. 23
24 E Slope-Deflection Method 24
25 E Slope-Deflection Method 25
26 E Slope-Deflection Method E 26
27 E Slope-Deflection Method E 27
28 E Slope-Deflection Method E 28
29 E Slope-Deflection Method E 29
30 E Slope-Deflection Method E 30
31 Slope-Deflection Method Tutorial 4.1 Determine the reaction to each support for the continuous beam in the figure below. The support at C accidently constructed 10mm below its intended position. Given E=210e 6 kn/m 4 and I=180e -6 m 4. 31
32 Slope-Deflection Method Tutorial 4.2 Determine the moment value and shear force to each support and draw shear force diagram (SFD) and bending moment diagram (BMD) for structure beam below. Assume EI is constant 32
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UNIT II 1. Sketch qualitatively the influence line for shear at D for the beam [M/J-15]
![UNIT II 1. Sketch qualitatively the influence line for shear at D for the beam [M/J-15]](/thumbs/95/124878500.jpg "UNIT II 1. Sketch qualitatively the influence line for shear at D for the beam [M/J-15]")
UNIT II 1. Sketch qualitatively the influence line for shear at D for the beam [M/J-15] 2. Draw the influence line for shear to the left of B for the overhanging beam shown in Fig. Q. No. 4 [M/J-15] 3.