1. Chp-6:Lecture Goals Serviceability Deflection calculation
Deflection example

2. Structural Design Profession is concerned with:
Limit States Philosophy:
Strength Limit State
(safety-fracture, fatigue, overturning buckling etc.-)
Serviceability
Performance of structures under normal service load and are concerned the uses and/or occupancy of structures

3. ;

6. λ
:Amplification factor ζ
:Time dependent Factor

9. Solution:

10. See Example 2.2
Ma: Max. Service
Load Moment

20. Reinforced Concrete Sections - Example
Given a doubly reinforced beam with h = 24 in, b = 12 in., d’ = 2.5 in. and d = 21.5 in. with 2# 7 bars in compression steel and 4 # 7 bars in tension steel. The material properties are fc = 4 ksi and fy= 60 ksi.
Determine Igt, Icr , Mcr(+), Mcr(-), and compare to the NA of the beam.

21. Reinforced Concrete Sections - Example
The components of the beam

22. Reinforced Concrete Sections - Example
The compute the n value and the centroid, I uncracked

23. Reinforced Concrete Sections - Example
The compute the centroid and I uncracked

24. Reinforced Concrete Sections - Example
The compute the centroid and I for a cracked doubly reinforced beam.

25. Reinforced Concrete Sections - Example
The compute the centroid for a cracked doubly reinforced beam.

26. Reinforced Concrete Sections - Example
The compute the moment of inertia for a cracked doubly reinforced beam.

27. Reinforced Concrete Sections - Example
The critical ratio of moment of inertia

28. Reinforced Concrete Sections - Example
Find the components of the beam

29. Reinforced Concrete Sections - Example
Find the components of the beam
The neutral axis

30. Reinforced Concrete Sections - Example
The strain of the steel
Note: At service loads, beams are assumed to act elastically.

31. Reinforced Concrete Sections - Example
Using a linearly varying e and s = Ee along the NA is the centroid of the area for an elastic center
The maximum tension stress in tension is

32. Reinforced Concrete Sections - Example
The uncracked moments for the beam

33. Calculate the Deflections
(1) Instantaneous (immediate) deflections
(2) Sustained load deflection
Instantaneous Deflections
due to dead loads( unfactored) , live, etc.

34. Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases

35. Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases

36. Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases

37. Calculate the Deflections
Instantaneous Deflections
Equations for calculating Dinst for common cases

38. Sustained Load Deflections
Creep causes an increase in concrete strain
Curvature increases
Compression steel present
Increase in compressive strains cause increase in stress in compression reinforcement (reduces creep strain in concrete)
Helps limit this effect.

39. Sustained Load Deflections
Sustain load deflection = l Di
Instantaneous deflection
ACI
at midspan for simple and continuous beams
at support for cantilever beams

40. Sustained Load Deflections
x = time dependent factor for sustained load
5 years or more 12 months months months
Also see Figure from ACI code

41. Sustained Load Deflections
For dead and live loads
DL and LL may have different x factors for LT ( long term ) D calculations

42. Sustained Load Deflections
The appropriate value of Ic must be used to calculate D at each load stage.
Some percentage of DL (if given)
Full DL and LL

43. Serviceability Load Deflections - Example
Show in the attached figure is a typical interior span of a floor beam spanning between the girders at locations A and C. Partition walls, which may be damaged by large deflections, are to be erected at this level. The interior beam shown in the attached figure will support one of these partition walls. The weight of the wall is included in the uniform dead load provided in the figure. Assume that 15 % of the distributed dead load is due to a superimposed dead load, which is applied to the beam after the partition wall is in place. Also assume that 40 % of the live load will be sustained for at least 6 months.

44. Serviceability Load Deflections - Example
fc = 5 ksi
fy = 60 ksi

45. Serviceability Load Deflections - Example
Part I
Determine whether the floor beam meets the ACI Code maximum permissible deflection criteria. (Note: it will be assumed that it is acceptable to consider the effective moments of inertia at location A and B when computing the average effective moment of inertia for the span in this example.)
Part II
Check the ACI Code crack width provisions at midspan of the beam.

46. Serviceability Load Deflections - Example
Deflection before glass partition is installed (85 % of DL)

47. Serviceability Load Deflections - Example
Compute the centroid and gross moment of inertia, Ig.

48. Serviceability Load Deflections - Example
The moment of inertia

49. Serviceability Load Deflections - Example
The moment capacity

50. Serviceability Load Deflections - Example
Determine bending moments due to initial load (0.85 DL) The ACI moment coefficients will be used to calculate the bending moments Since the loading is not patterned in this case, This is slightly conservative

51. Serviceability Load Deflections - Example
The moments at the two locations

52. Serviceability Load Deflections - Example
Moment at C will be set equal to Ma for simplicity, as given in the problem statement.

53. Serviceability Load Deflections - Example
Assume Rectangular Section Behavior and calculate the areas of steel and ratio of Modulus of Elasticity

54. Serviceability Load Deflections - Example
Calculate the center of the T-beam

55. Serviceability Load Deflections - Example
The centroid is located at the A’s < 4.5 in. = tf Use rectangular section behavior

56. Serviceability Load Deflections - Example
The moment of inertia at midspan

57. Serviceability Load Deflections - Example
Calculate average effective moment of inertia, Ie(avg) for interior span (for 0.85 DL) For beam with two ends continuous and use Ig for the two ends.

58. Serviceability Load Deflections - Example
Calculate instantaneous deflection due to 0.85 DL:
Use the deflection equation for a fixed-fixed beam but use the span length from the centerline support to centerline support to reasonably approximate the actual deflection.

59. Serviceability Load Deflections - Example
Calculate additional short-term Deflections (full DL & LL)

60. Serviceability Load Deflections - Example
Calculate additional short-term Deflections (full DL & LL)
Let Mc = Ma = k-in for simplicity see problem statement

61. Serviceability Load Deflections - Example
Assume beam is fully cracked under full DL + LL, therefore I= Icr (do not calculate Ie for now).
Icr for supports

62. Serviceability Load Deflections - Example
Class formula using doubly reinforced rectangular section behavior.

63. Serviceability Load Deflections - Example
Class formula using doubly reinforced rectangular section behavior.

64. Serviceability Load Deflections - Example
Calculate moment of inertia.

65. Serviceability Load Deflections - Example
Weighted Icr

66. Serviceability Load Deflections - Example
Instantaneous Dead and Live Load Deflection.

67. Serviceability Load Deflections - Example
Long term Deflection at the midspan
Dead Load (Duration > 5 years)

68. Serviceability Load Deflections - Example
Long term Deflection use the midspan information
Live Load (40 % sustained 6 months)

69. Serviceability Load Deflections - Example
Total Deflection after Installation of Glass Partition Wall.

70. Serviceability Load Deflections - Example
Check whether modifying Icr to Ie will give an acceptable deflection:

71. Serviceability Load Deflections - Example
Check whether modifying Icr to Ie will give an acceptable deflection:

72. Serviceability Load Deflections - Example
Floor Beam meets the ACI Code Maximum permissible Deflection Criteria. Adjust deflections:

73. Serviceability Load Deflections - Example
Adjust deflections:

74. Serviceability Load Deflections - Example
Part II: Check crack midspan

75. Serviceability Load Deflections - Example
Assume
For interior exposure, the crack midspan is acceptable.