3. CALCULATED vs MEASURED

4. ENERGY DISSIPATION

8. IMPLICIT NONLINEAR BEHAVIOR

9. STEEL STRESS STRAIN RELATIONSHIPS

10. INELASTIC WORK DONE!

11. CAPACITY DESIGN STRONG COLUMNS & WEAK BEAMS IN FRAMES
REDUCED BEAM SECTIONS
LINK BEAMS IN ECCENTRICALLY BRACED FRAMES
BUCKLING RESISTANT BRACES AS FUSES
RUBBER-LEAD BASE ISOLATORS
HINGED BRIDGE COLUMNS
HINGES AT THE BASE LEVEL OF SHEAR WALLS
ROCKING FOUNDATIONS
OVERDESIGNED COUPLING BEAMS
OTHER SACRIFICIAL ELEMENTS

12. STRUCTURAL COMPONENTS

13. MOMENT ROTATION RELATIONSHIP

14. IDEALIZED MOMENT ROTATION

15. PERFORMANCE LEVELS

16. IDEALIZED FORCE DEFORMATION CURVE

17. ASCE 41 BEAM MODEL
These capacities are for compact, laterally braced beams. The capacities are smaller for non-compact beams.
The capacities in ASCE 41 are the same for all steel strengths and all depth-to-span ratios. Intuitively this looks like an over-simplification. The ASCE 41 recommendations are an excellent starting point, but the values will probably be revised as more experience is gained.
Note that these capacities assume a symmetrical beam with no transverse load. They must be used with caution if the beam is unsymmetrical or if there are significant transverse loads.

18. ASCE 41 ASSESSMENT OPTIONS
Linear Static Analysis
Linear Dynamic Analysis (Response Spectrum or Time History Analysis)
Nonlinear Static Analysis (Pushover Analysis)
Nonlinear Dynamic Time History Analysis
(NDI or FNA)

19. STRENGTH vs DEFORMATION
ELASTIC STRENGTH DESIGN - KEY STEPS
CHOSE DESIGN CODE AND EARTHQUAKE LOADS
DESIGN CHECK PARAMETERS Stress/BEAM MOMENT
GET ALLOWABLE STRESSES/ULTIMATE– PHI FACTORS
CALCULATE STRESSES – Load Factors (ST RS TH)
CALCULATE STRESS RATIOS
INELASTIC DEFORMATION BASED DESIGN -- KEY STEPS
CHOSE PERFORMANCE LEVEL AND DESIGN LOADS – ASCE 41
DEMAND CAPACITY MEASURES – DRIFT/HINGE ROTATION/SHEAR
GET DEFORMATION AND FORCE CAPACITIES
CALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH)
CALCULATE D/C RATIOS – LIMIT STATES

20. STRUCTURE and MEMBERS For a structure, F = load, D = deflection.
For a component, F depends on the component type, D is the corresponding deformation.
The component F-D relationships must be known.
The structure F-D relationship is obtained by structural analysis.

21. FRAME COMPONENTS For each component type we need :
Reasonably accurate nonlinear F-D relationships.
Reasonable deformation and/or strength capacities.
We must choose realistic demand-capacity measures, and it must be feasible to calculate the demand values.
The best model is the simplest model that will do the job.

22. F-D RELATIONSHIP

23. DUCTILITY
LATERAL
LOAD
Partially Ductile
Ductile
Brittle
DRIFT

24. ASCE 41 - DUCTILE AND BRITTLE

25. FORCE AND DEFORMATION CONTROL

26. BACKBONE CURVE

27. HYSTERESIS LOOP MODELS

28. STRENGTH DEGRADATION

29. ASCE 41 DEFORMATION CAPACITIES
This can be used for components of all types.
It can be used if experimental results are available.
ASCE 41 gives capacities for many different components.
For beams and columns, ASCE 41 gives capacities only for the chord rotation model.

30. BEAM END ROTATION MODEL

31. PLASTIC HINGE MODEL
It is assumed that all inelastic deformation is concentrated in zero-length plastic hinges.
The deformation measure for D/C is hinge rotation.

32. PLASTIC ZONE MODEL
The inelastic behavior occurs in finite length plastic zones.
Actual plastic zones usually change length, but models that have variable lengths are too complex.
The deformation measure for D/C can be :
- Average curvature in plastic zone.
- Rotation over plastic zone ( = average curvature x plastic zone length).

33. ASCE 41 CHORD ROTATION CAPACITIESLS CP
Steel Beam
qp/qy = 1
qp/qy = 6
qp/qy = 8
RC Beam
Low shear
High shear
qp = 0.01
qp = 0.005
qp = 0.02
qp = 0.025

34. COLUMN AXIAL-BENDING MODEL34

35. STEEL COLUMN AXIAL-BENDING

36. CONCRETE COLUMN AXIAL-BENDING

37. SHEAR HINGE MODEL

38. PANEL ZONE ELEMENT
Deformation, D = spring rotation = shear strain in panel zone.
Force, F = moment in spring = moment transferred from beam to column elements.
Also, F = (panel zone horizontal shear force) x (beam depth).

39. BUCKLING-RESTRAINED BRACE
The BRB element includes “isotropic” hardening.
The D-C measure is axial deformation.

40. BEAM/COLUMN FIBER MODEL
The cross section is represented by a number of uni-axial fibers.
The M-y relationship follows from the fiber properties, areas and locations.
Stress-strain relationships reflect the effects of confinement

41. WALL FIBER MODEL

42. ALTERNATIVE MEASURE - STRAIN

43. NONLINEAR SOLUTION SCHEMESƒ
iteration ∆ ƒ u 1 2
iteration
CONSTANT STIFFNESS
ITERATION 3 4 5 6 1 2 ∆ ƒ ∆ u u
NEWTON – RAPHSON
ITERATION

44. CIRCULAR FREQUENCY +Y Y q
Radius, R -Y

45. THE D, V & A RELATIONSHIP u . t
ut1 ..
Slope = ut1 t1

46. UNDAMPED FREE VIBRATION)
cos( t u w = + k m
&
where

47. RESPONSE MAXIMA u t )
cos( w = 2 -
&
sin(
max

48. BASIC DYNAMICS WITH DAMPINGu
& - = w + x 2 M Ku C t C K g u
&

49. RESPONSE FROM GROUND MOTIONug .. 2
ug2
ug1 1 t1 t2
Time t
& A B t u g = + - x w

50. DAMPED RESPONSE { [ & ] cos ( )] sin } e u B t A = - + w Bt x [u ) xw1 2 Bt x [u ) 3

51. SDOF DAMPED RESPONSE

52. RESPONSE SPECTRUM GENERATION
Earthquake Record
0.40
0.20 16
GROUND ACC, g
0.00 14
-0.20 12
-0.40 10
0.00
1.00
2.00
3.00
4.00
5.00
6.00
DISPLACEMENT, inches 8
TIME, SECONDS 6
T= 0.6 sec 4
4.00 2
2.00
DISPL, in.
0.00 2 4 6 8 10
-2.00
PERIOD, Seconds
-4.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
T= 2.0 sec
8.00
Displacement
Response Spectrum
5% damping
4.00
DISPL, in.
0.00
-4.00
-8.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00

53. SPECTRAL PARAMETERS S PS w = V d a v DISPLACEMENT, in. PERIOD, sec4 8 12 16 2 6 10
PERIOD, sec
DISPLACEMENT, in. 20 30 40
VELOCITY, in/sec
0.00
0.20
0.40
0.60
0.80
1.00
ACCELERATION, g d V S PS w = v a

54. Spectral Acceleration, Sa
THE ADRS SPECTRUM
ADRS Curve
Spectral Displacement, Sd
Spectral Acceleration, Sa
2.0 Seconds
1.0 Seconds
0.5 Seconds
Period, T
RS Curve

55. THE ADRS SPECTRUM

56. ASCE 7 RESPONSE SPECTRUM

57. PUSHOVER

58. THE LINEAR PUSHOVER

59. EQUIVALENT LINEARIZATION
How far to push? The Target Point!

60. DISPLACEMENT MODIFICATION

61. DISPLACEMENT MODIFICATION
Calculating the Target Displacement
d = C0 C1 C2 Sa Te2 / (4p2)
C0 Relates spectral to roof displacement
C1 Modifier for inelastic displacement
C2 Modifier for hysteresis loop shape

62. LOAD CONTROL AND DISPLACEMENT CONTROL

63. P-DELTA ANALYSIS

64. P-DELTA DEGRADES STRENGTH

65. P-DELTA EFFECTS ON F-D CURVES

66. THE FAST NONLINEAR ANALYSIS METHOD (FNA)
NON LINEAR FRAME AND SHEAR WALL HINGES
BASE ISOLATORS (RUBBER & FRICTION)
STRUCTURAL DAMPERS
STRUCTURAL UPLIFT
STRUCTURAL POUNDING
BUCKLING RESTRAINED BRACES

67. RITZ VECTORS

68. FNA KEY POINT
The Ritz modes generated by the nonlinear deformation loads are used to modify the basic structural modes whenever the nonlinear elements go nonlinear.

69. ARTIFICIAL EARTHQUAKES
CREATING HISTORIES TO MATCH A SPECTRUM
FREQUENCY CONTENTS OF EARTHQUAKES
FOURIER TRANSFORMS

70. ARTIFICIAL EARTHQUAKES

71. MESH REFINEMENT

72. APPROXIMATING BENDING BEHAVIOR
This is for a coupling beam. A slender pier is similar.

73. ACCURACY OF MESH REFINEMENT

74. PIER / SPANDREL MODELS

75. STRAIN & ROTATION MEASURES

76. STRAIN CONCENTRATION STUDY
Compare calculated strains and rotations for the 3 cases.

77. STRAIN CONCENTRATION STUDY
No. of elems
Roof drift
Strain in bottom element
Strain over story height
Rotation over story height 1
2.32%
2.39%
1.99% 2
3.66%
2.36%
1.97% 3
4.17%
2.35%
1.96%
The strain over the story height is insensitive to the number of elements.
Also, the rotation over the story height is insensitive to the number of elements.
Therefore these are good choices for D/C measures.

78. DISCONTINUOUS SHEAR WALLS

79. PIER AND SPANDREL FIBER MODELS
Vertical and horizontal fiber models

80. RAYLEIGH DAMPING
The aM dampers connect the masses to the ground. They exert external damping forces. Units of a are 1/T.
The bK dampers act in parallel with the elements. They exert internal damping forces. Units of b are T.
The damping matrix is C = aM + bK.

81. RAYLEIGH DAMPING
For linear analysis, the coefficients a and b can be chosen to give essentially constant damping over a range of mode periods, as shown.
A possible method is as follows :
Choose TB = 0.9 times the first mode period.
Choose TA = 0.2 times the first mode period.
Calculate a and b to give 5% damping at these two values.
The damping ratio is essentially constant in the first few modes.
The damping ratio is higher for the higher modes.

82. & & & √ √ √ √ & & + & & ( b ) & a b a b u M + C u Ku = u M + a M + K u
RAYLEIGH DAMPING t u M
& + C u
& + Ku = t u M
& +
& ( a M + b K ) u + Ku = u
& C u
& K + + u = M M u
& + u
& 2 2 x w + w u = w x 2 M = C ; a M b K x C C C = = = = + 2 M w √ M K √ 2 M 2 K M 2 √ M K 2 √ M K a b w x = + 2 w 2

83. Higher Modes (high w) = b
RAYLEIGH DAMPING
Higher Modes (high w) = b
Lower Modes (low w) = a
To get z from a & b for any w √ M K = 2p T = ; w
To get a & b from two values of z1& z2 z1 = a
2w1 +
b w1 2
Solve for a & b z2 = a
2w2 +
b w2 2

84. DAMPING COEFFICIENT FROM HYSTERESIS

85. DAMPING COEFFICIENT FROM HYSTERESIS

86. A BIG THANK YOU!!!