Elastic and Plastic Analysis

1
Case Studies of Plastic Analysis

• Elastic and Plastic Analysis
• Using Dr. Frame

2
Basic Concept ?

• Elastic Analysis
• Element is assumed to obey Hookes law and
  recover to original state on removal of load.
• Elastic Analysis determining the maximum load
  that cause one point in any member section to
  reach the yield stress or elastic critical
  buckling stress.

• Plastic Analysis
• Stress remains constant while the strain
  increases.
• The plastic hinge forms.
• Collapse occurs when sufficient plastic hinges
  have formed to convert the structure to a
  mechanism.
• Plastic analysis determining the minimum load
  that causes the structure to collapse.

3
Basic Concept ??

• Plastic Moment
• Simple plastic moment formula Mp ZFy
• Effect of the shear force
• 1. Tresca and Von-Mises yield criteria
• 2. In I-beams, the reduction in the plastic
  moment is negligible until the shear force is
  over 50 shear capacity of the web

• Effect of the axial force
• 1. Both tensile and compressive force reduce the
  plastic moment
• 2. Reduced plastic moment formula

4
About Dr. Frame

• Plastic Hinge Objects
• Installation and removal of plastic hinges
• Before yielding blue rectangles
• After yielding solid red circles during loading,
  open red circles during unloading
• Simple linear interaction relation to account for
  axial force effect
• Second-Order Analysis
• Load-Dependent EI

5
Case Model

• Beam W 310 x 60

• Column W 250 x 67

6
Analysis Result Case 1

• Combined mechanism

• ?e 5.35 ?c 7.70

7
Analysis Result Case 2

• Combined mechanism

• ?e 3.11 ?c 4.73

8
Analysis Result Case 3

• Sway mechanism

• ?e 1.43 ?c 1.95

9
Analysis Result Case 4

• Sway mechanism

• ?e 2.07 ?c 2.96

10
Analysis Result Case 5

• Sway mechanism

• ?e 2.12 ?c 3.06

11
Discussion ?

• Reduction in plastic moment capacity due to axial
  force
• Column 1 in all cases are investigated.
• W250x67 Np 2992.5 kN
• Mp 315.35 kNm

• The bigger axial force, the bigger reduction in
  plastic moment capacity.

12
Discussion ??

• Comparison of linear analysis and second-order
  analysis in Case 1, Case 3
• The collapse load factor reduces.
• The P-? effect is more significant in multistory
  frame.

13
Discussion ???

• A handy plastic analysis tool -- Dr. Frame
• Model and observe plastic behavior in structure
  interactively
• Get the order of plastic hinges formation and the
  final collapse mechanism easily and visually

14
Plastic Hinge Formation ?
15
Plastic Hinge Formation ??
16
Conclusion

• The plastic moment reduction due to axial force
  must be considered in the plastic analysis,
  especially when the axial force N exceeds 15 Np.
• For more accurate result, one needs to do the
  second-order analysis accounting for the P-?
  effects and load-dependent EI.

• The location of plastic hinges in frames to form
  a mechanism varies with the type of loading,
  shape and other physical properties of the frame.
• Dr. Frame is a very handy program to perform
  plastic analysis for framed structures of
  complicated shape and with a high degree of
  redundancy.