PLASTIC ANALYSIS OF THE STRUCTURES DR. AZRUL A. MUTALIB UNIVERSITI KEBANGSAAN MALAYSIA
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Elastic Analysis Example 1
Determine the maximum moment at the supports and mid span
• Solution
Conventional stress-strain diagram
Introduction • Suitable for structures that match with plastic theory • Mostly used for steel structures • Plastic Analysis vs. Elastic Analysis Plastic analysis is based on ULS (ultimate stress) Elastic analysis is based on SLS (yield stress) Elastic analysis is overdesigned and not economical In the building life cycle, the life load might be increased caused the ultimate stress > yield stress hence the building is failed. Therefore, plastic analysis can be used.
Plastic Analysis 1. The ultimate load is the load that can caused the failure of the structures 2. The concept of the analysis is to determine the collapse load & collapse mechanisms 3. The collapse mechanisms can be divided to independent mechanism or combine mechanisms
Example 2
Example 3
Plastic Analysis • Only the behavior of beams and frames in the plastic range will be discussed • In the plastic analysis, this diagram can be idealized as an ideal elastic-plastic / perfectly elasto-plastic materials for easy calculation σy σw
Fully plastic moment Elastic region
Plastic region
Example 4 : Analysis of rectangular cross section
Load Factor & Shape Factor • The additional load work, Wk from the elastic conditionj to plastic condition can be stated as λWk • λ = load factor Example 5
λWk L/2
L/2
M
My Mp
• Load factor increases, bending moment is increases from elastic condition to the yield condition • Load factor increases, the elastic area decreases & at the end all under load area will be in the plastic area & plastic hinge is develop
• The ratio of the plastic moment to yield moment is called shape factor, φ
• Φ = 1.5 = The section can carry 50% more moment than My before it reaches Mp
• Different section has different shape factor!!
Example 5 310 mm 32 mm 293 mm 650 mm 293 mm
18 mm 32 mm
Determine the load shape and load factor for the I-beam as above Solution
General Collapse Condition The collapse mechanisms must satisfy 3 conditions; 1.Equilibrium Bending moment distribution must be in equilibrium with the external imposed loads. StaJc equilibrium equaJon can be used i.e. ∑ M = 0, ∑ V = 0 & ∑ H = 0 to get any forces. 2.Mechanism The number of plastic hinge should be sufficient in order for a structure to form a mechanism.
General Collapse Condition 3. Yield Bending moment should not exceed Mp In practice 3 theorems are of considerable use in plastic collapse; 1. Lower bound theorem • Also known as static theorem or safe theorem • Load system calculated or collapse load obtained is smaller or equal to the true collapse conditions W < Wp
• Fulfill equilibrium & yield conditions
General Collapse Condition 2. Upper bound theorem • Also known as kinematic theorem or unsafe theorem • Load system calculated or collapse load obtained is greater or equal to the true collapse conditions W > Wp
• Fulfill only mechanism conditions.
General Collapse Condition 3. Uniqueness theorem • Load system calculated or collapse load obtained is true collapse conditions W = Wp
• Fulfill only equilibrium, yield & mechanism conditions.
Plastic Hinges Development Normal hinge • No moment (M=0) • Allowed rotation M=0
VS.
Plastic hinge • Caused by Mp • M = Mp
M=Mp M=Mp
Plastic Hinges Development Plastic hinge develop at certain area; 1. Internal support
2. Under the loads
3. Fixed supports
Collapse mechanism Example 6
P
P
Mp
Mp
P
Mp
Mp
Mp P
Mp
Collapse mechanism P
Mp
Mp
2P
First collapse mechanism
Mp
Second collapse mechanism
Mp
General Collapse Condition • Structural problem condition solution in plastic theory can be carried out using various method. • Two methods that normally used are; i. Graphical method ii. Virtual work method Graphical method • Combination of free bending moment diagram and rotation moment
General Collapse Condition i.
Free bending moment diagram By assuming every spans are simply supported
ii. Reaction moment diagram Resistance moment value at fixed end moment Collapse load is obtained by solving the moment equilibrium equations at any section along the span Example 7
