8. Failure Criteria
Mode of failure
1. By excessive deflection
2. By general yielding
3. By fracture
4. By instability (Bucking)
- Yielding of Brittle Metals
- Yielding of Ductile Metals
- Alternative Yield Criteria
9. 1. Maximum Principal Stress Theory
𝜎𝑒 = max 𝜎1 , 𝜎2 , 𝜎3
𝑓 = 𝜎𝑒 − 𝑌
Yielding
of Brittle
Metals
3. Strain Energy Density Criterion
𝜎𝑒 = 𝜎1
2
+ 𝜎2
2
+ 𝜎2
3
− 2𝜈(𝜎1 𝜎2 + 𝜎2 𝜎3 + 𝜎1 𝜎3)
𝑓 = 𝜎𝑒
2
− 𝑌2
2. Maximum Principal Stain Theory
𝜎𝑒 = max
𝑖>𝑗>𝑘
𝜎𝑖 − 𝜈𝜎𝑗 − 𝜈𝜎 𝑘
𝑓 = 𝜎𝑒 − 𝑌
16. Stiffness Method
Flexibility Method
1. Equilibrium of force
2. Compatibility of deformations
3. Hooke’s low
1. Equilibrium of force
2. Displacement continuity
3. Linear force-displacement
relationship
17. Stiffness Method
𝛿 = 𝐾 −1
𝐹
𝐹� = 𝐾� 𝛿̅
𝐹 = 𝐾 𝛿
𝑆 = 𝜎 𝛿
หา 𝐾� ใน Local coordinate System
Transform 𝐾� เป็น global coordinate
System รวมเป็น 𝐾
หา Nodal Displacement
หา Internal force จาก Stress matrix
24. System of linear equations
A x b
Uniqueness of solution
0
6
0 10
x1
x2
0
6
0 10
x1
x2
0
6
0 10
x1
x2
No Solution
A = 0
Infinite Solution
A = 0
Ill-condition
A → 0
25. Eigenvalue Problems
Ax = λ x
Eigenvalues Eigenvectors A − λI x = 0
Trivial Solution
x = 0
Nontrivial Solutions
from
det(A − λ𝐼) = 0Standard Form
Ax1 = λ1x1
Ax2 = λ2x2
Ax3 = λ3x3
⋮
Axn = λnxn
27. Interpolation
Assumption:
Data are accurate and distinct.
To do:
Find a smooth curve through the data points.
Curve fitting & Interpolation
Curve fitting
Assumption:
Data contain scatter(noise), usually due to
measurement errors.
To do:
Find a smooth curve that approximates
the data.
40. Method of Plastic Analysis
1. Must have enough hinges to form mechanism
2. Moment diagram must be in equilibrium with applied load
3. lMl ≤ l 𝑀 𝑃l everywhere
42. Torsion of I-Shape member
𝑇 = 𝑇𝑠𝑠 + 𝑇 𝑤
𝑇𝑠𝑠 = The portion of T resisted by torsional shear stresses
𝑇 𝑤 = The portion of T resisted as a result warping restraint
48. Admixture in Concrete
Mineral Admixture
inexpensive and eco-friendly.
Chemical
control setting or water requirement
Mineral
improve some properties