Comprehensive

Examination

ประธานกรรมการ

ผศ.ดร. ธรพงศ จนทรเพง

กรรมการรวม

อ.ดร. รงสรรค วงศจรภทร

ผแทนบณฑตวยาลย

อ.ดร. ปารเมศ กาแหงฤทธรงค

Coursework Outline

Advanced Solid Mechanics

Mathematics Methods in Structural

Engineering

Advanced Reinforced Concrete

Matrix Methods in Structural Analysis

Bridge Design

Advanced Steel Structures

Advanced Concrete Technology

วชาเอกบงคบ

วชาเอกเลอก

Advanced

Solid Mechanics

Basic of Analysis

Equilibriam

Constitutive Law

Kinematics

Basic of Analysis

Displacement

External Forces

and Moments Strains

Internal Forces

And Moment Stress

Static Equivalency

ε =∆LL

σ = Eε

σ =Mc

I

Stress-strain relationship

Stress Tensor

𝑇 =𝜎𝑥𝑥 𝜎𝑥𝑥 𝜎𝑥𝑥𝜎𝑥𝑥 𝜎𝑥𝑥 𝜎𝑥𝑥𝜎𝑥𝑥 𝜎𝑥𝑥 𝜎𝑥𝑥

Shear stress

Normal stress

𝜎𝑥𝑥

Plane-x Direction-y

Strain Tensor

𝑇 =𝜀𝑥𝑥 𝜀𝑥𝑥 𝜀𝑥𝑥𝜀𝑥𝑥 𝜀𝑥𝑥 𝜀𝑥𝑥𝜀𝑥𝑥 𝜀𝑥𝑥 𝜀𝑥𝑥

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

1. Maximum Principal Stress Theory 𝜎𝑒 = max 𝜎1 , 𝜎2 , 𝜎3

𝑓 = 𝜎𝑒 − 𝑌

Yielding

of Brittle

Metals

3. Strain Energy Density Criterion

𝜎𝑒 = 𝜎12 + 𝜎22 + 𝜎23 − 2𝜈(𝜎1𝜎2 + 𝜎2𝜎3 + 𝜎1𝜎3)

𝑓 = 𝜎𝑒2 − 𝑌2

2. Maximum Principal Stain Theory 𝜎𝑒 = max

𝑖>𝑗>𝑘𝜎𝑖 − 𝜈𝜎𝑗 − 𝜈𝜎𝑘

𝑓 = 𝜎𝑒 − 𝑌

Yielding

of Ductile

Metals

1. Maximum Shear-Stress Theory

𝜎𝑒 = max (𝜏1, 𝜏2, 𝜏3) ,𝑓 = 𝜎𝑒2 − 𝑌2

2. Distortional Energy Density Criterion (Von mises)

𝜎𝑒 = 12

[ 𝜎1 − 𝜎2 2 + 𝜎2 − 𝜎3 2 + 𝜎3 − 𝜎1 2] ,

𝑓 = 𝜎𝑒2 − 𝑌2

1. Mohr-Coulomb Yield Criterion 𝑓 = 𝜎1 − 𝜎3 + 𝜎1 + 𝜎3 sin𝜙 − 2𝑐 cos𝜙

2. Drucker-Prager Yield Criterion

𝑓 = 𝛼𝐼1 + 𝐽2 − 𝐾 𝛼 = 2 sin 𝜙3(3−sin 𝜙)

𝐾 = 6𝑐 cos 𝜙3(3−sin 𝜙)

Alternative

Yield

Criteria

Shear center

𝐹𝑙 = � 𝑞 𝑑𝑙𝑏

0 𝑞 =

𝑉𝑥𝑄𝐼𝑥

𝑒 =𝐹𝑙ℎ𝑉′

Linear Elastic

1. Symmetrical Bending : 𝜎𝑥𝑥 = 𝑀𝑥𝑌𝐼𝑥

, 𝜎𝑥𝑥 = 𝑀𝑦𝑋𝐼𝑦

2. Unsymmetrical Bending

– Load : 𝜎𝑥𝑥 = 𝑀𝑥𝑌𝐼𝑥

− 𝑀𝑦𝑋𝐼𝑦

– Geometry : 𝜎𝑥𝑥 = 𝑀𝑥(𝑥−𝑥 tan 𝛼)𝐼𝑥−𝐼𝑥𝑦 tan 𝛼

, tan𝛼 = 𝐼𝑥𝑦−𝐼𝑥 cot 𝜃𝐼𝑦−𝐼𝑥𝑦 cot 𝜃

Fully Plastic

𝑀𝑝 =𝑌𝑏ℎ2

4

Bending

Torsion

Linear Elastic

•Circular Cross Section

•Noncircular cross section

Inelastic

• Elastic-Plastic

• Fully Plastic

Matrix Methods in Structural

Analysis

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

Stiffness Method

𝛿 = 𝐾 −1 𝐹

𝐹� = 𝐾� 𝛿

𝐹 = 𝐾 𝛿

𝑆 = 𝜎 𝛿

หา 𝐾� ใน Local coordinate System

Transform 𝐾� เปน global coordinate

System รวมเปน 𝐾

หา Nodal Displacement

หา Internal force จาก Stress matrix

Flexibility Method

𝑆 = 𝐵0 𝐹 + 𝐵1 𝑋

𝑆 = 𝑍 𝐹

𝑍 = 𝐵0 − 𝐵1(𝐵1𝑇𝐴𝐵1)−1𝐵1𝑇𝐴𝐵0

𝛿 = 𝑍 𝑇 𝐴 𝑆 = 𝐵0 𝑇 𝐴 𝑆

หา 𝐵0 และ 𝐵1จากการตดโครงสรางใหเปน

Primary structure ทมเสถยรภาพ

หา Internal force

หา Nodal Displacement

Mathematics

Methods in Structural

Engineering

Differential Equation

Ordinary differential equations (ODE)

Partial differential equations (PDE)

Algebraic equations Differential equations

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑚𝑑2𝑦(𝑥)𝑑𝑥2 + 𝑐

𝑑𝑦(𝑥)𝑑𝑥 + y x

= sin(x)

Find the numerical values Find the functions

Ordinary differentiale

quations

(ODE)

Homogeneous Nonhomogeneous

0 = f(x) =

𝑦 = 𝑦ℎ 𝑦 = 𝑦ℎ + 𝑦𝑝

General solution General solution

+

Particular solution

Partial differential

equations

(PDE)

Fourier Series

𝑎0 = 1𝑇 ∫ 𝑓 𝑡 𝑑𝑡𝑇

0

𝑎𝑛 = 2𝑇 ∫ 𝑓 𝑡 cos 𝑛𝜔�𝑡 𝑑𝑡𝑇

0

𝑏𝑛 = 2𝑇 ∫ 𝑓 𝑡 sin 𝑛𝜔�𝑡 𝑑𝑡𝑇

0

𝑓 𝑡 = 𝑎0 + �𝑎𝑛 cos 𝑛𝜔�𝑡∞

𝑛=1

+ �𝑏𝑛 sin 𝑛𝜔�𝑡∞

𝑛=1

Matrix Algebra

Vector – Row vector : ∎ ∎ ∎ 1×𝑛

– Column vector : ∎∎∎ 𝑛×1

Matrix – Square Matrix : ∎ ∎∎ ∎ 𝑛×𝑛

– Rectangular Matrix : ∎ ∎ ∎∎ ∎ ∎ 𝑚×𝑛

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

Eigenvalue Problems

Ax = λ x

Eigenvalues Eigenvectors A − λI x = 0

Trivial Solution x = 0

Nontrivial Solutions

from det(A − λ𝐼) = 0 Standard Form

Ax1 = λ1x1

Ax2 = λ2x2

Ax3 = λ3x3

⋮

Axn = λnxn

Roots of Equations

Bracketing methods

Bisection :

𝑥𝑟 = 𝑥𝑙+𝑥𝑢2

False-position :

𝑥𝑟 = 𝑥𝑢 −𝑓(𝑥𝑢)(𝑥𝑙−𝑥𝑢)𝑓 𝑥𝑙 −𝑓(𝑥𝑢)

Open methods

Newton-Raphson Method

𝑥𝑖+1 = 𝑥𝑖 −𝑓(𝑥𝑖)𝑓′(𝑥𝑖)

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.

Numerical Differential & Integration

Differentiation

1. Finite difference approximations

2. Derivative by interpolation

Integration

1. Newton-Cote formulas

2. Gaussian integration

Advanced

Reinforced Concrete

พฤตกรรมการรบแรงดด

1. ชวงกอนการแตกราว

2. ชวงคอนกรตใตแกนสะเทนราวบางสวน

3. จดท เหลกเสรมถงจดคราก

4. ชวงหลกการเรมตนการครากของเหลก

ความสมพนธ M−𝛟 แบบประมาณ

ทจดเรมตนการคราก

ทกาลงประลย

ความสมพนธ M−𝛟 แบบประมาณ

𝑀𝑥 = 𝐴𝑠𝑓𝑥(𝑑 −𝑘𝑑3 ) ≤ 𝑀𝑢

𝜙𝑥 =𝜀𝑥

𝑑 − 𝑘𝑑

𝑀𝑢 = 𝐴𝑠𝑓𝑥𝑑(1 − 0.59𝜌𝑓𝑥𝑓′𝑐

)

𝜙𝑢 =0.003𝑘𝑢𝑑

ทจดเรมตนการคราก ทกาลงประลย

พฤตกรรมการรบแรงอด

เสาเหลกปลอกเกลยว เสาเหลกปลอกเดยว

พฤตกรรมการรบแรงอดรวมกนแรงดด

พฤตกรรมการรบแรงของเสาชะลด

1. คานชวงการเฉอนยาวมาก : a d⁄ = 6

2. คานชวงการเฉอนยาว : a d⁄ = 3 หรอ 4

3. คานชวงการเฉอนสน : 2.5 > a d > 1⁄

4. คานลก : a d ≤ 1⁄

พฤตกรรมการรบแรงเฉอน

พฤตกรรมการรบแรงบด

Advanced

Steel Structures

Beam Behavior– Flexure

Method of Plastic Analysis

1. Must have enough hinges to form mechanism

2. Moment diagram must be in equilibrium with applied load

3. lMl ≤ l𝑀𝑃 l everywhere

Stiffeners

Torsion of I-Shape member

𝑇 = 𝑇𝑠𝑠 + 𝑇𝑤

𝑇𝑠𝑠 = The portion of T resisted by torsional shear stresses

𝑇𝑤 = The portion of T resisted as a result warping restraint

Bridge Design

Bridge Design

1. Aesthetics

2. Bridge types

- Concrete bridge

- Steel bridge

- Wood bridge

- Composite bridge

3. General design considerations

4. Loads

5. Bridge/system analysis

Advanced Concrete Technology

Cement hydration

Cement microstructure

Admixture in Concrete

Mineral Admixture

inexpensive and eco-friendly.

Chemical

control setting or water requirement

Mineral improve some properties

Carbonation

Attack

Chloride

Attack

Sulfate

Attack

Alkali

Aggregate

Reactions

Thank you for your

attention

You can find me at:

Anirut_decha@hotmail.com