1. ECIV 520 A Structural Analysis II
Lecture 4 – Basic Relationships

2. Process of Matrix/FEM Analysis
Reliability of Solution depends on choice of Mathematical Model
Accurate Approximations of Solutions

3. Process of Matrix/FEM Analysis
Reliability of Solution depends on choice of Mathematical Model

4. Process of Matrix/FEM Analysis
Accurate Approximations of Solutions

5. Process of Matrix/FEM Analysis

6. Process of Matrix/FEM Analysis
Reliability of Solution depends on choice of Mathematical Model
Accurate Approximations of Solutions

7. Stress
Resultant Force and Moment represent the resultant effects of the actual distribution of force acting over sectioned area

8. Stress Assumptions Consider a finite but very small area
Material is continuous
Material is cohesive
Force can be replaced by the three components
DFx, DFy (tangent) DFz (normal)
Quotient of force and area is constant
Indication of intensity of force

9. Normal & Shear Stress Normal Stress Shear Stress
Intensity of force acting normal to DA
Shear Stress
Intensity of force acting tangent to DA

10. General State of Stress
Set of stress components depend on orientation of cube

11. Basic Relationships of Elasticity Theory
Concentrated
Distributed on Surface
Distributed in Volume

12. Equilibrium Equilibrium SFx=0 SFy=0 SFz=0
Write Equations of Equilibrium
SFx=0
SFy=0
SFz=0

13. Equilibrium
Equilibrium - X X

14. Equilibrium Equilibrium SFx=0 SFy=0 SFz=0
Write Equations of Equilibrium
SFx=0
SFy=0
SFz=0

15. Boundary Conditions
Prescribed Displacements

16. Boundary Conditions
Equilibrium at Surface

17. Deformation Forces applied on bodies tend to change the body’s
Intensity of Internal Loads is specified using the concept of
Normal and Shear STRESS
Forces applied on bodies tend to change the body’s
SHAPE and SIZE
Body Deforms

18. Deformation Deformation of body is not uniform throughout volume
To study deformational changes in a uniform manner consider very short line segments within the body (almost straight)
Deformation is described by changes in length of short line segments and the changes in angles between them

19. Deformation is specified using the concept of Normal and Shear STRAIN

20. Normal Strain - Definition
Normal Strain: Elongation or Contraction of a line segment per unit of length

21. Normal Strain - Units Dimensionless Quantity: Ratio of Length Units
Common Practice US
in/in SI
m/m
mm/m
(micrometer/meter)
Experimental Work:
Percent
0.001 m/m = 0.1%
e=480x10-6: 480x10-6 in/in
480 mm/m
m (micros)

22. Shear Strain – Definition
Shear Strain: Change in angle that occurs between two line segments that were originally perpendicular to one another

23. Cartesian Strain Components
Normal Strains: Change Volume
Shear Strains: Change Size

24. Small Strain Analysis
Most engineering design involves application for which only small deformations are allowed
DO NOT CONFUSE
Small Deformations with Small Deflections
Small Deformations => e<<1
Small Strain Analysis: First order approximations are made about size

25. Strain-Displacement Relations
For each face of the cube
Assumption
Small Deformations

26. Stress-Strain (Constitutive)Relations
Isotropic Material:
E, n
Generalized Hooke’s Law

27. Stress-Strain (Constitutive)Relations
Note that:
Equations (a) can be solved for s...

28. Stress-Strain (Constitutive)Relations
Or in matrix form

29. Stress-Strain (Constitutive)Relations

30. Stress-Strain: Material Matrix

31. Special Cases s = E e One Dimensional: v=0 No Poisson Effect
Reduces to:
s = E e

32. Special Cases Two Dimensional – Plane Stress
Thin Planar Bodies subjected to in plane loading

33. Special Cases Two Dimensional – Plane Strain
Long Bodies Uniform Cross Section subjected to transverse loading

34. Special Cases Two Dimensional – Plane Stress Orthotropic Material
Dm=
For other situations such as inostropy obtain the appropriate material matrix

35. Strain Energy
During material deformation energy is stored (strain energy)
e.g. Normal Stress
Strain Energy Density

36. Strain Energy
In the general state of stress for conservative systems

37. Principle of Virtual Work
Load Applied Gradually
Due to another Force

38. PVW Concept Apply Virtual Load Apply Real Loads Internal Real Virtual
Forces
Real
Deformns

39. Principle of Virtual Work
A body is in equilibrium if the internal virtual work equals the external virtual work for every kinematically admissible displacement field

40. Principle of Virtual Work