ABSTRACT
Consider an infinitesimal parallelepiped element shown in Figure 1.1. The state of stress of this element
is defined by nine stress components or tensors (s11, s12, s13, s21, s22, s23, s31, s32, and s33), of which six (s11, s22, s33, s12¼ s21, s23¼s32, and s13¼ s31) are independent. The stress components that act normal to the planes of the parallelepiped (s11, s22, s33) are called normal stresses, and the stress components that act tangential to the planes of the parallelepiped (s12¼s21, s23¼s32, s13¼ s31) are called shear stresses. The first subscript of each stress component refers to the face on which the stress
acts, and the second subscript refers to the direction in which the stress acts. Thus, sij represents a stress acting on the i face in the j direction. A face is considered positive if a unit vector drawn perpendicular to
the face directing outward from the inside of the element is pointing in the positive direction as defined
1.1 Stresses ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 Stress Components and Tractions Stress on an Arbitrary Surface Stress Transformation Principal Stresses and Principal Planes Octahedral, Mean, and Deviatoric Stresses Maximum Shear Stresses
1.2 Strains .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Strain Components Strain-Displacement Relationships Strain Analysis
1.3 Equilibrium and Compatibility .. .. . . . . . . . . . . . . . . . . . . . . . . . . 1-10
1.4 Stress-Strain Relationship .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11 Linear Elastic Behavior Nonlinear Elastic Behavior Inelastic Behavior Hardening Rules Effective Stress and Effective Plastic Strain
1.5 Stress Resultants .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-20
1.6 Types of Analyses.. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21 First-Order versus Second-Order Analysis Elastic versus Inelastic Analysis Plastic Hinge versus Plastic Zone Analysis Stability Analysis Static versus Dynamics Analysis
1.7 Structural Analysis and Design... .. . . . . . . . . . . . . . . . . . . . . . . . . 1-23
Glossary... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23
References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24
Further Reading ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25
by the Cartesian coordinate system (x1, x2, x3). A stress is considered positive if it acts on a positive
face in the positive direction or if it acts on a negative face in the negative direction. It is considered
negative if it acts on a positive face in the negative direction or if it acts on a negative face in the positive
direction.
