Branko S. Bedenik

2 – Definitions and basic concepts

This chapter provides an account of the sign conventions for right-handed co-ordinate systems. It discusses forces and moments. A force vector has magnitude and direction and can be represented by a single straight line in space. Vectors, which can be represented as straight lines, are physical quantities, having magnitude and direction. It describes the equilibrium of a body. From Newtons second law, the following must be true if the body is to be at rest: the resultant force must be zero and the resultant moment must be zero. A generalized displacement consists of two vectors and cannot be represented by a single straight line in space. If the normal to the plane is in the positive co-ordinate axis direction, then the positive normal and shear stresses act in the direction of the corresponding positive co-ordinate axis and vice versa . The chapter discusses specific deformations (strains). Shear strains (rotations!) are functions of shear stresses and are independent of any normal stresses. When considering structural elements as discrete elements, it is of interest to find the overall element deformations and the corresponding joint displacements.

12 – A simple bridge analysis

This chapter presents a simple bridge analysis. A bridge on a secondary road is bridging a deep valley and is founded in the riverbed on a rock; both outside supports are resting on deep piles such that no displacements at the supports are possible. The chapter discusses the geometrical properties of the bridge. It calculates the cross-section areas, centers of gravity, second moments of area (moments of inertia) for the bridge deck (beam) and for the column. The chapter presents the analysis by the force method and the moment distribution method.

4 – Kinematics of structures

This chapter discusses the kinematics of structures. Members of structures are connected to each other through rigid connections or by hinges. A hinged connection enables independent rotation of all connected elements but the displacement is common to all elements at the joint. If more than two elements are connected in a hinge, then there are double, triple, etc., hinges. A support is a member of the structure that disables a displacement of the structure in at least one direction. As the displacement or rotation is disabled, it can only be done so by the reactions at the supports. The supports can be fixed (3 reactions), pinned (2 reactions), roller (1 reaction), swinging (1 reaction), guided (2 reactions) or elastic (1-3 reactions). Elementary displacement of a body is performed, if the displacements are small such that a chord can be substituted by a tangent.

9 – The Displacement Method

This chapter discusses the displacement method. Statically indeterminate structures are solved by the displacement method as if unknown displacements and rotations were chosen. From a system of equilibrium equations, the deformations can be calculated from which internal forces and reactions are calculated. The displacement method is superior to the force method when the number of unknown forces exceeds the number of unknown displacements and rotations. This chapter discusses two simplified longhand methods: a classical deformational method and a moment distribution method (Crosss method). The chapter presents a new method for influence line determination using ψ functions as a further development of well-known ω numbers.

11 – Inelastic material behaviour in structures

It is important to be able to predict the initiation of the inelastic response of materials that are subjected to various stress states. The term, “inelastic” is used to define the material response in relation to the stress-strain diagram that is non-linear and that retains a permanent strain or returns to an unstrained state on complete unloading. The term, “plastic” or “plasticity” is used to describe the inelastic behavior of a material that retains a permanent set on complete unloading. This chapter discusses the condition for the initiation of yield in ductile metals, such as structural steels. The use of uniaxial stress-strain data and their limitations are discussed together with a general description of non-linear material behavior. The stress-strain relationship might be greatly affected by the rate at which a load is applied. If a normal ductile material is considered, then the stress-strain relationship has an elastic range followed by a non-linear inelastic or plastic range. If the loading rate is very high, then the magnitude of the inelastic strain that precedes fracture can be reduced compared to that from normal load rates that are experienced under test conditions. High load rates result in an apparent increase in yield stress and modulus of elasticity. The material response is also less ductile under such conditions and, in the case of extremely high load rates, the response resembles brittleness.

3 – Statically determinate structures

A plane structure is considered if it lies in one plane in space. Usually, loads act in the same plane, but that is not always the case, as in floor plates carrying loads normal to its plane. On a plane a movement of a point is defined by three components of a displacement in the Cartesian co-ordinate system. The displacements and forces are conjugate quantities, which means the following: if the displacement in a direction is zero, then a force in the same direction must exist to prevent that displacement. These forces are called reaction forces or simply reactions. Supports are the points on a structure that do not permit rigid body movement, sometimes forces at these points would be called constraints.

5 – Basic concepts of structural analysis

A displacement at a point, caused by a force, is linearly dependent on the magnitude of a force. That assumption is based on the proportionality and reversibility of stresses and deformations at any small particle of the material, from which the structure is made. This phenomenon is called elasticity. This chapter highlights that the assumption of elasticity is not always true as the deflections of a structure can significantly change the geometry and thereby change the manner in which potential energy is stored in the deflected structure. The principle of superposition may be used if loads and deflections are linearly dependent, which in general relates linear dependence between stresses and strains. This example of non-linear behaviour in compression is of great structural importance and is called buckling. The principle of superposition is valid if all deformations are so small that no significant change of the structure occurs, as even small changes in geometry can have a considerable influence on structural behavior. According to the principle of conservation of energy, when a structure is gradually loaded, the kinetic energy is zero and the work done by external loads W is equal to the strain energy. The energy is related to the internal forces and deformations they cause and is stored in the structure as a potential energy due to axial forces, bending moments, shearing forces, and twisting moments (torsion).

The Force Method ( Method of consistent deformations )

A degree of static indeterminacy can be calculated from a number of equilibrium equations and a number of unknown forces on a structure. Equilibrium equations are set for joints as free bodies so that in rigid joints there are three equations and only two equations in a pinned joint. Unknown forces appear at element ends where a structure is connected together. The statics equations must be supplemented by a number of equations of geometry equal to the degree of redundancy of the structure. A primary structure is obtained in such a way that an indeterminate structure is transformed into determinate structure by removal of redundant forces, which can be reactive forces at supports or internal forces at an arbitrary section. The reduction statement states that deformations of an indeterminate system can be determined if bending moments of an indeterminate system are known and if virtual bending moments of the primary system are known. The method of forces is a useful tool for the analysis of simple structures and gives a good understanding of structural behavior through deformation calculations. However, there is no independent control of the calculated unknowns, since the equilibrium of the whole structure is established using unknowns as external loads. The reduction statement offers a simple and effective control of the calculated results. A new primary structure is determined that has to be different from the one from which unknowns were determined.