For many structures, because of the number of finite elements and nodes, analysis by a finite-element method requires mathematical treatment of large amounts of data and solution of numerous simultaneous equations. For this purpose, the use of computers is advisable. The mathematics of such analyses is usually simpler and more compact when the data are handled in matrix for. (See also Art. 5.10.7.) 5.13.1 Force and Displacement Methods The methods used for analyzing structures generally may be classified as force (flexibility) or displacement (stiffness) methods. In analysis of statically indeterminate structures by force methods, forces are chosen as redundants, or unknowns. The choice is made in such a way that equilibrium is satisfied. These forces are then determined from the solution of equations that ensure compatibility of all displacements of elements at each node. After the redundants have been computed, stresses and strains throughout the structure can be found from equilibrium equations and stress-strain relations. In displacement methods, displacements are chosen as unknowns. The choice is made in such a way that geometric compatibility is satisfied. These displacements are then determined from the solution of equations that ensure that forces acting at each node are in equilibrium. After the unknowns have been computed, stresses and stains throughout the structure can be found from equilibrium equations and stress-strain relations. In choosing a method, the following should be kept in mind: In force methods, the number of unknowns equals the degree of indeterminacy. In displacement methods, the number of unknowns equals the degrees of freedom of displacement at nodes. The fewer the unknowns, the fewer the calculations required. Both methods are based on the force-displacement relations and utilize the stiffness and flexibility matrices described in Art. 5.10.7. In these methods, displacements and external forces are resolved into componentsusually horizontal, vertical, and rotationalat nodes, or points of connection of the finite elements. In accordance with Eq. (5.103a), the stiffness matrix transforms displacements into forces. Similarly, in accordance with Eq. (5.103b), the flexibility matrix transforms forces into displacements. To accomplish the transformation, the nodal forces and displacements must be assembled into correspondingly positioned elements of force and displacement vectors. Depending on whether the displacement or the force method is chosen, stiffness or flexibility matrices are then established for each of the finite elements and these matrices are assembled to form a square matrix, from which the stiffness or flexibility matrix for the structure as a whole is derived. With that matrix known and substituted into equilibrium and compatibility equations for the structure, all nodal forces and displacements of the finite elements can be determined from the solution of the equations. Internal stresses and strains in the elements can be computed from the now known nodal forces and displacements. 5.13.2 Element Flexibility and Stiffness Matrices The relationship between independent forces and displacements at nodes of finite elements comprising a structure is determined by flexibility matrices f or stiffness FIGURE 5.92 Beam subjected to end moments and shears. matrices k of the elements. In some cases, the components of these matrices can be developed from the defining equations: The jth column of a flexibility matrix of a finite element contains all the nodal displacements of the element when one force Sj is set equal to unity and all other independent forces are set equal to zero. The jth column of a stiffness matrix of a finite element consists of the forces acting at the nodes of the element to produce a unit displacement of the node at which displacement j occurs and in the direction of j but no other nodal displacements

An excellent example of the application of Daltons law of partial pressures is the use of a liquid barometer to indicate atmospheric pressure. The barometer level indicates the sum of the partial pressure of water vapor and the partial pressure of the air. Partial pressures of air and water vapor are of great importance in psychrometry and are used to calculate the degree of saturation of the air or relative humidity at a specific dry-bulb temperature. 13.2.11 Relative Humidity and Specific Humidity Relative humidity is sometimes defined by the use of mole fractions, a difficult definition for psychrometric use. Hence, a more usable definition is desired. For this purpose, relative humidity may be closely determined by the ratio of the partial pressure of the water vapor in the air to the saturation pressure of water vapor at the same temperature, usually expressed as a percentage. Thus, dry air is indicated as 0% relative humidity and fully saturated air is termed 100% relative humidity. Computation of relative humidity by use of humidity ratios is also often done, but with somewhat less accuracy. Humidity ratio, or specific humidity Wa, at a specific temperature is the weight, lb, of water vapor in air per pound of dry air. If Ws represents the humidity ratio of saturated air at the same temperature (Table 13.2), then relative humidity can be calculated approximately from the equation W RH  a  100 (13.13)

Most commonly this staining can arise from the products of the setting reaction of Portland cement, which are leached out of concrete elements such as sills, lintels, copings, cement render or from insufficiently-dense mortar. They combine with carbon dioxide from the atmosphere to form white deposits that are insoluble in water but soluble in dilute hydrochloric acid. The combination of clay from the mortar with calcium and silica residues from the cement forms calcium silicate, which could also produce the insoluble white scum. Calcium silicate is highly insoluble in most acids and is white in colour. When wet these calcium deposits are invisible. Kaolin, a clay mineral present in most bricklaying sands, can also form a hard deposit. It is insoluble in most acids except hydrofluoric acid (which is a restricted product). However, the main cause of these insoluble white deposits is bad cleaning practice; allowing mortar made with sand containing too much clay to remain too long on the surface of the bricks and