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Ponchos, vino, aguardiente, frutas secas, lienzos y algod6n. Entre ellos. Las consecuencias de estas medidas en la activi- al dad mercantil se hicieron sentir de inmediato. En se abonaron sisa por 7. La D venta y, por ende, las existencias de tucuyos en Ia ciudad de Salta en no parece haber sido exigua al comparar esta cifra con el promedio anual de la Libro de Alcabalas.

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Finalmente en Salta, esfu- diando en su recientemente creada escuela de primeras letrasr se encuentra el menor de sus tres hijos, Juan de Dos, residiendo en casa de Miguel Francisco Ataoz, un comerciante tucumano avecindado en Salta. Alcabalas de Salta Ignacio de l,a Torre. Expedientes Colo. AGN, Sala X. Libro de Alcabalas de Salta. Los valles, antiguos centros productores e invemadores de ganado vacuno o productores de harina y vino quedaron relegados.

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Samuel y obos historiadores del "History Workshop". Y es esto lo que diferencia la nueva historia local de la vieja. Evidentemente, en este punto discrepo de las ideas de al Giruburg sobre lo singular en la historia. Stone y oaos De acuerdo con Ph. Para poder comprender el enigma crucial de Barrington Moore baste conocer su papel en el desarrollo del nacional-sindicalismo. Wallerstein, Perry Anderson, Eric Wolf y otros.

Peter Worsley Ahora bien, los resultados han parecido a muchos tan pretenciosos como discutibles. The third term has a sign fluctuation thus the previously derived relationships apply and the following is the solution:.

If we divided a polynomial of nth degree by a quadratic equation, the result will be a polynomial of n2 degree plus some remainder. This remainder can be used to give a closer approximation of the root quadratic equation.

When the remainder is zero, the quadratic is a root equation. Bairstows method involves using the remainders from double synthetic division to approximate the error in an assumed quadratic root equation of a polynomial. Look at the process of factoring a polynomial into a quadratic equation times a polynomial of two degrees less than the original polynomial as follows:.

Divide the resulting polynomial by the quadratic equation in order to derive an equation that has something to do with the derivative of the original equation. This will be a polynomial four degrees less than the. The following is the form of the second polynomial with the terms in the brackets being the remainder:.

The solution may be set up in synthetic division form shown in Table 1. Bairstows method a0. The two roots are as follows by the quadratic equation:. Now, repeat the process using the revised values for u and v shown in Table 1. Repeat the process using the revised values for u and v shown in Table 1. Substitute into the quadratic equation to find the roots. The remaining values are the coefficients of the factored polynomial. The remaining polynomial may be solved using the same method.

Repeat the process using the revised values for u and v as shown in Table 1. Substitute them into the quadratic equation to find the roots. Applied Aerodynamics. London, England: Longmans, Green and Co.

Descartes, R. Discours de la mthode pour bien conduire sa raison, et chercher la vrit dans les sciences. Lieden, Netherlands. James, M.

Smith, and J. Applied Numerical Methods for Digital Computations. Newton, I. De analysi per aequationes numero terminorum infinitas, London, England. Taylor, B. Methodus Incrementorum Directa et Inversa.

London, England. Solutions of Simultaneous Linear Algebraic Equations Using Matrix Algebra Matrix algebra is commonly utilized in structural analysis as a method of solving simultaneous equations. Each motion, translation or rotation, at each discrete location in a structure is normally the desired variable. This chapter explores matrix terminology, matrix algebra, and various methods of linear algebra to determine solutions to simultaneous equations.

These solutions are generally linear algebraic equations. A typical linear algebraic equation with n unknowns is as follows, where a is the coefficient, x is the unknown, and C is the constant.

In some cases, x, y, and z are used in lieu of x1, x2, etc. Homogeneous equation sets are those in which all the Cs are zero and all other equation sets are known as non-homogeneous.

A unique solution to a non-homogeneous set exists only if the equations are independent or non-singular determinant is non-zero , and a non-trivial solution set exists to a homogeneous set only if the equations are not independent determinant is zero.

The determinant is further discussed in. Section 2. In comparison to a non-trivial solution, a trivial solution is one where all the unknowns in the equations are equal to zero.

A matrix consisting of n rows and m columns is defined as a matrix of order n m. The relationship between the number of rows and the number of columns is arbitrary in a general matrix.

Many types of matrices exist, such as row, column, diagonal, square, triangular, identity, and invert. These are discussed in the following sub-sections. Square matrices are unique because they are the only matrix that has a reciprocal or invert as described later in this section.

Several types of square matrices exist such as the diagonal matrix, the identity matrix, the triangular matrix, and the invert matrix. The diagonal running from the upper left corner of the array to the lower right corner is considered the principal diagonal.

There are two types of triangular matrices, upper and lower. An upper triangular matrix, [U], is when all of the elements below the principal diagonal are zero, and a lower triangular matrix, [L], occurs when all of the elements above the principal diagonal are zero.

These matrices are called non-singular, which implies that reciprocals of rectangular matrices do exist. Theinverse of a matrix is defined as follows:. It is used in the computation of the determinant. For example, the minor, [A22] is shown in the following. Addition is achieved by adding the terms with the same row and column position.

Forexample, if matrix [A] and [B] were to be added to obtain matrix [C], the following equations would be valid:. Example 2. In order for two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting product consists of the same number of rows as the first matrix and the same number of columns as the second matrix. If A is the determinant of the matrix [A], then the following equations are valid when row k and column k are expanded: n.

The basket weave method may be used for a three-by-three determinant only. Take the sum of the products of the three down-right diagonals minus the sum of the product of the three up-right diagonals shown as follows: a11 a21 a One is the elimination of unknowns by elementary row operations and the second involves the use of determinates. One of the methods involving determinates is known as Cramers rule.

This method was published in by Gabriel Cramer As you might see, A1 is the original coefficient matrix, [A], with columnone replaced with the constant column matrix, [c]. The cofactor and adjoint matrices are very helpful in finding this invert by utilizing determinants. The cofactor matrix is one where the elements of the matrix are cofactors. Each term in the cofactor. Therefore, given the matrix [A], the cofactor matrix is shown as follows. The solutions to the simultaneous equations are now found from matrix multiplication.

Similar to the cofactor method, the method of adjoints, Adj[A], is another common way to solve for the invert of a matrix. The adjoint matrix, Adj[A], is simply the transpose of the cofactor matrix. This can be expressed in a few ways. It is noted that the subscripts of the adjoint matrix are the reverse of the cofactor matrix.

The main difference is that the transpose is performed during the operation of taking the adjoint, while in the cofactor method is done at the end.

The determinants are shown by row expansion for the 4 4 matrix and by the basket weave for all the 3 3 matrices in Table 2. The last step in Table 2. Use the basket weave method for determinants. It was referenced by the Chinese as early as Anon Gaussian elimination is a method for solving matrix equations by composing an augmented matrix, [A C], and then utilizing elementary row operations to reduce this matrix into upper triangular form, [U D]. Theelementary row operations used to reduce the matrix into upper triangular form consist of multiplying an equation by a scalar, or adding two equations to form another equation.

The equation used to eliminate terms in other equations is referred to as the pivot equation. The coefficient of the pivot equation that lies in the column of terms to be eliminated is called the pivot coefficient or pivot element. If this coefficient of the pivot element is zero, the pivot row must be exchanged with another row.

This exchange is called partial pivoting. If the row with the largest element in the pivot column is exchanged to the pivot row, accuracy is increased. Partial pivoting is when the rows are interchanged and full pivoting is when both the rows and columns are reordered to place a particular element in the diagonal position prior to a particular operation. Whenever partial pivoting is utilized, the determinant changes sign with each pivot unless done before reduction starts.

However, the value of the determinant of the matrix is not affected by elementary row operations. Once the matrix is reduced to an equivalent upper triangular matrix, the solutions are found by solving equations by back substitution. With the following variables: ak1 original elements ak new elements i row n j column m k pivotal row number The following is the back substitution procedure in algorithmic form:. The numbers to the right of each row outside the augmented matrix, [A C], are the reduction multipliers.

The pivot row is multiplied by these numbers to reduce the rows below the pivot row. As an example, the first row is multiplied by 8 and added to the second row, producing a zero in the first column of the second row. Then, the first row is multiplied by 1 and added to the third row, producing a zero in the first column of the third row. Last, the first row is multiplied by0 and added to the last row, producing a zero in the first column of the last row.

The result is the reduction of all the values below the pivot element in the first column to zero. The second column is then reduced to zeros below the pivot element, and lastly the third column is reduced to zeros below the pivot element. The solution vector, [x], is also shown.

These values for x are solved after the final reduction. From row three, the following equation is valid: 3 x3 1.

Include a determinant check for uniqueness. Again, the numbers to the right of each row outside the augmented matrix, [A C], are the reduction multipliers. Also noted is the partial pivoting. Note that the second row has the largest number in the first column. Therefore, that row is swapped with the first row, placing the largest element in the pivot element position.

Reduction is then performed on the first column. After reduction of the first column, the largest number in the second column is in the third row. That row is swapped with the second row, placing the largest number in the pivot position.

Reduction is then performed on the second column. The third column does not require partial pivoting, since the largest number in the third column is already in the pivot position.

Back substitution is performed to determine the solution vector, [x], which is shown in Table 2. This method was described by Wilhelm Jordan in Clasen When an unknown is eliminated, it is eliminated from equations preceding the pivot equation as well as those following the pivot equation.

The result is a diagonal matrix and eliminates the need for back substitution. In fact, if the pivot elements are changed to ones by dividing each row by the pivot element, the last column will contain the solution. The disadvantage to the GaussJordan elimination method is that two matrices are required for elimination, however, they do get smaller as the process continues.

The previous pivot is moved to the bottom, with a new pivot on top. The GaussJordan process is shown in Example 2. The first step is to divide the first equation of the set by the coefficient of the first unknown in that equation, 2.

Equation 2. Next, Equation 2. E quation 2. Now, Equation 2. Just as the previous cycle, Equation 2. The same result may be obtained by working with just the coefficients and constants of the equations.

This is beneficial if the amount of space available on the computer is limited. The algorithm for this improved method is as follows: akj.

In other words, normalize the matrix then utilize partial pivoting to reduce the matrix. However, there is no need to reduce the elements under the pivot. Reducing up and down is easier without a need to reorder the rows. This is how Example 2. From partial pivoting, the first row can be swapped with the fourth row to form the following matrix: 81 Now elimination may be performed as shown in Table 2.

Note that for each column reduction, elements are reduced to zero below and above the pivot position. Once it reduces to a diagonal matrix, the solution is foundby dividing each row by the pivot element.

Cholesky decomposition changes the original augmented equation to an equivalent upper and lower triangular set. If a set of three simultaneous equations exist, they can be represented as follows: a11 a 21 a If [A] represents the coefficient matrix, [x] represents the column matrix of the unknowns, and [C] represents the column matrix of the constants, the previous can be expressed as the following equation:. If the original system of equations is reduced into an equivalent system in upper triangular form, the following is true: 1 u12 0 1.

The order of the solution process is as follows with each producing an equation involving only one unknown: 1. Obtain column 1 of [L] by multiplying each row of [L] by column1 of [U] to get column 1 of [A]. That is, use a11, a21, a31 to get l11, l21, l Obtain row 1 of [U] by multiplying row 1 of [L] times each column of [U] to get row 1 of [A], excluding column 1 of [U].

That is, use a12, a13, c1 to get u12, u13, d1. Obtain column 2 of [L] by multiplying each row of [L] times column 2 of [U] to get column 2 of [A], excluding row 1 of [L], That is, use a22, a32 to get l22, l Obtain row 2 of [U] by multiplying row 2 of [L] times each column of [U] to get row 2 of [A], excluding columns 1 and 2 of [U]. That is, use a23, c2 to get u23, d2.

Obtain column 3 of [L] by multiplying each row of [L] times column 3 of [U] to get column 3 of [A], excluding row 1 and 2 of [L]. That is, use a33 to get l Obtain row 3 of [U] by multiplying row 3 of [L] times each column of [U] to get row 3 of [A], excluding columns 1, 2, and 3 of [U]. That is, use c3 to get d3. The reduced lower triangular matrix is shown in Table 2. Table 2. The upper triangular matrix is found at the same time and is shown in Table 2.

Finally, the solution is calculated from the [U D] matrix using back substitution and shown in Table 2. Consider the set of equations as follows:. These are then added to the approximate solution and the process is repeated until accuracy is achieved.

From here, the coefficient matrix is reduced until the identity matrix is on the left and the original identity on the right becomes the invert of A. If partial pivoting is used during the reduction, the columns in the invert must be swapped back in reverse order as the rows were swapped during reduction. The constant vector must also be reordered in the same way.

Include partial pivoting during the reduction. The matrix is shown in Table 2. Note that partial pivoting should be performed and row three is now swapped with row four. Columns 3 and 4 are then eliminated as shown in Table 2. Now the third and fourth columns are swapped back for the inverse matrix shown in Table 2. A matrix normally can be considered sparse if approximately two-thirds or more of the entries in a matrix are zero. The GaussSeidel iteration method was developed for such systems.

This method is an iteration method in which the last calculated values are used to determine a more accurate solution. Typically, all unknown x values are assumed to be zero to begin the iteration. This method mainly works best with a diagonal system in which the largest values lie on the diagonal.

The elastic stiffness matrix used to analyze structures is a typical example of a diagonal system and will be presented in Chapter 4.

A diagonal system is sufficient, but not necessary to provide convergence. During the process, each row is used to find a better approximation of the variable corresponding to the row using all other variables as known. After completing the first cycle, start with the first equation using the new values and find a closer approximation for each unknown.

Also, check the difference between the new values and the previous values to determine if the desired accuracy is achieved. The process can be stopped when each value has changed less than e or when a cycle results in each value changing less than e. The typical set of n homogeneous equations with n unknown solution sets is as follows:.

Let us consider the solution of eigenvalue problems. In other words, there are exactly n roots that satisfy this equation. These roots are known as eigenvalues of A. The non-trivial solution exists if the determinant of the coefficient matrix is zero. We use this so that Cramers rule can be used to find the eigenvalues.

From linear algebra, the trace of a matrix is the sum of the diagonal terms. The process for determining the characteristic polynomial is as follows:. The characteristic polynomial is found from the trace values.

It may also be used to find intermediate eigenvalues and eigenvectors using a sweeping technique. James, G. The steps of procedure are as follows: 1. Multiply the coefficient matrix times the vector [A][x]. Normalize the right hand side of the equation as follows l[x]:. Divide all the xs by first x value. Divide all the xs by the largest x. Normalize to a unit length. Use the components of the normalized vector as improved values of x.

Repeat steps 2 through 4 until the previous values differ from the new values by less than some small value e. A common structural problem is the modal node analysis of a multi-story frame. The equation can be rewritten as follows: 1. The [B] and [M] matrices are for a four-story single mass structural model. The [B] and [M] matrices can be combined by matrix multiplication. The iteration process and the final solution are shown in Table 2.

The xvector was normalized to the top value. The Nine Chapters on the Mathematical Art. Clasen, B. Sur une nouvelle method de resolution de equations lineaires et sul lapplication de cette method au calcil des determinates. Brussels, Belgium. Commandant Benoit. Note sur une mthode de rsolution des quations normales provenant de lapplication de la mthode des moindres carrs un systme dquations linaires en nombre infrieur celui des inconnues, Bulletin Godsique.

Heidelberg, Germany. Cramer, G. Introduction lanalyse des lignes courbes algbriques. Geneva, Switzerland. Gauss, C. Werke 9, Gttingen, Germany. Le Verrier, U.

Thesis on the secular variations of the orbits of the planets. Paris, France: Academy of Sciences. Arithmetica Universalis. Numerical Integration and Differentiation The integration of a continuous function is used to find the area under the function and to evaluate integral relationships of functions.

Differentiation evaluates the rate of change of one variable with respect to another. Examples of structural engineering problems involving integration and differentiation include geometrical properties of centroids of areas and volumes; moment of inertia; relationships between load, shear, moment, rotation, and deflection of beams using the equation of the elastic curve; and other strain energy relationships of structures involving shear, torsion, and axial forces. Many methods exist to solve such types of problems with varying levels of exactness.

These and other problems will be covered in the following chapters. One approximation of the area under the curve is to apply the trapezoidal rule by dividing the area into n strips of width Dx.

Then, approximate the area of each strip as a trapezoid. Example 3. If a function can be defined as a continuous mathe matical expression having continuous derivative f x and f x , the error of the trapezoidal rule is shown in Figure3.

The first term on the right side in Equation 3. The exact integral, I,can be derived using this error relation from two separate approximate integrals. This is a defined as a first-order extrapolation. If two first-order extrapolations are performed, then their results can be combined into a secondorder relationship with the following: I h 2 2 I h1 4.

The general nth order extrapolation would take the following form with n being the order of extrapolation: I. The trapezoidal integration for one strip is as follows in Table3. Two strips are shown in Table3. Four strips are shown in Table3. Eight strips are shown in Table3. An approximation of the area under the curve between these two points would be to pass a parabola through the points and zero three points. If we performed the error truncation to obtain Rombergs integration, the following occurs: I.

Similarly, Simpsons three-eighths rule can be derived using three strips and a third-degree parabola. Solve the four equations with four unknowns and then substitute these back into Equation 3. For an odd number of strips, both the one-third and three-eighths rules must be used. The three-eighths rule is used to obtain the area contained in three strips under the curve and then the one-third rule is used for the remaining n3 strips.

Perform Romberg extrapolation with these two integrations to get a more exact solution as follows:. Simpsons three-eighths rule is set up in Table3. Simpsons one-third rule is set up in Table3. Instead, a central point is used to determine the best places to evaluate the function.

The Gauss points indicate how far from the central point to go and then each point is weighted. The derivation of this method is not included here, but can be found in many advanced mathematics textbooks. The method is named for Carl Friedrich Gauss The following is a general equation that shows the process for n Gauss points for integration: b. The number of points used should closely match the degree of the equation to integrate.

Gaussian quadrature xi 0 Two points are shown in Table3. Three points are shown in Table3. Four points are shown in Table3. The following is a general weighting array for four strips: 1 4 2 4 1 4 16 8 16 4 2 8 4 8 2 4 16 8 16 4 1 4 2 4 1 For two strips, the weighting array is the following: 1 4 1 4 16 4 1 4 1 Any even set of strips will follow the same pattern and this could also be done using any other type of integration.

Using the trapezoidal rule would be less accurate, but could do any number of strips and Simpsons three-eighths rule would require a multiple of three strips in each direction.

The summation of the weighting array multiplied by f x,y is used in the following equation to obtain the volume. In this case, the Gauss equation is applied in both directions and then multiplied by the weighting factors. The same Gauss points and weights from Section 3.

Often in structural engineering, there is a need to find differential relationships. One simple way to easily evaluate transcendental equations is to use polynomial expansion developed for the Taylor series.

This is often referred to as the power series. By taking successive derivatives of the function then evaluating them, the coefficients of the polynomial may be found.

This is how most digital equipment like computers and calculators find values for transcendental equations.

Check by calculating sin Check by calculating e1. The value of the e1 can be evaluated to check the accuracy of the approximation. If Equation 3. This equation is known as the central-difference approximation of yi at xi with errors, order of h2. These are the central difference expressions with error order h2. Higher order expressions can be derived if we include more terms in each expansion. The following are the central, forward, and backward difference expressions of varying error order.

They can also be written in a reverse graphical form that is sometimes used and compiled from Numerical Methods in Engineering, by Salvadori and Baron One common relationship is that of the equation of the elastic curve. The equation of the elastic curve relatesthe deflected shape of a beam to the rotation, moment, shear, and load on thebeam. To solve the problem, a sketch of the beam and the assumed deflected shape is created. To use central difference operators, the model must go beyond the boundaries of the physical beam.

The deflected shape must. If the general equation is continuous at the boundaries, then this type of model is appropriate. If not, then the model should end at the boundaries and forward or central difference operators must be used. By symmetry of the model, only four specific values of the deflection are unknown. The central-difference expressions with error of order h2 will be used to solve for the values.

Since the load is known, we will use the fourth derivative relationship between load and deflection. Placing the central difference operator on y0, the first equation can be written from Figure3. For the second equation, the central difference operator is placed on y1 and is shown in Figure3. The third and fourth equations can be written by placing the central difference operator on y2 and y3.

These four equations constitute a non-homogeneous linear algebraic set and can be written in matrix form. First, the load is uniform and all the values of q are the same. Second, the deflection at point 0 is known to be zero, so the first row and column can be eliminated since they correspond to those values. This can be solved by many of the methods presented in Chapter 2. The method of cofactors is used here, since the solution is small enough to solve determinants directly.

The procedure is the same as the operator is laid upon each of the values that are unknown then the corresponding equation may be written. The large error at the end is due to the fact that the shear drops there, which creates a discontinuity in the equation.

Using more segments would reduce the error, but it would still be more inaccurate than the other values. The next example is similar to the previous one, but is included to show differences in modeling and accuracy. It also uses the higher order smaller error of error equations for more accuracy. The primary difference in the simply supported beam in Example3. The solution to this example is very similar to Example 3.

The central difference expressions with error of order h4 will be used to solve for the values. Placing the central difference operator on all four unknown points in the model, the linear non-homogeneous solution set can be obtained and solved. This is more error than the deflection found in Example 3. This is due to the fact that this physical model has more variation in deflection than that of the simply supported beam. The large error at the end is due to the fact that the moment drops there, which creates a discontinuity in the equation.

The large error at the end is due to the fact that the shear and moment drop dramatically at that point, which creates a discontinuity in the equation. The final example of using difference operators to solve differential equations is the critical buckling load of a column. The critical buckling load of a pinned end column is sometimes included in strength of materials, but will be derived here. The derivations start with the differential equation of the elastic curve similar to a beam.

The deflected column under the critical load is shown in Figure3. This is used in the equation of the elastic curve as follows: Z. The constants A and B can be evaluated using the boundary condition. The other higher modes can also be found with the other values of n. This is known as the Euler buckling stress and was derived by Leonhard Euler in This problem has the same model as Example 3.

The central difference expressions with error of order h2 will be used to solve for the values. This can be found Z. The solution process is similar to Example 3. The value at point 0 of y0 is zero and can be eliminated from the solutions. This becomes a homogeneous linear algebraic solution set. L2 L2 L2 A non-trivial solution to a homogeneous linear algebraic set exists if the determinant of the coefficient matrix is zero.

Therefore, we can find Q by setting the determinant of the coefficient matrix equal to zero. This will be done using the basket weave method for a 33 matrix. This central difference solution has an error of 8. One example is the bending of a plate under uniform lateral load. Theory of Plates and Shells by Timoshenko and Woinowsky-Krieger contains an exact solution to general plates. The differential relationship for plate bending uses partial differential equations.

Their difference operators can be derived from the basic difference operators using two basic principles. If you add two differential. To take the product of two differential equations, you must multiply the value of the operator at iof one differential to each of the terms in the other differential. Partial differentials also require both the x and y direction, so they will be written horizontally and vertically.