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Matris kuvvet metodu

Matris kuvvet metodu

##### Dosyalar

##### Tarih

1990

##### Yazarlar

Karasulu, Mehmet

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Matris-kuvvet metodu, kafes-kiriş sistemlerine etkiyen dxş kuvvetlerin sonucunda oluşan iç kuvvetlerin ve deformas yonların, kompüter yardımı ile elde edilmesinde kolaylık sağlar. Her bir elemana etkiyen kuvvetler denge denklemle ri yardımı ile saptanabildiği taktirde sistem statikçe be lirlidir. Stabilitenin sağlanabilmesi için fazla sayıda dış destek veya eleman gerekiyorsa, bu sistem statikçe be lirsiz olarak adlandırılır. Bu elemanlar tarafından taşınan fazla kuvvetler fazlalıklar olarak tanımlanır. Aşağıdaki bilgiler açığa çıktığı taktirde problemin ke sin bir çözümü yapılabilir. 1- Denge: Oluşturulan dış ve iç kuvvetler her noktada dengededir. 2- Uygunluk: Elemanlar o kadar deforme olmuşlardır ki, hepsi birbirine uyabilir. 3- Kuvvet-dönme ilişkisi: tç kuvvetler ve deformasyon lar elemanın stress-strain ilişkisini sağlar. Fazla kuvvetler, matris kuvvet metodunda problemin bi linmeyenidirler. İç kuvvetler, dış ve fazla kuvvetlere ba ğlı olarak açıklanabilirler Stress-strain ilişkisini kulla narak, elemanların deformasyonları, dış ve fazla kuvvetler bakımından belirtilebilir. Sonuçta uygunluk kriteri kulla narak, deformasyona uğrayan elemanlar eşitlenmelidirler. Lineer eşitliklerin formüllerinin oluşturulmasıyla fazla kuvvetlerin değerleri ortaya konur. Esneklik matrisi (F) yardımıyla bağlantı kurulan dış kuvvetlerin yönlerindeki noktaların yer değiştirmeleriyle, cismin elemanlarındaki gerilmeler hesaplanabilir.

In this text, we shall consider the effect of external loading (either static or dynamic) on an elastic system which consists of an assembly of elastic elements connected at a finite number of joints (or nodal points). The system is said to be statically determinate when the forces acting on each of the elementb can be determined by means of the equations of equilibrium. Should the system possess more elements or external supports than are necessary for stabilty then it is decribed as being statically indeterminade. The excess forces carried by these elements and supports are described as redumdants. We can be certain that the correct solution of our prob lem has been found if the following requirements are satisfied. 1- Equilibrium: The external forces and the internal forces they induce are in equilibrium at each joint. 2- Compatibility: The elements are so deformed that they can all fit together. 3- Force deflection relationship: The internal forces and deformations satisfy the stress-strain relationship of the element. In our further considerations this relation will be assumed to be linear. Two basic methods of analysis are available for solving the problem, and are referred to as the force method and the displacement method. In the force method the redundant forces are taken as the unknowns of the problem, so that all the internal forces can be expressed in terms of the external and the redundant forces. Then by using the stress-strain relationship the deformations of the elements can also be expressed in terms of the external and redundant forces. Finally, by applying tha compatibility criterion that the deformed elements must fit together, it is possible to formulate a set of linear simultaneoub equation which yield the values of the redundant forces. We may then calculate the stresses in the elements of the structure as well as the displacements of its joints in the directions of the external which are related by the flexibility matrix F. Basic Theory of The Force Method The system to be analyzed consist of s individual ele: ments that are assembled in some definite manner. This structure is subjected to a set of m external forces repre sented by the column vector.,'-(, *f ¦ "^ F2... pm and the displacements of these forces along their lines of action are represented by the column vector > k d_... dm 2 m The internal forces of the elements are represented by the column vector t < P. [Px p2... Pi} ît requires more than one internal force to specify the state of stress of an element, and herce e conclude that IV S Finally system is n times redundant, and forces chosen as the redundancies are represented by the column vector. «x -[xx x2... xnj The mtermal forces expressed in terms of the external and redundant forces. Let the external redundant supports be removed and the structure be cut so that the system is reduced to a statically determinate structure is called the basic system. In addition to the external forces f, the redundants x are applied at the cust and the redundant supports, Then the ihternal forces p can be expressed in terms in terms of the forces f and x according to the equation. (p=['b"o -«' ¦ is pla ced in block 10, and in block 11 is placed the product of blocks 10 and 8. Block 11 is repeated in blocks 11 a. Block 12 is formed as the product of block 2 and 11a, and matrix 13 is then comhleted by adding block 1 and 12. Block 14 is formed as the product of blocks 7 and lib, and the flexibility matrix in block 15 is formed adding blocks 14 and 6. Finally the values of the external loads are placed in column 16 so that the values of the internal forces in column 17 arofound by multiplying block 13 and column 16 and the values of the displacements in column 18 are by multiplying block 15 and column 16. Table: 1. Matric Calculations in Tabular Form. Bf -F0f Several points of interest should now be discussed. First of all, we notice that all the calculation are made using the basic matrices B,B and F. The external load matrix f comes into the caîculation only at the end, so that change in the elements of f involes only the minor cal culations indicated in block 17 and 18. Should we desire to apply an additional load at some point, the increase of com putational labor is still not large as its point of applica tion coxncides with one of the joints of the structure. The matrix B is increased by an extra columin from m to m + 1 columns.0 The modifet matrix calculations are illustrated in Table 2 with the shaded parts remaining unchanged, while the additional columns and rows completed according to the previously described. The additional work involves no more than matrix multiplication. The most difficult operation involed in the calculation is the inversion of the Dn matrix The order of this matrix is equal to the number of redundan cies and is independent of the loading. Table JL. The modified matrix calculations Some further Observations on the Matrix force method: Although the presentation of the basic there are several po ints of interest that can be profitably brought to light. By the application of the dummy-load theorem it is pos sible to find the relation that exists between the displace ments d and the deformation of the elements v. Under the action of the applied forces f the actual displacements are d and the deformations v, Now apply dummy external forces fj for xhich the corresponding internal forces are P,. The from the dummy- load theorem d XI f,d = p,v Jd' But so that >d - Bfd fdd=fdB'V However, since this equation must be satisfied for any choice of dummy load f, (it is independent of d and v)- it follows that d = Bv (14) The same process can be reoeated, this however, using a set df internal dummy forces p,* that only has to be sta- ticall equivalent to the dummy load f^. m^A" *.* - 4-^° ^mm, load theorem gives This time the dummy fdd= pfl*'v A suitable set of internal dummy forces is given by loading the statically determinate structure so hat * P 6 o d Substituting this in the dummy- load equation, we obtain d=B v o (15) Equation (14) and (15) give the rather surprising result that the displacements d can be obtained by premultiplying the deformations v by either B or B. In fact, any B* that corresponds to a statically equivalent set of internal for ces will give the same result. Equating the displacements d of Eqs. (14) and (15) gives the result B v= B v o (16) From this equation it is possible to fxnd the expression for X, but it is left as an exercise for those who would like to do it. However, let us continue with eq. Eq (4) (16) by setting, from / ( / / B- BQ * X Bx xii > I I l Then B v+X B v = B v o i o / / from which x=B v = 0 (17) However, since it is assumed that x^ 0. it follows that if Eq. (17) is to be satisfied for all v, then column vector resulting from the product B v must be zero, that is, B^O (18) From this result it can se shown that the relative disp lacements of the redundant forces at the cust made in the rudundant structure must be zero. When the redundant struc ture is loaded by the forces f, the deformations of the elements are v and we assume that the relative displacements at the cust are e. Now apply dummy loads x, at the cust. Then the internal dummy forces are given by Pd = Bixd Now the dummy- load theorem stateb that the work done by the dummy forces x, in moving through the actual displa cements e is equal to the work done by the dummy internal forces in moving through the deformation v, that is, II ill R.v = X,e or x,B v= x,e o d did But since this must be true for all dummy x,, we cone lude that the relative displacements at the cust are zero. Besides the physical interpretation, the discussion o of Eq. (18) shows how quickly we would have obtained this compation by means df the dummy- load theorem if we had star ted out with the condition that e=0 Let us close oun general remarks on the matrix force method by stating by stating with reference to the first paragraph that the following theree equations from the ba sic of this method. p = (BQ + B X)f equilibrium (3) B v= 0 compatibility (18) i v=F p stress-strain relationship (15) xm

In this text, we shall consider the effect of external loading (either static or dynamic) on an elastic system which consists of an assembly of elastic elements connected at a finite number of joints (or nodal points). The system is said to be statically determinate when the forces acting on each of the elementb can be determined by means of the equations of equilibrium. Should the system possess more elements or external supports than are necessary for stabilty then it is decribed as being statically indeterminade. The excess forces carried by these elements and supports are described as redumdants. We can be certain that the correct solution of our prob lem has been found if the following requirements are satisfied. 1- Equilibrium: The external forces and the internal forces they induce are in equilibrium at each joint. 2- Compatibility: The elements are so deformed that they can all fit together. 3- Force deflection relationship: The internal forces and deformations satisfy the stress-strain relationship of the element. In our further considerations this relation will be assumed to be linear. Two basic methods of analysis are available for solving the problem, and are referred to as the force method and the displacement method. In the force method the redundant forces are taken as the unknowns of the problem, so that all the internal forces can be expressed in terms of the external and the redundant forces. Then by using the stress-strain relationship the deformations of the elements can also be expressed in terms of the external and redundant forces. Finally, by applying tha compatibility criterion that the deformed elements must fit together, it is possible to formulate a set of linear simultaneoub equation which yield the values of the redundant forces. We may then calculate the stresses in the elements of the structure as well as the displacements of its joints in the directions of the external which are related by the flexibility matrix F. Basic Theory of The Force Method The system to be analyzed consist of s individual ele: ments that are assembled in some definite manner. This structure is subjected to a set of m external forces repre sented by the column vector.,'-(, *f ¦ "^ F2... pm and the displacements of these forces along their lines of action are represented by the column vector > k d_... dm 2 m The internal forces of the elements are represented by the column vector t < P. [Px p2... Pi} ît requires more than one internal force to specify the state of stress of an element, and herce e conclude that IV S Finally system is n times redundant, and forces chosen as the redundancies are represented by the column vector. «x -[xx x2... xnj The mtermal forces expressed in terms of the external and redundant forces. Let the external redundant supports be removed and the structure be cut so that the system is reduced to a statically determinate structure is called the basic system. In addition to the external forces f, the redundants x are applied at the cust and the redundant supports, Then the ihternal forces p can be expressed in terms in terms of the forces f and x according to the equation. (p=['b"o -«' ¦ is pla ced in block 10, and in block 11 is placed the product of blocks 10 and 8. Block 11 is repeated in blocks 11 a. Block 12 is formed as the product of block 2 and 11a, and matrix 13 is then comhleted by adding block 1 and 12. Block 14 is formed as the product of blocks 7 and lib, and the flexibility matrix in block 15 is formed adding blocks 14 and 6. Finally the values of the external loads are placed in column 16 so that the values of the internal forces in column 17 arofound by multiplying block 13 and column 16 and the values of the displacements in column 18 are by multiplying block 15 and column 16. Table: 1. Matric Calculations in Tabular Form. Bf -F0f Several points of interest should now be discussed. First of all, we notice that all the calculation are made using the basic matrices B,B and F. The external load matrix f comes into the caîculation only at the end, so that change in the elements of f involes only the minor cal culations indicated in block 17 and 18. Should we desire to apply an additional load at some point, the increase of com putational labor is still not large as its point of applica tion coxncides with one of the joints of the structure. The matrix B is increased by an extra columin from m to m + 1 columns.0 The modifet matrix calculations are illustrated in Table 2 with the shaded parts remaining unchanged, while the additional columns and rows completed according to the previously described. The additional work involves no more than matrix multiplication. The most difficult operation involed in the calculation is the inversion of the Dn matrix The order of this matrix is equal to the number of redundan cies and is independent of the loading. Table JL. The modified matrix calculations Some further Observations on the Matrix force method: Although the presentation of the basic there are several po ints of interest that can be profitably brought to light. By the application of the dummy-load theorem it is pos sible to find the relation that exists between the displace ments d and the deformation of the elements v. Under the action of the applied forces f the actual displacements are d and the deformations v, Now apply dummy external forces fj for xhich the corresponding internal forces are P,. The from the dummy- load theorem d XI f,d = p,v Jd' But so that >d - Bfd fdd=fdB'V However, since this equation must be satisfied for any choice of dummy load f, (it is independent of d and v)- it follows that d = Bv (14) The same process can be reoeated, this however, using a set df internal dummy forces p,* that only has to be sta- ticall equivalent to the dummy load f^. m^A" *.* - 4-^° ^mm, load theorem gives This time the dummy fdd= pfl*'v A suitable set of internal dummy forces is given by loading the statically determinate structure so hat * P 6 o d Substituting this in the dummy- load equation, we obtain d=B v o (15) Equation (14) and (15) give the rather surprising result that the displacements d can be obtained by premultiplying the deformations v by either B or B. In fact, any B* that corresponds to a statically equivalent set of internal for ces will give the same result. Equating the displacements d of Eqs. (14) and (15) gives the result B v= B v o (16) From this equation it is possible to fxnd the expression for X, but it is left as an exercise for those who would like to do it. However, let us continue with eq. Eq (4) (16) by setting, from / ( / / B- BQ * X Bx xii > I I l Then B v+X B v = B v o i o / / from which x=B v = 0 (17) However, since it is assumed that x^ 0. it follows that if Eq. (17) is to be satisfied for all v, then column vector resulting from the product B v must be zero, that is, B^O (18) From this result it can se shown that the relative disp lacements of the redundant forces at the cust made in the rudundant structure must be zero. When the redundant struc ture is loaded by the forces f, the deformations of the elements are v and we assume that the relative displacements at the cust are e. Now apply dummy loads x, at the cust. Then the internal dummy forces are given by Pd = Bixd Now the dummy- load theorem stateb that the work done by the dummy forces x, in moving through the actual displa cements e is equal to the work done by the dummy internal forces in moving through the deformation v, that is, II ill R.v = X,e or x,B v= x,e o d did But since this must be true for all dummy x,, we cone lude that the relative displacements at the cust are zero. Besides the physical interpretation, the discussion o of Eq. (18) shows how quickly we would have obtained this compation by means df the dummy- load theorem if we had star ted out with the condition that e=0 Let us close oun general remarks on the matrix force method by stating by stating with reference to the first paragraph that the following theree equations from the ba sic of this method. p = (BQ + B X)f equilibrium (3) B v= 0 compatibility (18) i v=F p stress-strain relationship (15) xm

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1990

##### Anahtar kelimeler

Kafes kirişler,
Matris kuvvet yöntemi,
Truss,
Matrix force method