Answer the following descriptive questions a. Write five properties of the element stiffness matrix. b. Write five properties of the structural stiffness matrix. c. Write five properties of the global stiffness matrix. d. Will subdividing a truss element into many smaller elements improve the accuracy of the solution? Explain. e. Explain when the element force p ( e ) is positive or negative. f. For a given spring element, explain why we cannot calculate nodal displacement u i and u j , when nodal forces f i ( e ) and f j ( e ) are given. g. Explain what “striking the rows” and “striking the columns” are. h. Once the global DOFs {Q} is solved, explain two methods of calculating nodal reaction forces. i. Explain why we need to define the local coordinate in 2D truss element. j. Explain how to determine the angle of 2D truss element. k. When the connectivity of an element is changed from i → j to j → i , will the global stiffness matrix be the same or different? Explain why. I. For the bar with fixed two ends in figure 1.22, when temperature is increased by T, which of the following strains are not zero? Total strain, mechanical strain, or thermal strain. Describe the finite element model you would use for a thin slender bar pinned at both ends with a transverse (perpendicular to the bar) concentrated load applied at the middle. (i) Draw a figure to show the elements, loads, and boundary conditions. (2) What type of elements il1 you use?

Question

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Answer 1

Textbook Question

Chapter 1, Problem 1E

Answer the following descriptive questions

a. Write five properties of the element stiffness matrix.

b. Write five properties of the structural stiffness matrix.

c. Write five properties of the global stiffness matrix.

d. Will subdividing a truss element into many smaller elements improve the accuracy of the solution? Explain.

e. Explain when the element force $p ( e )$ is positive or negative.

f. For a given spring element, explain why we cannot calculate nodal displacement $u i$ and $u j$ , when nodal forces $f i ( e )$ and $f j ( e )$ are given.

g. Explain what “striking the rows” and “striking the columns” are.

h. Once the global DOFs {Q} is solved, explain two methods of calculating nodal reaction forces.

i. Explain why we need to define the local coordinate in 2D truss element.

j. Explain how to determine the angle of 2D truss element.

k. When the connectivity of an element is changed from $i → j$ to $j → i$ , will the global stiffness matrix be the same or different? Explain why.

I. For the bar with fixed two ends in figure 1.22, when temperature is increased by T, which of the following strains are not zero? Total strain, mechanical strain, or thermal strain. Describe the finite element model you would use for a thin slender bar pinned at both ends with a transverse (perpendicular to the bar) concentrated load applied at the middle. (i) Draw a figure to show the elements, loads, and boundary conditions. (2) What type of elements il1 you use?

a.

Expert Solution

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

b.

Expert Solution

To determine

To find:The properties description of the structural stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

c.

Expert Solution

To determine

To find:The properties description of the global stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

d.

Expert Solution

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

e.

Expert Solution

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

f.

Expert Solution

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

g.

Expert Solution

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

h.

Expert Solution

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

i.

Expert Solution

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

j.

Expert Solution

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

k.

Expert Solution

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

l.

Expert Solution

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

m.

Expert Solution

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

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Answer 2

Textbook Question

Chapter 1, Problem 1E

Answer the following descriptive questions

a. Write five properties of the element stiffness matrix.

b. Write five properties of the structural stiffness matrix.

c. Write five properties of the global stiffness matrix.

d. Will subdividing a truss element into many smaller elements improve the accuracy of the solution? Explain.

e. Explain when the element force $p ( e )$ is positive or negative.

f. For a given spring element, explain why we cannot calculate nodal displacement $u i$ and $u j$ , when nodal forces $f i ( e )$ and $f j ( e )$ are given.

g. Explain what “striking the rows” and “striking the columns” are.

h. Once the global DOFs {Q} is solved, explain two methods of calculating nodal reaction forces.

i. Explain why we need to define the local coordinate in 2D truss element.

j. Explain how to determine the angle of 2D truss element.

k. When the connectivity of an element is changed from $i → j$ to $j → i$ , will the global stiffness matrix be the same or different? Explain why.

I. For the bar with fixed two ends in figure 1.22, when temperature is increased by T, which of the following strains are not zero? Total strain, mechanical strain, or thermal strain. Describe the finite element model you would use for a thin slender bar pinned at both ends with a transverse (perpendicular to the bar) concentrated load applied at the middle. (i) Draw a figure to show the elements, loads, and boundary conditions. (2) What type of elements il1 you use?

a.

Expert Solution

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

b.

Expert Solution

To determine

To find:The properties description of the structural stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

c.

Expert Solution

To determine

To find:The properties description of the global stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

d.

Expert Solution

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

e.

Expert Solution

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

f.

Expert Solution

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

g.

Expert Solution

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

h.

Expert Solution

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

i.

Expert Solution

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

j.

Expert Solution

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

k.

Expert Solution

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

l.

Expert Solution

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

m.

Expert Solution

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

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Answer 3

Textbook Question

Chapter 1, Problem 1E

Answer the following descriptive questions

a. Write five properties of the element stiffness matrix.

b. Write five properties of the structural stiffness matrix.

c. Write five properties of the global stiffness matrix.

d. Will subdividing a truss element into many smaller elements improve the accuracy of the solution? Explain.

e. Explain when the element force $p ( e )$ is positive or negative.

f. For a given spring element, explain why we cannot calculate nodal displacement $u i$ and $u j$ , when nodal forces $f i ( e )$ and $f j ( e )$ are given.

g. Explain what “striking the rows” and “striking the columns” are.

h. Once the global DOFs {Q} is solved, explain two methods of calculating nodal reaction forces.

i. Explain why we need to define the local coordinate in 2D truss element.

j. Explain how to determine the angle of 2D truss element.

k. When the connectivity of an element is changed from $i → j$ to $j → i$ , will the global stiffness matrix be the same or different? Explain why.

I. For the bar with fixed two ends in figure 1.22, when temperature is increased by T, which of the following strains are not zero? Total strain, mechanical strain, or thermal strain. Describe the finite element model you would use for a thin slender bar pinned at both ends with a transverse (perpendicular to the bar) concentrated load applied at the middle. (i) Draw a figure to show the elements, loads, and boundary conditions. (2) What type of elements il1 you use?

a.

Expert Solution

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

b.

Expert Solution

To determine

To find:The properties description of the structural stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

c.

Expert Solution

To determine

To find:The properties description of the global stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

d.

Expert Solution

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

e.

Expert Solution

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

f.

Expert Solution

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

g.

Expert Solution

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

h.

Expert Solution

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

i.

Expert Solution

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

j.

Expert Solution

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

k.

Expert Solution

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

l.

Expert Solution

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

m.

Expert Solution

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

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Answer 4

Textbook Question

Chapter 1, Problem 1E

Answer the following descriptive questions

a. Write five properties of the element stiffness matrix.

b. Write five properties of the structural stiffness matrix.

c. Write five properties of the global stiffness matrix.

d. Will subdividing a truss element into many smaller elements improve the accuracy of the solution? Explain.

e. Explain when the element force $p ( e )$ is positive or negative.

f. For a given spring element, explain why we cannot calculate nodal displacement $u i$ and $u j$ , when nodal forces $f i ( e )$ and $f j ( e )$ are given.

g. Explain what “striking the rows” and “striking the columns” are.

h. Once the global DOFs {Q} is solved, explain two methods of calculating nodal reaction forces.

i. Explain why we need to define the local coordinate in 2D truss element.

j. Explain how to determine the angle of 2D truss element.

k. When the connectivity of an element is changed from $i → j$ to $j → i$ , will the global stiffness matrix be the same or different? Explain why.

I. For the bar with fixed two ends in figure 1.22, when temperature is increased by T, which of the following strains are not zero? Total strain, mechanical strain, or thermal strain. Describe the finite element model you would use for a thin slender bar pinned at both ends with a transverse (perpendicular to the bar) concentrated load applied at the middle. (i) Draw a figure to show the elements, loads, and boundary conditions. (2) What type of elements il1 you use?

a.

Expert Solution

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

b.

Expert Solution

To determine

To find:The properties description of the structural stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

c.

Expert Solution

To determine

To find:The properties description of the global stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

d.

Expert Solution

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

e.

Expert Solution

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

f.

Expert Solution

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

g.

Expert Solution

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

h.

Expert Solution

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

i.

Expert Solution

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

j.

Expert Solution

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

k.

Expert Solution

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

l.

Expert Solution

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

m.

Expert Solution

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

a.

Expert Solution

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

b.

Expert Solution

To determine

To find:The properties description of the structural stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

c.

Expert Solution

To determine

To find:The properties description of the global stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

d.

Expert Solution

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

e.

Expert Solution

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

f.

Expert Solution

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

g.

Expert Solution

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

h.

Expert Solution

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

i.

Expert Solution

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

j.

Expert Solution

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

k.

Expert Solution

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

l.

Expert Solution

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

m.

Expert Solution

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

Answer 8

a.

Expert Solution

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

The properties description of the element stiffness matrix.

Answer 13

Explanation of Solution

Properties of stiffness matrix describes as shown below.

In order of stiffness matrix shows the total Degree of freedoms (DOFs).
In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
It includes the for an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Its includes the Symmetric stiffness matrix signified the force-displacement are proportional to each other.
It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

Answer 14

b.

Expert Solution

To determine

To find:The properties description of the structural stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

Answer 15

b.

Expert Solution

Answer 16

b.

Expert Solution

Answer 17

Expert Solution

Answer 18

To determine

To find:The properties description of the structural stiffness matrix.

Answer 19

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
Its includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

Answer 20

c.

Expert Solution

To determine

To find:The properties description of the global stiffness matrix.

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

Answer 21

c.

Expert Solution

Answer 22

c.

Expert Solution

Answer 23

Expert Solution

Answer 24

To determine

To find:The properties description of the global stiffness matrix.

Answer 25

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
Its includes each element depicted the two local nodes with convertible to the global node. Further the element, the matchingwith the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

Answer 26

d.

Expert Solution

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

Answer 27

d.

Expert Solution

Answer 28

d.

Expert Solution

Answer 29

Expert Solution

Answer 30

To determine

To find: improvement justification of the truss element solution by Finite element method (FEM) method.

Answer 31

Explanation of Solution

The subdividing of a structure in to suitable number of lesserparts called as discretization. These smaller components are collected and process of bondingthe elements combined is called assemblage is further used to progress the accuracy of Finite element method.

In the order of polynomial estimate for all elements is kept constant and the numbers ofelements are increased. In numbers of elements are kept constant and the polynomial order guesstimate of element is amplified.

Answer 32

e.

Expert Solution

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

Answer 33

e.

Expert Solution

Answer 34

e.

Expert Solution

Answer 35

Expert Solution

Answer 36

To determine

To find:The element force condition of positive and negative characteristics discussed in the problem.

Answer 37

Explanation of Solution

For an equilibrium set of nodal force for each column stiffness matrix is required single Degree of freedoms. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

As the finite element mesh comprises discrete element with using the nodes, element stiffness matrix for each element made distinctly. For the global matrix, element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix, which is maximum, and minimum global node numbers.

Answer 38

f.

Expert Solution

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

Answer 39

f.

Expert Solution

Answer 40

f.

Expert Solution

Answer 41

Expert Solution

Answer 42

To determine

To find:Nodal displacement for the nodal forces given condition shows in the problem.

Answer 43

Explanation of Solution

As the finite element comprises discrete element with mesh using the nodes, each element for the element stiffness matrix for made distinctly. For the global matrix further using each of the element stiffness matrices of individual element are added to suitable places. With each element depicted the two local nodes with convertible to the global node. Further the element, the matching with the global node numbers getting a matrix which is maximum and minimum global node numbers.

Displacement of the nodes is theunknown problem which solved with the use of the Displacement or stiffness method. Using with two methods, displacement method isNeeded to find the solution and also used for computer program for displayingin graphical form to the user.

Answer 44

g.

Expert Solution

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

Answer 45

g.

Expert Solution

Answer 46

g.

Expert Solution

Answer 47

Expert Solution

Answer 48

To determine

To find:The striking the rows and striking the column condition shown in the problem.

Answer 49

Explanation of Solution

The truss elements are shows the truss structure relatedorganizedwith the use of point joint by transmits for the axial force to element show a matrix with "strike-out lines" on some of its rows and columns.

Further methods are available for swapping every matrix fromglobal and local matrix change would be too time-consuming. In addition, FEM and other way to get the strikeout effect which resulting in a matrix consistent with the rest of the matrices.

Answer 50

h.

Expert Solution

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

Answer 51

h.

Expert Solution

Answer 52

h.

Expert Solution

Answer 53

Expert Solution

Answer 54

To determine

To find:The two methods of nodal reaction forces for the global matrix solution shown in the problem.

Answer 55

Explanation of Solution

Here using each type of the finite element has aexact structural shape which is linkedby the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

$two term approximationu−(x)=(1−x)+c1ϕ1(x)+c2ϕ2(x) =(1−x)+c1x1(1−x)+c2x22(1−x)$

$using boundary value condition to the problemc1x1+c2x22=0c1x1(0)+c2x22(0)=0$

Answer 56

i.

Expert Solution

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

Answer 57

i.

Expert Solution

Answer 58

i.

Expert Solution

Answer 59

Expert Solution

Answer 60

To determine

To find:Explanation of the local coordinate in the 2D truss element problem.

Answer 61

Explanation of Solution

Two dimensional elements shows the three or more nodes for the 2D plane and basic element used 2D studywhich is triangular element. Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. For an equilibrium set of nodal force for each column stiffness matrix is required single DOF. Symmetric stiffness matrix signified the force-displacement are proportional to each other. Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

Answer 62

j.

Expert Solution

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

Answer 63

j.

Expert Solution

Answer 64

j.

Expert Solution

Answer 65

Expert Solution

Answer 66

To determine

To find:The examination of the angle of the 2D truss element shown in the problem.

Answer 67

Explanation of Solution

The points in which the wholeassembly are shown for coordinates system signified the global coordinate system and in the natural coordinate system only shows the point element using the number with dimensionless and unity magnitude, which very useful in assembling of stiffness matrices.

local conventionalfor an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

Answer 68

k.

Expert Solution

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

Answer 69

k.

Expert Solution

Answer 70

k.

Expert Solution

Answer 71

Expert Solution

Answer 72

To determine

To find: connectivity of an element change condition and global matrix condition discussed in the problem.

Answer 73

Explanation of Solution

Here using the Each type of the finite element has a exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly shows for two method of nodal reaction forces for the global matrix solution shown in the problem.

Answer 74

l.

Expert Solution

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

Answer 75

l.

Expert Solution

Answer 76

l.

Expert Solution

Answer 77

Expert Solution

Answer 78

To determine

To find:Strain characteristics of the given bar with fixed condition shown in the problem.

Answer 79

Explanation of Solution

The strain state of strain normal to plane and the shear strains are taken to be zero with the Plane stress which the normal stress and shear stress that perpendicular to the plane finds zero.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

Answer 80

m.

Expert Solution

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.

Answer 81

m.

Expert Solution

Answer 82

m.

Expert Solution

Answer 83

Expert Solution

Answer 84

To determine

To find: FEM model for the thin slender bar pinned condition for given load condition with given boundary condition shown in the problem.

Answer 85

Explanation of Solution

A differential equation describes a boundary value problem reliant on variable with BVP to progress the accuracy in the solution of FE method. And also polynomial estimate for all elements is kept constant and the No of elements are amplified.

Here using the Each type of the finite element has exact structural shape which is linked by the adjacent element with the nodal point or nodes. For the nodes, degrees of freedom are placed and forces act on the nodes element. The methods mainly show for two method of nodal reaction forces for the global matrix solution shown in the problem. local conventional for an element or element level that change orientation of the element direction differs the element. Global conventional for the whole system that shows the same in direction for all elements with also the differently oriented.