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 $f_i{}^{\left(e\right)}$ and $f_j{}^{\left(e\right)}$ 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 $i\to j$ to $j\to i$, 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?

To determine

The properties description of the element stiffness matrix.

Explanation of Solution

Properties of stiffness matrix describes as shown below.

1. In order of stiffness matrix shows the total Degree of freedoms (DOFs).
2. In addition to the singular stiffness matrix also known as structure is unrestrained and rigid body movement.
3. 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.
4. It includes the Diagonal terms of the matrix are always positive force directed for left direction with not any displacement in right direction.
5. Diagonal terms will be zero or negative for the structure is not stable or unstable condition.

To determine

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

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

1. Properties of structurestiffness matrix mainly describes that an order of structure stiffness matrix shows the structure of the totalDOFs.
2. In addition to the singular structurestiffness matrix also known as structure is unrestrained and rigid body movement.
3. Its includes the for an equilibrium set of nodal force for each column of structurestiffness matrix is required greater than single DOF.
4. Its includes the Symmetric structurestiffness matrix signified the force-displacement are proportional to each other.
5. 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.

To determine

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

Explanation of Solution

Properties of structure stiffness matrix describes as shown below.

1. The global and localterms signified the global and local coordinate systems which is used denoted the member of a system
2. Its including with stiffness matrix associated member denotes the local stiffness matrix and also denotes the global stiffness matrix with overall mechanical system.
3. It includes theprocess the collecting the global stiffness matrix used for solution of the finite element solution.
4. Its includes the finite element mesh comprisesdiscrete element with using the nodes, element stiffness matrix for each element made distinctly,
5. For the global matrix, element stiffness matrices of individual element are added to suitableplaces.
6. 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.

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.

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.

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.

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.

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.

  $\begin{array}{l}\text{two term approximation}\\ \stackrel{-}{u}(x)=(1-x)+c_1{\varphi }_1(x)+c_2{\varphi }_2(x)\\ \text{ =}(1-x)+c_1{x}_1(1-x)+c_2{x}_2{}^2(1-x)\end{array}$

  $\begin{array}{l}\text{using boundary value condition to the problem}\\ c_1{x}_1+c_2{x}_2{}^2=0\\ c_1{x}_1(0)+c_2{x}_2{}^2(0)=0\end{array}$

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.

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.

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.

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.

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.