CHAPTER 1

INTRODUCTION

The advent of computer software packages has been proven in recent years to be a powerful tool for educational purposes. The lack of freely available, user friendly and highly visualized web-based software packages for Structural Analysis at the educational level has been the motivation for the development of the Stiffness Matrix Method [1-3] (SMM) module. The (SMM) module is being developed for the CEE 310 Structural Analysis I, a junior-level course required for the BS in Civil Engineering program. The module will be implemented and assessed during the Spring 2008 semester with features like computer randomly generated questions, accepting students’ answers, automatically grading students’ performance and emailing students’ scores to both the students and the instructor.

The SMM module includes brief reading sections on the various components of the SMM process and the theoretical backgrounds behind the developed formulas adopted for calculations. The reading sections are followed by an interactive application unit that includes the computation of the structural responses (such as nodal displacements, member-end-actions and support reactions), visualization and animation (such as plots of un-deformed and deformed structures) under the FLASH computer environment [4-5], and high-lighted observations to enhance student learning. Students are then assigned exercises that will require both hand calculations and the use of the interactive unit. Figure 1.1 shows an example of an interactive (and visual) application unit for the “pre-processing” phase (to create the structural model using the developed SMM module).This pre-processing phase will be followed by structural “analysis/computation” (to calculate the structural responses) and “post-processing” (to display the structural responses in the “graphical” forms) phases.

Un-deflected Shape

Deflected Shape

Figure 1.1 Sample of Pre-Processing and Post-Processing Phase of SMM Module

The objectives and the outcomes of the module and their mapping are shown in Table 1.1. The table also includes the level of achievement each outcome targets in relation to Bloom’s taxonomy.

Table 1.1 Objectives and Outcomes of the Developed SMM Module

Objectives and Outcomes	Bloom’s Level of Achievement	Objective 1: Students are capable to compute/verify the structural responses (using SMM module learned from CEE- 310 course)	Objective 2: Students can conduct “what if” studies, and are familiar with modern computer software/hardware technology
Outcome 1: Students can create the structural model/problem (in a user-friendly, interactive, visual environment)	Knowledge	X	X
Outcome 2: Students can compute “element” stiffness matrices, in both local & global references	Comprehension (level 1)	X
Outcome 3: Students can “assemble” the “global” stiffness matrix and load vector.	Comprehension (level 2)	X
Outcome 4: Students can impose proper “boundary conditions”,compute and visualize the structural responses	Application/ Analysis	X	X
Outcome 5: Students can conduct “what if” studies, interpret the results and identify/fix potential errors made in earlier phase (such as creating an unstable/improper structural model)	Analysis/ Synthesis	X	X

Figure 1.2 displays the layout of the module’s structure. It also shows how the various components of the module contribute to the outcomes as well as the practicality, hierarchical, connectivity and the viscompana characteristics.

Figure 1.2 Layout of Stiffness Matrix Method (SMM) Module

The CEE 310 course will be offered only once during the active period of this project. Therefore, we will have a limited opportunity to assess the module being developed (the developed SMM module will be implemented for the first time in the spring 2008 semester). It is planned to establish the baseline data by giving a test on the subject SMM “before” (Spring 2007) and “after” (Spring 2008) the SMM module has been introduced to a similar group of students. Furthermore, since Spring 2007 test scores on different topics of the CEE 310 course (such as Virtual Works, Slope Deflection, Moment Distribution methods, etc.) were available, the “indirect” impacts of the developed SMM module (which can also be used to verify the students’ solutions) on students’ performance can be evaluated as additional measurements. The comparison between pre-module and post-module test results will demonstrate how successful the module is. The assessment rubric shown in Table 1.2 will be used to prepare and grade these tests. In addition, carefully designed SMM module surveys will be conducted in Spring 2008.

Table 1.2 Assessment Rubric for the Stiffness Matrix Method (SMM) Module

Out- come	Unacceptable	Marginal	Acceptable	Excellent
1	Little/no knowledge of what data are required to create a structural model/problem	Can identify some data required to create a structural model.	Can identify most data required, but have difficulty to follow interactive instructions to (visually) create a structural model.	Can identify all data required and to visually/interactively create a structural model.
2	Inadequate ability to identify the degree-of-freedom (dof), size and rotational matrices associated with a particular truss/beam/frame element	Know to identify the dof and size of element matrices, but can’t compute numerical values of element stiffness matrix in local references	Can compute the element stiffness matrix in local references.	Know to transform element stiffness matrix from local to global references.
Out- come	Unacceptable	Marginal	Acceptable	Excellent
3	Inadequate ability to determine the locations of element stiffness within the “structural” stiffness matrix. Also does not know how to impose “boundary conditions”.	Know to place the locations of element stiffness matrices in a structural stiffness matrix. However, still confuse to handle “over-lap” terms.	Know how to “assemble” the structural stiffness matrix. Still have some difficulty to impose “boundary conditions”.	Completely understand the assembly process, including properly imposed “boundary conditions”.
4	Can’t recognize the roles of linear equation solver (to solve for nodal displacements). Have no ideas to compute member-end-actions, support reactions. Have no abilities to interpret the obtained results.	Know to compute the nodal displacements, and member-end-actions.	Know to compute all structural responses. However, still have some difficulty to interpret the computed results.	Know to compute all structural responses, and have abilities to interpret the computed results.
5	Can’t identify important parameters that have impacts on the structural responses. No abilities to apply the SMM software to conduct “what if” studies. Can’t identify/fix errors made in preparing the structural model.	Can identify some important parameters for conducting “what if” studies.	Can identify most (or all) important parameters for “what if” studies.	Can conduct all “what if” studies, interpret the computed results and be able to identify/fix potential errors made in earlier phase (such as preparing the input structural model).

CHAPTER 2

THEORETICAL BACKGROUND FOR THE STIFFNESS MATRIX

METHOD

The entire Stiffness Matrix Method (SMM) will include the following major

components (also refer to Figure 1.2):

(a) Element local matrices

(b) Element global matrices

(c) Assembly process

(d) Boundary conditions

(e) Solution of system of linear equations

(f) Structural responses

Details of the above key components will be explained in the following sub-sections. To simplify the discussion, plane (2-D) truss elements will be derived. Generalized ideas for space (3-D) problems and for other element types, such as beam and frame structures, will follow the same procedures.

2.1 Element Local Matrices

A general orientation 2-D truss element is shown in Figure 2.1

Figure 2.1 Development of 2-D Truss Element “Local” Stiffness Matrix

Both ends of a truss element is assumed to be fixed by pin supports. A positive unit axial displacement is applied at node “j”. The axial forces developed at nodes j and i can be easily computed as

(2.1)

or, (2.2)

Equilibrium conditions along the element local x_L axis give:

(2.3)

Similarly, if one applies a positive, unit axial displacement at node “i”, one obtains:

(2.4)

(2.5)

Equations (2.2-2.5) can be conveniently expressed in matrix notation as:

(2.6)

The first portion of Equation (2.6) can be expressed as where represents the “relative” axial deformation of a truss member when unit (and hence ).Thus, this equation is identical to Equation 2.4.The first and second columns of the 2x2 “stiffness” matrix ( shown in Equation 2.6), therefore, represent Equations (2.4,2.5) and Equations (2.2,2.3), respectively. For reasons which will be explained in Section 2.2, the matrix Equation (2.6) can be expanded as:

(2.7)

Equation (2.6) or (2.7) can be symbolically expressed as:

(2.8)

where

2.2 Element Global Matrices:

Having computed the “element” matrix equations (shown in equation 2.7) these element matrix equations need to be assembled (or combined/glued together) to form the “system” , global matrix equations. Since each truss element, in general, will have different local reference axis x_L and y_L, one needs to transform Equation (2.7) into the global axis x and y before computing the assembled, system matrix (see Section 2.3). This process can be done by 2 different approaches.

2.2.1 Engineering Approach:

Applying a positive unit displacement ∆_xi = 1 inch along the global x-axis, the developed reaction forces F_ix, F_iy, F_jx and F_jy at nodes “i and j” can be easily computed as described in the following paragraphs.

Figure 2.2 Truss Element “Global” Stiffness Matrix

First, the projected axial displacement at node “i” (=∆_i ) can be computed as:

(2.9)

Using Equation (2.4), the “local” axial forces f_i and f_j can be computed as:

(2.10)

(2.11)

The global x and y components f_ix and f_iy of f_i can be computed as:

(2.12)

(2.13) (2.14)

(2.15)

(2.16)

(2.17)

Equations (2.14-2.17) can be conveniently expressed in matrix notation as:

(2.18)

where

and (2.19)

Similarly, the 2^nd, 3^rd and 4^th columns of the matrix equation (2.18) can be computed by applying the unit displacement ∆_iy=1 inch, then ∆_jx=1 inch and ∆_jy=1 inch, respectively. The incomplete Equation (2.18) becomes:

(2.20)

Equation (2.20) can be symbolically expressed as:

(2.21)

2.2.2 Mathematical Approach

The “local” coordinates (A_xL, A_yL) vector can be related to the “global” coordinates (A_x, A_y) by the following equations:

Figure 2.3 Local-Global Vector Transformations

(2.22)

(2.23)

Using the notations introduced in Equation (2.19), Equations (2.22, 2.23) can be expressed as:

(2.24)

Equation (2.24) is quite general, since the vector may represent either the nodal displacement vector, nodal force vector or any other engineering variables.

Starting from Equation (2.9), one has

(2.25)

Using Equation (2.24), the “local” nodal displacement vector can be expressed in terms of the “global” nodal displacement vector as:

(2.26)

Similarly, one has:

(2.27)

Substituting Equations (2.26, 2.27) into Equation (2.25), one obtains:

(2.28)

Pre-multiplying both sides of Equations (2.28) by, one gets:

(2.29)

Recalled from Equation (2.24), one gets

(2.30)

(2.31)

Since, see Equation (2.31), Equation (2.29) becomes:

(2.32)

In the expanded form, Equation (30) can be expressed as:

(2.33)

where:

(2.34)

Substituting Equation (2.34) and Equation (7) into Equation (2.33), one gets:

(2.35)

The triple matrix product, shown in Equation (2.33) or Equation (2.35), will be “exactly identical” to the coefficient matrix, see Equations (2.20, 2.21).

Thus, one obtains the following relationship:

(2.36)

Remarks:

1. For a 3-D truss element, since each node may have 3 translational displacements (each element will have 6 degree-of-freedoms or 6 dof) the dimensions of the above equation become:

(2.37)

where:

(2.38)

(2.39)

or direction cosines of the “local” x_i axis and the “global” x_j axis, where (x₁, x₂, x₃) axis (x,y,z) axis

(2.40)

In practice, for a (2-D, or 3-D) truss element (connected by 2 end nodes “i and j”, with length L), c_ij can be computed as [1]:

For a general 3-D truss element orientation

(2.41)

for a “vertical” ( parallel to global y-axis) 3-D truss element:

(2.42)

where:

;; ; (2.43)

2. For a typical 2-D frame element (shown in Figure 2.4 ),each node will have 3 dof (including axial deformation).Thus, the 6x6 element stiffness matrix can be given in Table 2.1[1].

Figure 2.4 Numbering System for a Plane Frame Member

(a) Local Reference Axis

(b) Global Reference Axis

Table 2.1 Plane Frame Stiffness Matrix for Member Axes (Figure 2.4a)

The transformation matrix [c] (between the “Local” and “Global” axis) can be given as:

(2.44)

The inflated rotation transformation matrix R for a plane frame member can be shown to take the familiar form (see Equation 2.34)

(2.45)

In Equation (2.45) the matrix c is the 3 X 3 rotation matrix given by Equation (2.44).

Having the matrix on hand, one may then calculate the member stiffness matrix for “global” axes (see Equation 2.36):

(2.46)

Table 2.2 Frame Member Stiffness Matrix for Structure Axes (Figure 2.4b)

3. For a typical 3-D frame element (shown in Figure 2.5), there are 6 dof per node. Thus, the 12 x12 element “local” stiffness matrix can be generated (with the aids from Figure 2.6) and is given in Table 2.3[1].

Figure 2.5 Restrained Member (in local axis)

Figure 2.6 Unit Displacement is Applied to Each of the 12 dof of a 3-D Frame Element

Table 2.3 Space Frame Member Stiffness Matrix

The 3-D frame element (global) stiffness matrix can be given by the familiar triple product:

(2.48)

where (2.47)

and (2.48)

However, if the 3-D frame member is VERTICAL (such as parallel to the “global” Y axis), then the Equation 2.48 should be replaced by Equation (2.49).

(2.49)

The angle α is defined in Figure 2.7

Figure 2.7 Rotation of a Space Frame Member (with an “I” cross sectional shape) about x_L axis

For convenient computation, if the “global” coordinates of point “p” (see Figure 2.7) are known, then the following two equations can be used (in conjunction with Equation 2.48).

(2.50)

(2.51)

Since Equation (2.51) can be applied for both “general” 3-D frame element orientation and “vertical” orientation, therefore:

(a) We do “NOT” need to use Equation (2.49)

(b) We can directly calculate sin(α) and cos(α) , in order to apply Equation (2.48), without any required knowledge about the angle α.

2.3 Assembly Process:

Figure 2.8 A 3-Bar Truss Structure

Table 2.8: Connectivity and Material Properties of a 3-Bar Truss Example

Member	Node i	Node j	Length(in)	E(k/in^2)	Area(in^2)
	1	2	10	29,000	1
	3	2	6	29,000	1
	3	1	8	29,000	1

To facilitate the discussion of the assembly process, a 3-bar truss structure with its element connectivity, material properties, global x-y axis etc. is shown in Figure 2.8 and Table 2.4.

Since each node of a 2-D truss structure has 2 translational degree-of-freedoms (dof), the 2 dof numbers associated with a typical i^th node are designated as 2i-1 and 2i, respectively.

(See ∆₁-∆₆ in Figure 2.8) Element (global) stiffness matrices for members 1-3 can be computed from Equations (2.35, 2.43) as:

For member 1,

(2.52)

For member 2,

(2.53)

For member 3,

(2.54)

The total, assembled stiffness matrix [K] for the entire truss structure can be obtained from the contribution of element stiffness matrices, as following:

(2.55)

In Equation (2.55), the numbers written inside the rectangles symbolically represent a particular element’s contribution to specific locations of the assembled stiffness matrix. The actual numerical values of [K] can be computed by substituting Equations (2.52-2.54) into Equation (2.55).Thus, one obtains the following “structural” system of equations:

(2.56)

where (2.57)

; (2.58)

2.4 Boundary Conditions

The “system” equation, shown in Equation (2.56), has several undesirable features, such as:

(a) The “system” stiffness matrix [K], shown in Equation(2.57), is singular.

(b) Portions of the nodal displacement vector {∆}are known, due to the prescribed Dirichlet boundary conditions (for example, ∆₄→∆₆ are known).

(c) Portions of the right-hand-side load vector {F} are the unknown reaction forces (such as R_2y, R_3x and R_3y).

By properly imposing boundary conditions according to the following procedures, the aforementioned problems can be alleviated. Equations (2.56-2.58) can be expressed as:

(2.59)

Due to the applied (known) boundary conditions (∆₄→∆₆), the above Equation (2.59) can be modified as:

(2.60)

The above matrix Equation (2.60) can be conveniently solved by any available direct solver because:

(a) The 6x6 coefficient stiffness matrix is now non-singular.

(b) The entire nodal displacement vector can be considered unknown.

(c) The entire nodal load vector can be considered as known quantities.

Having solved for ∆₁→∆₆ from Equation (2.60), the unknown reaction forces R_2y, R_3x and R_3y can be calculated by applying the bottom 3 equations of Equation (2.59).

2.5 Solution of System of Linear Equations

The “system” stiffness matrix equations, with appropriated boundary conditions (see Equation 2.60) can be symbolically solved by the following 3-step procedures.

[K]{∆}={F} (2.61)

Step 1: Factorization Phase

For most practical engineering applications the coefficient stiffness matrix [K] is sparse, symmetrical and positive definite. Hence, efficient Choleski factorization algorithms can be employed here.

(2.62)

where [U] = an upper triangular matrix

Step2: Forward Solution Phase

Substituting Equation (2.62) into Equation (2.61), one gets

(2.63)

(2.64)

where:

(2.65)

Step 3: Backward Solution Phase

(2.65, repeated)

Detailed formulas and procedures to obtain the factorized matrix [U], forward solution vector {y} and backward solution vector {∆} in steps 1-3, respectively , will be explained by the following simple example.

For the given 3x3 stiffness matrix [K], see Equations (2.61, 2.62), one has:

(2.66)

The product of right-hand-side of Equation (2.66) when equated to the upper triangular portion of left-hand-side matrix [K] will give the following 6 equations

(2.67)

For a general square matrix [K]_NxN, the following 2 formulas can be used to compute the “diagonal” and “off-diagonal” terms of the factorized, upper triangular matrix [U], respectively.

(2.68)

(2.69)

Having obtained the factorized matrix [U] from Equation (2.67), the forward solution vector {y} can be obtained from:

(2.70)

Thus, one can solve:

(2.71)

Equation (2.71) can be generalized to any matrix size NxN, as:

(2.72)

Finally, the backward solution vector {∆} can be obtained from:

(2.73)

Thus, one can solve:

(2.74)

Equation (2.74) can be generalized to any matrix size NxN as:

(2.75)

Remarks

The state-of-the-art mixed Parallel-Vector Direct-Iterative Sparse Solver [6-8] can be utilized for solving large-scale systems of linear equations.

2.6 Structural Responses

Having obtained the system nodal displacement vector {∆} by solving Equation (2.60),according to the 3-step procedure explained in Section 2.5, the support reaction forces (such as ) can be found from Equation(2.59).Finally , member-end-actions(or members’ axial forces in the case of truss structures) can be found in the following step-by-step procedure:

Step 1: For each 2-dimension k^th truss member, connected by 2 nodes “i and j”, the four associated global dof are 2i-1, 2i, 2j-1 and 2j, respectively. These 4 “element” global dof displacement vectors {∆^(k)}_4x1 can be extracted from the “system” displacement vector {∆}_Nx1 as shown in Equation(2.60).Thus ,{∆^(k)} is a sub-set of {∆}.

Step 2: The element nodal displacement vector {∆^(k)} in global reference can be related to its local references as (see Equation 2.24):

(2.76)

where the transformation matrix [R] has already been defined in Equations (2.34, 2.43).

Step 3: Element (local) nodal forces can be computed, see Equation (2.25), as:

(2.25, repeated)

(2.77)

and element stiffness matrix [k_L] (in the local references) has already been defined earlier in Equations(2.7,2.8). If the axial component of {f_L} at node “j” is positive (see Figure 2.3), then the truss member is considered to be in “Tension”. Once the axial force of any truss member has been computed (see Equations 2.77), axial stress () can be easily computed and will be compared with the member allowable axial stress σ_allowed.

2.7 Support Reactions and Member-End-Actions (MEA) due to Combined Joint Loads Applied at Fixed (Support Constraint) Degree-of-Freedom (dof)

For a 2-D truss structure with the applied joint loads (such as Q₄ and Q₆) associated with the fixed dof (such as Δ₄ and Δ₆, due to support boundary conditions), shown in Figure 2.9, the following procedures should be used to compute the unknown reactions and MEA.

Figure 2.9: Loads Applied at the Joints Associated with Fixed dof (due to supports’ boundary conditions)

Step 1: Assuming the combined (system) joint load vector has already been computed as:

(2.78)

Step2: After imposing the boundary conditions, one obtains the system (global) equations as:

(2.79)

Step 3: Reactions can be computed as:

(2.80)

where (2.81)

step 4: Member-End-Actions (MEA) can be computed as ( say, for member 2)

(2.82)

where:

(2.68, repeated)

(2.83)

Remarks

(a) If there are loads acting on the members, then both reaction and MEA computation might be added (such as including the, as shown in Equations (2.80), (2.82)).

(b) If there are loads acting directly on the fixed support, boundary dof then only reaction computation will be changed. (Such as including the vector, as shown in Equation (2.80)).

CHAPTER 3

ADDITIONAL FORMULATION/FEATURES/CAPABILITIES

FOR TRUSS STRUCTURES

3.1 Loads Acting on Members, Temperature, Fabrication errors, Support Settlements

For simple cases where the truss structure is subjected to loads applied only at the joints, the “system” global equations, shown in Equation (2.51), can be partitioned as:

(3.1)

In equation (3.1), the subscripts F and R represent the “Free” dof and “Restrained” dof, respectively.

If one also has the case where the applied loads act on the member, see Figure 3.1, then Equation (3.1) can be symbolically generalized to:

(3.2)

where is the “actions” (=forces and/or moments) in the restrained structure (= shown where all joints are assumed to be fixed, see Figure 3.1) corresponding to the unknown joint displacements and caused by the loads (all loads except those corresponding to the unknown joint displacements).

Figure 3.1: Truss End-Actions Due to Loads Acting On the Member

Figure 3.2: Truss End-Actions Due to Applied Temperature and Pre-strain

Loads acting on the truss member due to temperature and fabrication error (or pre-strain) can also be included, see Figure 3.2, Equation (3.2), therefore, becomes:

(3.3)

(3.4)

(3.5)

where the combined, equivalent joint load vector is defined as:

(3.6)

Remarks:

If one reverses the sign of the truss end-actions (shown in Figures 3.1,3.2), then one will get the “equivalent” joint loads due to all loads acting on the member.

The support reactions can be computed from the bottom portions of Equation (3.1):

(3.7)

where denotes the combined loads (actual and equivalent) applied directly to the restrained supports.

Finally, the member-end-actions can be added to Equation (2.8) as:

(3.8)

where is a vector of fixed-end-actions due to loads applied to the members (see Figure 3.1 and 3.2 ,if applicable).

Since Equation (3.8) has been expressed in terms of the element’s “local reference”, (see Equations 2.26, 2.33, 2.34):

(3.9)

where [R] has been defined earlier in Equation (2.34) and is a subset of the system nodal displacement vector, expressed in the “global reference” (see Equation 2.52).Finally, if support settlements do occur, then the known/prescribed (non-zero) values for the vector (see Equation 3.1), or (see Equation 2.52) can be used.

3.2 Inclined Roller Supports, Multi-Point Constraints (MPC)[9]

The system equilibrium equations, see Equation (2.52), can be symbolically expressed as:

(3.10)

It can be easily shown [6, 10] that the above equilibrium equations can be re-cast as the solution of the following unconstrained optimization problem (Find the vector to minimize the total potential energy of the system)[10]:

(3.11)

The necessary conditions for minimizing Equation (3.11) can be given as:

(3.12)

Equation (3.12) is the same as that given in Equation (3.10)

The inclined rollers ( see Figure 3.3) can be handled as a special case of multi-point constraints(MPC’s).Consider a case where there exists a known constrained relationship between two different degrees of freedom and . Let the equation that represents the relationship be

(3.13)

where, and are known constants.

The MPC Equation (3.13) can be incorporated (as the penalty term) into Equation (3.11) to get:

minimize (3.14)

where λ is a large penalty (positive) number.

Note:

The equality constraint (shown inside the parentheses) needs to be squared to make sure that the sign of the penalty term is positive.

Figure 3.3: Inclined Roller Support

or

or

(3.13, repeated)

where

Note that takes on a minimum value when is zero (or, numerically very small). Using the theorem of minimum potential energy, yields:

(3.15)

(3.16)

(3.17)

or, (3.18)

The usual and the modified terms are:

(3.19)

(3.20)

Note that the modified equations containing terms from Equations (3.19) and (3.20) are still symmetric and positive definite. A popular choice that seems to work effectively is to make the large (positive) penalty constant λ a function of the largest element in the structural stiffness matrix:

(3.21)

The above MPC relations, shown in Equations(3.13-3.20), between any 2 dof and can be generalized to any 3 (or more) dof for which Equation(3.19) and Equation(3.20) will become a 3x3 matrix and 3x1 vector, respectively.

3.3 Adding the Translational Springs into Truss Structures

For some real-life applications, some dof of the truss may not be totally fixed at the support boundary conditions (such as dof , shown in Figure 2.4), or may not be totally free ( such as dof , shown in Figure 2.4).

For the above cases, Figure 2.4 can be represented as shown in Figure 3.4.

Figure 3.4: Three-Bar Truss Structure with Translational Springs

For truss structures, the translational spring stiffness can be represented as:

(3.22)

Thus, if joint 2 in figure 3.4 is totally free to move in the y-direction, then (or set A=0, in Equation 3.22). On the other hand, if it is totally fixed in the y-direction (for example, by a roller support as shown in Figure 2.4), then (or set A = = very large value, in Equation 3.22). In general (shown in Figure 3.4) can be assigned any value between these 2 extreme limits [] to represent the “partial support” boundary conditions. Inclusion of the translational springs can be incorporated by simply adding the spring stiffness constants (such as) into the appropriated “diagonal locations” of Equation (2.51).

CHAPTER 4

COMPUTER IMPLEMENTATION UNDER MACROMEDIA

FLASH ENVIRONMENT

Based upon the theoretical developments mentioned in the previous sections, general truss analysis with different material properties , different loading conditions ( such as loads directly applied at the joints, and/or concentrated forces/ moments , uniformly distributed loads acting on the members), applied temperature, prescribed support settlements ( such as earthquake ground motions), applied pre-strain ( or fabrication errors) , inclined roller supports, user’s specified Multi-Point Constraint (MPC) relations between any 2 (or more) dof, partial support boundary conditions ( such as a user’s applied translational springs) can all be incorporated into the following step-by-step procedures for practical, user-friendly computer implementation:

4.1 User Input Data

Step 1: User-friendly, visualized, menu-driven input data phase. In this step, the user will “draw” the picture of a 2-D truss, select the proper supports (pins, rollers, or springs) from the ICONS MENU, provide material properties (E= young modulus, coefficient of thermal expansion), cross-sectional properties (areas), pre-strain (or fabrication error) values for different truss members, applied (Joint’s and/or members’) loads, MPC relations, spring stiffness constraints. Joint coordinates and member connectivity’s information are automatically generated by the developed software, while the user draws the truss structures.

4.2 Element’s Local Stiffness Matrix and its Transformation Matrix

Step 2: After computing the element stiffness matrix (in local reference), the “element” local-global transformation matrix [R] can be computed from Equations (2.38, 2.40, 2.41, 2.42 and 2.43)

4.3 Element’s Global Stiffness Matrix

Step 3: Element stiffness matrices in the “global” reference can now be computed from Equations (2.36, 2.7 and 2.8).

4.4 Equivalent Joint Load Vector

Step 4: Load acting on the members ( see Figure 3.1), including the applied temperature and pre-strain ( see Figure 3.2) can be converted into the “equivalent joint loads” and stored in the vectors , respectively ( see Equation 3.4).

4.5 Assembled Global System Matrix

Step 5: The assembled “system” stiffness matrix (see Equation 2.49) can be obtained as discussed in section 2.3 (see Equation 2.47).Finally, the combined equivalent joint load vector can be constructed as shown in Equation (3.6).

4.6 Incorporating Boundary Conditions, Spring Supports and MPC

Step 6: The Dirichlet prescribed boundary conditions are first incorporated by modifying the assembled system stiffness matrix [K] and the combined equivalent joint load vector (see Equation 2.51) into the form shown in Equation (2.52).Partial support boundary conditions (in the form of translational spring constants) are then incorporated in the appropriated “diagonal locations” of the system stiffness matrix ( see section 3.2) Finally,MPC relations can be integrated by the procedures explained through Equations (3.10-3.21).

4.7 Solving System of Simultaneous Equations

Step 7:Having imposed all proper boundary conditions on the system stiffness matrix [K] and nodal load vector {F}, the system nodal displacement vector {}, shown in Equation(2.52), can be efficiently solved by efficient sparse algorithms[6-8] which have been summarized in section 2.5 ( see Equations 2.54,2.56 and 2.57).

4.8 Computation of Support Reactions and Member-End-Actions

Step 8: Support reactions can be computed from Equation (3.7) and member-end-actions (such as axial forces in truss members) can be found from Equations (3.8, 3.9).Once truss members’ axial forces are computed, axial stresses can be easily computed and compared with the allowable axial stresses.

4.9 Post-Processing

Step 9: In this output phase, the structural deformed shape is plotted for visualization purposes. Support reactions and member-end-actions’ results can also be displayed by clicking the computer mouse at the appropriated ICONS. Member stresses are plotted with easily recognizable color codes. For example, GREEN, YELLOW and RED colors represent members’ stresses below, near and above the allowable stresses, respectively.

CHAPTER 5

Numerical Simulation and Visualization

5.1 Problems’ Descriptions:

In this section, the following examples are used to validate accuracy and to demonstrate general capabilities of the developed (user-friendly, interactive, visualized) software package.

Example 1: Statically Determinate Truss with Applied Nodal Loads[11]

The geometry of the 2-D truss structures (with E = 20.7 x 10³ kN/cm² and

having following areas

Area of Member 1-2 (connected from joint 1 to joint 2) = 10 cm²

Area of Member 2-3 (connected from joint 2 to joint 3) = 10 cm²

Area of Member 3-4 (connected from joint 3 to joint 4) = 10 cm²

Area of Member 2-5 (connected from joint 2 to joint 5) = 5 cm²

Area of Member 3-6 (connected from joint 3 to joint 6) = 5 cm²

Area of Member 4-7 (connected from joint 4 to joint 7) = 5 cm²

Area of Member 5-6 (connected from joint 5 to joint 6) = 10 cm²

Area of Member 6-7 (connected from joint 6 to joint 7) = 10 cm²

Area of Member 7-8 (connected from joint 7 to joint 8) = 10 cm²

Area of Member 1-5 (connected from joint 1 to joint 5) = 12.5 cm²

Area of Member 5-3 (connected from joint 5 to joint 3) = 12.5 cm²

Area of Member 3-7 (connected from joint 3 to joint 7) = 12.5 cm²

Area of Member 4-8 ⁽connected from joint 4 to joint 8) = 12.5 cm²),

Joint numbers, members’ lengths, applied nodal load (at joint 2, in the negative Y_G direction, with magnitude 100 kN) and support boundary conditions (at joints 1 and 3) are shown in Figure 5.1

Figure 5.1 Determinate Truss with Applied Joint Load

Example 2: Determinate Truss with Support Settlements only. [11]

For the same truss shown in Figure 5.1(without the applied joint load), the following support settlements are given:

At joint1, horizontal displacement = 0.0050 m to left

At joint1, vertical displacement = 0.0075 m down

At joint 3, vertical displacement = 0.0025 m down

Example 3: Indeterminate (extra supports) Truss with Joint Loads (see Figure 5.2).

E= 30 x 10³ kips/in²

Area of Member 1-2 (connected from joint 1 to joint 2) = 10 in²

Area of Member 2-3 (connected from joint 2 to joint 3) = 10 in²

Area of Member 2-4 (connected from joint 2 to joint 4) = 10 in²

Area of Member 1-4 (connected from joint 1 to joint 4) = 12.5 in²

Area of Member 4-3 (connected from joint 4 to joint 3) = 12.5 in²

Figure 5.2 Indeterminate (extra supports) Truss with Joint Loads

Example 4: Indeterminate (extra members) Truss with Joint Loads (see Figure 5.3) [11]

E and A are constants for all members.

Figure 5.3 Indeterminate (extra members) Truss with Joint Loads

Example 5: Indeterminate Truss with Applied Temperatures (see Figure 5.4) [11]

Compute bas forces due to an increase of 60⁰ F in the temperature at bars 1-5, 5-6 and 6-4. No change in the temperature of any other bars. α_t=1/150,000 per ⁰ F. E=30x 10³ kips/in².

Area of Member 1-2 (connected from joint 1 to joint 2) = 10 in²

Area of Member 2-3 (connected from joint 2 to joint 3) = 10 in²

Area of Member 3-4 (connected from joint 3 to joint 4) = 10 in²

Area of Member 2-5 (connected from joint 2 to joint 5) = 10 in²

Area of Member 3-6 (connected from joint 3 to joint 6) = 10 in²

Area of Member 2-6 (connected from joint 2 to joint 6) = 10 in²

Area of Member 3-5 (connected from joint 3 to joint 5) = 10 in²

Area of Member 1-5 (connected from joint 1 to joint 5) = 10 in²

Area of Member 5-6 (connected from joint 5 to joint 6) = 10 in²

Area of Member 6-4 (connected from joint 6 to joint 4) = 10 in²

Figure 5.4 Indeterminate Truss with Applied Temperatures

Example 6: Indeterminate Truss with Pre-strain Effects[11]

Compute the bar forces of the two-hinged trussed arch in Figure 5.5 caused by forcing member 2-3 into place even though it had been fabricated 0.01 m too short.

E=20.7 x 10³ kN/cm²

Area of Member 1-4 (connected from joint 1 to joint 4) = 10 cm²

Area of Member 4-5 (connected from joint 4 to joint 5) = 10 cm²

Area of Member 5-6 (connected from joint 5 to joint 6) = 10 cm²

Area of Member 6-3 (connected from joint 6 to joint 3) = 10 cm²

Area of Member 5-2 (connected from joint 5 to joint 2) = 5 cm²

Area of Member 1-2 (connected from joint 1 to joint 2) = 12.5 cm²

Area of Member 2-4 (connected from joint 2 to joint 4) = 12.5 cm²

Area of Member 2-6 (connected from joint 2 to joint 6) = 12.5 cm²

Area of Member 2-3 (connected from joint 2 to joint 3) = 12.5 cm²

Case (a) without the applied force 30kN at joint 5

Case (b) with the applied force 30kN at joint 5

Figure 5.5 Indeterminate Truss with Pre-strain Effects

Example 7: Indeterminate Truss with Support Settlements (see Figure 5.6) [11]

Compute the bar forces in this truss due to the following support movements

E=30x 10³ kips/in².

Support at joint 1 vertical displacement = 0.24 in. down

Support at joint 3 vertical displacement = 0.48 in. down

Support at joint 5 vertical displacement = 0.36 in. down

Area of Member 1-2 (connected from joint 1 to joint 2) = 10 in²

Area of Member 2-3 (connected from joint 2 to joint 3) = 10 in²

Area of Member 3-4 (connected from joint 3 to joint 4) = 10 in²

Area of Member 4-5 (connected from joint 4 to joint 5) = 10 in²

Area of Member 2-6 (connected from joint 2 to joint 6) = 10 in²

Area of Member 3-7 (connected from joint 3 to joint 7) = 10 in²

Area of Member 4-8 (connected from joint 4 to joint 8) = 10 in²

Area of Member 6-7 (connected from joint 6 to joint 7) = 10 in²

Area of Member 7-8 (connected from joint 7 to joint 8) = 10 in²

Area of Member 1-6 (connected from joint 1 to joint 6) = 10 in²

Area of Member 6-3 (connected from joint 6 to joint 3) = 10 in²

Area of Member 3-8 (connected from joint 3 to joint 8) = 10 in²

Area of Member 8-5 (connected from joint 8 to joint 5) = 10 in²

Figure 5.6 Indeterminate Truss with Support Settlements

Example 8: Gere’s and Weaver’s Truss Structure with General Loading[1,pages 244-249]

The plane truss (shown in Figure 5.7) is to be analyzed using the SMM module described in the previous sections. The truss is restrained at joints 3 and 4 by hinge supports, that prevent translations in both x and y directions. The loads on the truss consist of both joint loads and member loads. E = 10² kN/cm², A=10 cm²

Figure 5.7 Gere’s and Weaver’s Truss Structure

Example 9: Indeterminate Truss with “All” Combined Effects

(a) Support Settlements

Support at joint 1 0.24 in. down

Support at joint 3 0.48 in. down

Support at joint 5 0.36 in. down

(b) Temperature in members 1-2,2-3,3-4 and 4-5 each is increased by 60⁰ F

(α_T= 1/150,000 per ⁰ F)

(c) Member 8-5 is 0.2” too long and member 78 is 0.3” too short.

(d) A joint, horizontal load 10 k applied at joint 7

(e) A uniformly distributed load (w= 2 k/ft) applied on member 1-2.

(f) A concentrated force 5k (in the – Y_G direction) and a concentrated (clockwise) moment (=18 k-ft) applied at mid-point of member 6-7.

(g) The inclined roller support (at 30⁰ with respect to the horizontal line) at node”5”.Thus the single point constraint( SPC) relation between the global x & y axis (at node 5) can be expressed as indicated in Figure 3.3 (equation 3.13) or:

Sin (30⁰)Δ_5x-Cos(30⁰)Δ_5y = 0

Thus,

(h) The following “hypothetical” MPC relation is imposed:

2Δ_3x+3 Δ_3y- 5Δ_4y =0.8

The above MPC equation can be expressed in the “general” form of Equation

(3.13) as:

where:

(i) The horizontal and vertical (partially) spring supports k₁= E(A=5 in²)/(L=30ft) and k₂= E(A=25 in²)/(L=40ft) are applied at nodes 6 and 4, respectively.

E=30x 10³ kips/in²

Area of Member 1-2 (connected from joint 1 to joint 2) = 15 in²

Area of Member 2-3 (connected from joint 2 to joint 3) = 15 in²

Area of Member 3-4 (connected from joint 3 to joint 4) = 15 in²

Area of Member 4-5 (connected from joint 4 to joint 5) = 15 in²

Area of Member 2-6 (connected from joint 2 to joint 6) = 5 in²

Area of Member 3-7 (connected from joint 3 to joint 7) = 5 in²

Area of Member 4-8 (connected from joint 4 to joint 8) = 5 in²

Area of Member 6-7 (connected from joint 6 to joint 7) = 15 in²

Area of Member 7-8 (connected from joint 7 to joint 8) = 15 in²

Area of Member 1-6 (connected from joint 1 to joint 6) = 25 in²

Area of Member 6-3 (connected from joint 6 to joint 3) = 25 in²

Area of Member 3-8 (connected from joint 3 to joint 8) = 25 in²

Area of Member 8-5 (connected from joint 8 to joint 5) = 25 in²

Figure 5.8 Indeterminate Truss with “All” Combined Effects

Example 10: [12]

The geometry of the 2-D beam structure (with E = 29 x 10⁶ psi and I=80 in⁴).

Joint numbers, members’ lengths, applied nodal load (at joint 3, in the negative Y_G direction, with magnitude 10 k) and support boundary conditions (at joints 1 and 2) are shown in Figure 5.9

Figure 5.9 2-D Beam Structure

Example 11: [12]

The geometry of the 2-D frame structure (with E = 29 x 10⁶ psi , I=100 in⁴,A = 10 in² for member 1 and E = 29 x 10⁶ psi , I=220 in⁴,A = 20 in²).

Joint numbers, members’ lengths, applied uniformly distributed load (on member, in the negative Y_G direction, with magnitude 2.4 k/ft) and support boundary conditions (at joints 1 and 3) are shown in Figure 5.10

Figure 5.10 2-D Frame Structure

5.2 Input and Output of the Developed VIS_SA Package

Using the developed Visualized Structural Analysis (VIS_SA) computer program (under the FLASH computer environment [4, 5]), the Input/Output (I/O) data for the above 9 examples (see Section 5.1) are summarized in this section:

Example 1:I/O for Determinate Truss with Applied Nodal Loads

Figure 5.11 Deflected Shape for Example 1

Figure 5.12 Deflections and Reactions for Example 1

Figure 5.13 Member Forces for Example 1

Example 2: Determinate Truss with Support Settlements only

Figure 5.14 Deflected Shape for Example 2

Figure 5.15 Deflections and Reactions for Example 2

Figure 5.16 Member Forces for Example 2

Example 3: Indeterminate (extra supports) Truss with Joint Loads.

Figure 5.17 Deflected Shape for Example 3

Figure 5.18 Deflections and Reactions for Example 3

Figure 5.19 Member Forces for Example 3

Example 4: Indeterminate (extra members) Truss with Joint Loads

Figure 5.20 Deflected Shape for Example 4

Figure 5.21 Deflections and Reactions for Example 4

Figure 5.22 Member Forces for Example 4

Example 5: Indeterminate Truss with Applied Temperatures

Figure 5.23 Deflected Shape for Example 5

Figure 5.24 Deflections and Reactions for Example 5

Figure 5.25 Member Forces for Example 5

Example 6: Indeterminate Truss with Pre-strain Effects

Case (a) without the applied force 30kN

Figure 5.26 Deflected Shape Example 6(a)

Figure 5.27 Deflections and Reactions for Example 6(a)

Figure 5.28 Member Forces for Example 6(a)

Case (b) with the applied force 30kN

Figure 5.29 Deflected Shape for Example 6(b)

Figure 5.30 Deflections and Reactions for Example 6(b)

Figure 5.31 Deflections and Reactions for Example 6(b)

Example 7: Indeterminate Truss with Support Settlements

Figure 5.32 Deflected Shape for Example 7

Figure 5.33 Deflections and Reactions for Example 7

Figure 5.34 Member Forces for Example 7

Example 8: Gere’s and Weaver’s Truss Structure with General Loading[1,pages 244-249]

Figure 5.35 Deflected Shape for Example 8

Figure 5.36 Deflections and Reactions for Example 8

Figure 5.37 Member Forces for Example 8

Example 9: Indeterminate Truss with “All” Combined Effects

Figure 5.38 Deflected Shape for Example 9

Figure 5.39 Deflections and Reactions for Example 9

Figure 5.40 Member Forces for Example 9

Example 10: 2-D beam structures with applied nodal load

Figure 5.41 Deflected Shape for Example 10

Figure 5.42 Deflections and Reactions for Example10

Figure 5.43 Member Forces for Example 10

Example 11: 2-D frame structures with applied uniformly distributed load

Figure 5.44 Deflected Shape for Example 11

Figure 5.45 Deflections and Reactions for Example 11

Figure 5.46 Member Forces for Example 10

CHAPTER 6

EDUCATIONAL AND RESEARCH VALUES OF THE

DEVELOPED SOFTWARE PACKAGE

The developed user-friendly, interactive, visualized software package “Visual Structural Analysis (VIS_SA)” had been based upon the Stiffness Matrix Method (SMM) and takes full advantage of the highly visualized and menu-driven capabilities of the Macromedia FLASH computer environment [4, 4]. Both the educational and research values offered from this work are summarized in the following paragraphs.

6.1 Educational Values of the Developed VIS_SA Package

(a) Special efforts have been made to present and explain the theoretical formulas (and all equations derived) in such a simple fashion so that students can read and understand the materials with little or no help from the instructor.

(b) By exploiting the graphical and menu-driven capabilities provided by the FLASH environment, students can easily learn how to use the powerful VIS_SA software in just a few minutes.

(c) For the specific SMM topic, not only do students get the final results( such as nodal displacement, support reactions, member-end-actions) to compare with his/her own (hand-calculated) results, but he/she can also compare the intermediate results in order to understand where he/she has made errors. For other topics in the Structural Analysis 1 course ( such as Virtual Work, Super-position methods for indeterminate structures, Slope Deflection, Moment Distribution methods), the VIS_SA code ( for SMM topics) is still useful for verifying students final solutions( with VIS_SA’s solutions)

(d) Students can quickly create the “extra” homework problems with known solutions (through VIS_SA) to further enhance their understanding of SMM (and other) topics. Different “what-if” scenarios (for analysis and optimal design) can be easily conducted. The students’ learning can be made more fun through extensive use of colorful, graphical and menu-driven capabilities provided by VIS_SA.

(e) The instructor for the Structural Analysis 1 course is equipped with the user-friendly, highly visualized, interactive software VIS_SA.Thus, creating new homework assignments, regular tests and final examination can be done in just a few minutes. Hence, the problems of giving the same homework/tests every year (to save the instructor’s preparation time) and students’ passing “old” homework/tests to other students in subsequent years can be eliminated.

(f) Through this work, both the instructor and students are also given a clear set of lecture notes that are presented in an attractive PowerPoint presentation and also made available online. Thus, students can freely download these instructional materials from the internet and study these topics at anytime and from anywhere.

6.2 Research Values of the Developed VIS_SA Package

The VIS_SA software package, is not only user-friendly, interactive and highly visualized, it also has a lot of powerful capabilities. Example #9 in Chapter 5 has clearly demonstrated that it can handle quite general and complicated truss structures. More advanced graduate courses (such as Finite Element Analysis and Sparse High Performance Computing), and newly created research algorithms can be quickly validated by the VIS_SA software on small to medium-scale tested examples before conducting more testing on larger-scale problems.

6.3 Existing Structural Analysis Software

2D Frame Analysis Dynamic Edition 1 [12]:

Frame Analysis Dynamic Edition is a civil engineering application that performs static and dynamic analysis of frames and trusses using the finite element method. Even though this software package provides immediate drawing of deformed shape, axial-shear forces, and bending moments diagrams, it has a have 5-node limitation and if the user wants the unlimited version then he/she has to pay $86.00 for the full package. This amount is quite a sum for an undergraduate student for only a few time uses and he/she might lose the software from the system due to a system crash.Futhermore, he/she will not have the flexibility to use the software anywhere, on any system since this software first needs to be downloaded and has a long, routine setup and registration process. The VIS_SA Package developed in a FLASH environment on the other hand, is just like a web page and anyone can use it from anywhere with any operating system without downloading the whole package (thus saving time and system memory) rather than downloading huge files for running only one analysis.

Dr. Frame2D 3.0 [13]:

This is also a good frame analysis software, but again, it has the same constraint of paying $69 to purchase an Educational License (2 year term) or $399 for a Professional License. It do not have the flexibility of analyzing beams, but it does provide another package (Dr. Beam) for that purpose.VIS_SA is considered to be 3 in 1 with the flexibility to analyze planar frames, trusses, and multi span beams in a single package, thus saving the user’s time , money and effort.

Static and Dynamic Structural Analysis of 2D and 3D Frames package [14]

This software compiles with the gcc compiler, the DJGPP gcc compiler, and the compiler. The biggest constraint is that the user needs to first develop an input file in Notepad with element connectivity, element length and element properties. The user then loads this file which is a time consuming job.VIS_SA does not require a complier to run and by offering simple mouse clicks (user-friendly environment), it reduces the chances for making errors during the input phase.

Frame Solver 2D [15]:

2D Frame Solver is used to analyze planar frames, trusses, and multi span beams. This software is built to be used on a 32-bit Windows platform and sells for $45.The demo version is limited to 4 elements. Additionally, member properties are not editable.The file download size of 5000 kb requires a good amount of disk space and time for installation. In the case of VIS_SA all the above constraints are taken care of from operating system compatibility to unlimited use of software for no cost.

6.4 Automated Self-Assessment Tests

6.4.1 “Legal” User Listing for Self-Assessment Test

Step1: The instructor manually types in the list of legal user IDs (see Figure 6.1) to the Self-Assessment test database file with a .FLA extension (Flash file with source code).

Step2: The saved .FLA file with new IDs is compiled (published in Flash) to extract the executable file and HTML page with .swf and .html extensions, respectively.

Step3: The SWF extension file and HTML page is uploaded to the ODU Lions directory.

Figure 6.1 Outline of Self-Assessment Test

6.4.2 Self-Assessment Test Layout

Step1: The website address http://www.lions.odu.edu/~amoha006/selfassessment.html is provided to the students by the instructor for an assessment test. Once the student goes to the above website, a login page is displayed on screen as shown in Figure 6.2.

Figure 6.2 Student’s Information and Selected Problem Type

Step2: The student provides the information requested and selects the test type to be taken (shown in Figure 6.2).Once the student clicks the “Enter” ICON at the bottom, the UIN is checked against the list of IDs provided earlier by the instructor. The student will gain access only if his/her UIN matches with one of the IDs from the instructor’s list.

Figure 6.3 Computer Randomly Generated Questions and Student Answers

Step3: After being given a randomly generated problem on the screen, as shown in Figure 6.3, the student’s hand calculated answers are typed into the input text boxes of the respective questions. Then, the student needs to click the submit ICON so that his/her answers will be evaluated. After evaluation, the email.php file automatically sends the score and correct answers to both the student and the instructor (as indicated in Table 6.1).

Table 6.1 Evaluated Students Answers

Student Answers (1):0 (2):833.33 (3):833.33 (4):0 (5):833.33 (6):1 (7):120 (8):0 (9):0 (10):0

VIS_SA’s correct Answers (1):320 (2):-180 (3):945 (4):0 (5):1 (6):0 (7):120 (8):0 (9):0 (10):0

Student’s Score** (1):35 (2):35 (3):35 (4):100 (5):35 (6):35 (7):100 (8):100 (9): 100 (10): 100

** (a) For automated grading policies, if the student’s answer is within 3% of the

VISA_SA software’s generated answer, then a score of 100 is given. Otherwise,

a score of 35 is given.

(b) The numbers given inside the parentheses, shown in Table 6.1, represent the

randomly generated question numbers.

CHAPTER 7

CONCLUSIONS AND FUTURE WORK

7.1 Conclusions

A general unified framework for developing simple, user-friendly, fun, interactive and highly visualized VIS_SA software for enhancing students’ learning capabilities for the SMM module is presented in this report. The developed software has taken advantage of menu-driven and graphical capabilities offered by Macromedia FLASH computer environment to make the learning process more exciting.

Numerous examples have been used to test different capabilities of VIS_SA. It is strongly believed that VIS_SA can offer significant educational and research benefits to both students and their instructors. An instructor can save a lot of time framing a question for a class test or assignment and can also avoid repetition of the same problems from previous years by coming up with a new problem for every semester for the test (by using the developed VIS_SA software). Finally, intelligent questions and automatically graded student answers for “self-assessment” purposes help the instructor to get feedback about students’ level of understanding.

7.2 Future Work

1. The 2-D element libraries (such as 2-D Truss, Beam and Frame) can be expanded into the corresponding 3-D elements for 3-D applications.

2. The possibility of developing a Structural Analysis software package in other languages (FORTRAN, C++) in conjunction with the FLASH web environment can be further explored.

3. Additional modules for other topics in the Structural Analysis I course (CEE-310) or other courses can be developed in the user-friendly Flash environment.

References

1. W.Weaver, Jr. and J.M Gere, Matrix Analysis of Framed Structures, Van Nostrand Reinhold Company Inc (1965)
2. R.C. Hibbeler, Structural Analysis, 6^th Edition , Prentice-Hall, 2006
3. K.M.Leet and C.M Wang, Fundamentals of Structural Analysis, 2^nd Edition McGraw-Hill, 2005
4. Macromedia FLASH-MX, 2004 ; Macromedia ,Inc., 600 Townsend St., San Francisco ,CA 94103
5. http://www.macromedia.com/support/flash/action_scripts/actionscript_dictionary/
6. D.T. Nguyen , Finite Element Methods: Parallel-Statics and Eigen-Solution, Springer Publisher (April 2006)
7. D.T. Nguyen, Parallel-Vector Equation Solvers for Finite Element Engineering Applications, Kluwer Academic/Plenum Publishers(2002)
8. O.O.Storaasli,J.M,Housner and D.T Nguyen, Parallel Computational Methods for Large-Scale Structural Analysis and Design, Guest Editors, Computing Systems in Engineering, an Intern. Journal, Vol. 4, Nos 4-6 (August/October/December, 1993), Pergamon
9. S.D.Rajan, Introduction to Structural Analysis and Design, John Wiley & Sons, Inc. (2001)
10. A.D.Belegundu and T.R.Chandrupatla,Optimization Concepts and Applications in Engineering, Prentice-Hall(1999)
11. C.H.Norris, J.B.Wilbur and Utku, Elementary Structural Analysis, 4^th Edition, McGraw-Hill,Inc.(1991)
12. Robert E.Sennett, Matrix Analysis of Structures, Waveland Press, Inc.(2000).
13. http://www.engissol.com/Products.htm
14. http://www.drsoftware-home.com/
15. http://www.duke.edu/~hpgavin/frame/
16. http://www.esads.com/

VITA

For

AHMED ALI MOHAMMED

AHMED was born in Hyderabad, India on April 13, 1984.He graduated from Osmania University in May 2005 with a Bachelor of Science degree in Civil Engineering. He began to pursue his Master of Science in Civil Engineering at Old Dominion University in the spring of 2006.He completed his Master of Science in August 2007