Frame Structural Analysis using the Stiffness Matrix Method in Microsoft Excel and ETABS

CE 155 Matrix Structural Analysis Project2nd Semester AY 2014-2015

Submitted by:Carl Chester Ragudo

Submitted to:Prof. Oscar Victor Antonio

I. INTRODUCTIONA planar structural element such as a beam or column element has six unknowns: two forces and a moment on each side. The total number of unknowns of a planar structure would then be equal to thrice the number of nodes in the structure. When 3-dimensional frames are considered, each node would present a total of 6 unknowns: forces and moments on each of the 3 axes. As the complexity of the structure increases, the computational effort to determine all the unknowns increases exponentially. Matrix structural analysis is now introduced to address this concern. Below is the given structure for this analysis project.Figure 1. Free body diagram of the structureThe structure consists of 20 nodes labeled A to Q with 31 members using the nodes as joints as shown in Figure 1. With 6 degrees of freedom for each node, a total of 120 degrees of freedom must be analyzed. Nodes A, D, H, K, N, and R are defined as fixed supports. These restraints make the displacements of the nodes equal to zero. The analysis is now left with 84 displacements and 36 support reactions to solve for. A manual computation was done to solve for these unknowns using the stiffness matrix method using Microsoft Excel. Another analysis was made using ETABS to compare the results from the manual computation with.II. METHODOLOGYa. Stiffness Matrix MethodThe goal of this method is to generate the global stiffness matrix which relates the displacements of each node to the forces applied on each node. The global stiffness matrix is assembled from the element stiffness matrix generated from each of the members using the matrix below. The matrix is dependent on the following parameters: length of the member (L), cross-sectional area (A), modulus of elasticity (E), modulus of rigidity (G), Poissons Ratio (), polar moment of inertia (J), and moments of inertia about the z and y axes (Iz and Iy). Information about the section properties needed were obtained from an online source.

The element matrices were then generated using the material and section properties for each member. The generated matrices now relate the displacements on their end nodes to the forces applied on them on its local coordinates. To transform the element matrix to the same coordinate system as the global stiffness matrix, a Transformation Matrix was used. The transformation matrix relates the local forces to its corresponding equivalent in the global coordinate system. It is constructed using the relative angles of the local coordinate system of the member with the global coordinate system. The definitions of the terms in the matrix are summarized in the table that follows.K =

Table 1. Relative angles of coordinate axesAngledefinition

xGlobal x to local x

yGlobal x to local y

zGlobal x to local z

xGlobal y to local x

yGlobal y to local y

zGlobal y to local z

xGlobal z to local x

yGlobal z to local y

zGlobal z to local z

The transformation matrix also applies to displacements thus {} = [] {} and {F} = [K] {}. Substituting {} will yield as proven in class. [K] now corresponds to all the factors multiplied to.

To further simply the computations, the terms attributed to the free nodes and support nodes could be segregated and the global stiffness matrix could be rearranged. To skip this step, the members were arranged starting from the members connected at the free nodes so that the nodes at the support will already be separated. The labeling started with member QG as member 1 and member GF as member 2 and so on. When the global stiffness matrix was assembled, the sections for the supports are already segregated. The global stiffness matrix is now arranged as follows:

Further expansion of the equations show that the applied loads could be represented as fixed-end forces and the previous equation becomes:

Where the f subscript correspond to the free nodes and s for support. Rearranging the equation, the displacements of the free nodes can be computed using

And support reactions from

Since there are no forces applied on the joints, can already be taken as zero. Likewise, there are no applied forces on the support nodes as well thus is also equal to zero.

b. ETABSAs a control, the same structure was also analyzed using a commercially available software. Although it is a paid software, its trial version contains enough features to analyze the structure as required. From the start up interface of the program, a grid system was defined on the x-y plane as dictated by the program. The project was defined to have the ground level of the structure on the x-z plane, however, the environment of the program constructs the structure on the x-y plane which it uses as well for its plan view. Adjustments for this matter were made later for the comparison of the results of the two methods. The grid was set-up with 3 nodes along the x-axis at 3745.45 mm and 2 nodes along the y-axis at 2746.66 mm. Three stories were created with ceiling elevations of 3500 mm, 7500 mm, and 11000 mm. For consistency, kN and kN/mm were used as units for force and moments as mm were used for lengths since displacements are generally on that scale. Three section properties were used for the structure: W14x43, W16x26, and W18x35. The easy interface of ETABS allows drawing of the members while picking its appropriate section properties. All members were set use A36 steel. All material and section properties were already pre-loaded on the program. The support nodes were set as fixed supports and the loading cases were entered using its guided dialogue box.When all of the necessary information had been loaded on the program, the analysis of the structure was run. ETABS displays it results as nodal displacements and support reactions drawn on the model with an option to display the deformed shape of the structure.

III. RESULTS AND DISCUSSIONCalculations made for the direct stiffness method were consolidated in an Excel file. Each element stiffness matrix was generated for the 31 members and assembled to form the global stiffness matrix. The distributed loads were translated to fixed end forces with a separate sheet on the file for their computations. The nodal forces and moments used for the analysis are listed below in Table 2.Table 2. Fixed-end forcesFxQ0.00FxF0.00FxP0.00FxE0.00FxO0.00

FyQ23.73FyF2.04FyP2.04FyE0.00FyO0.00

FzQ0.00FzF0.00FzP0.00FzE0.00FzO0.00

MxQ-3789.75MxF394.87MxP-394.87MxE0.00MxO0.00

MyQ0.00MyF0.00MyP0.00MyE0.00MyO0.00

MzQ-9645.03MzF734.27MzP734.27MzE0.00MzO0.00

FxG0.00FxJ0.00FxT0.00FxI0.00FxS0.00

FyG23.73FyJ2.04FyT2.04FyI0.00FyS0.00

FzG0.00FzJ0.00FzT0.00FzI0.00FzS0.00

MxG3789.75MxJ394.87MxT-394.87MxI0.00MxS0.00

MyG0.00MyJ0.00MyT0.00MyI0.00MyS0.00

MzG-9645.03MzJ-734.27MzT-734.27MzI0.00MzS0.00

FxC-19.07FxM0.00FxB-41.21FxL0.00

FyC23.73FyM23.73FyB0.00FyL0.00

FzC0.00FzM0.00FzB0.00FzL0.00

MxC3789.75MxM-3789.75MxB0.00MxL0.00

MyC0.00MyM0.00MyB0.00MyL0.00

MzC-11547.51MzM9645.03MzB35320.90MzL0.00

The forces in Table 2 together with the [Kff] portion of the global stiffness matrix were used to compute for the displacements of the free nodes. This is possible because the displacements at the supports are equal to zero. Table 3 lists the displacements from this operation.Table 3. Displacements of free nodes from Direct Stiffness MethoduQ0.84624uF5.47708uP0.82420uE3.03364uO0.31137

vQ-0.15906vF-0.07885vP-0.11172vE-0.03531vO-0.05042

wQ-0.29747wF-0.00646wP-0.00689wE0.05295wO0.05294

xQ0.00009xF-0.00002xP-0.00001xE0.00000xO0.00000

yQ0.00154yF0.00103yP0.00102yE0.00057yO0.00057

zQ0.00020zF-0.00018zP-0.00004zE-0.00036zO-0.00006

uG5.54723uJ5.45873uT0.82141uI2.99335uS0.31360

vG-0.12463vJ-0.07108vT-0.03394vI-0.05059vS-0.02158

wG-0.29792wJ-1.98338wT-1.98335wI-1.01677wS-1.01681

xG-0.00012xJ-0.00001xT0.00001xI-0.00002xS-0.00002

yG0.00148yJ0.00122yT0.00123yI0.00070yS0.00070

zG0.00020zJ-0.00021zT-0.00003zI-0.00060zS-0.00008

uC5.53739uM0.81990uB3.14721uL0.30522

vC-0.08734vM-0.09255vB-0.02484vL-0.03915

wC1.98727wM1.98755wB0.96392wL0.96360

xC-0.00009xM0.00013xB0.00003xL0.00002

yC0.00122yM0.00126yB0.00081yL0.00081

zC0.00000zM-0.00021zB-0.00114zL-0.00008

Displacements are represented in millimeters and rotations are in radians. The obtained displacements was then used to compute for the support reactions using the [Ksf] portion of the global stiffness matrix. Table 4 lists the obtained support reactions of the structure.Table 4. Support Reactions from Direct Stiffness MethodFxA-11.4962FxH-19.3145FxN-2.13565

FyA11.53697FyH23.50004FyN23.42166

FzA-0.95591FzH1.030621FzN-0.0646

MxA-1713.08MxH1823.759MxN-108.214

MyA-8.02922MyH-6.9618MyN-5.60537

MzA31719.83MzH39945.92MzN4302.834

FxD-23.9292FxK-1.7115FxR-1.69402

FyD16.40281FyK18.18656FyR10.0248

FzD-0.06381FzK-0.9749FzR1.028606

MxD-108.863MxK-1729.77MxR1824.195

MyD-5.62397MyK-8.02913MyR-6.96248

MzD45564.17MzK3772.362MzR3800.436

The support reactions are in terms of kN and kN/mm for forces and moments respectively. The maximum displacements are recorded at 5.5 mm and as low as 0.006 mm. rotations are fairly unnoticeable ranging around 0.0008 rad. As for support reactions, majority of the forces are on the positive y-axis resisting the download distributed load while the forces on the x-axis resist the horizontal point load and horizontal component of the distributed load on the inclined member.Since ETABS is an integrated program, results of the analysis was spot on. Table 5 and 6 lists the results of the analysis from ETABS. Table 5. Displacements from ETABSuQ6.2uF6.1uP1uE3.4uO0.4

vQ-0.2vF-0.1vP-0.2vE-0.1vO-0.1

wQ-0.3wF0.02211wP0.02168wE0.1wO0.1

xQ-0.00013xF-2.8E-05xP-5E-06xE-1.1E-05xO0.000004

yQ0.001681yF0.001137yP0.001131yE0.000633yO0.000633

zQ0.000217zF-0.0002zP-0.00005zE-0.00041zO-0.00067

uG1uJ6.1uT1uI3.4uS0.4

vG-0.2vJ-0.1vT-0.1vI-0.1vS-0.03659

wG-0.3wJ-2.2wT-2.2wI-1.1wS-1.1

xG0.000104xJ-2.7E-05xT0.000013xI-4.2E-05xS-2.8E-05

yG0.00176yJ0.001365yT0.001366yI0.000788yS0.000788

zG0.000218zJ-0.00024zT-2.6E-05zI-0.00065zS-0.00009

uC6.2uM1uB3.5uL0.4

vC-0.1vM-0.1vB-0.04232vL-0.1

wC2.2wM2.2wB1.1wL1.1

xC-9.4E-05xM0.000141xB0.000036xL0.000041

yC0.001386yM0.001433yB0.0009yL0.0009

zC-3.1E-05zM-0.00025zB-0.00122zL-9.6E-05

Table 6. Support Reactions from ETABSFxA-11.9453FxH-19.2226FxN-2.2504

FyA20.7498FyH30.3184FyN35.6122

FzA-1.0511FzH1.1101FzN-0.0631

MxA-1878.13MxH1988.23MxN-115.08

MyA-8.64MyH-7.75MyN-6.07

MzA33342.5MzH40242.73MzN4620.15

FxD-23.1801FxK-1.7622FxR-1.9204

FyD29.3123FyK26.6795FyR18.0888

FzD-0.091FzK-1.0413FzR1.1365

MxD-147.41MxK-1866.53MxR2018.93

MyD-6.08MyK-8.64MyR-7.57

MzD44702.3MzK4060.75MzR4281.28

To compare the results of the two methods, the relative deviation was computed. In general, there is no conclusive relationship between the results of the two methods. Although there are results that with deviation almost equal to zero, there are those that have more than 50% deviation. Interestingly enough, although the deviation of the displacements have a wide range, the deviation of the results of the manual computation for the support reactions have relatively closer results with deviations of about 3-5%.ETABS has a feature to display the free body diagram of the structure together with the computed support reactions. As shown in Figure 2 and 3, the support reactions for each of the support joints are presented as directed arrows in the joints. Note that the drawn structure on the program had a different coordinate orientation though it is still consistent with the given structure. To clearly visualize the results, the deformed shape of the structure can also be generated. Since the displacements of the joints are insignificant with respect to the actual lengths of the members, the deformed shape is exaggerated to visualize the relative magnitudes of the displacements with respect to each other. Figure 4 shows the deformed shape of the structure.Figure 2. Support Reactions (Forces)Figure 3. Support Reactions (Moments)

Figure 4. Deformed shape of the structureSeveral factors may contribute to the difference in the answers obtained from the two methods. The most probable error can be attributed to clerical error. Because of the scale of the amount of information to be processed, there might have been a tiny error in the assembly of the global stiffness matrix. Although it can be assured that the procedure has been thoroughly checked, some errors that can be attributed to data handling, such as rounding off and values of Excel can produce small errors that can cause the results to greatly deviate.

IV. CONCLUSIONThis project aimed to demonstrate the process of how matrix structural analysis is used to compute for the large amount of unknowns normally encountered in more complex structures. Although the basic concepts of analysis can be done easily with pen and paper, its real life application would require more computing power as the difficulty of the analysis grows exponentially with increasing number of elements in the structure. Using Microsoft Excel, it has been shown that manual computations for matrix analysis using the direct stiffness method can be done. Excel can easily reproduce repetitive calculations such as the computation of the element stiffness matrices that must be done for all members of the structure. Although not entirely automated, this project has shown the great flexibility of Excel in processing vast amounts of data all the while motivating the user to come up with ways to optimize how the computations can be handled thus little by little automatizing the procedure to just entering the bare inputs. But because of the scale of the computations, it should be noted that analysis requires great care as small errors in the procedure can produce big deviation from desired quantities. This also has been a great opportunity to be familiar with structural software such as ETABS. The integration of the analysis and the design components of the program can be useful. The analysis it can make can be made as close as possible to reality because of the versatility it offers with the attributes and properties of materials already encoded in the program.

ReferencesMcGuire, W., Gallagher, R. H., & Zeimiam, R. D. (n.d.). Matrix Structural Analysis.