FEA_Final_Paper-2.5.51

MENG 5136FEA Spring 2014

May 6, 2014, Statesboro, GA

MENG 5136 - AB

NUMERICAL ANALYSIS OF DEFORMATION ALONG LOADED CANTILEVERED BEAMS

Thomas Elliott FranklinUndergraduate Student

Mechanical Engineering DepartmentGeorgia Southern University

Statesboro, Georgia 30460 – [email protected]

Vishal LachhawaniUndergraduate Student



Charlie DrakeUndergraduate Student



Ibrahim AhmedUndergraduate Student



ABSTRACTStructural analysis for cantilevered beams is necessary

for many real world engineering applications. These projects range from residential scale items like overhanging balconies to industrial applications as huge cantilevered bridges. In these applications the structural material of the supports can be subject to variable loads depending on the purpose. Though not addressed in this project, temperature and chemical composition of the loaded beams also contribute to the structural integrity of the object, and it is necessary to keep these variables in mind when assessing lifetime and maintenance of the project.

This project details the structural support of an overhanging balcony as an addition to a pre-existing structure.Household additions are great to add space for relaxation or hobbies, manufactured in all shapes and sizes to serve a particular purpose for the client. The balcony project that will be examined in this project is located as a third story of a coastal home to provide a view of the ocean.

With any addition to property, there will be different structural conditions. Finite element analysis (FEA) provides a cost effective method of numerical computations that simulate real world mechanics. Through this process, resources are conserved until a final design has been tested, reducing overhead and increasing the structural integrity of any household addition.

INTRODUCTION

The idea that was chosen for the finite element analysis final project was running analysis on the metal beams of a third story balcony. The balcony was built onto a house in Tybee Island, GA by Alan Drake, Charlie Drake’s father. The original design was applied with four inch channel iron, seventeen years ago, and through the years the power coating on the metal was worn off by the salty air. A redesign was necessary since the metal used had a great amount of deflection in it when it was originally placed. A civil engineering company named William Hunter Saussy III PC was contacted and given the dimensions, in which Mr. Hunter Saussy ran the calculations and determined the optimal metal shape and size to use would be 5”x5” square tubing with a wall thickness of 0.3125”. The data analysis Mr. Saussy did was not shared with us.

It was decided that our group would build the design, and test that metal shape and size and also test the original metal shape and size, which Elliott did testing on. Ibrahim analyzed the 5”x5”x.3125” square tubing which is what was actually applied to the balcony substructure. We also found two other metal shapes and sizes that were chosen to be used, which were a 4”x4” I beam and Charlie did the analysis on this beam, and a 6”x4”x.25” rectangular tube that Vishal did analysis on. Finite

element analysis was run on all four metal shapes and sizes to determine if there may have been a better one to use, or if the advice that was given was the best route to take in this application.

A limitation we must consider is that the balcony cannot be mounted to structural code. When building a cantilevered balcony, it is recommended to have twice the overhang length inside the house and mounted in the subfloor of the home. In this instance, due to the pitch of the roof on the second story going above the floor line on the third story, a riser floor had to be built to hide the supports. With 8’4” being the longest point on the 9’4” wide balcony with an arched face, the length mounted in the subfloor of the home should have been 16’8”. That mounting length was impossible to achieve due to the previous limitations described, so to compensate for being able to only mount 7’8” inside the house we had to use a very strong material shape and size.

Also, we could not install trussed beams under the cantilevered balcony due to the roofline valley runs up to about two feet below the bottom of the balcony. If it was possible to add trussed sections under the balcony, a smaller sized beam could have been used and given us the small amount of deformation that needed to be achieved.

A trussed beam that Elliott designed was added and it is mounted to the wall above the balcony, and runs down to the end of each side of the balcony. This was not added in the actual application, it was just for testing purposes in ANSYS. The idea behind adding the truss was to minimize overall deflection of the balcony. In theory, adding truss beams to each side will minimize deflection greatly since the tensile strength of steel is greater than the directional deformation of steel.

Safety factor is a part of ANSYS that takes into account the material strength and divides that by the load applied to the cantilevered part of the balcony. We look for a safety factor greater than one in most cases, so it will be considered that any design with a safety factor less than one would not be adequate to install on the house.

NOMENCLATURE

A Area, inches squared (in2)F Force, (lb/in2)σe equivalent stress, (psi)p local pressure, (psi)Fs Safety FactorMs Margin of SafetyYs Yield Strength E Modulus of Elasticity, (psi)I Moment of Inertia (in4)y Deflection (inches)x distance, (inches)l length, (inches)ε Strain

NUMERICAL MODELGoverning Equations

The beam governing equations assisting in the verification of this study are shown below:

Load Intensity:

qEI

=d4 ydx 4 (1)

Shear Force:

VEI

=(∂3 y∂ x3 )

(2)Moment:

MEI

=(∂2 y∂ x2 )

(3)Slope:

θ=( ∂ y∂ x

) (4)

Deflection:y=f (x ) (5)

Given the load intensity, we can solve for the deflection of the beam. Using method of sections, boundary conditions are applied to a beam with a constant cross section and a general equation for deflection is achieved.

d2 yd x2 =

MEI

=−FxEI

Integrating 2x:

y=−F x3

6 EI+C1 x+C2

Boundary Conditions:y=0@ x=Ldydx

=0@x=L

Substitution yields a general deflection equation:

y= F6 EI

(−x3+3 L2 x−2L3)

It is convenient to define a von Mises stress, σe, which is used to predict yielding of materials under multiaxial loading conditions.

σ ε=√(σ1−σ 2)2+(σ 2−σ3 )2+(σ1−σ2)

2

2

Strain can be defined as deformation of a solid due to stress, and is given by:

ε= σE

Methodology

2 Copyright © 2013 by ASME

ANSYS Workbench was used to solve this real world application of a statically overloaded cantilever. The cantilever balcony was modeled in Solidworks as individual parts and fused together in an assembly. This assembly was then imported into ANSYS using the appropriate file extension (.STEP). The balcony structure was then subjected to predetermined constraints and loadings given initial construction values.

Structural Steel properties:

IPE Wood properties:

Mesh Refinement:

SOLID187 element is a higher order 3-D, 10-node element. SOLID187 has a quadratic displacement behavior and is well suited to modeling irregular meshes (such as those produced from various CAD/CAM systems). The element is defined by 10 nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. The element has plasticity, hyper elasticity, creep, stress stiffening, large deflection, and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyper elastic materials.

CONTA174 is used to represent contact and sliding between 3-D "target" surfaces (TARGE170) and a deformable surface, defined by this element. This element has three degrees of freedom at each node: translations in the nodal x, y, and z directions. This element is located on the surfaces of 3-D solid or shell elements with midside nodes (SOLID92, SOLID95, HYPER158, VISCO89, SHELL91, SHELL93, SHELL99 and MATRIX50). It has the same geometric characteristics as the solid or shell element face with which it is connected (see Figure 4.174-1). Contact occurs when the element surface penetrates one of the target segment elements (TARGE170) on a specified target surface. Coulomb and shear stress friction is allowed.

TARGE170 is used to represent various 3-D "target" surfaces for the associated contact elements (CONTA173 and CONTA174; see Sections 4.173 and 4.174). The contact elements themselves overlay the solid elements describing the boundary of a deformable body and are potentially in contact with the target surface, defined by TARGE170. This target surface is discretized by a set of target segment elements (TARGE170) and is paired with its associated contact surface via a shared real constant set. You can impose any translational or rotational displacement on the target segment element. You can also impose forces and moments on target elements. For rigid target surfaces, these elements can easily model complex target shapes. For flexible targets, these elements will overlay the solid elements describing the boundary of the deformable target body.


SURF154 may be used for various load and surface effect ap-plications. It may be overlaid onto an area face of any 3-D ele-ment. The element is applicable to three-dimensional structural analyses. Various loads and surface effects may exist simulta-neously. The geometry, node locations, and the coordinate sys-tem for this element are shown in Figure 4.154-1. The element is defined by four to eight nodes and the material properties.

RESULTS AND DISCUSSION

Deformation of 5”x5” cantilever beams with a .3125” wall thickness without a truss beam.

The total deformation for the cantilever beams are shown in the following figures. The coloration corresponds to the uniform pressure applied across the platform of the balcony. A distributive pressure of 2.82 psi units was applied.

Fig. 1: Pressure of 2.82 psi was applied at the center

Fig. 2: A Fixed support was applied to the bottom 3 surfaces

Table 1: Fine Mesh element selection for balcony

without a truss beam

Fig. 3: Fine Mesh selection of balcony without a truss beam


Fig. 4: Total Deformation of Fine Mesh selection of balcony without a truss beam

Table 2: Medium Mesh element selection for balcony

without a truss beam

Fig. 5: Medium Mesh selection of balcony without a truss beam

Fig. 6: Total Deformation of Medium Mesh selection of balcony without a truss beam

Table 3: Coarse Mesh element selection for balcony without a truss beam

Fig. 7: Coarse Mesh selection of balcony without a truss beam


Fig. 8: Total Deformation of Coarse Mesh selection of balcony without truss beam

Deformation of 5”x5” cantilever beams with a .3125” wall thickness with a truss beam.

Table 4: Fine Mesh element selection for balcony without a truss beam

Fig. 9: Fine Mesh selection of balcony with a truss beam

Fig. 10: Total Deformation of Fine Mesh selection of balcony with a truss beam

Table 5: Medium Mesh element selection for balcony with a truss beam

Fig. 11: Medium Mesh selection of balcony with a truss beam


Fig. 12: Total Deformation of Medium Mesh selection of balcony with a truss beam

Table 6: Coarse Mesh element selection for balcony with a truss beam

Fig. 13: Coarse Mesh selection of balcony with a truss beam

Fig. 14: Total Deformation of Coarse Mesh selection of balcony with a truss beam

Table 7: Total Deformation according to Mesh selection along with number of nodes and elements of balcony without a truss

beam

Table 8: Total Deformation according to Mesh selection along with number of nodes and elements of balcony with a truss

beam

Deformation and Safety Factor of a 4”x6” Rectangular beam with a 0.25” wall thickness without a truss beam.


1 7484 SOLID187 2 35398 SOLID187 3 2318 CONTA174 4 2318 TARGE170 5 2464 SURF154





1 4088 SOLID187 2 15394 SOLID187

3 1268 CONTA174 4 1268 TARGE170

5 1320 SURF154 --- Number of total nodes = 37866 --- Number of contact elements = 3856 --- Number of solid elements = 19482 --- Number of total elements = 23338

Table 10: Medium Mesh element selection for balcony without a truss beam



1 1436 SOLID187 2 7746 SOLID187 3 592 CONTA174 4 592 TARGE170 5 466 SURF154 --- Number of total nodes = 17844

--- Number of contact elements = 1650 --- Number of solid elements = 9182 --- Number of total elements = 10832




Fig. 20: Total Deformation of Coarse Mesh selection of balcony without a truss beam

Deformation of 4”x6” cantilever beams with a 0.25” wall thickness with a truss beam.

1 2911 SOLID187 2 2911 SOLID187 3 6779 SOLID187 4 32041 SOLID187 5 38 CONTA174 6 38 TARGE170 7 56 CONTA174 8 56 TARGE170 9 35 CONTA174 10 35 TARGE170 11 57 CONTA174 12 57 TARGE170

13 1981 CONTA174 14 1981 TARGE170 15 2244 SURF154

--- Number of total nodes = 85974 --- Number of contact elements = 6578 --- Number of solid elements = 44642 --- Number of total elements = 51220


Fig. 21: Fine Mesh selection of balcony with a truss beam

Fig. 22: Total Deformation of Fine Mesh selection of balcony with a truss beam

1 2644 SOLID187 2 2644 SOLID187 3 3982 SOLID187 4 15119 SOLID187 5 35 CONTA174 6 35 TARGE170 7 40 CONTA174 8 40 TARGE170 9 34 CONTA174 10 34 TARGE170 11 40 CONTA174 12 40 TARGE170

13 1254 CONTA174 14 1254 TARGE170

15 1294 SURF154--- Number of total nodes = 47798

--- Number of contact elements = 4100 --- Number of solid elements = 24389 --- Number of total elements = 28489




Fig. 24: Total Deformation of Medium Mesh selection of balcony with a truss beam.

1 1398 SOLID187 2 1398 SOLID187 3 1346 SOLID187 4 7572 SOLID187 5 23 CONTA174 6 23 TARGE170 7 30 CONTA174 8 30 TARGE170 9 23 CONTA174 10 23 TARGE170 11 30 CONTA174 12 30 TARGE170 13 556 CONTA174 14 556 TARGE170

15 442 SURF154

--- Number of total nodes = 23078 --- Number of contact elements = 1766 --- Number of solid elements = 11714 --- Number of total elements = 13480



Fig. 26: Total Deformation of Coarse Mesh selection of balcony with a truss beam

Table 15: Total Deformation according to Mesh selection along with number of nodes and elements of balcony without a truss beam

Table 16: Total Deformation according to Mesh selection along with number of nodes and elements of balcony with a truss beam

Deformation of 4” “C” Channel cantilever beams with a variable wall thickness without a truss beam.





Table 18: Medium Mesh element selection for balcony without a truss beam






Fig. 32: Total Deformation of Coarse Mesh selection of balcony without a truss beam

Deformation of 4” “C” Channel cantilever beams with a variable wall thickness with a truss beam.

[MISSING TABLE]Table 20: Fine Mesh element selection for balcony with a truss

beam

[MISSING Figure]Fig. 33: Fine Mesh selection of balcony with a truss beam

[MISSING Figure]Fig. 34: Total Deformation of Fine Mesh selection of balcony

with a truss beam.



Fig. 36: Total Deformation of Medium Mesh selection of balcony with a truss beam.




Fig. 38: Total Deformation of Coarse Mesh selection of balcony with a truss beam.

Mesh Type Nodes Elements Total Deformation (in)Coarse 17071 9599 2.5552Medium 35595 20675 2.8064Fine 70099 41270 2.8807Table 23: Summary of mesh and deformation of balcony

Mesh Type Nodes Elements Total Deformation (in)

Coarse 22713 12578 0.99445

Medium 46220 26265 1.0914

Fine

Table 24: Summary of mesh and deformation of balcony with a truss beam.

Deformation of 4” cantilever I beams without truss supports


1 7737 SOLID187 2 40052 SOLID187

3 2351 CONTA174 4 2351 TARGE170 5 2546 SURF154

6 2180 SURF154Number of total nodes = 93494

Number of contact elements = 9428 Number of solid elements = 47789 Number of total elements = 57217

Table 25: Fine mesh element selection for balcony without truss support

Fig. 39: Fine mesh selection of balcony

Fig. 40: Total deformation of balcony

1 3976 SOLID187 2 22224 SOLID187 3 1304 CONTA174 4 1304 TARGE170 5 1298 SURF154 6 990 SURF154

Number of total nodes = 51754 Number of contact elements = 4896 Number of spring elements = 0 Number of solid elements = 26200 Number of total elements = 31096

Table 26: Medium mesh element selection for balcony without truss support


Fig. 41: Medium mesh selection of balcony

Fig. 42 Total deformation of balcony

1 1263 SOLID187 2 13141 SOLID187 3 653 CONTA174 4 653 TARGE170 5 408 SURF154 6 696 SURF154

Number of total nodes = 27863 Number of contact elements = 2410 Number of spring elements = 0 Number of solid elements = 14404 Number of total elements = 16814

Table 27: Coarse mesh element selection for balcony without truss support

Fig. 43: Coarse mesh selection of balcony

Fig. 44: Total deformation of balcony

Deformation of 4” cantilever I beams with truss supportsThe total deformation for the cantilever beams with truss

supports are shown in the following figures. The coloration corresponds to the uniform pressure applied across the platform of the balcony. A distributive pressure of 2.82 psi units was applied.

1 2919 SOLID187 2 2919 SOLID187 3 7238 SOLID187 4 37239 SOLID187 5 39 CONTA174 6 39 TARGE170 7 39 CONTA174 8 39 TARGE170 9 39 CONTA174 10 39 TARGE170 11 39 CONTA174 12 39 TARGE170 13 2086 CONTA174 14 2086 TARGE170 15 2358 SURF154 Number of total nodes = 98256 Number of contact elements = 6842 Number of spring elements = 0 Number of solid elements = 50315 Number of total elements = 57157

Table 28: Fine mesh element selection for balcony with truss supports

Fig. 45: Fine mesh selection of balcony with truss beams


Fig. 46: Total deformation of balcony with truss supports


Table 29: Medium mesh element selection for balcony with truss supports

Fig. 47: Medium mesh selection of balcony with truss beams

Fig. 48: Total deformation of balcony with truss beams


Table 30: Coarse mesh element selection for balcony with truss supports

Fig. 49: Coarse mesh selection of balcony with truss beams


Fig. 50: Total deformation of balcony with truss beams

Fig. 51: .74167 Safety factor of coarse mesh selection of 4” “I” beam balcony with truss beams

Fig. 52. 1.4049 Factor of safety for Coarse Mesh selection of square beam balcony with a truss beam.

Fig. 53: 2.3156 Factor of safety for Coarse Mesh selection of rectangular beam balcony with a truss beam.

Mesh Type # of Nodes # of Elements

Total Deformation

Fine 93494 57217 0.98536 inMedium 51754 31096 0.96038 inCoarse 27863 16814 0.83906 inTable 31: Summary of mesh and deformation of balcony

Mesh Type # of Nodes # of Elements

Total Deformation

Fine 98256 57157 0.72261 inMedium 61477 35053 0.70112 inCoarse 33459 19160 0.64401 in

Table 32: Summary of mesh and deformation of balcony with truss supports

Fine Medium Coarse5”x5”

Square Tube0.724 0.68752 0.61562

4”x6” Rectangular

Tube

0.66434 0.62773 0.58592

4” I beam 0.98536 0.96038 0.83906

4” C-Channel

2.8807 in 2.8064 in 2.5552 in

Table 33. Total deformation comparison between all beams without truss beams

Fine Medium Coarse5”x5”

Square Beam

0.53706 in 0.47481 in 0.41005 in


4”x6” Rectangular

Beam

0.30598 in 0.28062 in 0.2557 in

4” I beam 0.72261 in 0.70112 in 0.64401 in

4” C-Channel

MISSING in 1.0914 in 0.99445 in

Table 34. Total deformation comparison between all beams with truss beams

CONCLUSIONSThis study that was conducted shows that the best beam to

use would be a 4”x6”x0.25” rectangular beam since the maximum deflection under a pressure of 2.82 psi was 0.66434 inch. The 5”x5”x0.3125” square beam would be the second best to use since the maximum deflection was 0.724 inch. With a difference in deflection of almost six hundredths of an inch, it is almost negligible which beam should be used.

With the truss supports added, the deflection drops on all the beams. The 4”x6”x0.25” rectangular beam’s maximum deflection is merely 0.30598 inch under the same pressure, and the 5”x5”x0.3125” square beam’s maximum deflection is 0.53706 inch. With a difference of a little more than two tenths of an inch, the next thing that should be observed would be the factor of safety. In comparison, the rectangular beam has a safety factor of 2.3156 and the square beam has a safety factor of 1.4049. This data concludes the best beam to use would be the 6”x4”x0.25”, since the safety factor is the greatest and the maximum deflection is the lowest.

The C-Channel beam and the I-Beam both had a safety factor less than one, so neither would be an adequate support in this application.

REFERENCES [1] Bénichou, N. et al, "Thermal Properties of Wood, Gypsum

and Insulation at Elevated Temperatures," IRC Internal Report No. 710, National Research Council of Canada, Ottawa, Canada, 2001.

[2] Thermal and mechanical finite element modelling of wood-floor assemblies subjected to furnace exposure. (n.d.). .Retrieved May 6, 2014,from http://www.ul.com/global/documents/offerings/industries/buildingmaterials/fireservice/DHS_Report_Modeling_Floors_1208_final.pdf

[3] ASTM, “Standard Methods of Fire Test of Building Construction and

Materials”, Test Method E119-01, American Society for Testing and Materials,West Conshohocken, PA., 2007.

[4] UL 263, Fire Tests of Building Construction and Materials, Underwriters Laboratories, Northbrook, IL, 2003

[5] "Ipe." The Wood Database. N.p., n.d. Web. 06 May 2014.

[6] "Beam" def. 1. Whitney, William Dwight, and Benjamin E. Smith. The Century dictionary and cyclopedia. vol, 1. New York: Century Co., 1901. 487. Print.

[7] ”Ching,Frank”. A visual dictionary of architecture. New York: Van Nostrand Reinhold, 1995. 8-9. Print.

[8] "Mechanical Properties of wood." . N.p., n.d.Web.6thMay2014. <http://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr190/chapter_05.pdf>.

[9] American Society of Testing and Materials. ASTM standard terminology of structural sandwich constructions (C274-99). West Conshohocken, PA: ASTM International 1999.

[10] The Naked Scientists. (n.d.). Building Bridges. Retrieved May 6, 2014, from http://www.thenakedscientists.com/HTML/content/kitchenscience/exp/building-bridges-the-science-of-beams/

[11] Young, W.: Roark's Formulas for Stress and Strain, 6th edition. McGraw-Hill, 1989.

[12] Timoshenko, S: Strength of Materials, Volume 1. Krieger, 1958.

[13] Statics eBook:. (n.d.). Statics eBook:. Retrieved May 6, 2014, from https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=me&chap_sec=04.2&page=case_sol

[14] Renovation Coach: Deck & Balcony safety- Atlanta INtown Paper. (n.d.). Atlanta INtown Paper. Retrieved May 6, 2014, from http://www.atlantaintownpaper.com/2011/08/renovation-coach-deck-balcony-safety/

[15] Beer, F, Johnston, R, Dewolf, J, & Mazurek, D. (2009). Mechanics of materials. New York: McGraw-Hill companies.

[16] Ronald L. Huston and Harold Josephs (2009), "Practical Stress Analysis in Engineering Design". 3rd edition, CRC Press, 634 pages.

[17] Wai-Fah Chen and Da-Jian Han (2007), "Plasticity for Structural Engineers". J. Ross Publishing

[18] Fundamentals of Materials Science and Engineering, William D. Callister, John Wiley and Sons, 2nd International edition (September 3, 2004)

[19] Finnemore, John, E. and Joseph B. Franzini (2002). Fluid Mechanics: With Engineering Applications. New York: McGraw Hill, Inc.


http://books.google.com/books?id=E8jptvNgADYC&pg=PA46

http://books.google.com/books?id=E8jptvNgADYC&pg=PA46

[20] Rees, David (2006). Basic Engineering Plasticity - An Introduction with Engineering and Manufacturing Applications. Butterworth-Heinemann.


FEA_Final_Paper-2.5.51

Documents

Transcript of FEA_Final_Paper-2.5.51