GEEN1125: Finite Element Applications

Resit Coursework Questions

Start date: 11:59 pm on 6^th July 2021 End Date: 11:59 pm on 17^th July 2021

Course Tutor: Dr. Michael Okereke Email: m.i.okereke@gre.ac.uk

Phone: 01634 88 3580

Instructions

1. Answer ALL questions in this Coursework booklet

2. You have access to software at Nelson 128 laboratories where you can do this coursework. Alternatively, you can also remote log to virtual desktops which has all required software needed for this laboratory.
3. All submissions will be via Turnitin on the Course Moodle Page in the section of the portal where 2021 Coursework Resit are specified.
4. Do not email the course tutor your submissions as this will not be accepted.

5. All your solutions will be in a Microsoft Word document which you should convert to a PDF file before uploading to the Course Moodle Page.
6. Support resources are available on the Course Moodle page to help you with revising and answering these questions.

Q1.

A three-noded planar triangular truss, shown in Figure Q1, is subjected to a concentrated ^{force, F = 10 kN at node, N}2^{. The truss is made of a cylindrical bar of diameter, d = 3.0 mm}and steel material of Young’s Modulus, E = 210 GPa. If the planar truss is fixed securely at ^{node N}1^{but has x- and y-axis roller supports at nodes N}2^{and N}3^{respectively.}

Using the principles of Direct Stiffness Method, and from first principles, undertake the following calculations:

1. Determine the three-member stiffness matrices, 𝑲^𝑒, 𝑲^𝑒 and 𝑲^𝑒 for the bars.

1 2 3

[10 marks]

2. Derive the structural stiffness matrix, K^Sfor this structure. [5 marks]

3. Using the appropriate boundary condition, show all the steps and then determine ^{the displacements at nodes N}2^{and N}3^{. [5 marks]}

Note that you must show in detail all the steps required to determine all stiffness matrices.

Y

x_X F =10kN

[0,0] _[2,0]

FigureQ1

Q2.

A two-step cylindrical bar of total length 4m is subjected to a uniaxial force, F as shown in Figure Q2. To determine its nonlinear deformation whilst reducing computational cost, you decide to develop a new element formulation based on a three-noded 1D element whose isoparametric natural coordinates are given in Figure Q2.

1. Derive the mapping function, ξ= f(x) needed to relate the xy-coordinate system to the isoparametric natural coordinates, ξ.

[2 mark]

2. ^{Derive the element’s three shape functions, namely N}1^,N2^{and N}3^.

[12 marks]

3. Derive the strain-displacement matrix, Band evaluate its value when ξ=0.5.

[6 marks]

y

F

ξ

^ξ=0 ξ=0.5 ^ξ=1.0

FigureQ2

Q3.

A 2D dogbone specimen, whose dimensions are given in Figure Q3, is made by extrude- cutting the gauge section of an initially rectangular specimen using the insert elliptical geometry. The specimen is made from aluminium of Young’s Modulus, E = 70GPa, Poisson ^{ratio, v=0.33, and yield stress, σ}yield ^{= 220 MPa. It has both gripped and loaded ends. The} test specimen is imposed with a Dirichlet uniform displacement of ΔL = 4mm. You are to undertake the mesh density study to determine the appropriate mesh for subsequent investigation.

1. Undertake tensile simulation of the specimen using seed sizes of 6,3, 0.5. Show the contour plots for plastic strain, PE for all simulations. Draw a convergence plot of Average PE versus number of elements. Comment on the implications of mesh density on simulation results. [10 marks]

ΔL = 4 mm

Gauge sect ion ext rude cut using

ellipse below

Loaded End, wit h prescribed

load, ΔL

Gripped End, f ixed securely

20 mm

40 mm

20 mm

40 mm

FigureQ3A

2. In order to prescribe a kinematic constraint to a typical finite element model, Multi- Freedom Constraint (MFC) equations are required. Write the ABAQUS-style linear constraint equation for the following equations:

a. 0.5u^N60  3.5u^N78  2u^N15  8.3u^N15  3.2u^N6  0.10u^N54

3 2 1 3 1 2

b. u^N4  3u^N2  2u^N5  3u^N16 whereu^N16  ce²u^N3  bu^N4 ;

forc,bbeingconstants

2 1 1 2 2 3 1

[3 marks]

3. An inverted T-section support structure, of dimensions given in FigureQ4, is designed to support a central column load on the load region zone. The two end faces of the horizontal bar are fixed securely in X, Y, and Z-axes. The bar is made of steel of Young’s Modulus, E=210GPa, Poisson ratio, v= 0.33and yield stress, σ_yield= 220MPa.

Run a simulation of the compressive behaviour of the structure by imposing concentrated load, U = 20 mm on the structure based on the following canonical equation:

𝑈^{𝐿𝑅𝑆𝑒𝑡} − 2.3log₁₀ (20)𝑈^{𝑅𝑃𝑆𝑒𝑡} = 0

2 2

Where LRSet and RPSet are nodal sets for load region and reference point respectively. Show contour plots for von Mises stress, Misesand in-plane shear stress, S12. [7 marks]

Load Region

y _z

x

125 mmFixed Face in XYZ

50 mm

200 mm

125 mmFixed Face in XYZ

50 mm

Ref erence Point

300 mm

50 mm

[- 500,- 500,- 500]

FigureQ3C,D

Q4.

The Rochester bridge consists of arched trusses with Figure Q4 showing a segment of the bridge. The total length of the bridge deck is 140 m with the span directly over the river measured at L =100m. Assume that the bridge is made of cylindrical

trusses of steel material with Young’s Modulus, E = 210 GPa and diameter, d = 40^{mm. The bridge is fixed securely at nodes N}1^{, N}2^{with roller supports at nodes N}7 ^{and N}8^.

The test load case under consideration represents a traffic pile-up of sedan cars of same sizes and weights which imposes a uniformly distributed load, ω = 3000 N/macting solely on the 100-m span of the bridge. The bridge deck has a rectangular cross-section of base, b=8mand thickness, h=5m.

The Rochester Bridge Trust has announced a competition for University of Greenwich MSc engineering students to undertake a detailed structural analysis of the current design with a view to identify probable improvements to extend the lifespan of the bride.

1. Using MATFESE^TMsolver, undertake the FE simulation of the bridge and determine the absolute maximum displacement, 𝑣_{𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙} of the bridge under the specified load case. Show contour plots of the deformed profile.

[8 marks]

2. In order to understand the benefit of the arched truss design in limiting the deflection over the 100-mspan of the bridge, calculate the deflection of the

5𝜔𝐿⁴

bridge without the arched trusses using the equation: 𝑣_{𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙} = _384𝐸𝐼

where L=bridge span, ω = distributed load, E=Young’s Modulus, and I=

𝑏ℎ³

area moment of inertia for rectangular section, i.e. 𝐼 =

.

12

[3 marks]

3. Calculate the percentage improvement in limiting the deflection of the bridge by introducing the arched truss. Note that percentage improvement is calculated using the equation:

%𝐼𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡 =

^𝑣𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 ^{− 𝑣}𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙

^𝑣𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙

× 100%

[3 marks]

4. As part of a review of your analysis, you noticed that the actual Rochester Bridge truss is made from an I-section wide-flange beam of designation W150x18 and cross-sectional area, A = 1730 mm². What is the percentage error introduced to your simulations by assuming a solid cylindrical truss configuration of diameter, d=20mm.

[2 marks]

5. Suggest and use MATFESE to demonstrate the nature of one improvement of the bridge to ensure extended life span.

[4 marks]

N₂

Figure Q4

Q5.

A unidirectional (UD) composite consists of circular E-glass fibre material embedded within a square-shaped polypropylene matrix. The UD composite has a 20% volumefraction of fibre with fibre diameters ranging from 1.5 um to 2.52 um. We want to use the principles of computational micromechanics to determine the effective properties of the test composite in a specific test direction.

For this study, three representative volume elements (RVEs) shown in Figure Q5Awere identified. The centre locations (in μm units) of the fibres within the matrix medium are given in Table Q5A. All three RVEs have a dimension of length, L = 10 μmand width, W=10μm. The fibre radius for each RVE type are given in TableQ5A. The properties of the fibre and polypropylene constituents are given in TableQ5B.

1. Generate the virtual domains for all three RVE types ensuring that you trim off any boundary fibres to preserve the 20% volume fraction. [5 marks]

2. Undertake an FEM simulation of tensile deformation along the X-axis for all three RVE types. Since this is a micromechanical study on a heterogeneous material, you MUSTusePeriodicBoundarycondition foryouranalysis.

Show contour plots of (a) Von-Mises Stress, Mises (b) displacement magnitude, UMagnitudeand (c) plastic strain, PEfor all three RVE types.

[8 marks]

3. ^{Determine the stress-strain (i.e. σ}11^{vs ε}11^{) graphs for the three RVEs showing the numerically determined Young’s Modulus, E}11,numerical^{and Yield strength, σ}Y,numerical^{of the test composite. [3 marks]}

4. Determine the analytically determined Young’s Modulus of the composite

based on the formula:

𝐸_𝑚𝐸_𝑓

^𝐸11,𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙 ⁼ _𝐸

𝑉 + 𝐸 𝑉

𝑚 𝑓 𝑓 𝑚

^{Where E}m^{and E}f^{are Young’s Moduli of the matrix and fibre respectively; V}f ^{and V}m ^{are volume fractions of the fibre and matrix constituents respectively.}

Compare the analytical and numerical Young’s modulus values for the three RVE types. Comment on the effect of number of fibres within the RVE window on the effective properties of the test composite. [4 mark]

Figure Q5

TableQ5A:LocationsofFibresinthreeRVEs(inμmunits)

	Radius=1.26μm	Radius=1.00μm	Radius=0.75μm
S/No	RVE 1	RVE2	RVE3
	X	Y	X	Y	X	Y
1	5	5	8.6	8.9	9.1	5.3
2			5.1	8.8	1.1	8.3
3			5.9	1.5	3.4	2.9
4			2.0	4.1	7.5	0.1
5			7.9	3.2	6.0	5.3
6			1.1	1.4	7.3	7.1
7					7.8	2.9
8					4.0	0.6
9					1.0	1.3
10					7.5	10.1
11					4.0	10.6

TableQ5B:Propertiesofinclusionsoftheunidirectionalcomposite

Material	Young’sModulus, E [GPa]	Poissonratio, ν	Yieldstress, y [MPa]	Plasticstrain, p
Glass fibre particles	73	0.20
Polypropylene	1.308	0.40	40	0.00

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