ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 1 – 25 points max.

ME 323 Final Exam SOLUTION

December 10, 2018 PROBLEM NO. 2 – 25 points max.

Given: Consider the structure above that is made up of rod segments BC and DH, a spring of stiffness k and rigid connectors C and D. The two rod segments are made up of the same material having a Young’s modulus of E. Segment BC has a tapered cross section, with the cross-sectional area varying linearly from 3A at B to A at C. Segment DH has a constant cross-sectional area of A over its length. The spring joining connectors C and D has a stiffness of k = 4EA / L . A load of P acts to the right on connector C. It is desired to set up and solve a finite element model of the system, with segments BC and DH each being represented by a single element. This model will have four nodes: B, C, D and H.

Find: For this problem,

a) Draw a free body diagram of the entire system (BC, DH and the spring). Include all forces acting on the system, including both applied and reaction forces.

b) Write down the stiffnesses of segments BC and DH in terms of E, A and L. c) Write down the (4x4) stiffness matrix [K] for the system corresponding to the axial

displacements of nodes B, C, D and H. d) Write down the forcing vector {F}, where {F} has a length of 4. e) Enforce the boundary conditions on [K] and {F}. f) Solve for the displacements of nodes C and D. g) Determine the stress in element DH.

B

C D

H

L 2L

3AA A

Pk

PFB FH

k1 =

12

EL

3A+ A( ) = 2 EAL

k2 = k = 4 EA

L

k3 =

12

EAL

Therefore,

K⎡⎣ ⎤⎦ =EAL

2 −2 0 0−2 6 −4 00 −4 9 / 2 −1/ 20 0 −1/ 2 1/ 2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

F{ } =−FB

P0

FH

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

Enforcing the boundary conditions ( uC = uD = 0 ) gives:

K⎡⎣ ⎤⎦ =EAL

6 −4−4 9 / 2

⎡

⎣⎢

⎤

⎦⎥

F{ } = P0

⎧⎨⎩⎪

⎫⎬⎭⎪

or:

6uC − 4uD = PLEA

−4uC + 92

uD = 0 ⇒ uC = 98

uD

Therefore:

6 9

8uD

⎛⎝⎜

⎞⎠⎟− 4uD = PL

EA⇒ uD = 4

11PLEA

⇒ uC = 922

PLEA

Load carried by DH:

FDH = −k3uD = − 1

2EAL

411

PLEA

⎛⎝⎜

⎞⎠⎟= − 2

11P compression( )

Stress in element DH:

σ DH =

FDHA

= − 211

PA

compression( )

ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 3 – 25 points max.

Given: The member shown below is subject to a distributed load with a magnitude P. It is known

that the member has a uniform cross section area 𝐴, second area moment 𝐼, and a shear shape factor 𝑓!. The member is composed of a material whose Young’s modulus is 𝐸 and shear modulus 𝐺.

Find: Using Castigliano’s theorem, determine the slope and vertical deflection at the free end “O” of the member. Please include the shear, normal and bending shear strain energies in your solution. Leave your answer in terms of, at most, the known parameters of 𝑤, 𝐿,𝐻,𝐸,𝐺,𝐴, 𝐼, and 𝑓!.

NOTE: For full credit, you need to draw all necessary free body diagrams and internal cuts, and write down all relevant equilibrium equations.

> >

(4)(4)

> >

> >

(8)(8)

> >

> >

(6)(6)

> >

(2)(2)

> >

(1)(1)

(9)(9)

(5)(5)

> >

(3)(3)

> >

(7)(7)

> >

restart;Nd 0; dNdMdd diff N, Md ; dNdFdd diff N, Fd ;

Nd 0dNdMdd 0dNdFdd 0

Vd P$xCFd; dVdMdd diff V, Md ; dVdFdd diff V, Fd ;Vd P xCFddVdMdd 0dVdFdd 1

MdMdK P$x2

2 KFd$x; dMdMdd diff M, Md ; dMdFdd diff M, Fd ;

MdMdK 12 P x2KFd x

dMdMdd 1dMdFddKx

N1d P$LCFd; dN1dMdd diff N1, Md ; dN1dFdd diff N1, Fd ;N1d P LCFddN1dMdd 0dN1dFdd 1

V1d 0; dV1dMdd diff V1, Md ; dV1dFdd diff V1, Fd ;V1d 0

dV1dMdd 0dV1dFdd 0

M1dMdK P$L2

2 KFd$L; dM1dMdd diff M1, Md ; dM1dFdd diff M1, Fd ;

M1dMdK 12 P L2KFd L

dM1dMdd 1dM1dFddKL

Mdd 0; Fdd 0;Mdd 0Fdd 0

deflMdxd 1EI

$int M$dMdMd, x = 0 ..L CfsAG

$int V$dVdMd, x = 0 ..L C1AE

$int N

$dNdMd, x = 0 ..L ;

deflMdxdKP L3

6 EI

deflMdyd 1EI

$int M1$dM1dMd, y = 0 ..H CfsAG

$int V1$dV1dMd, y = 0 ..H C1AE

$int N1

$dN1dMd, y = 0 ..H ;

deflMdydKP L2 H2 EI

> >

> >

> >

(13)(13)

(11)(11)

> >

(12)(12)

> >

(10)(10)

> >

delfMdd deflMdxCdeflMdy;

delfMddKP L3

6 EI KP L2 H2 EI

deflFdxd 1EI

$int M$dMdFd, x = 0 ..L CfsAG

$int V$dVdFd, x = 0 ..L C1AE

$int N$dNdFd, x

= 0 ..L ;

deflFdxd P L4

8 EI Cfs P L2

2 AG

deflFdyd 1EI

$int M1$dM1dFd, y = 0 ..H CfsAG

$int V1$dV1dFd, y = 0 ..H C1AE

$int N1

$dN1dFd, y = 0 ..H ;

deflFdyd P L3 H2 EI C

P L HAE

delfFdd deflFdxCdeflFdy;

delfFdd P L4

8 EI Cfs P L2

2 AG CP L3 H2 EI C

P L HAE

ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 4 - PART A: 10 points max. A slab is brought to failure subjected to the loading configuration depicted in the following figure.

Circle the correct answer in the following statements: (a) The internal resultant for cross section between the two applied loads

correspond to: (i) An unloaded configuration

(ii) Pure bending (iii) Axial loading (iv) Pure shear (v) A combined loading condition comprised of shear force and bending

moment (b) For any cross section between the two loads, the state of stress of a point on the top face of the

slab is represented by the following Mohr’s circle:

(i) (ii) (iii) (iv) (c) For any cross section between the two loads, the state of stress of a point on the bottom face of

the specimen is represented by the following Mohr’s circle:

(i) (ii) (iii) (iv)

L L L

P P

ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 4 - PART A (continued) (d) If the slab is made of steel, a ductile material, then failure will occur first

between the two loads and on: (i) the top face of the specimen at a cross section between the two loads

(ii) the bottom face of the specimen at a cross section between the two loads

(iii) the top and bottom faces of the specimen simultaneously

(iv) the neutral surface of the specimen

(e) If the specimen is made of concrete, a brittle material with an ultimate

compressive strength larger than the ultimate tensile strength, then failure will occur first between the two loads and on: (i) the top face of the specimen at a cross section between the two loads

(ii) the bottom face of the specimen at a cross section between the two loads

(iii) the top and bottom faces of the specimen simultaneously

(iv) the neutral surface of the specimen

ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 4 - PART B: 9 points max.

A column of height 25t and rectangular cross section of dimensions 4t by t is subject to a compressive load P. The column is made of a material with Young’s modulus E and yield strength 𝜎! = 𝐸/100. Determine the critical buckling load following the steps below.

(a) The lowest buckling load corresponds to a [circle the correct answer]:

(i) Pinned-pinned column

(ii) Pinned-fixed column

(iii) Fixed-fixed column

(iv) Pinned-free column

P P

4tt

25t

Front viewSide viewx

ycross

section

ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 4 - PART B (continued)

(b) Determine the cross-sectional area, the second area moments with respect to the x- and y-axes of the column.

A = 4t2

I yy =

4t( ) t( )312

= t4

3

Ixx =

t( ) 4t( )312

= 16t4

3

Notice that buckling about the y-axis will require a lower critical load than the buckling about the x-axis.

(c) The column buckles following: (i) Euler’s theory (ii) Johnson’s theory.[circle the correct statement]

Sr =Leff

r=

Leff

I yy / A= 0.7L

t4 / 3( ) / 4t2= 60.62

Sr( )c =Leff

r⎛

⎝⎜

⎞

⎠⎟

c

= π 2EσY

= 10 2π = 44.4

Since Sr > Sr( )c ⇒ Euler theory to be used

(d) Based on the above assessment, determine the critical buckling load of the column, Pcr.

Pcr = π2 EI yy

Leff2 = 0.0107 Et2

ME 323 Final Exam SOLUTION December 10, 2018 PROBLEM NO. 4 - PART C: 6 points max.

Page 13 of 13

ME 323 Final Exam Name ___________________________________

December 10, 2018 PROBLEM NO. 4 - PART C: 6 points max. Fill in the blanks with words. Please do not use any symbols or numbers. (a) One can define only _______________ independent elastic properties to characterize a material.

For example, if the ___________________ and the _________________ are given, then the

__________________ is determined as ! = !/2(1+ !).

(b) The mechanical design of any mechanical members must _______________ the structure from

buckling and from plastic deformations or sudden fracture, depending on whether the material is

_____________ or _____________, respectively. _____________ and ____________ failure

modes occur at a critical point in the structure, and they are determined from the

_______________ __________________ at such critical point. In contrast, buckling is greatly

affected by the overall _____________________ of the structure.

(c) The internal loads at a certain cross section of a three-dimensional deformable solid are

conveniently decomposed into __________ internal resultants, namely ____________________

_____________________________________________________________________________

and ______________________________.

(d) For pure ______________, Euler-Bernoulli beam theory assumes that cross sections remain

_______________ and __________________ to the deflection curve of the deformed beam. In

addition, it assumes that the ______________________ remains free of _________________

as the beam deforms.

twoYoung'smodulus Poisson'sratio

shearmodules

prevent

ductile brittle Ductile brittle

state ofstressgeometrydimensions

six axialforcetwoshear forces two bending moments

torque

bending

plane perpendicularneutralsurface strain