SKAA 4223/ SAB4223

@uffiMl**''*va\{
FINALEXAMNATION
SEMESTERII, SESSION
20T6120T7
COURSECODE : SKAA 4223/ SAB 42'23

COURSE : STRUCTURALANALYSIS

PROGRAMME : SKAW/SAW

DURATION : 2 HOURS30 MINUTES

DATE : JLINE,2017

II{STRUCTION TO CANDIDATES:
Answerthree(3) questions
only

WARNNG!
Studentscaughtcopying/cheatingduring theexaminationwill be liablefor disciplinary
actionsand thefaculty mayrecomnxend thestudentto beexpelled-fro* thestudv.

This examinationquestionconsistsof ( 8 ) printed pagesincluding this page.

 SKAA 4223/ SAB 4223

Ql' Figure Ql showsa trussstructure

ABCD is subjectedto nodal loadsof 55
and 36 kN at node A' The truss is supported kN
by pinned supportsat nodesB, c
and [f' The global coordinate(X,
n in unit meter of nodesA,B, C and D is
indicatedin the figure' The modulus
of elasticity,Eaird cross-section
al arca,A
of all members of the truss is 200 kN/mm2
and l05B mmr, ,esp""tively. By
using the flexibility method,determine:

(a) the vertical and horizontal displacements

at node A (assumingmember
AC as redundant).

(21 marks)

(b) member axial forces for the truss. and

(6 marks)
(c) draw the deformedshapefor the truss.

(6 mark)

55 kN (33 marks)

c ( 1 . 50,
(3,0)

FIGURE 01
 SKAA 4223/ SAB 12'3 I 3
I
Q2' Figure Q2 shows a beam with overhang
is subjectedto a point load of 60 kN
at A and a uniformly distributedload
of 108 kN/m on spanBC. The beam
supportedby a roller at B and fixed is
at c. A downward settlementof l0
occurredat supportB. By using the mm
stiffnessrnethod,

(a) Determinethe displacementat A

and r.tations at B.
(23 marla)
(b) Draw the shearforce and bendingrnoment
diagramsof the beam.
(10 marks)
T a k eE = 2 x 108kN/rn', I :5.06 x
rO-ama andA = 0.0105m2 for all
members.

(33 marks)

60 kN

108kN/rn

FIGUREO?
 SKAA 4223/ SAB 4223

Q3. Figure Q3 shows a frame structure subjectedto a point load of 80 kN on

member 1 and a uniformly distributed loacl of 40 kN/m on member 2. The
frame is fixed at node 1. and pinned at node 3. If the axial deformation of
member I (beanr)is neglected,and by using the stiffnessmethod,

(a) salculatethe rotations(slopes)at nodes2 and3,

(25 marks)
(b) calculatethe reactingrnomentat node 1, and
(4 marl<s)
(c) draw the deflectedshapeof the frame.
(4 marlrs)

G i v e n ,E : 2 x 1 0 8k N / m t ,I : 4 . 2 5 x l O - am a a n dA, : 0.0102 m2 for all

members.
(33 marks)

o 80 kN

T-
ca
II 40 kN/m

FIGURE 03
 SKAA 4223/ SAB 4223

Q4' (a) With the appropriatediagrarns,discussthree factorsthat influenoe

on the
accuracyof the finite elementmethod.
(9 marks)

(b) Sketchthe following elements:

(i) Threenodal, one dimensionalelement
(ii) six nodal, triangurarcurve-sidedelement
(iii) Eight nodal, squarestraight-sicred
element
(iv) Four nodal, tetrahedralelernent
(v) Eight nodal, hexahedralelement
(10 marks)

(c) Figure Q4(c) shows a tluee-nodeone:dimensionalelement

of 15 mm
length. Derive the displacement fuirction (u) of the element
in x
direction.

7.5*
C, a Zgr,** _>x
123 (14 marks)
FIGUBE Q4(c)
(33 marks)
 SKAA 42231SAB 4223

FORMULA
The symbols indicateparametersusually used,

(a) Flexibility method

' -tl
f n=l !l fortruss fo : J-l *,. beams
andframes
LAE) 6ErL-1 2 )

F.*= bllb^ or ftrf T^= Kro

Fo=btrfhr or TtrfTr=Ko
F^^= blfb^ or rf .f T*= K^^
F",=blfb* or T{fTr=K*
X = -rrlF"oR or x - -K;trKr*a'

If rx +0, X:F;rr* -n;|fr R or If Ax *0, X = KrtrL*- K*K,RR

F : F^^- FftxF;.f
FxR or K = K ^,,- t<ut<;lt< ,*
r=FR or A,-KR

<
['l > = t[F** F*ll <[Rl> or i
I a I> = lI K^* ,<*Il <[.n]
L',J LF,"r",l Lxj Ln"JlKr* ,K,',-]lxj
rnl
e=[b^ n"iJni or F-Iro&r1;l
or
i;t
b =bn -n"n;frr or T =Tn-'TrK;'rKr^

Q:bR or F=TR

Ifrxl0,Q-bRlbxF*)r* ol' L**0,F=TR+T.rK;lLr

(b) Truss analysisusing the stiffnessmethod

l,:cosd,-nt -tt or t r , = * ; o r ; 9- ,x u - x "

L Ln,

l ; , = c o s d ,*, ! r --!N or l , = c o s d u- ! a - ! e
L L,ou
 SKAA 4223/ SAB 4223

-A,' -7,111
i A*' A,1,,
^,^,. 7,,' -1,1, -7r'
k = AEI-^,' *1,A, t
L A,' A,A,
| I
-Ar,' h,tr, A,,' l
l-1r1,,

[q-I =[K,, K,,lIu,,'l or fql> = l[K,, K,,l1n,l

tq,,lL*,,K,,lt"-
{ l< }
I L4'J LKu Kr,) [.A*
J

i"*l Io*I
-)",,
Qr=#l-^,-1,,A,,,]l;:f or.f,=#l-^" )"" 'rlill
[D.,J [4",-l

(c) Frames and beamsanalysisusing the stiffnessmethod

For beamsand frameswithout axial and sheareffectri

'1 Bz
,
N, F,
| 4il zu1
k=?Ellztl"'
l-:-
or
K'=l L L- |ln'
L Ll 2),, | 2'Er 4Er I B,
l-T Ll
Member stiffiressmatrix for framesin global system

.V,.or I, .\'u or l.r, N A, F, ot B, F' or 8., I ' . c , tB ,

"or
( ,qn^, l2EI ^,\ (es nu\"" - 6 E n'
r" ( lg ", tzEI ",\ (en nu..'^" bEl
') /r- t ----;- /".. 1
-t - ---
r/t../ 'u'z
\.l, L', [;- t )^,n' t \r L'') I r t )" t" ' L'
( en -----
rzEt\^ /"." ( eg rzEI
^"\
" , * --'----:-r''l 6EI" ( a n t z n t \" . ( lt
-[; " , + t z E I t"i
^,\ -b E I ''^
l/, |- A- | /,, ----"-
l/,-./ ni
\l,L' \T' L' / L,, \r L')' ,; .) L'
6EI , 6EI ^ 4EI 6EI 6EI , 28,!
_-\_ A, -t " -
k o rK ' = t UL U t", L
( e t , " , r z E I^ , \ ( ,rc - ----;-
lzEI\ ^ ^ 6EI " ( , , q n" , I T E I ^ , \ ( er rzEI\. bEI .
n-'r--'--;-/v.. t
') 1/v,,/"..
----;- lv. n;) -----;- "
1/v-t"..
-'-:-- t-.
\Z l; \T L' l'(. r ' ; * t \z L')'' l-!

-1( *t e -';?
nu\^. -
( lt ", rzEI^,\
- -r -----i-
') -
6EI"
--'-'7-
tzlr\ " .
,qn -----;- ( et ", tzEI ^,\
l-/u-.+ /". - -6
T 'E
/ ,l,
\. ,.
1i.i',TJ
/
|

\z
iv,. /!- |
r'
t
"..
\r L')'
ll""r".
\t'
"
L' )
| rr
6EI .
**t 6EI 2EI 6EI 6EI , 4EI
n" ------4.." ---- "
4. --- t"-
rL L" T" L

Iql_[o,,K,,'l{o,,}
il]{3;i
{S:}=[[:, [4,J LK,Krr)lA^J
 SKAA 4223/ SAB 4223

I )"]' o 0 00
-)"u .2_,. 0 0 00
0 0t 0 00
T=
0 00 ),, 1, 0
0 00 -)"y tr* 0
0 00 001

Q= k ' T D of f -krL for frarnes

Q =kI) or f * K'A for beams

Mernberstiffnessmatrix for framesin local system.

t
I
AE
L
00 -AE
L
o 0
AE
L
00 - -,4E
T00
I
I
0 -T
ILEI
-tr
6EI
0 -yy 6EI
----- 0 *7-
lzEl 6EI
--7T
ILEI
-tr
6EI

I 6EI 4EI
t t 7'
o-g 2EI
I 0 t, 0
6EI 4EI
0 -*tr
6EI zEI
k ' =I Ll. t L TL L
AE
i - ALE 00
L
0 0 _T
AE
00
AE
00
T
0 --T-
I2E1 --T
6EI ^
"_F
12EI -*tr
6ET
0
ILEI --:-
-----=- 6EI
0 -7-
I2EI
- -6Er
n I
6EI
L' L' LI I
2E]
7L o-9 !Et -tr1
6EI zEI
--tr
6EI 4Er I
t ,t
tl

(d) Fixed-endmoments,ff

Foruniformdistributecl
loads,Mo =twE f tZ
Forpointloads,MF =tPah'?fE or Mr =tpbat
lE

(e) Finite ElementMethod

Lagrangefunction

N,,,:nf-t4
- *
t:'* (q,,
D*t 1
r)