Exercise 1 - 2019(1)

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY NTNU
FACULTY OF ENGINEERING SCIENCE AND TECHNOLOGY
DEPARTMENT OF STRUCTURAL ENGINEERING

TKT4192 FINITE ELEMENT METHOD IN STRENGTH ANALYSIS
Autumn 2019
EXERCISE 1

Problem 1

y

3 x

b

12

a

Figure 1

Figure 1 shows the geometry for a three-node triangle. Assume that the field quantity φ can be
written as

φ =a1 + a2 x + a3 y ,

where the ai are generalized d.o.f. For the particular shape of triangle shown in Figure 1, express
φ in the form

φ = f1φ1 + f2φ 2 + f3φ3 ,

where the fi are functions of x, y, a and b .

Hint: Obtain three equations for the generalized d.o.f. ai from the conditions:

φ (=x 0, =y 0=) φ1
φ (=x a, =y 0=) φ2
φ (=x 0, =y b=) φ3

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY NTNU
FACULTY OF ENGINEERING SCIENCE AND TECHNOLOGY
DEPARTMENT OF STRUCTURAL ENGINEERING

Problem 2

k3

k1 F3 k1 F3
F1 u3 u3
F1
k2 F2 k2 F2
u2 u2
u1 u1

a) b)

Figure 2

a) In Figure 2a, three rigid blocks are connected by linear springs. Establish the equilibrium
equations
Kr = R ,

in terms of the spring stiffnesses ki , displacement d.o.f. ui , and applied loads Fi.

b) An extra spring is attached to the system and connected to a rigid wall, see Figure 2b.
Establish the equilibrium equations
Kr = R ,

in terms of the spring stiffnesses ki , displacement d.o.f. ui , and applied loads Fi.

c) Do the two systems shown in Figure 2a and 2b have a unique solution? Substantiate you
answer. Give a physical interpretation on why they eventually are different.

d) Set the spring stiffnesses=ki 1=0, i 1, 2,3 for the system shown in Figure 2b, and assume
that=: u3 0=, F1 0.5 and=F2 1.0. Calculate the response in terms of u1, u2 and F3.

e) Assume that the following quantities have been prescribed: u1 =−0.1, u2 =0.07 and F3 =0.
Calculate the response in terms of u3, F1 and F2, when=ki 1=0, i 1, 2,3.

NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY NTNU
FACULTY OF ENGINEERING SCIENCE AND TECHNOLOGY
DEPARTMENT OF STRUCTURAL ENGINEERING

Problem 3

y,v

v1 vA v2 vB
x

k1 k2

LL

Figure 3

Figure 3 shows a plane structure consisting of a rigid, weightless bar and two linear springs of
stiffnesses k1 and k2 . Only small vertical displacements are permitted. The stiffness matrix K
of the structure shown in Figure 3 is 2× 2 but can have various forms, depending on the choice
of d.o.f. Determine K for each of the following choices of lateral translation d.o.f.:

a) v1 at x = 0 , and v2 at x = L.
b) v1 at x = 0 , and vA at x = L / 2.
c) v2 at x = L , and vB at x = 2L.

Due date for Exercise 1 - Friday August 30, 15:00