Problem 2.5-3
Given: Two collinear cantilever beams are
connected by a frictionless hinge. Flexural
stiffness EI z is the same for both beams. Load
P and deformations are confined to the xy plane.
Write the stiffness matrix that operates on the
"active" d
Problem 3.2-1.
Given: In Fig. 3.2-2b, let: x 1 := 0 x 2 := 2 x 3 := 3 Then use this data in the following: Find: a) Verify numerically that shape functions sum to unity b) What should the sum of the x derivatives of the shape function be? Verify the prope
Problem 6.4-1
Given: Consider two adjacent plane quadratic elements, as shown. Show that the shape functions provide interelement continuity of the field quanity along the boundary. Solution:
Along the right edge of the left element:
:= 1
1 1 1 2 2 N2 =
Problem 6.1-1
(a) Complete calculations begun in Eq. 6.1-2, and obtain shape functions Ni. (b) Verify that the same results is provided by Lagrange interpolation Solution: (a)
a1 2 x = 1 a2 a 3
(
)
a1 2 u = 1 a2 a 3
(
)
Eq. 6.1-1
x 1 a1 x 2 = A a2 x a
Problem 6.4-4
Sketch an eight-node plane element for which J is a function of but not , if (a) the element is rectangular, and (b) the element has two curved sides 1 1 1 2 2 N1 = ( 1 ) ( 1 ) 1 ( 1 ) ( 1 ) 1 4 4 4 1 1 2 N1 = ( 1 ) + ( 1 ) + 1 4 2 4
( (
) )
Problem 3.1-2.
Imagine that stresses in the xy plane are reported to be:
x = 6 a 1 x
2
y = 12 a 1 x
2
xy = 12 a 1 y
2
where a1 is a constant. Consider the square region 0<= x <=b, 0<=y <=b. Find: a) expressions for the tractions x and y on each side of th
Problem 6.2-6
Consider the elements that are square and two units on a side. Node numberins shown for two such elements create difficulties. For each, determine [J] and J, using the shape function of Eq. 6.2-3. What do the node numbers shown imply about t
Problem 3.4-3 (a) Two nodes of an isosceles CST element are fixed, as shown. The =0, and determine the 2x2 stiffness matrix associated with d.o.f. at the unrestrained node. (b) A plane square region of uniform thickness is divided into eight congruent CST
Problem 3.6-4
Element j is a Q4 element that is to be attachted to structure nodes 19,20,30, and 31.
What i the numerical contribution of element j to the single coefficient of [K] at the intersection of
a) row 48 and column 39
b) row 37 and column 37
c)
Problem 6.3-6
Find: Solutions to Integrals using 1, 2 and 3 point quadrature.
Solution:
a)
Iexact
1
2
3
2
d 0.66667
f ( )
3
1
0.
One Point Rule:
W 2.
1
1
I1 Iexact
I1 W f 0
1
1
Two Point Rule:
Iexact
1
1
W 1
1
3
0.6
W
1
1
5
2
Iexact
W
Problem 6.7-3
Given: The sketch shows two uniform beam elements connected at node 2. Prior to making the connection,
d.o.f. z2 in element 1-2 is condensed, so that node 2 is a hinge connection. Starting with the stiffness matrix
of Eq. 2.3-5, obtain the c
Problem 2.3-7
a)
From Eq. 2.3-6, with z1 and z2 the only
active d.o.f. we get:
4 + 2
2 4 +
z1
z2
M0 ( 1 + y) L
E Iz
=
M0 ( 1 + y) L
E Iz
Write out as two equations and solve.
(4 + y) z1 + (2 y) z2 = M0 (1 + y) E I
L
Given
z
(2 y) z1 + (4 + y) z
Problem 16.5-2
Given:
(a) Define terms in Jacobian matrix [J] in terms of , N i, Ni, N i, nodal coordinates, and components of
V3i.
(b) Specialize the results of part (a) for an element that is flat, of uniform thickness, and whose midsurface
coincides wi
Problem 14.2-2(a)
Given: The sketch represents three nodes on a z=constant face of an axisymmetirc quadratic element.
Nodes shown are uniformly spaced. Determine the consistent nodal load vector if z-direction traction z is
applied to the face as follows:
Problem 2.5-2
Given: The structure shown consists of a 2-node element A, a 3-node element B, and a 4-node
element C. There is one d.o.f. per node. Place letters A, B, and C in the appropriate positions in
arrays [K] and cfw_R to indicate the locations to
Form the stiffness matrix for a Q4 element using Gauss Quadrature
Please note that the lower bound for arrays is set to 1 for this MathCad worksheet
Select node coordinates:
Define a function to compute
the Jacobian matrix
Define a function to compute the
Problem 3.11-2
If load q is constant, the integrand in Eq. 3.11-6 becomes [N] T q dx. Using the integral, verify that
the total force F on a straight side with a midside node is allocated F/6, 2F/3, and F/6.
Solution:
The nodal coordinates are:
x 4 = a
y4
Problem 15.1-4
Given: An isotropic thin rectangular plate has dimension a parallel to the x axis and dimension b parallel to
the y axis. The x-parallel edges are simply supported; the y-parallel edges are free. Uniform downward
pressure p is applied to th
Problem 3.9-3a
The bar element shown are uniform and have nodes 1
and 2. Let the axial displacement field be:
2
u = a1 a2 x
Find: [B] and [k] and determine defects in the element
Solution:
a1
u = 1 x
2
a
2
At node 1, x = 0 and
u = u1
u = u2
At node 2,
Problem 2.4-2 (b)
Given: the d.o.f. as shown in the figure for a bar
element.
Find: the 4 x 4 stiffness matrix.
Solution:
c = cos( )
c
s
T=
0
0
s = sin( )
s 0 0
c 0 0
0 1 0
0 0 1
c
s
T
k = T k' T =
0
0
s 0 0
c 0 0
0 1 0
0 0 1
T
1
A E 0
L 1
0
0 1 0
c
Truss or Bar Elements
Consider a tapered bar with an axial force applied on the
right side and fixed on the left. Let the bar be aligned with
the x axis and let the total length of be bar be LT. We could
approximate this problem by three elements with con
Problem 2.9-4
y
Given: Cantilever beam with a distributed load q.
Find: Find tip deflection and root moment with
consistent and lumped nodal loads for a) one
element and b) two elements.
Solution:
q
x
Lt
First lets get an exact solution using elementary b
Problem 3.3-3.
Given: A uniform bar elemen of length L has a node at
each end and a node at the middle as shown.
Find: the element stiffness matrix that operates on the
nodal d.o.f. u1 , u2 , u3 .
Solution:
x1 = 0
x2 =
L
x3 = L
2
L
x2 x x3 x 2 x ( L x)
N1
Problem 2.6-2
Given: 2-D beam element undergoes a 180 degree rotation about node 1. Why is [k]cfw_d not equal to zero?
6
E 10 10
E A
0
L
12 E Iz
0
3
L
6 E Iz
0
2
L
k
E A
0
L
12 E Iz
0
3
L
6 E Iz
0
2
L
L 20
0
6 E Iz
A 1
E A
L
0
2
L
4 E Iz
L
0
6 E I
Hints for Problem C6.1(a)
Make a mesh with elements that look like the ones illustrated below. The actual dimensions and load
values are not important. Apply uniform distributed loads as shown.
Draw a rectangle 2 inches by 2 inches. Make a surface. Repeat
Problem 2.2-1
(a) Consider a two-node bar element but let the corss-sectional area A vary lineary with x from A0 at x=0
to cA0 at x=L, where c is a constant. Write the element stiffness matrix, first as in Eq. 2.2-1 using the
average A, then using the exa
Problem 6.8-1
In a 3 node bar element, let x3 - x2 = L/3 and x2 - x1 = 2L/3. Use one-point Gauss Quadrature to
determine the 3 by 3 stiffness matrix of an element with uniform A and E.
Solution:
N=
B=
1 2 1 2 1 2
2
2
Eq. 6.1-4
1
1
( 1 2 ) 2
( 1 2 )
Problem 6.2-6
Consider the elements that are square and two units on a side. Node numberings shown for two such
elements create difficulties. For each, determine [J] and J, using the shape function of Eq. 6.2-3. What
do the node numbers shown imply about