VERIFICATION of the MAESTRO QUAD4 element (isotropic)

Rectangular Plate Under Lateral Load

How to use MSC/NASTRAN For Windows V3.0 QUADR element

Initially, the QUAD4 element in MAESTRO was the same as the QUAD4 element in COSMIC Nastran, but since then it has received two major improvements:

1) It has all six degrees of freedom at each node, including in-plane rotational stiffness. This overcomes the traditional problems associated with in-plane rotational mechanisms. This improvement was developed by the MacNeal-Schwendler Corporation for the QUAD4 element in MSC Nastran [1]. The same theory and method of implementation have been used for the MAESTRO QUAD4 element.

2) The MAESTRO QUAD4 element can have structural orthotropy, whereby it can represent a stiffened panel. This feature is verified later. The present chapter addresses the isotropic properties of the element.

The MAESTRO QUAD4 element has been tested using the patch tests and all of the other standard test problems published by MacNeal and Harder [2]. As shown in th following, the results show either similar or better level of accuracy as the results from Nastran and/or from ABAQUS.

1.1 Patch Test

For the patch tests, there are five files in all. Three of them are set up for the constant in plane strains, corresponding to tensions in X, Y direction and shear in X-Y plane. The other two files are for constant bending curvature, first in the Y direction and then in the X and Y directions, which involves some twist. Table 1.1 gives the boundary conditions, loads and the calculated strains and stresses for the patch elements for the five cases.

Fig 1.1 Patch test for plates, a = 0.12; b = 0.24; t = 0.001; E = 1.0E6 ; v = 0.25

Location of inner nodes:

Node	X	Y
5	0.04	0.02
6	0.18	0.03
7	0.16	0.08
8	0.08	0.08

Table 1.1 Patch Test of MAESTRO Quad4 Element vs MSC/NASTRAN For Windows V3.0 QUADR Element

BCs

Loads

Strains

(output)

Stresses

(out put)

Maestro COM Solver

MSC/NASTRAN For Window V3.0

Input Data File

Constant strain in

X-direction

1: 111001

2: 011000

4: 101000

2: P_x = 0.06

M_z = -1.2e-3

3: P_x = 0.06

M_z = 1.2e-3

e_x = 10^-3

e_y = 0.25e-3

e_xy = 0

s_x = 1000

s_y = 0

s_xy = 0

U2=0.000289978

U3=0.000287805

V3=-0.000123489

SIGX=1013.8

U2=0.00028987

U3=0.00028765

V3=-0.00012322

SIGX=1012.42

Constant

strain in

Y-direction

1: 111001

2: 011000

4: 101000

3: P_y = 0.12

M_z = -4.8e-3

4: P_y = 0.12

M_z = 4.8e-3

2: M_z=4.8e-3

e_x = 0.25e-3

e_y = 10^-3

e_xy = 0

s_x = 0

s_y = 1000

s_xy = 0

V3=0.00012

V4=0.00012

U3=-6e-005

SIGVM=1000

V3=0.00012

V4=0.00012

U3=-6e-005

SIGVM=1000

Constant

shear in

X-Y plane

1: 111011

4: 101000

2:P_x = -0.048

P_y = 0.024

3:P_x = 0.048

P_y = 0.024

4:P_y = -0.024

e_x = 0

e_y = 0

e_xy = 10^-3

s_x = 0

s_y = 0

s_xy = 400

V3=0.00024

Sxy=400

V3=0.00012

Sxy=400

Constant Bending,

m_y = 8.889e-8

1: 111111

2: 000100

3: 000100

4: 101111

2:M_y = 5.33e-9

3:M_y = 5.33e-9

Curvature: 1.e-3

Slopes:

2:q_y = 2.4e-3

3:q_y = 2.4e-3

Surface Stress:

s_x = 0.533

q3_y = 0.00023985

Constant Biaxial Bending,

m_xy = 3.33e-8

1: 111111

2: 011010

4: 101101

2&3: M_x = 0.2e-8

3&4: M_y = -0.4e-8

Twist: 0.5e-3

Slopes:

2:q_x = 1.2e-3

q_y = 0

3:q_x = 1.2e-4

q_y = -6.0e-5

4:q_y = -6.0e-5

Surface Stress:

s_xy = 0.2

q3_x = 0.00012

q3_y =-6e-005

W3=1.44e-005

q3_x = 0.00012

q3_y =-6e-005

W3=1.44e-005

1.2 Cantilever Beam

The second test is a cantilever beam modeled with six trapezoidal (or parallelogram) shell elements. The dimensions and the material properties are given in Figure 1.2. Three loads are applied at the free end of the beam: a unit force in the Y direction (in the plane of the element), a unit force in the Z direction (out of plane) and a unit twisting moment. The in-plane force causes in-plane shear. The out of plane force causes shell bending. The critical part of this test is the in-plane shear. The MSC NASTRAN QUAD4 element completely failed the in-plane test because of shear locking. The MAESTRO element is only 1.2% different from the theoretical value. For the out of plane bending, the errors are 2.3% and 1.6% respectively. For the twist the theoretical value is 0.0233 radians and not 0.0321 as given in [2] and MAESTRO matches this value exactly. Table 1.2 presents the results.

Fig 1.2 Straight cantilever beam. Length = 6.0; height = 0.2; depth = 0.1; E = 1.0 E7; v = 0.3; mesh = 6 ´ 1

Table 1.2 Cantilever Beam Results

	BCs	Loads	Masetro COM Solver	MSC/Nastran V3.0	Input Data File
In Plane Shear	Clamped at one end of the beam	Unit force in Y-direction at free end	v = 0.1068 (MAESTRO) v = 0.1081 (theory) relative error = 1.2%	v = 0.1068	trapz.mdl (Load Case 1) trapz1.dat trapz1.mod trapz1.f06
Out of Plane Shear		Unit force in Z-direction at free end	w = 0.4252 (MAESTRO) w = 0.4321 (theory) relative error = 1.6%	w = 0.42642	trapz.mdl Load Case 2 trapz2.dat trapz2.mod trapz2.f06
Twist		Unit twisting moment at free end	q_x = 0.00233 radians (MAESTRO) q_x = 0.00233 radians (theory) VALUE GIVEN IN [2] WAS WRONG	q_x = 0.00306	trapz.mdl Load Case 3 trapz3.dat trapz3.mod trapz3.f06

1.3 Curved Beam

The third test is the curved beam problem. The geometry, dimensions, material properties, and loading conditions are shown in Fig. 1.3. The element shape in this test is not exactly rectangular and so this test includes the effect of a small irregularity in the element. Table 1.3 presents the results of this test, which shows that the MAESTRO QUAD4 element has about the same accuracy as the MSC QUAD4 element, but distributed differently.

Fig 1.3 Curved beam. Inner radius = 4.12; outer = 4.32; arc = 90⁰; thickness = 0.1 ;

E = 1.0E7; v = 0.25; mesh = 6 ´ 1; Loading: unit forces at tip.

Table 1.3 Curved Beam Results

BCs

Loads

Maestro COM Solver

MSC/NASTRAN

Data File

In Plane, Shear

Clamped at one end of the curved beam

Unit force in Y direction at the free end

v = 0.0873 (theory)

v = 0.0880 (MAESTRO)

MAESTRO error = 0.8 %

MSC QUAD4 error = 0.8%

v = 0.087978

Out of Plane, Bending

Clamped at one end of the curved beam

Unit force in X direction at the free end

u = 0.5022 (theory)

u = 0.4492 (MAESTRO)

MAESTRO error = 10.5 %

MSC QUAD4 error = 4.9 %

u = 0.48664

The fourth test is the twisted beam problem. The size of the beam, element mesh and material properties are given in Figure 1.4. The purpose of this test is to study the ability of an element to treat the coupling of in plane and out of plane strain when there is a warped element mesh. In this test, the warp of each element is 7.5⁰. As shown in Table 1.4, the MAESTRO QUAD4 gives results that are in good agreement with theoretical results.

Fig 1.4 Twisted beam. Length = 12.0; width = 1.1; depth = 0.32; twist = 90⁰ (root to tip);

E = 29.0E6; v = 0.22; mesh = 12 ´ 2; Loading: unit forces at tip.

Table 1.4 Twisted Beam Results

BCs

Loads

Displacement

MSC/NASTRAN

Input Data File

Out of Plane Shear

Clamped at one end of the beam

Unit force in Y direction at the free end

V = 1.728e-3 (MAESTRO)

V = 1.754e-3 (theory)

Error = 0.8 %

V = 1.727e-3

twsb.mdl (Load Case 1)

twsb1.mod

twsb1.nas

twsb1.f06

In Plane Shear

Clamped at one end of the beam

Unit force in Z direction at the free end

W = 5.382e-3 (MAESTRO)

W = 5.424e-3 (theory)

Error = 1.5 %

W = 5.388e-3

twsb.mdl (Load Case 2)

twsb2.mod

twsb2.nas

twsb2.f06

1.5 Rectangular Plate Under Lateral Load

The fifth test investigates the accuracy of the elements plate bending response for the case of a rectangular plate. A lateral load is applied to a rectangular plate of a given aspect ratio. The plate is tested separately for each of the two types of loads: a uniform pressure of 1.E-4 and a central point load of 4.E-4. It is tested for two types of boundary conditions: edges simply supported and edges clamped, and two aspect ratios, thus yielding a total of eight test problems. Because of symmetry only one quarter of the plate is modeled, and the model has two elements in each direction, for a total of four elements. The results are presented in Table 1.5, showing that the MAESTRO QUAD4 element gives quite good results.