## 13.181. SHELL181 - 4-Node Shell Matrix or VectorShape Functions Integration Points
Stiffness Matrix; and Thermal Load Vector Equation 11–69, Equation 11–70, Equation 11–71, Equation 11–72, Equation 11–73, and Equation 11–74

In-plane:
1 x 1 (KEYOPT(3) = 0)
2 x 2 (KEYOPT(3) = 2)

Thru-the-thickness:
1, 3, 5, 7, or 9 per layer for
data input for section general
shell option (KEYOPT(1) = 0)

Thru-the-thickness:
1 per layer for section data
input for membrane shell
option (KEYOPT(1) = 1)

Consistent Mass and Stress Stiffness Matrices Equation 11–69, Equation 11–70, Equation 11–71, Equation 11–72, Equation 11–73, and Equation 11–74 Closed form integration
Lumped Mass Matrix Equation 11–69, Equation 11–70, Equation 11–71 Closed form integration
Transverse Pressure Load Vector Equation 11–71 2 x 2
Edge Pressure Load Vector Equation 11–69 and Equation 11–70 specialized to the edge 2
Element TemperatureBilinear in plane of element, linear thru each layer
Nodal TemperatureBilinear in plane of element, constant thru thickness
PressureBilinear in plane of element and linear along each edge

References: Ahmad(), Cook(), Dvorkin(), Dvorkin(), Bathe and Dvorkin(), Allman(), Cook(), MacNeal and Harder()

### 13.181.1. Other Applicable Sections

Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations.

### 13.181.2. Assumptions and Restrictions

Normals to the centerplane are assumed to remain straight after deformation, but not necessarily normal to the centerplane.

Each set of integration points thru a layer (in the r direction) is assumed to have the same element (material) orientation.

### 13.181.3. Assumed Displacement Shape Functions

The assumed displacement and transverse shear strain shape functions are given in Shape Functions. The basic functions for the transverse shear strain have been changed to avoid shear locking (Dvorkin(), Dvorkin(), Bathe and Dvorkin()).

### 13.181.4. Membrane Option

A membrane option is available for SHELL181 if KEYOPT(1) = 1. For this option, there is no bending stiffness or rotational degrees of freedom. There is only one integration point per layer, regardless of other input.

### 13.181.5. Warping

A warping factor is computed as: (13–346)

where:

 D = component of the vector from the first node to the fourth node parallel to the element normal t = average thickness of the element

If φ > 1.0, a warning message is printed.

### 13.181.6. Shear Correction

The element uses an equivalent energy method to compute shear-correction factors. These factors are predetermined based on the section lay-up at the start of solution.