## Calculation of transverse shear stiffness

### Definition

The effective transverse shear stiffness of the section of a shear flexible beam is defined in Abaqus as

${\overline K _{\alpha 3}} = f_p^\alpha {K_{\alpha 3}}$

where $${\overline K _{\alpha 3}}$$ is the section shear stiffness in the $$\alpha$$-direction;  $${K_{\alpha 3}}$$ is the actual shear stiffness of the section having units of force; and $$\alpha = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} 2$$ are the local directions of the cross-section.  $$f_p^\alpha$$ is a dimensionless factor used to prevent the shear stiffness from becoming too large in slender beam elements and is always included in the calculation of transverse shear stiffness defined as

$f_p^\alpha = \frac{1}{{1 + \xi \cdot SCF\frac{{{l^2}A}}{{12{I_{\alpha \alpha }}}}}}$

where l is the length of the elementA is the cross-sectional area,  $${{I_{\alpha \alpha }}}$$ is the inertia in the $$\alpha$$-direction, SCF is the slenderness compensation factor (with a default value of 0.25), and $$\xi$$ is a constant of value 1.0 for first-order elements and 10-4 for second-order elements.

## Beam elements in ABAQUS

Beam elements in ABAQUS make me a little confused, especially about the shear stiffness and elements for open sections. In this post, I want to make a summary of the elements in ABAQUS and give an example about the settings of beam elements.

## Beam element library

Beam elements in a plane only have active degrees of freedom 1, 2, 6. Beam elements in space have active degrees of freedom 1, 2, 3, 4, 5, 6. Open section beams in space B31OS, B31OSH, B32OS, B32OSH have active degrees of freedom 1, 2, 3, 4, 5, 6, 7. The beam elements in ABAQUS Standard and Explicit (v6.14) are summarized below.

## An example of progressive collapse analysis – Steel frame

This is an example of progressive collapse analysis and can be a benchmark for analysis of real structures. The steel frame is the same as the Frame 2 in the paper ‘Effect of span length on progressive collapse behaviour of steel moment resisting frames’ of Rezvani et al. (2015)

## Model description

The model of 4×4 span, 6 story frame is figured below. The beam section IPE500 is adopted for stories 1 to 4 and IPE450 is used for stories 5 and 6. The column sections differ every two floors and listed in the following table. The steel (Steel02 in OpenSees materials) with yield strength 235MPa, elastic modulus 202GPa and a post-yield stiffness of 2% of initial stiffness is adopted for all the members.

## Analysis of discrete systems

The essence of a lumped-parameter mathematical model is that the state of the system can be described directly with adequate precision by the magnitudes of a finite (and usually small) number of state variables. The solution requires the following steps:

1. System idealization: the actual system is idealized as an assemblage of elements
2. Element equilibrium: the equilibrium requirements of each element are established in terms of state variables
3. Element assemblage: the element interconnection requirements are invoked to establish a set of simultaneous equations for the unknown state variables
4. Calculation of response: the simultaneous equations are solved for the state variables, and using the element equilibrium requirements, the response of each element is calculated.

## Special Matrix

symmetric matrix

identity matrix / unite matrix

symmetric banded matrix

the following matrix is a symmetric banded matrix of order 5 and the half-bandwidth is 2.

${\rm{A}} = \left[ {\begin{array}{*{20}{c}} 3&2&1&0&0\\ 2&3&4&1&0\\ 1&4&5&6&1\\ 0&1&6&7&4\\ 0&0&1&4&3 \end{array}} \right]$

diagonal matrix: nonzero elements only on the diagonal of the matrix

upper half of the matrix

inverse matrix, the inverse of a matrix

partitioning of matrix

the trace and determinant of a matrix: only defined if the matrix is square

tr(A) = sum of elements on the diagonal

det A = determinant

det (BC…F) = (det B)(det C)…(det F)

orthogonal matrix:${{\rm{P}}^T} = {{\rm{P}}^{ – 1}}$

## 2D frame

This is a plane steel frame with box sections subjected to a point load at one node figured as below.

An dynamic implicit analysis step was defined and the following lines were added at the end of input file.

**
*STEP
*MATRIX GENERATE, STIFFNESS, MASS, LOAD, Structural Damping,Viscous Damping
*MATRIX OUTPUT, STIFFNESS, MASS, LOAD, Structural Damping,Viscous Damping, FORMAT=COORDINATE
*END STEP

## Time and Amplitude in ABAQUS

I am recently using the dynamic analysis in Abaqus, so I decided to summarize the understanding of time and amplitude in it so that I can refer to this article later if needed.

## Time

• Step Time: the time period in *STEP definition.
• Total Time: the sum of all the Step Time.
• Natural time: actual time that one thing takes.
• The time in Abaqus/Standard does not have any actual meaning, it can be understood as the relative time.
• The time in Abaqus/Explicit is the actual time, that is the time that loads applied on the structure.

## Amplitude definition

• Select Step time for time that is measured from the beginning of each step.
• Select Total time for total time accumulated over all non-perturbation analysis steps.

## Matrix in ABAQUS

Matrices: Stiffness, Mass, Viscous Damping, Structural Damping, Load

Formats:

• FORMAT=MATRIX INPUT (default) to specify that the output use the matrix input text format that is consistent with the format used by the matrix definition technique in Abaqus/Standard.
• FORMAT=LABELS to specify that the output use the standard labeling format.
• FORMAT=COORDINATE to specify that the output use the common mathematical coordinate format.