Element Stiffness Data

January 13, 2022
BLOG Tips & Tutorials

OVERVIEW


Material property and section (or thickness) data are necessary to compute the stiffnesses of elements. Material property data are entered through Model>Properties>Material, and section data are entered through Model> Properties>Section or Thickness.

Table 1 shows the relevant commands for calculating the stiffnesses of various elements.

 

Element

Material
property data

Section or
thickness data

Remarks

Truss element

Material

Section

Note 1

Tension-only
element

Material

Section

Note 1

Compression-only element

Material

Section

Note 1

Beam element

Material

Section

Note 2

Plane stress
element

Material

Thickness

Note 3

Plate element

Material

Thickness

Note 3

Plane strain
element

Material

-

Note 4

Axisymmetric
element

Material

-

Note 4

Solid element

Material

-

Note 5

Wall element

Material

Thickness

Note 3

Table 1. Commands for Computing Element Stiffness Data

 

 

* Note

  1. For truss elements, only cross-sectional areas are required for analysis. However, the section shape data should be additionally entered for the purposes of design and graphic display of the members.
  2. When a beam element is used to model a Steel-Reinforced Concrete (SRC) composite member, the program automatically calculates the equivalent stiffness reflecting the composite action.
  1. The thickness should be specified for planar elements.
  2. No section/thickness data are required for plane strain and axisymmetric elements as the program automatically assigns the unit width (1.0) and unit angle (1.0 rad) respectively.
  3. The program determines the element size from the corner nodes, and as such no section/thickness data are required for solid elements.

 

 

√ Table of Contents 

1. Area (Cross-Sectional Area)

2. Effective Shear Areas (Asy, Asz)

3. Torsional Resistance (Ixx)

4. Area Moment of Inertia (Iyy, Izz)

5. Area Product Moment of Inertia (Iyz)

6. First Moment of Area (Qy, Qz)

7. Shear Factor for Shear Stress (Qyb, Qzb)

8. Stiffness of Composite Sections

 

 

 

Definitions of section properties for line elements and their calculation methods are as follows:

The user may directly calculate and enter the section properties for line elements such as truss elements, beam elements, etc. However, cautions shall be exercised as to their effects of the properties on the structural behavior. In some instances, the effects of corrosions and wears may be taken into account when computing section properties

MIDAS/Gen offers the following three options to specify section properties:

  1. MIDAS/Gen automatically computes the section properties when the user simply enters the main dimensions of the section.
  2. The user calculates and enters all the required section properties.
  3. The user specifies nominal section designations contained in the database of AISC, BS, Eurocode3, JIS, etc.

In specifying section properties, you can assign individual ID numbers for prismatic, tapered, combined and composite sections. In the case of a construction section, two separate predefined sections are used in combination. Section properties for composite construction sections composed of steel and reinforced concrete vary with construction stages reflecting the concrete pour and maturity.

The following outlines the methods of calculating section properties and the pertinent items to be considered in the process:

 

 

 

Area (Cross-Sectional Area)


The cross-sectional area of a member is used to compute axial stiffness and stress when the member is subjected to a compression or tension force. Figure 1 illustrates the calculation procedure.

Cross-sectional areas could be reduced due to member openings and bolt or rivet holes for connections. MIDAS/Gen does not consider such reductions. Therefore, if necessary, the user is required to modify the values using option 2 above and his/her judgment.

 
Figure 1. Example of Cross-Sectional Area Calculation
Figure 1. Example of Cross-Sectional Area Calculation
 
  

 

Effective Shear Areas (Asy, Asz)


The effective shear areas of a member are used to formulate the shear stiffness in the y and z-axis directions of the cross-section. If the effective shear areas are omitted, the shear deformations in the corresponding directions are neglected.

When MIDAS/Gen computes the section properties by option 1 or 3, the corresponding shear stiffness components are automatically calculated. Figure 2 outlines the calculation methods.

Asy: Effective shear area in the ECS y-axis direction

Asz: Effective shear area in the ECS z-axis direction

 
Figure 2. Effective Shear Areas by Section Shape
Figure 2. Effective Shear Areas by Section Shape

 

 

 

Torsional Resistance (Ixx)


Torsional resistance refers to the stiffness resisting torsional moments. It is expressed as

<Eq. 1>

where,

Ixx: Torsional resistance

T: Torsional moment or torque

θ: Angle of twist

 

The torsional stiffness expressed in <Eq. 1> must not be confused with the polar moment of inertia that determines the torsional shear stresses. However, they are identical to one another in the cases of circular or thick cylindrical sections.

No general equation exists to satisfactorily calculate the torsional resistance applicable for all section types. The calculation methods widely vary for open and closed sections and thin and thick thickness sections.

For calculating the torsional resistance of an open section, an approximate method is used; the section is divided into several rectangular sub-sections and then their resistances are summed into a total resistance, Ixx, calculated by the equation below.

<Eq. 2>

where,

ixx: Torsional resistance of a (rectangular) sub-section

2a: Length of the longer side of a sub-section

2b: Length of the shorter side of a sub-section

Figure 3 illustrates the equation for calculating the torsional resistance of a thin-walled, tube-shaped, closed section.

 

<Eq. 3>

where,

A: Cross-sectional area of the tube

dS: Infinitesimal length of thickness centerline at a given point

t: Thickness of tube at a given point

 

For those sections such as bridge box girders, which retain the form of thick-walled tubes, the torsional stiffness can be obtained by combining the above two equations, <Eq. 1> and <Eq. 3>.

 
Figure 3. Torsional Resistance of a Thin-Walled, Tube-Shaped, Closed Section
Figure 3. Torsional Resistance of a Thin-Walled, Tube-Shaped, Closed Section
 
 
 
Figure 4. Torsional Resistance of Solid Sections
Figure 4. Torsional Resistance of Solid Sections
 
 
 
Figure 5. Torsional Resistance of Thin-Walled, Closed Sections
Figure 5. Torsional Resistance of Thin-Walled, Closed Sections
 
 
 
Figure 6. Torsional Resistance of Thick-Walled, Open Sections
Figure 6. Torsional Resistance of Thick-Walled, Open Sections
 

 

 
 
Figure 7. Torsional Resistance of Thin-Walled, Open SectionsFigure 7. Torsional Resistance of Thin-Walled, Open Sections
 

In practice, combined sections often exist. A combined built-up section may include both closed and open sections. In such a case, the stiffness calculation is performed for each part, and their torsional stiffness is summed to establish the total stiffness for the built-up section.

For example, a double I-section as shown in Figure 8 (a) consists of a closed section in the middle and two open sections, one on each side.

 

-The torsional resistance of the closed section (hatched part)

<Eq. 4>

-The torsional resistance of the open sections (unhatched parts)

<Eq. 5>

-The total resistance of the built-up section

<Eq. 6>

 
 

Figure 8 (b) shows a built-up section made up of an I-shaped section reinforced with two web plates, forming two closed sections. In this case, the torsional resistance for the section is computed as follows:

If the torsional resistance contributed by the flange tips is negligible relative to the total section, the torsional property may be calculated solely on the basis of the outer closed section (hatched section) as expressed in <Eq. 7>.

<Eq. 7>

If the torsional resistance of the open sections is too large to ignore, then it should be included in the total resistance.

 
Figure 8. Torsional Resistance of Built-Up Sections
Figure 8. Torsional Resistance of Built-Up Sections
 

 

 

 

 

Area Moment of Inertia (Iyy, Izz)


The area moment of inertia is used to compute the flexural stiffness resisting bending moments. It is calculated relative to the centroid of the section.

-Area moment of inertia about the ECS y-axis

<Eq. 8>

-Area moment of inertia about the ECS z-axis

<Eq. 9>

 
Element Stiffness DataFigure 9. Example of Calculating Area Moments of InertiaFigure 9. Example of Calculating Area Moments of Inertia
Figure 9. Example of Calculating Area Moments of Inertia
 

 

 

 

 

Area Product Moment of Inertia (Iyz)


The area product moment of inertia is used to compute stresses for non-symmetrical sections, which is defined as follows:

<Eq. 10>

 

Sections that have at least one axis of symmetry produce Iyz=0. Typical symmetrical sections include I, pipe, box, channel and tee shapes, which are symmetrical about at least one of their local axes, y and z. However, for non-symmetrical sections such as angle-shaped sections, where Iyz¹0, the area product moment of inertia should be considered for obtaining stress components.

The area product moment of inertia for an angle is calculated as shown in Figure 10.

 
Figure 10. Area Product Moment of Inertia for an Angle
Figure 10. Area Product Moment of Inertia for an Angle
 
 
 
 
Figure 11. Bending Stress Distribution of a Non-Symmetrical Section
Figure 11. Bending Stress Distribution of a Non-Symmetrical Section
 

The neutral axis represents an axis along which bending stress is 0 (zero). As illustrated on the right-hand side of Figure 1.57, the n-axis represents the neutral axis, to which the m-axis is perpendicular. Since the bending stress is zero at the neutral axis, the direction of the neutral axis can be obtained from the relation defined as

<Eq. 11>

The following represents a general equation applied to calculate the bending stress of a section:

<Eq. 12>

In the case of I shaped section, Iyz=0, hence the equation can be simplified as:

<Eq. 13>

where,

Iyy: Area moment of inertia about the ECS y-axis

Izz: Area moment of inertia about the ECS z-axis

Iyz: Area product moment of inertia

y: Distance from the neutral axis to the location of bending stress calculation in the ECS y-axis direction

z: Distance from the neutral axis to the location of bending stress calculation in the ECS z-axis direction

My: Bending moment about the ECS y-axis

Mz: Bending moment about the ECS z-axis

The general expressions for calculating shear stresses in the ECS y and z-axes are:

<Eq. 14>

<Eq. 15>

where,

Vy: Shear force in the ECS y-axis direction

Vz: Shear force in the ECS z-axis direction

Qy: First moment of area about the ECS y-axis

Qz: First moment of area about the ECS z-axis

by: Thickness of the section at which shear stress is calculated, in the direction normal to the ECS z-axis

bz: Thickness of the section at which shear stress is calculated, in the direction normal to the ECS y-axis

 

 

 

First Moment of Area (Qy, Qz)


The first moment of area is used to compute the shear stress at a particular point on a section. It is defined as follows:

<Eq. 16>

<Eq. 17>

When a section is symmetrical about at least one of the y and z-axes, the shear stresses at a particular point are:

<Eq. 18>

<Eq. 19>

where,

Vy: Shear force acting in the ECS y-axis direction

Vz: Shear force acting in the ECS z-axis direction

Iyy: Area moment of inertia about the ECS y-axis

Izz: Area moment of inertia about the ECS z-axis

by: Thickness of the section at the point of shear stress calculation in the ECS y-axis direction

bz: Thickness of the section at the point of shear stress calculation in the ECS z-axis direction

 

 

 

Shear Factor for Shear Stress (Qyb, Qzb)


The shear factor is used to compute the shear stress at a particular point on a section, which is obtained by dividing the first moment of the area by the thickness of the section.

<Eq. 20>

<Eq. 21>

 
Figure 12. Example of Calculating a Shear Factor
Figure 12. Example of Calculating a Shear Factor

 

 

 

Stiffness of Composite Sections


MIDAS/Gen calculates the stiffness for a full composite action of structural steel and reinforced concrete. Reinforcing bars are presumed to be included in the concrete section. The composite action is transformed into equivalent section properties.

The program uses the elastic moduli of the steel (Es) and concrete (Ec) defined in the SSRC79 (Structural Stability Research Council, 1979, USA) for calculating the equivalent section properties. In addition, the Ec value is decreased by 20% in accordance with the EUROCODE 4.

Equivalent cross-sectional area

Equivalent effective shear area

Equivalent effective shear area

Equivalent area moment of inertia

Equivalent area moment of inertia

where,

Ast1: Area of structural steel

Acon: Area of concrete

Asst1: Effective shear area of structural steel

Ascon: Effective shear area of concrete

Ist1: Area moment of inertia of structural steel

Icon: Area moment of inertia of concrete

REN: Modular ratio (elasticity modular ratio of the structural steel to the concrete, Es/Ec)

 

 

 

 

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Author Information
Young-Wook Hur | Technical Manager, MIDAS IT

Young-Wook has over 8+ years of experience in building design, especially apartment, complex and palnt structures. He is currently working at MIDAS IT as a technical manager, mainly assisting with various projects related to structural engineering.

E-BOOK Element Stiffness Data

For computing the stiffnesses of elements, material property and section (or thickness)
data are needed. Midas Gen can automatically calculate the data.