Curved Beam Strength User Manual

Ver. 3.0

Centre Composite

December 2016

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Curved Beam Strength, Ver. 3.0

User Manual

“Centre Composite”, 2016 (software@composite.lv)

Table of Contents Introduction ......................................................................................................................................................... 1

System requirements .......................................................................................................................................... 1

Theoretical background ....................................................................................................................................... 2

Loads ................................................................................................................................................................ 2

Boundary conditions........................................................................................................................................ 2

Literature ......................................................................................................................................................... 3

User interface ...................................................................................................................................................... 3

Materials .......................................................................................................................................................... 3

Laminate .......................................................................................................................................................... 3

Loads ................................................................................................................................................................ 6

Plot ................................................................................................................................................................... 7

Report generation ........................................................................................................................................... 8

Strength calculation ......................................................................................................................................... 9

Point data ...................................................................................................................................................... 10

Options .......................................................................................................................................................... 10

Appendix A ........................................................................................................................................................ 11

Appendix B ......................................................................................................................................................... 12

Introduction

The CBS (“Curved Beam Strength”) program solves the problem of the bending of the layered

orthotropic curved beams. Any number of layers with different orientations and any number of

materials are allowed. The solution gives the full 3D stress-strain state in all layers in global and

local coordinate systems.

System requirements

Microsoft .Net Framework ver. 4 or higher is required to run this software.

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Theoretical background

The solution is based on the approach described in [1] and modified to account for arbitrary

orientation angles of laminate layers. Internally the CBS program uses the Lekhnitskii’s [2] solution

for stress function of homogeneous curved beam for each layer and the coefficients of stress

function are determined to satisfy the boundary conditions for multilayer beam. Therefore the

obtained solution is exact, except the regions where load is applied and keeping in mind applied

plane strain conditions, as will be discussed later.

The CBS program finds the stresses of the layered curved beam loaded by the force or bending

moment as shown in Figure 1. The beam may consist of any number of layers, made from isotropic

or orthotropic materials with arbitrary angle to the X coordinate axis (see Figure 2) so that 0 fibers

wrap around the corner and 90 fibers are straight.

Figure 1 Geometry of the curved beam with

loads.

Figure 2 Direction of coordinate axes.

Loads

Any combination of bending moment, M, radial (PZ) and tangential (PX) forces components and

temperature load, T, is allowed. In addition the user may specify axial deformation 0. It is

assumed that positive moment opens the angle.

Boundary conditions

Internal and external surfaces of the beam are stress free. Generalized plane strain conditions are

assumed by default, i.e. 0;0;0 xyyzyy . However, accurate 3D finite element analysis

reveal that in the middle part of the long (in axial direction) curved beam the stress state is close to

the next one: 0;0;0 xzyzyy . These conditions will be applied if you chose “3D Beam”

option. It should be noted, that this option have little effect on the stresses in the “XZ” plane and

usually can be ignored in engineering analysis.

By default (“Fixed ends”) the axial deformation of the beam is constant and equal to the user

defined, 0. The “Free ends” option solves the problem for free lying beam, where the total axial

force is assumed to be equal zero and axial deformation is unknown. The axial deformation will be

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determined during the course of the solution and stored as yy. The axial deformation defined by

the user as 0 is ignored in this case.

Literature

1. Peck S.O. „Curved Laminated Beams Subjected to Shear Loads, Moments and Temperature

Changes”, Composite Structures: Theory and Practice, ASTM STP 1383, P. Grant and C.Q.

Rousseau, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000,

pp. 437-451.

2. S.G. Lekhnitskii, Anisotropic plates. Gordon and Breach Science publishers, 1984.

User interface

Graphical user interface is used to specify material properties, apply loads and view the results. The

main window has four tabs:

Materials – allows to specify elastic and strength properties of each material

Laminate – allows to specify laminate lay-up

Loads – allows to specify geometry and loads

Plot – allows to visualize the results

Materials

Figure 3 Table with material’s elastic and strength data.

In the “Materials” tab the materials properties are defined. Each material has a name and

sequential number (in the first column). User has to input elastic properties of each material.

Coefficients of thermal expansion also required if the thermal load will be used. Strength properties

are required only for strength plot, in other case they can be omitted.

Laminate

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Figure 4 User interface for the definition of the laminate layup.

In the “Laminate” tab the material, thickness and orientation angle for each ply have to be defined.

The user may choose the material name from the drop-down menu or input its sequential number

in the “Mat. Nr.” column.

To simplify laminate definition, several additional options are present in this tab. ”Symmetry” drop-

down menu allows to create symmetric or anti-symmetric laminates. The current laminate will be

mirrored and added to the existing plies to form symmetric or anti-symmetric laminate. If

“Symmetric / Mid-Ply” or “Anti-symmetric / Mid-ply” options are used, current last ply will be used

as mid-ply of new laminate. The “Layer thickness” option can be used to change the thickness of all

layers. ”Rotate” option will rotate all layers by specified angle.

Additionally, composite layup stacking sequence may be entered as text string in compressed

format, as shown in Figure 5. The notation for layup stacking sequence definition follows ASTM

standard D6507-11 (except material codes). Two options are available:

1. By pressing “Get” button, current laminate stacking sequence is analyzed and written in

text-box as string in compressed form.

2. By pressing “Apply” button, the text string from text-box is expanded and resulting stacking

sequence will be set as current. The material number and thickness of the first layer will be

used for all layers in this case.

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Figure 5 Laminate layup definition via layers stacking sequence notation.

Ply directions and number of layers are indicated using the laminate orientation code as follows:

[1m1/2m2/…]ns (or [1:m1/2:m2/…]:ns in computer format), where

1 – ply orientations (degrees) of the laminate staking sequence m1 – number of plies at each particular orientation (not used for single ply) n – number of repetitions of the bracketed group of plies s – indication of geometric symmetry …\ – indication of the last ply for mid-plane symmetry

+-1 – short format for /1/-1/ sequence (…) – group of plies (nested groups are allowed).

Examples:

Stacking sequence Compressed form 0/90/90/0 [0/90]:s

0/0/90/0/0 [0:2/90\]:s 45/-45/-45/45 [+-45]:s 45/0/90/90/0/0/90/90/0/-45 [45/((0/90):s):2/-45]

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Loads

Figure 6 User interface for the definition of the loads and boundary conditions.

In the “Loads” tab the geometrical and load parameters can be defined.

User has to enter internal radius of the beam, and, for L-shaped beam, the length of the arm, where

the force is applied. The thickness will be determined based on the laminate lay-up. It is assumed

that the load is applied at the mid-line.

The next group of parameters defines the loads applied to the beam. Any combination of bending

moment, radial and tangential force, temperature difference and axial deformation is allowed. All

loads are per unit length.

The last options define the general boundary conditions: “Plane strain” or “3D beam” and “Fixed

ends” or “Free ends”. These options are discussed in the “Theoretical background” section.

Simple mathematical expressions with trigonometric functions can be used in this form, as shown

in Figure 6. “Sqrt” function is not supported, but it can be replaced by power 0.5, for example

“(3^2+4^2)^0.5”.

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Plot

Figure 7 Example of the solid plot with distribution of the radial stress.

The “Plot” tab allows to visualize the results and generate report file.

All six components of stress or strain in global or local (ply) coordinate systems can be plotted.

Please note, that engineering strain notation is used. The global coordinate system is cylindrical so

that S_ZZ is radial stress and S_XX is circumferential stress. In local (ply) coordinate system the

fibers are oriented along the X axis.

Please note, that even so the solution is exact, stresses in this plot are calculated only at midplane

of the layer, therefore for laminate with few thick layers such plot shows only approximate stress

distribution. You can subdivide each thick layer into several layers to increase accuracy.

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Figure 8 Example of the chart plot with through thickness stress distribution.

To create chart user has to enter cross-section angle and check the “Chart” check-box. Then choose

the stress/strain component you want to plot. The stress state for this plot is evaluated at three

points for each layer.

Report generation

Report can be generated with input data and results for specific cross-section. User has to enter the

cross-section angle for which report will be generated. The report file includes:

Input data (loads and boundary conditions) used for calculation

Material data

Laminate layup

Maximum radial stress for given cross-section

Final angle of the curved beam

Stress/strain data for each layer in given cross-section

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Figure 9 Window with generated report.

Strength calculation

The program calculates the beam strength based on three failure criteria: Max. Stress, Hashin and

Chang-Springer. The lamina strength properties need to be defined in “Materials” section before

strength calculations. The program calculates one or several failure indexes (depending on the

failure criteria), finds the maximal index in each point and plots this value. The expressions for

failure indexes used for calculation are presented in the Appendix A.

At least three positive strength values have to be defined to calculate strength indexes: 11+, 22+,

12. If some of the remaining strength values are missing or negative, next assumptions will be

used:

11- = 11+

22- = 22+

33+ = 22+

33- = 22-

13 = 12

23 = 12 If there are insufficient data for strength calculation for some criterion (see App. A), the value

“9999” will be used as strength index.

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Point data

Figure 10 Example of the point data results.

Any time solid plot (with stress/strain or strength data) is shown on the screen, the user may click

on the beam and get the report for the clicked point. The point data report includes the layer

number and coordinates of the point, full stress/strain state in global and local coordinate systems

and failure indexes for three strength criteria. The expressions used for failure indexes calculations

are shown in Appendix A.

If one of the failure indexes is equal “9999”, error was encountered during strength indexes

calculation; most likely some of the material’s strength data is missing.

Options

In the “Options” menu user can change the format of the solid plot (color or grayscale) and

background color of the images. Also the number of points per layer, where stresses will be

calculated for the report, can be changed.

Figure 11 Options dialog.

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Appendix A

According to the Maximum Stress criterion, six failure indexes Fi are calculated as follows:

0,

0,

11

111

x

x

x

x

F

0,

0,

22

222

y

y

y

y

F

0,

0,

33

333

zz

zz

F

12

4

xyF

23

5

yzF

13

6

xzF

For the Hashin criterion four failure indexes are calculated:

0,0

0,1 22

2

12

2

111

x

xxzxyx

F

0,0

0,112

x

x

x

F

0,0

0,111 22

2

12

2

2

23

2

2

223

zy

zyxzxyzyyzzyF

0,0

0,1

1

4

11

2

1

22

2

12

2

2

23

2

23

2

23

22

22

4

zy

zyxzxy

zyyzzyzy

F

For Chang-Springer criterion one failure index is calculated:

0,0

0,

2

13

2

331

z

z

xzzz

F

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Appendix B

To check the correctness of the used analytical solution, the CBS tool results were compared with

accurate 3D finite element solution. In the FE solution each physical layer of the laminate was

represented by 3 layers of the finite elements. 8-node solid elements were used.

Step 1: open new project and enter material data in the first tab as shown in Figure 12.

Elastic material data for lamina:

Material h

mm E11

MPa E22

MPa

E33

MPa

G12

MPa

G13

MPa

G23

MPa

CFRP 0.19 143e3 9300 9300 0.29 0.29 0.4 7600 7600 3321

Coefficient of thermal expansion:

Material

CFRP -7.6e-7 36e-6 36e-6

Figure 12 Material data for test problem.

Step 2: enter the lay-up of the laminate as shown in Figure 13.

Laminate lay-up:

Material Lay-up

CFRP [-45/45/-45/90/45/0/45/90/-45/45/-45]3

Figure 13 Laminate lay-up.

Step 3: enter the geometry and load data.

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Let’s find the stress state in the beam with internal radius 5 mm, loaded by end force equal 5 N per

unit length, applied at distance 20mm from curved portion of the beam with direction 45 degrees,

as shown in Figure 14a. The beam is also loaded by axial deformation equal -0.06%. The

corresponding input data for CBS tool is shown in Figure 14b.

a) b)

Figure 14 Loading scheme (a) and input parameters for CBS tool (b).

Step 4: results visualization.

After all necessary input parameters are entered, results can be plotted in the “Plot” tab.

Figure 15 Radial stress distribution in the beam (solid plot).

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Figure 16 Radial stress distribution in the cross-section with angle 45 degrees (chart plot).

Figure 15 and Figure 16 show the radial stress distribution in the curved portion of the beam. User

can enter the cross-section angle and press “Print” to get the whole stress/strain state in the text

format, which can be imported in Excel (or any other spreadsheet software) to generate

customized plots. Figure 17 and Figure 18 show the comparison of the analytical solution for

stresses with 3D finite element solution.

Figure 17 Comparison of the radial stresses obtained by CBS tool with finite element solution.

0

0.5

1

1.5

2

2.5

3

3.5

5 6 7 8 9 10 11 12

z,

MP

a

z, mm

CBS tool

FEM

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Figure 18 Comparison of the xx and yy stresses obtained by CBS tool with finite element solution.

“Spring-In” effect

The CBS tool can be used to assess the severity of so called “spring-in” effect for the given laminate.

User can apply cooling of the laminate as the only loading factor, and the CBS program will

calculate the final angle of the cooled laminate (with the assumption, that the initial angle equal 90

degrees). Using thermal expansion coefficients of the lamina as shown in Figure 12 and cooling by

100 degrees for the curved laminate with the internal radius equal 20 mm (Figure 19a), the

resulting radial stresses are show in Figure 19b (analytical solution and FEM).

a) b)

Figure 19 Input data (a) and radial stresses (b) for the thermal cooling problem.

-100

-80

-60

-40

-20

0

20

40

60

80

5 6 7 8 9 10 11 12

Stre

ss, M

Pa

z, mm

sx CBS tool

sy CBS tool

sx FEM

sy FEM

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

20 21 22 23 24 25 26 27

Stre

ss, M

Pa

z, mm

Sz (FEM)

Sz (CBS)

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The obtained results show minimal level of the radial stresses due to cooling, however the “spring-

in” angle is equal 0.41 (90-89.59). The angle of the laminate after cooling can be found in the

report, as shown in Figure 20 (scroll down the report to find the calculated angle of laminate).

Figure 20 L-shaped laminate angle due to cooling by 100 degrees.

As a next step user can estimate the induced stresses in the part in the fitted position. Apply the

opening moment equal 190 N*mm per unit length to deform the cooled laminate back to the initial

angle, as shown in Figure 21. Resulting radial stresses are shown in Figure 22.

Figure 21 L-shaped laminate angle after applying compensating moment.

Figure 22 distribution of the radial stresses in the cooled laminate due to applied compensating

moment.