This workbook performs an interlaminar strength analysis of bends in composite laminates per the method of ESDU 94019, "Through-the-Thickness Stresses and Failure in the Corner Radius of a Laminated Composite Section".   Both interlaminar shear (ILS) and interlaminar tension (ILT) are taken into account.

Features and Capabilities

• An iterative solution of the interaction of interlaminar shear and tension (or compression) is performed to find the location with the lowest margin of safety.
• ILS and ILT are graphed vs both radial position and angle around the bend to provide a sanity check of the results.
• A balanced freebody diagram of the bend being analyzed is generated on the fly.
• Individual or multiple copies of the worksheet can be incorporated into existing workbooks.
• The VBA module that performs the calculations can be used independently of the worksheet.

Usage

The Bend ILS/ILT workbook consists primarily of a VBA module that performs all the calculations and a worksheet that aids in visualizing and verifying those calculations.  The VBA module can be copied into any other workbook independently of the visualization sheet.  It is recommended that the visualization sheet accompany the module as verification can frequently be more time consuming than the actual calculations.

The inputs are identified in blue on the worksheet and are shown in Figure 1.  Inputs consist of geometry, applied loads, laminate properties, and interlaminar allowables.

The geometry inputs are illustrated in Figure 2.  The bend angle, BA, is measured in degrees.  The inner radius (bend radius), IR or a, and the laminate thickness, T, are measured in inches or mm.

Loads are applied on the θ=0 face with the sign convention shown in Figure 2.  M₀, the running moment, is measured in in·lb/in or N·mm/mm.  N₀, the running axial load, and S₀, the running shear, are measured in lb/in or N/mm.

The interlaminar allowables are all input in psi or kPa.  f_s-all is the shear (ILS) allowable.  f_t-all is the tensile (ILT) allowable.  f_c-all is the compressive (ILC) allowable.

The laminate properties are entered in Msi or GPa for the moduli (E & G).  The Poisson's ratios, PR, are nondimensional.  The x, r & t subscripts represent the axial, radial and tangential directions as shown in Figure 2.  Two subscripts used together refer to the plane defined by those vectors.

The outputs are identified in violet on the worksheet and are shown in Figure 3.  Outputs consist of reaction loads, critical location information, and margin of safety parameters.

The reaction loads are located at the θ=BA face and are plotted in the free body diagram in Figure 4.  M_BA, N_BA, and S_BA have the same units as their corresponding applied loads.

The critical location is defined by r_crit and θ_crit and is plotted as a red dot in Figure 4.  The interlaminar stresses at the critical location, σ_crit and τ_crit, are displayed in psi or kPa.  The critical radial location, %T, is displayed as a thru-the-thickness percentage.

The stress ratios used to calculate the reserve factor and margin of safety are R_radial and R_shear.
R_radial=σ_crit/(f_t-all or f_c-all).  R_shear=τ_crit/f_s-all.  R_eq is the equivalent stress ratio based on a circular interaction, R_eq²=R_radial² + R_shear².  RF is the reserve factor, RF=1/R_eq.  MS is the margin of safety, MS=RF-1.

The various parameters used in ESDU 94019 are summarized in the "Intermediate Calculations" table shown in Figure 5.

The VBA function iterates to find the minimum RF at θ=0, θ=BA, and θs in between.  The blocks at the bottom of the worksheet display seven iterations in each of these ranges.  The results of these iterations are summarized in the "Comparison of Worksheet Calcs to Function Results" table, shown in Figure 6, along with the results from the VBA function.  The worksheet and function values should match.

The solution parameters are plotted vs. %Thickness and θ, as a secondary verification that the critical location has been found.  The objective of the iterations is to find the values of r and θ where the equivalent stress ratio, R_eq is a maximum.  Since R_eq is an interaction of R_radial and R_shear such that
R_eq² = R_radial² + R_shear²
it was easier to iterate on R_eq² (designated RR).  The Newton-Raphson method, which uses the parameters ∂RR/∂r and ∂RR/∂θ, is executed.  ∂²RR/∂r² and ∂RR²/∂θ² are also calculated to determine if a point is a min or max.  The ratio of each of these parameters against their maximum value is plotted, as shown in Figures 7 & 8.  σ and τ are also included.

If the critical RF occurs where 0<θ<BA, ∂RR/∂r and ∂RR/∂θ will be zero.

VBA and Name Considerations

If you plan to incorporate this worksheet into another workbook, here are some issues to keep in mind.

1. The freebody diagram on the visualization sheet employs VBA macros to maintain the correct aspect ratio when updated.  Also, some of the calculations are performed by VBA functions.  If this worksheet is copied into other workbooks, the following code will have to accompany it.
1. The VBA code contained in ThisWorkbook under 'Microsoft Excel Objects' must be copied to ThisWorkbook in the destination workbook.  It identifies the charts that will need to be rescaled on update.  It only runs when the workbook is initially opened.
2. The VBA class module clmChartEventClass must be copied to the destination workbook (just drag and drop the module).  It performs the rescaling of the diagrams.
3. The VBA module ILS_ILT... must be copied to the destination workbook (just drag and drop the module).  It contains functions that perform the iterative solution.  Function clmBendILS_ILT takes all the geometry, material, and loads inputs for the bend and returns the reserve factor, critical stresses, and critical location.  Function clmMaxMag returns the value with the largest magnitude from a list of values.
2. The following defined names are referenced by VBA code.  Don't change them.  The named ranges can be reviewed with the NAME MANAGER in the FORMULA ribbon.
   1. EnvFBD is a named range for the cells containing the bounds of the freebody diagram.
   2. StressExtents is a named range for the cells containing the bounds of the stress axes in the BendAngleDiagram and the ThicknessDiagram.
   3. FreeBodyDiagram is the chart on which the freebody diagram is plotted.
   4. BendAngleDiagram is the chart on which the solution parameters are plotted versus bend angle.
   5. ThicknessDiagram is the chart on which the solution parameters are plotted versus thickness.