Christos C. Chamis and Pappu L.N. Murthy

Glenn Research Center, Cleveland, Ohio

Buckling and Vibration of Fiber Reinforced

Composite Plates With Nanofi ber Reinforced

Matrices

NASA/TM—2011-217282

December 2011
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Christos C. Chamis and Pappu L.N. Murthy

Glenn Research Center, Cleveland, Ohio

Buckling and Vibration of Fiber Reinforced

Composite Plates With Nanofi ber Reinforced

Matrices

NASA/TM—2011-217282

December 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2011-217282 1

Buckling and Vibration of Fiber Reinforced Composite Plates 

With Nanofiber Reinforced Matrices 

Christos C. Chamis and Pappu L.N. Murthy 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Anisotropic composite plates were evaluated with nanofiber reinforced matrices (NFRM). The 

nanofiber reinforcement volumes ratio in the matrix was 0.01. The plate dimensions were 20 by 10 by 

1.0 in. (508 by 254 by 25.4 mm). Seven different loading condition cases were evaluated for buckling: 

three for uniaxial loading, three for pairs of combined loading, and one with three combined loadings. 

The anisotropy arose from the unidirectional plates having been at 30° from the structural axis. The 

anisotropy had a full 6 by 6 rigidities matrix which were satisfied and solved by a Galerkin buckling 

algorithm. For vibration the same conditions were used with the applied cods about a small fraction of the 

buckling loads. The buckling and vibration results showed that the NFRM plates increased about twice 

those with conventional matrix. 

Introduction 

Considerable research activities in nanocomposites have been actively pursued recently. One aspect 

of this research is on nanofiber reinforced matrices (NFRM) which can subsequently be used as matrices 

in conventional reinforced fiber composites. As of to date, the buckling of composite plates with NFRM 

has not been investigated yet. This is the objective of the present paper, to use a commercial composites 

loaded at 30° (Fig.1) from the material axis for seven different cases. The results are compared with fiber 

composite without the NFRM that is an AS/IMHS (AS graphite fiber/intermediate modulus high strength 

matrix) for all the loading conditions and for both buckling and vibration. Because the 30° off-axis 

composite has a full 6×6 bending stiffness matrix, a Galerkin algorithm is used to evaluate the buckling 

factor. The bending stiffness required for the buckling of off-axis loaded anisotropic composite were 

generated by the in-house integrated computer code in composite mechanics. The comparison of results 

without and with the NFRM show about twice the buckling factor for all loading conditions with only 

1 percent of nanofiber reinforced matrix. Results are presented for the buckling factor mode and its 

corresponding buckling factor for each loading condition. The graphical results shown will be for the type 

of loading condition, the 3–D buckling factor mode shape, a contour plot of the buckling factor mode and 

a convergence plot of the buckling algorithm. One interesting result is that the shear buckling factor does 

not converge and requires a tiny small loading in one or both structural axes to enhance the convergence. 

The theory was developed when the first author was a graduate student at Case Western Reserve 

University and it is described in Reference 1. The application to the anisotropic plates was performed 

when he started working at NASA Lewis Research Center (now Glenn Research Center) Reference 2. 

The programming details source code and several sample cases are in the appendices of Reference 2. The 

brief description of the theoretical background is from Reference 2 also. 

The underlying theory for buckling factors of anisotropic plates is described in References 1 and 2 

with pertinent discussions in References 3 to 5. Briefly, this theory consists of expressing the potential 

energy of plate in terms of displacement variables. Taking the variation of the potential energy function 

yields the field equation and the corresponding boundary conditions. The resulting system then is solved 

by the assumed mode technique in conjunction with the Galerkin method (Fig. 2). 
NASA/TM—2011-217282 2

The equation resulting after the variation of the energy function is 

  

 











    





   

   

  

a

0 y

b

0 xy23yy22xx21

b

0 x

a

0 xy13yy12xx11

xyyy23 yyyxyxyxxX

xxxx11 12 33 yyyy22xxyy xyxx13

a

0

b

0

0dxw,wD2wDwD

dyw,wD2wDwD

dxdywwNwN2wNwD4

 w,D4wDwD2D2wD

,, ,

,, ,

, , ,,

, , ,

(1) 

(The notation is defined in Appendix A of Reference 2). The area integral represents the field 

equation, and the line integrals represent the boundary conditions. 

The assumed buckling factor mode described in Reference 2 is represented by a Fourier double sine 

series. This mode satisfies the imposed boundary conditions, but it does not satisfy the natural boundary 

conditions if the material and structural axes do not coincide. However, the mode is forced to satisfy the 

natural boundary conditions approximately through the Galerkin method as discussed in Reference 2. 

Substituting the assumed mode in Equation (1), applying the Galerkin method, and carrying out the 

algebra result in a set of linear equations for buckling 

[K] {w} = λ [L] {w} (2) 

and for vibration is 

       wwL-KwM  (3) 

where K is the stiffness matrix, λ is the eigenvalue [L] is the loading matrix [M] is the mass matrix, w 

is the acceleration and   w is the displacement vector. This system is coupled for either a combination of 

shear and normal loads and/or noncoincident material and structural axes. 

The eigenvalue problem is solved by using the Power method, which is a highly effective iterative 

numerical technique in seeking the largest eigenvalue of the system. The indicial equations which were 

used to generate this system and the Power method are given in Appendix B of Reference 2 in outline 

form. 

The source code has been reprogrammed by Dr. Murthy in Metlab. The results for both presented 

subsequently are from the Metlab reprogrammed. There are two portions to the results: (1) is the 

simulation of the NFRM which is performed by a composite mechanics code ICAN/JAVA in order to 

obtain the nanofiber reinforcement in the matrix and (2) using the NFRM in the mechanics computer code 

again to simulate the effects of the reinforced matrix in the buckling and vibration behavior of the 

composite plate for different combination of loads. 

The nanofiber reinforced matrix (NFRM) is simulated by using the continuous solid nanofiber 

properties shown in Table 1 in an intermediate modules high strength matrix in Table 2 assuming a quasi 

isotropic fiber arrangement. The properties of this lay-up arrangement are shown in Table 3. Next, the 

conventional fiber in Table 4 was used with the NFRM to obtain the bending rigidities shown in Table 5. 
NASA/TM—2011-217282 3

Buckling Results and Discussion 

The composite plate shown in Figure 1 was selected for the buckling evaluation. The fiber 

reinforcement was 0.6 volume ration in a NFRM. The bending rigidities shown in Table 5 were 

transformed to 30° from the structural x-axis to obtain the anisotropic composite plate. The transformed 

rigidities were input into the buckling routine with the plate geometry (20×10×1.0) length, width, 

thickness and the results were obtained for seven different cases of combined in plane loads. For 

convenience all the inplane loads were assumed to be of equal magnitude. Two different buckling results 

are shown for conventional matrix Table 6 and for NFRM Table 7. 

The buckling factor conditions indicated in each table with a check mark. Note that the structural axis 

has only one check mark in the first column. The y axis has one check mark in the second column. The xy loading has a check mark in the third column combined load cases leave check marks in the appropriate 

column to indicate the combined loading. 

Comparing the buckling results of the two tables it is seen that the plates with the NFRM is about 

twice of the conventional matrix. 

Buckling Factor Mode Shapes 

The buckling factor mode shapes of anisotropic plates is very interesting. In this evaluation buckling 

factor mode shapes were evaluated for each loading condition that means there are seven charts of 

buckling factor mode shapen. Each chart has four parts in it. The top left (a) is a contour part of the modal 

shape. The upper left (b) is a 3–D view of the buckling factor mode shape. The lower left (c) is a buckling 

factor convergence plot. The lower right (d) is a box with the following information in it: first line loading 

conditioning, second line, load axis; third line the buckling factor of that case. 

Part (a) in Figure 3(a) depicts the contour plot of the buckling factor mode shape for buckling factor 

in the x direction. Part (b) depicts a 3–D presentation of the buckling factor mode shape. Part (c) depicts 

the buckling factor convergence in 8 interactions. Part (d) is the box with all the information of the mode 

shape. Note that the plate buckled in two modes. The buckling factor is 3673 lb (163.38 kN). 

Figure 4 depicts analogous information or the buckling factor in the transverse direction. Part (a) is 

the contour plot; part (b) is the 3–D view of the mode shape; part (c) is the convergence plot in 6 

interactions and part (d) is the buckling information box. The buckling factor is 1097 lb (48.79 kN). 

Figure 5 depicts similar information of the shear load buckling shape. Note that this mode shape 

symmetric about the material axis 30° from the x axis. Note that buckling factor converged in 7 

interactions. Note also that the very small load of 1 lb (4.448 N) in the x- and y-axis as was previous 

explained. The buckling factor is 2220 lb (98.75 kN). 

Figure 6 depicts the buckling factor mode shape of bi-axial combined loads. Note that this contour 

plot depicts a symmetric mode shape about the material axes also. Note also the buckling factor 

converged in 6 iterations as well. The buckling factor is 875 lb (3.892 kN). 

Figure 7 is the buckling factor mode shape of combined buckling factor axial and shear. The contour 

plot of this mode shape also is symmetric about the material axis. Note the buckling factor converged in 

7 iterations. The buckling factor is 1520 lb (6.761 kN). 

Figure 8 is the buckling factor mode shape of a y-axis and shear load. As was the case previously the 

contour plot show symmetry about the material axis. The buckling factor converged in 8 iterations with a 

value of 808 lb (35.94 kN). 

Figure 9 depicts the buckling factor mode shape of combined loads x, y, and xy. The buckling factor 

converged in 8 iterations to a value of 680 lb (30.25 kN). 
NASA/TM—2011-217282 4

Vibration Results and Discussion 

The composite plate shown in Figure 1 was selected for the dynamic evaluation. The fiber 

reinforcement was 0.6 volume ration in a NFRM. The bending rigidities shown in Table 5 were 

transformed to 30° from the structural x-axis to obtain the anisotropic composite plate. The transformed 

rigidities were input into the frequency routine with the plate geometry (20×10×1.0) length, width, 

thickness and the results were obtained for seven different cases of combined in plane loads. For 

convenience all the inplane loads were assumed to be of equal magnitude. Two different frequency results 

are shown for conventional matrix Table 6 and for NFRM Table 7. 

The vibration frequency evaluations indicated in each table with a check mark. Note that the 

structural axis has only one check mark in the first column. The y axis has one check mark in the second 

column. The x-y loading has a check mark in the third column combined load cases leave check marks in 

the appropriate column to indicate the combined loading. 

Comparing the frequency results of the two tables it is seen that the plates with the NFRM is about 

60-percent of the conventional matrix. 

Vibration Frequencies Mode Shapes 

The vibration frequencies mode shapes of anisotropic plates is very interesting. In this evaluation 

vibration frequencies mode shapes were evaluated for each loading case that means there are seven charts 

of vibration frequencies mode shapes. Each chart has four parts in it. The top is a counter part of the 

modal shapes. The center is a 3–D view of the buckling factor and the vibration frequencies mode shape. 

The lower part shows the angle of the anisotropy. 

The top part in Figure 10 depicts the counter plot of the load factor and the vibration frequencies 

mode shape for load factor and frequency in the x direction. The center part is a 3–D mode shape. The 

lower part is the box with all the information of the mode shape. Note that the plate buckled in two 

modes. The load factor is 3673 lb (163.38 kN) and the frequency is 2005 Hz. 

Figure 11 depicts analogous information of the load factor and the transverse direction and the 

corresponding frequency. The load factor is 1097 lb (48.79 kN) and the frequency is 1818 Hz. 

Figure 12 depicts similar information of the shear load frequency shape of the load factor. This mode 

shape is symmetric about the material axis 30° from the x axis. Note that frequency converged. Note that 

the very small load of 1 lb (4.448 N) in the x and y axis as was previous explained. The load factor is 

2220 lb (98.75 kN) and the frequency is 2245 Hz. 

Figure 13 depicts the frequency mode shape of bi-axial combined loads. Note that this counter plot 

depicts a symmetric mode shape about the material axes also. The load factor is 875 lb (3.892 kN). The 

frequency is 1819 Hz. 

Figure 14 is the load factor and frequency mode shape of combined loads axial and shear. The load 

factor counter plot shows symmetric about the material axis. The load factor is 1520 lb (6.761 kN) and the 

frequency is 2121 Hz. 

Figure 15 is the load factor and frequency mode shape of a y-axis and shear load. As was the case 

previously, the counter plot show symmetry about the material axis. The frequency converged in 

8 iterations with a value of load factor 808 lb (35.94 kN) and the frequency is 1909 Hz. 

Figure 16 depicts the load factor and frequency mode shape of combined loads x, y, and xy. The load 

factor is 680 lb (30.25 kN) and the frequency is 1898 Hz. 

Figure 17 shows the first six lowest frequencies of a square composite plate. Note that as the number 

of frequencies increases the mode shapes increase also. 

Figure 18 is a similar graph of a rectangular anisotropic composite plate. Note that the two plates 

vibrate differently. For example, the four vibration modes occur in the fourth frequency modes in the 

rectangular plate whereas the four modes occur in the fifth vibration frequency mode in the square plate. 

Figure 19 depicts the two lower modes as a function of load. Interesting to note that as the load 

increases both frequencies decrease with the first frequency decreasing at a faster rate at higher load. 
NASA/TM—2011-217282 5

Figure 20 is a graphical presentation of the orientation angle effects on the first frequency. Note that 

the frequency decreases nonlinearly at higher angles. 

Figure 21 shows that the error decreases nonlinearly and at a slower rate as the number of terms in the 

Galerkin procedure increases. 

Figure 22 shows the highly nonlinear effects of increasing load in the natural frequencies reaching 

zero at some load which corresponds to the buckling load. Note that the 45 curve reaches “zero” before 

the 60 curve. It also shows that at higher ages (>45) the frequencies reach “zero” progressively. 

Concluding Remarks 

The concluding remarks of a buckling of anisotropic composite plates with NFRM are as follows. 

The nanofiber reinforcement has a major effect in the buckling factor. All the cases investigated for 

combined loads showed the same about twice benefit compared to conventional matrix. All buckling 

factor mode shape were single except the one loaded alone the structure x-axis which had double mode. 

All buckling factor mode contour plot showed symmetry about the matrix axis except the first case which 

showed symmetry about 45° to the structural x-axis. The shear loading only case required a small load in 

the two other orthogonal axes to expedite convergence. All buckling factor converged between 6 and 8 

iterations. The buckling algorithm is very effective and its convergence is efficient. The simulation of the 

nanofiber reinforced matrix is also an effective means to obtain the influence of nanofibers in a matrix 

first and then combine the NFRF with the conventional fibers for other structural evaluations. 

The concluding remarks of frequency response of anisotropic composite plates with NFRM are as 

follows. The nanofiber reinforcement has a major effect in the vibration frequencies. All the cases 

investigated for combined loads showed the same about 60 percent benefit compared to conventional 

matrix. All vibration frequencies of the mode shape are single. All vibration frequencies mode counter 

plot showed symmetry about the matrix axis except the first case which showed symmetry about 45° to 

the structural x-axis. The shear loading only case required a small load in the two other orthogonal axes to 

expedite convergence. All vibration frequencies converged between 6 and 8 iterations. The frequency 

algorithm is very effective and its convergence is efficient. The simulation of the nanofiber reinforced 

matrix is also an effective means to obtain the influence of nanofibers in a matrix first and then combine 

the NFRF with the conventional fibers for other structure evaluation loading. The loading has severe 

effects on the frequency. All frequencies derease nonlinearly with increasing load reaching “zero” at 

higher loads <45 and the “zero” load decreases at loads >45. Frequencies decrease nonlinearly with 

increasing orientation angle. Mode shapes of square and rectangular plates differ at which frequency 

certain mode shapes will occur at a lower load. 

References 

1. Chamis, C.C.: Buckling of Anisotropic Composite Plates. J. Structural Div., ASCE, vol. 95, 

no. ST10, Oct. 1969, pp. 2119–2139. 

2. Chamis, C.C.: Theoretical Buckling Loads of Boron/Aluminum and Graphite/Resis Fiber-Composite 

Anisotropic Plates. NASA TN D-6572, December 1971. 

3. Wang, James T.-S.: On the Solution of Plates of Composite Materials. J. Composite Mat., vol. 3, 

July 1969, pp. 590–592. 

4. Hsu, T.M.: Buckling of Anisotropic Plates, Discussion J. Structural Div., Am. Soc. Civil Eng., 

vol. 96, no. ST7, July 1970, p. 1604. 

5. Chamis, C.C.: Buckling of Anisotropic Plates, Closure and Errata. J. Structural Div., Am. Soc. Civil 

Eng., vol. 97, no. ST3, Mar. 1971, p. 960. 
NASA/TM—2011-217282 6

TABLE 1.—NANOFIBER (TRIAL FIBER) 

[TRIAL FIBER MODULUS 123M FOR SINGLE FIBER 3–D STRESS ANALYSIS.] 

Description Symbol Value Value in SI units 

Number of fibers per end Nf 100.0 100 

Filament equivalent diameter df 4.0×10–10 in. 1.016×10–8 mm 

Weight density Rhof 0.06 lb/in.3 1.661×10–6 kg/mm3

Normal moduli (11) Ef11 1.0×109 psi 689.5×1010 Pa 

Normal moduli (22) Ef22 41.0×107 psi 282.7×1010 Pa 

Poisson’s ratio (12) Nuf12 0.2 non-dim. 0.2 non-dim. 

Poisson’s ratio (23) Nuf23 0.35 non-dim. 0.35 non-dim. 

Shear moduli (12) Gf12 2.0×107 psi 13.79×1010 Pa 

Shear moduli (23) Gf23 1.5×107 psi 10.34×1010 Pa 

Thermal expansion coefficient (11) Alfaf11 –6.0×10–8 in./in./F -10.8×10–8 /C 

Thermal expansion coefficient (22) Alfaf22 6.0×10–6 in./in./F 10.8×10–6 /C 

Heat conductivity (11) Kf11 400.0 BTU/hr/in./F 8.3075 W/mm/C 

Heat conductivity (12) Kf12 40.0 BTU/hr/in./F 0.83075 W/mm/C 

Heat capacity Cf 0.17 BTU/lb/F 712.3 J/kg/C 

Dielectric strength (11) Kef11 0.0 V/in. 0.0 V/mm 

Dielectric strength (22) Kef22 0.0 V/in. 0.0 V/mm 

Dielectric constant (11) Gammaf11 0.0 in./V 0.0 mm/V 

Dielectric constant (22) Gammaf22 0.0 in./V 0.0 mm/V 

Capacitance Cef 0.0 V 0.0 V 

Resistivity Ref 0.0 Ω-in. 0.0 Ω-mm 

Tensile strength SfT 1.0E7 6895×106 Pa 

Compressive strength SfC 9.0E6 psi 6205×106 Pa 

Shear strength SfS 5.0E6 psi 3447×106 Pa 

Normal damping capacity (11) psi11f 0.03%Energy 0.03%Energy 

Normal damping capacity (12) psi22f 0.4%Energy 0.4%Energy 

Shear damping capacity (12) psi12f 0.4%Energy 0.4%Energy 

Shear damping capacity (23) psi23f 0.8%Energy 0.8%Energy 

Melting temperature TMf 6000.0 F 3315.6 C 
NASA/TM—2011-217282 7

TABLE 2.—CONVENTIONAL MATRIX (IMHS) 

[INTERMEDIATE MODULUS 123M FOR SINGLE FIBER 3–D STRESS ANALYSIS.] 

Description Symbol Value Value in SI units 

Weight density Rhom 0.044 lb/in.3 1.218×10–6 kg/mm3

Normal modulus Em 5.0E5 68.94 GPa 

Poisson’s ratio Num 0.35 non-dim. 0.35 non-dim. 

Thermal expansion coefficient Alfa m 3.6×10–5 in./in./F 6.48×10–5/C 

Heat conductivity Km 0.008681 BTU/hr/in./F 1.803×10–4W/mm/C 

Heat capacity Cm 0.25 BTU/lb/F 1048 J/kg/C 

Dielectric strength Kem 0.0 V/in. 0.0 V/mm 

Dielectric constant Gammam 0.0 in./V 0.0 V/mm 

Capacitance Cem 0.0 V 0.0 V 

Resistivity Rem 0.0 Ω-in. 0.0 Ω-mm 

Moisture expansion coefficient Betam 0.0033 in./in./%Moisture 0.0033/%Moisture 

Diffusivity Dm 2.16×10–7 in.2/hr 1.394×10–4 mm2/hr 

Saturation Mm 0.0%Moisture 0.0%Moisture 

Tensile strength SmT 15000.0 psi 103.4 MPa 

Compressive strength SmC 35000.0 psi 241.3 MPa 

Shear strength SmS 13000.0 psi 89.63 MPa 

Allowable tensile strain eps mT 0.02 in./in. 0.02 mm/mm 

Allowable compr. strain eps mC 0.05 in./in. 0.05 mm/mm 

Allowable shear strain eps mS 0.035 in./in. 0.035 mm/mm 

Allowable torsional strain eps mTOR 0.035 in./in. 0.035 mm/mm 

Normal damping capacity psiNm 6.6%Energy 6.6%Energy 

Shear damping capacity psiSm 6.9%Energy 6.9%Energy 

Void heat conductivity Kv 0.225 BTU/hr/in./F 4.673×10–3 W/mm/C 

Glass transition temperature Tgdr 420.0 F 215.6 C 

Melting temperature TMm 600 F –17.78 C 

TABLE 3.—NANOFIBER REINFORCED MATRIX 

(NFRM—0.01 FIBER REINFORCEMENT) 

Description Symbol Value Value in SI units 

Weight density Rhom 0.047 lb/in.3 1.301×10–6 kg/mm3

Normal modulus Em 6.774×107 psi 467.1 GPa 

Poisson’s ratio Num 0.3323 non-dim. 0.3323 non-dim 

Thermal expansion coefficient Alfa m 5.6×10–9 in./in./F 10.08×10–9/C 

Heat conductivity Km 444.0 BTU/hr/in./F 9.221 W/mm/C 

Heat capacity Cm 0.227 BTU/lb/F 951.1 J/kg/C 

Dielectric strength Kem 0.0 V/in. 0.0 V/mm. 

Dielectric constant Gammam 0.0 in./V 0.0 V/mm. 

Capacitance Cem 0.0 V 0.0 V 

Resistivity Rem 0.0 Ω-in. 0.0 Ω-mm. 

Moisture expansion coefficient Betam 5.5×10–11 in./in./%Moisture 5.5×10–11/%Moisture 

Diffusivity Dm 5.4×10–11 in.2/hr 3.484×10–8 mm2/hr 

Saturation Mm 0.0%Moisture 0.0 %Moisture 

Tensile strength SmT 800000.0 psi 5.516 GPa 

Compressive strength SmC 600000.0 psi 4.137 GPa 

Shear strength SmS 400000.0 psi 2.758 GPa 

Allowable tensile strain eps mT 0.0070 in./in. 0.0070 mm/mm 

Allowable compr. strain eps mC 0.0060 in./in. 0.0060 mm/mm 

Allowable shear strain eps mS 0.0040 in./in. 0.0040 mm/mm 

Allowable torsional strain eps mTOR 0.0040 in./in. 0.0040 mm/mm 

Normal damping capacity psiNm 0.4%Energy 0.4%Energy 

Shear damping capacity psiSm 0.4%Energy 0.4%Energy 

Void heat conductivity Kv 0.225 BTU/hr/in./F 4.673×10–3 W/mm/C 

Glass transition temperature Tgdr 420.0 F 215.6 C 

Melting temperature TMm 550.0 F 287.8 C 

 
NASA/TM—2011-217282 8

TABLE 4.—CONVENTIONAL FIBER (AS00) 

Description Symbol Value Value in SI units 

Number of fibers per end Nf 10000.0 10000.0 

Filament equivalent diameter df 3.0×10–4 in. 76.2×10–4 mm 

Weight density Rhof 0.063 lb/in.3 1.744×10–6 kg/mm3

Normal moduli (11) Ef11 3.2×107 psi 22.06×1010 Pa 

Normal moduli (22) Ef22 2000000.0 psi 13.79 GPa 

Poisson’s ratio (12) Nuf12 0.2 non-dim. 0.2 non-dim. 

Poisson’s ratio (23) Nuf23 0.25 non-dim. 0.25 non-dim. 

Shear moduli (12) Gf12 2000000.0 psi 13.79 GPa 

Shear moduli (23) Gf23 1000000.0 psi 6.895 GPa 

Thermal expansion coefficient (11) Alfaf11 5.5×10–7 in./in./F 9.9×10–7/C 

Thermal expansion coefficient (22) Alfaf22 5.5×10–4 in./in./F 9.9×10–4/C 

Heat conductivity (11) Kf11 4.03 BTU/hr/in./F 0.08370 W/mm/C 

Heat conductivity (12) Kf12 0.403 BTU/hr/in./F 0.008370 W/mm/C 

Heat capacity Cf 0.17 BTU/lb/F 712.3 J/kg/C 

Dielectric strength (11) Kef11 0.0 V/in. 0.0 V/mm 

Dielectric strength (22) Kef22 0.0 V/in. 0.0 V/mm 

Dielectric constant (11) Gammaf11 0.0 in./V 0.0 mm/V 

Dielectric constant (22) Gammaf22 0.0 in./V 0.0 mm/V 

Capacitance Cef 0.0 V 0.0 V 

Resistivity Ref 0.0 Ω-in. 0.0 Ω-mm 

Tensile strength SfT 400000.0 psi 2.758 GPa 

Compressive strength SfC 350000.0 psi 2.413 GPa 

Shear strength SfS 250000.0 psi 1.724 GPa 

Normal damping capacity (11) psi11f 0.05%Energy 0.05%Energy 

Normal damping capacity (12) psi22f 0.25%Energy 0.25%Energy 

Shear damping capacity (12) psi12f 0.35%Energy 0.35%Energy 

Shear damping capacity (23) psi23f 0.5%Energy 0.5%Energy 

Melting temperature TMf 6000.0 F 3315.6 C 

TABLE 5.—BENDING RIGIDITIES OF UNIDIRECTIONAL 

COMPOSITE FOR BUCKLING VIBRATION FREQUENCIES ANALYSIS 

θ D11 D12 D13 D22 D23 D33 Comment 

0 3.87 + 6 5.41 + 4 0 2.14 + 5 0 2.10 + 5 With NFRM; kf = 0.01 

0 1.62 + 6 2.60 + 4 0 9.98 + 4 0 5.19 + 4 Without NFRM 

kPa 27.10 + 6 37.33 + 4 0 14.94 + 5 0 14.66 + 5 With NFRM; kf = 0.01 

kPa 11.31 + 6 18.15 + 5 0 69.60 + 4 0 36.23 + 4 Without NFRM 
NASA/TM—2011-217282 9

TABLE 8.—VIBRATION 

FREQUENCIES OF ANISOTROPIC 

COMPOSITE PLATES WITH NFRM 

AND WITH CONVENTIONAL 

MATRIX (CM) AND COMBINED 

LOADING 

Load conditions and frequency factor, Hz 

Combined loading NFRM CM 

 0 0 2005 1390 

0  0 1818 1270 

0 0  2245 1576 

  0 1819 1270 

 0  2122 1490 

0   1909 1339 

   1898 1331 

TABLE 6.—BUCKLING FACTOR 

RESULTS ON ANISOTROPIC 

COMPOSITE PLATES WITH 

CONVENTIONAL MATRIX 

Load conditions in vibration frequencies 

 0 0 –1793 

0  0 –529 

0 0  –1023 

  0 –422 

 0  –708 

0   –383 

   –323 

TABLE 7.—BUCKLING FACTOR 

RESULTS OF ANISOTROPIC 

COMPOSITE PLATES WITH NFRM 

Load conditions in vibration frequencies 

 0 0 –3673 

0  0 –1097 

0 0  –2220 

  0 –875 

 0  –1523 

0   –808 

   –680 
NASA/TM—2011-217282 10
NASA/TM—2011-217282 11

Buckling Mode Shapes 
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NASA/TM—2011-217282 25
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

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1. REPORT DATE (DD-MM-YYYY)

01-12-2011 

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Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Buckling and Vibration of Fiber Reinforced Composite Plates With Nanofiber Reinforced 

Matrices 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Chamis, Christos, C.; Murthy, Pappu, L.N. 

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5f. WORK UNIT NUMBER

WBS 561581.02.08.03.13.05 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

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E-18037 

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NASA/TM-2011-217282 

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Subject Category: 24 

Available electronically at http://www.sti.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Anisotropic composite plates were evaluated with nanofiber reinforced matrices (NFRM). The nanofiber reinforcement volumes ratio in the 

matrix was 0.01. The plate dimensions were 20 by 10 by 1.0 in. (508 by 254 by 25.4 mm). Seven different loading condition cases were 

evaluated for buckling: three for uniaxial loading, three for pairs of combined loading, and one with three combined loadings. The 

anisotropy arose from the unidirectional plates having been at 30° from the structural axis. The anisotropy had a full 6 by 6 rigidities matrix 

which were satisfied and solved by a Galerkin buckling algorithm. For vibration the same conditions were used with the applied cods about 

a small fraction of the buckling loads. The buckling and vibration results showed that the NFRM plates buckled at about twice those with 

conventional matrix. 

15. SUBJECT TERMS

30° loading axis; 0.1 nano fibers; Combined loads; Unidirectional plates 

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