Strength of the Thesis: The thesis reports an excellent work on the linear/nonlinear bending and buckling analysis of laminated and sandwich plates/curved panels using different shear deformation theories with emphasis on non-polynomial higher order shear deformation theory. The governing equations are solved using finite element method. For the validation of the results, 3D and Closed form solutions are also obtained wherever possible by the author. The nonlinear equations for equilibrium path are solved using arc length continuation technique. The results are presented for mechanical and hygrothermal loading. The extensive validation of the proposed solution methodology is presented. Many new and useful results are presented. The entire work requires thorough understanding of most challenging solution technique in the field of computational mechanics of composites. The thesis is recommended for the award of the Ph.D. degree provided the author addresses the following comments: 1. Page 2, 12th and 13th lines from the top: Replace “material has” with “material have”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 2. Page 2, 5th line from bottom: Replace “relay” with “reply”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 3. Page 4, 13th line from bottom: Replace “Clough and its” with “Clough and his”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 4. Page 6, 10th line from bottom: Replace “shows shown” with “shows”. Response: Suggested change has been made in the updated thesis. 5. Page 6, 3rd and 2nd line from bottom: The author has mentioned that imperfect plates do not posses any critical point. However, it may be noted that critical points are classified as limit points and bifurcations. An imperfect plate may not posses bifurcation point but it may posses limit point. Please correct the statement in the thesis. Response: As per the reviewer suggestion, particular type of critical point is specified in the sentence to remove the ambiguity, and same changes have been highlighted with magenta color in the updated thesis. 6. Page 9, second paragraph from bottom: Can you explain as to why shells depict greater critical buckling load than plates? Why shells depict unstable postbuckling response? 1 Response: Shell depict greater critical buckling load than plates due to curvature effect in the linear strain, i.e., w/Ry . Regarding the unstable posttbuckling reponse, panel can deflect in either direction. The direction in which it increases the effective curvature, postbukling response increases. While in opposite direction, effective radius decreases can eventually become plate and then exhibits increasing postbuckling response with further increases in the deflection. 7. Page 13, 2nd line from bottom: Replaced “are” with “should be of”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 8. Page 16, 2nd and 1st line from bottom: It should be mentioned that the bending stretching coupling in shell modeling through flat plate elements comes through the transformation element level governing equation to a common coordinate system. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 9. Page 17, 16th line from top: Replace “convergent” with “convergence” Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 10. Page 17, 10th line from bottom: Delete “are”. Response: Suggested change has been made in the updated thesis. 11. Page 19, 18th line from bottom: Replace “Newton-Raphson type” with “load control type”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 12. Page 44, 9th line from top: Replace “principle” with “principal”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 13. Page 45, 1st line from top: Replace “state of plane stress” with “state of generalized stress”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 14. Page 46, Expression for εNL : Check the equation. Term v0 /Rx (at three places) should be v0 /Ry 2 Response: The typographical mistakes made in the Equations related to εNL have been corrected. 15. Page 47, Eq. (3.4): Check the equation. Term v0 /Rx should be v0 /Ry . Also ε ∗ needs to be defined. It will be different from what is given in Page 46. Response: The typographical mistakes made in the Eq. (3.4) have been corrected, and the explicit relation of ε ∗ should be ε∗ = ∗ ∂w − Ru0x ∂∂wx ∂ x v0 ∂ w∗ ∂w − Ry ∂ y ∂y v0 u0 ∂ w ∗ ∂ w∗ ∂ w ∂w ∂x ∂ y − Ry + ∂ x − Rx ∂y 0 0 However, the same expression of strain due to imperfection as given in Page 46 is taken. The reason which author thinks that strain due to imperfection contributes as constant linear strain which is systematical derived. Further, terms neglected from Green-Lagrange nonlinearity does not guarantee to be vanished when used in the strain due to imperfection in von Karman nonlinearity. 16. Page 47, 5th and 6th lines after Eq. (3.4): Can the plane stress conditions be satisfied along with the inextensibility condition? Response: There was an ambiguity in the statement which is corrected in the updated thesis and the same is highlighted with magenta color. In this line, the thermal expansion term αzz is not considered due to plane stress condition as its contribution will come out to be zero in the principle of virtual work. 17. Page 50: Replace “standard finite element” with “standard Mechanics of Composites”. Response: Suggested change has been made in the updated thesis and the same is highlighted with magenta color. 18. Page 61: Add integration symbol in first equation. Also, how did you carry out integration to evaluate A and B? Gauss quadrature? Response: The missing integration symbol is placed correctly. The integration of A and B has been carried out using 2 × 2 Gauss-Legendre quadrature rule which is written just before the numerical results in Page 65. However, for clarity, 2 × 2 Gauss points have been specified in the Section 3.6. 3 19. Page 65, 14th line from bottom: Does linear stiffness matrix include linear stiffness contribution due to transverse shear strain energy terms? Response: The “bending” word has been added in between “linear stiffness” word phrase. There was a typographical mistake. Full integration is only used for linear bending stiffness matrix and force vector; and rest terms including linear transverse shear stiffness matrix. 20. Page 65. function f (z): The function used are odd functions. Theory based on this will be applicable for symmetric laminates/panels. Please comment on this. You may refer to the work of Prof. Tarun Kant on higher order shear deformation theories. Response: The theory developed by Prof. Tarun Kant is a generalized HSDT will 9 DOFs and can be used for general laminate. However, it is computational expensive. Moreover, C1 NPSDT is computational efficient. Regarding the applicability of HSDT for general laminate, different f (z) have been developed to give reliable and accurate result with less computation cost with C1 HSDT. In this work, a comparison is make to shown the limitation of IHSDT for anti-symmetric cross-ply laminated plates while IHTSDT give equivalent result to TSDT and perform better than IHSDT. However, the understanding of odd and even function is not taken during the comparison study and which will be explored by the author in his future work. 21. Page 67, 5th line from top: Why is γ taken as zero? what is the theory corresponding to γ = 0? Will the resulting theory satisfy zero transverse shear strain condition on top/bottom of the laminate/panel? Response: The importance of penalty parameter (or artificial stiffness matrix) is highlighted by showing the difference in the results. As most the work on C1 HSDT will C0 FEM model, penalty stiffness matrix is neglected. To justify the importance, the results using γ = 0 are obtained. The theory corresponds to γ = 0 will be unconstrained theory which does not satisfy the traction free boundary condition for both thick and thin plates with significant and small percentage error, respectively. 22. Page 67, right side third equation from top: check and correct left hand side of the equation. Response: Left hand side of the bottom right equation is modified correctly and the same is highlighted with magenta color 23. Page 69, 8th line from top: Replace “ results for both” with “results with γ = 107 for both”. Response: Suggested change has been made in the updated thesis and the same has been highlighted with magenta color. 4 24. Page 69, 8th line from top: Replace “in compare” with “in comparison”. Response: Suggested change has been made in the updated thesis and the same has been highlighted with magenta color. 25. Page 69, first paragraph: The second last sentence of the paragraph is not true for τxz . Please correct it accordingly. Response: As the reviewer suggestion, the exception for τxz has been added in the second last line of the first paragraph. 26. Page 69, section 4.2.2: Are all the advantages mentioned applicable for single flat plate? Is the shear deformation theory applicable for anti-symmetric cross-ply plates? Response: Based on the numerical study and the comparison with 3D elasticity solution, the applicability and accuracy of different NPSDT is assessed in this problem and advantages are drawn for anti-symmetric cross-ply laminated plate. The motivation behind taking this problem was to emphasis the assessment of various NPSDTs for anti-symmetric cross-ply laminated plates which is generally neglected in Navier solution methodology. 27. Page 72, At higher loads for thick plates, stresses will exceed failure limit and laminates will fail. Nonlinear effects are important for thin plates before failure. For thin plates, von Karman and Green-Lagrange nonlinearities will yield almost the same results. Thus the conclusion drawn is misleading. Response: Yes, at higher loads for thick plates, laminates may fail due to stress failure limit. Moreover, the present study is limited in the sense of failure analysis. However, the motivation of the present study is to extended the applicability of NPSDT model from linear analysis to nonlinear analysis thorough validation and comparison with well established, benchmark numerical results. In most of the nonlinear bending analysis, failure analysis is not study and the scope of the present research work was also not align with that. Hence, failure analysis was not carried out. However, author keeps the note of the valuable comment of reviewer and will incorporate the same in his future work. Moreover, in nonlinear thermal bending analysis, even for thin plate, considerable difference is observed between results using von Karman and Green-Lagrange nonlinearities which emphasis the importance of Green-Lagrange nonlinearity for thin plates. 28. Page 73: Why did you not present transverse shear stresses evaluated based on the stress equations of equilibrium? The presented results of transverse shear stresses do not satisfy continuity at the layer interfaces? Response: The primary focus of this problem is to show the incorporating of GreenLagrange nonlinearity and effect of per-applied traction free boundary condition in 5 linear analysis and then extending it to nonlinear analysis. Yes, equilibrium approach gives correct and reliable solution for the distribution of transverse stresses. Moreover, calculating the distribution of transverse stresses in nonlinear bending analysis is also computationally expensive after apply SPR technique for 12x12 mesh for each layer and each nodes. Hence, its calculation is avoided in this problem. 29. Page 77, 6th line from top: Why is reduced integration used for evaluating geometric stiffness matrix? Response: As stresses or generalized stresses are less erroneous at the reduced Gauss quadrature according to SPR technique. Hence, reduced integration is used for evaluating geometric stiffness matrix. 30. Page 79, Table 4.5: For convergence study, number of elements should at least be doubled in each mesh refinement step. What is UNTSDT? Response: Yes, h-refinement should be done with at-least double mesh size. However, incremental refinement is shown as done in several literature. Moreover, more results can be shown upto 28 × 28 mesh size. However, less than 0.1% error was observed after 16 × 16. Hence, results after 16 × 16 mes are not shown for the sake of brevity. The UNTSDT is unconstrained third-order shear deformation theory as utilized by Adhikari and Singh [63]. 31. Page 80, 1st line from top: Replace “stiffing” with “stiffening”. Response: Suggested change has been made in the updated thesis and the same is highlighted with magenta color. 32. Page 80, 2nd and 3rd lines: Please check the statement. Critical buckling load obtained from von Karman and Green-Lagrange nonlinearities are almost the same. Response: There is very less difference between the normalized buckling load. However, significant difference will be see in the dimensionalized value (shown in upcoming problems). 33. Page 80, Table 4.7: Is the eigenvalue buckling analysis valid for anti-symmetric crossply plates? Response: Eigenvalue buckling analysis is not reliable for anti-symmetric cross-ply plates and the same observation is also observed in nonlinear buckling analysis. The same point is highlighted in the updated thesis with magenta color. 34. Page 81, 2nd line from top: Replace “To obtained with “To obtain”. Response: Suggested change has been made in the updated thesis and the same is highlighted with magenta color. 6 35. Page 81, 3rd line from bottom: Replace “in compares” with “in comparison”. Response: Suggested change has been made in the updated thesis and the same is highlighted with magenta color. 36. Page 82: Please check and correct the titles of Tables 4.9. Response: As per reviewer’s suggestion, the title of the Table 4.9 has been modified according to the relevance of the subsection 4.4.3 and the same is highlighted with magenta in the updated thesis. 37. Page 83/84: Please check and correct the titles of Tables 4.10, 4.11 and 4.12. Response: The titles of the Table 4.10, 4.11 and 4.12 have been modified according to the relevance of the subsection 4.4.4 and the same is highlighted with magenta in the updated thesis. 38. Page 85, Table 4.13: Before buckling, plate material will undergo material failure for a/h = 5, 10. Please check and comment appropriately. Response: Yes, it may happen that plate material will undergo material failure for a/h = 5, 10. However, present study is limited to geometrically nonlinear analysis and does not account failure criteria. Thus, accordingly buckling strength is predicted. However, for better design and analysis of buckling strength, material nonlinearity and failure criteria must be used. Moreover, as per reviewer’s suggestion, a one line discussion has been added in the Section 4.4.5 about material failure. The same changes have been highlighted in the updated thesis with magenta color. 39. Page 89, Figure 4.7: What is the difference between prebuckling eigenvalue approach and SSSS1 eigenvalue approach? Response: In prebuckling eigenvalue approach, prebuckling analysis is carried out by considering prebuckling boundary condition. While in SSSS1 eigenvalue approach, prebuckling analysis is carried out using SSSS1. Then for both approaches, eigenvalue analysis is done with SSSS1 boundary condition. However, to make it clear, three linear buckling approach, i.e., assumed stress approach, prebuckling approach, and linear buckling approach have been used in the updated thesis and highlighted with magenta color. In linear buckling approach, prebuckling analysis is performed using SSSS1 boundary condition. 40. Page 91, Figures 4.8: How did you switch to postbuckling path in nonlinear analysis of perfect plates? Response: A simple branch switching technique which is based on permutation technique has been used to trace the postbuckling path in the nonlinear buckling approach. 7 41. Page 91, 9th line from bottom: Replace “to carried” with “to carry”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 42. Page 94, 6th line from bottom: Please state or cite an appropriate reference for DuhamelNewmann’s law. Response: As per reviewer’s suggestion, an appropriate reference for Duhamel-Newmann’s law has been stated in the text. 43. Page 96, 10th and 9th lines from bottom: Please check. T0 , T1 , T2 ,C0 ,C1 ,C2 will not not be function of x and y. x and y dependence is explicitly given in the equation. Response: The typographical mistakes made in the Section 5.1.2 have been corrected. 44. Page 98, 8th line from top: Replace “in compare” with “in comparison”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 45. Page 100, Table 5.3: Why have you considered without transformed thermal expansion coefficients? It is obvious that one should consider transformed coefficients. Response: Yes, one should consider transformed coefficients. However, many studies have considered carried out without transformed via Navier type analytical approach. To validate the accuracy of the present solutions, the comparison has been made with results without transformed thermal coefficients. Moreover in Table 5.4, a comparison has been also made between present obtained Navier, FEM, and 3D elasticity solution. Further, the significance of transformed thermal coefficients has been highlighted in the thermal bending analysis of composite plates using both Tables 5.3 and 5.4. 46. Page 101, Table 5.4: Are CFS-IHSDT your results? If yes, then put * else cite appropriate reference. Response: In Table 5.4, the CFS-IHSDT results are present obtained solutions using transformed thermal coefficients and results are marked with * to specify the present solution. 47. Page 102, 12th line from top: Delete hanging “and”. Response: The suggested change has been made in the updated thesis. 48. Page 103, Table 5.6: How many terms are used in the Navier’s approach? Please state. Response: One term (m = n = 1) has been used to obtain the Navier solution corresponding to sinusoidal distribution. The same point has been added in the updated thesis (including Appendix A.4) and highlighted with magenta color. 8 49. Page 104, 16th line from bottom: Is the lamination scheme mentioned correct? It does not match with title of Table 5.7. Response: There was a typographical mistake in the text. The correct lamination 0 0 0 0 scheme is 0 /90 /0 /90 . The same has been corrected in the updated manuscript and highlighted with magenta color. 50. Page 110, 3rd line from top: Validation of accuracy is clear how is efficiency validated? Response: There is a typographical mistake in the 3rd line from top. The correct word should “efficacy” instead of “efficiency”. 51. Page 112/113, Table 5.11/5.12: For thick plates, before reaching such high temperature, significant degradation in the material properties will take place and either material will fail or burn before buckling. Please justify your reported results. Response: Yes, it may happen that either material will fail or burn before buckling at such high temperature reported in the Tables 5.11 and 5.12. However, these problems are selected from literature to validated the present obtained solution for angle-ply laminated composite plates. In these particular problems, the aspect of material degradation and damage was not considered while predicting the initial buckling strength as also done by other investigators. Moreover, the above mentioned points by reviewer have been added in the updated thesis and highlighted with magenta color. 52. Page 112, 10th line from bottom: Replace “results reveals” with “results reveal”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 53. Page 121, 2nd line from top: Replace “in compare to” with “in comparison to”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 54. Page 121, 5th and 4th lines from bottom: Replace “geometrical” with “geometrically”. Response: The suggested change has been made in the updated thesis and the same is highlighted with magenta color. 55. Page 130, 2nd line from bottom: Replace “compare to” with “comparison to”. Response: The suggested change has been made in the updated thesis and the same is highlighted with magenta color. 56. Page 132, Figure 6.4: It will be informative if deformed shapes of the panels are given corresponding to few important points in the equilibrium path. 9 1 10-3 8 0.08 0.5 6 0.06 0 4 0.04 -0.5 2 0.02 -1 1 0 1 0 1 0.5 1 0.5 1 1 0.5 0.5 0 0.5 0 0 (a) Mode shape using nonlinear buckling approach at wmax /h = 0 0.5 0 0 (b) Mode shape using nonlinear buckling approach at wmax /h = 0.06 0.01 0 (c) Mode shape using nonlinear buckling approach at wmax /h = 0.6 0.06 0.04 0.005 0.02 0 1 0 1 0.5 1 0.5 1 0.5 0 0.5 0 0 (d) Mode using nonlinear eigenvalue approach at wmax /h = 0.1 0 (e) Mode using nonlinear eigenvalue approach at wmax /h = 0.5 Figure 1: Mode shapes along the equilibrium path of the buckling response As per the reviewer’s suggestion, mode shapes are plotted for section 4.5.2 instead of section 4.5.1. In section 4.5.2, both nonlinear eigenvalue and nonlinear buckling approaches are utilized in compare to section 4.5.1 in which only nonlinear buckling approach is utilized. To give more clarity about the deformed shapes of the plate along equilibrium path, the mode shapes (deflection) pertaining to section 4.5.2 are illustrated in 1. In 1, mode shapes are obtained by both nonlinear buckling approach and nonlinear eigenvalue approaches at different wmax /h. 57. Page 134, Table 6.5: Is eigenvalue buckling analysis valid? What is the corresponding buckling mode. Response: For anti-symmetric laminated curved panel, linear eigenvalue approach is not reliable for predicting the initial buckling strength of the cylindrical panel subjected to axial loads. However, the corresponding problem is taken to validated the present model with assumed/inplane stress distribution as carried out in the literature 10 0 -0.005 -0.01 -0.015 -0.02 1 0.5 1 0.5 0 0 Figure 2: First mode shape of cylindrical panel under uniaxial inplane load by various investigators. The corresponding buckling modes for this problem is also plotted in Section 6.4.1. 58. Page 140, Section 6.5: How can you identify whether the bifurcation bulking is exhibited or not? Response: Bifurcation buckling can be identify from change in the sign of the determent of tangent stiffness matrix, i.e., positive to negative. The critical point will be at the singular tangent stiffness matrix. If the predicted solution in the arclength method is having negative determent of the tangent stiffness matrix then small perturbation (mostly probably first eigenmode) is added to displacement. If equilibrium path changes its direction, then we can say that the critical point is the bifurcation point otherwise it is a limit point. 59. Page 141, Figure 6.6: Is the response given for flat panel a postbuckling response? Where is the bifurcation/limit point? Response: In this problem, the lamination is 00 /900 which contributes to bendingstretching coupling. Thus, no critical point is observed. Moreover, the response of the flat panel is similar to response in Figure 4.6. 60. Page 148, 2nd and 3rd lines from top: Please check the x, y dependence of various terms. 11 Response: The typographic mistake in Section 7.1.2 has been corrected in the updated thesis. The same is highlighted with magenta color. 61. Page 149, Table 7.1 title: Delete buckling load. Response: The typographical mistake in the title of Table 7.1 is corrected in the updated thesis. 62. Page 150, 16th line from bottom: Replace “reliable of” with “reliability of”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 63. Page 150, 15th line from bottom: Replace “be check” with “be checked”. Response: Suggested change has been made in the updated thesis and highlighted with magenta color. 64. Section 7.3 and 7.4: In the Figures reference number should be cited for the results from literature. Response: As per the reviewer’s suggestion, reference number has been added for the results from literature. Weaknesses of the Thesis: There are no obvious weakness in the thesis. However, it would have been more informative if at least first-ply failure load was identified in the results presented. The comparison of buckling load/temperature from eigenvalue analysis and the one esteemed from nonlinear analysis would have given more insight in the solution methodology adopted. The presentation of deformed shape in nonlinear analysis is another useful aspect missing the thesis. Some of the Questions to be asked during the Thesis Defense: 1. Can you define and explain critical points of equilibrium path. Explain the difference between limit points and bifurcation points. 2. Can you explain as to why shells depicts greater critical buckling load than plates? Why shell depict unstable postbuckling response? 3. How is bending stretching coupling in shell modelling through flat plate accounted for? Please explain. 4. Can the plane stress conditions be satisfied along with inextensibility condition? 12 5. The function, f (z) used in the thesis are odd functions. Theory based on this will be applicable for symmetric laminates/panels. Please comment on this. 6. Why is γ taken as zero? What is the theory corresponding to γ = 0? Will the resulting theory satisfy zero transverse shear strain condition on top/bottom of the laminate/panel? 7. Are all the advantage mentioned for anti-symmetric laminates applicable for single flat plate? Is the shear deformation theory used applicable for anti-symmetric crossply plates? 8. At higher loads for thick plates, stresses will exceed failure. For thin plates, von Karman and Green-Lagrange nonlinearities will yield almost the same results. Please comment on this. 9. Why did you not present transverse shear stresses evaluated based on the stress equations of equilibrium? The presented results of transverse shear stresses do not satisfy continuity at the layer interfaces. 10. Why is reduced integration used for evaluating geometric stiffness matrix? 11. Is the eigenvalue buckling valid for anti-symmetric cross-ply plates? 12. Before buckling, thick plates/panels will undergo material failure. Please comment. 13. Can you explain the difference between prebuckling eigenvalue approach and eigenvalue approach? 14. How did you switch to post buckling path in nonlinear analysis of perfect plates? 15. Please state or explain Duhamel-Newmann’s law. 16. Why did yo consider without transformed thermal expansion coefficients? 17. For thick plates, before reaching reported high critical buckling temperature, significant degradation in the material properties will take place and either the material will fail or burn before buckling. Please justify your reported results. 18. Is eigenvalue buckling analysis valid for curved panels? Justify your choice. 19. How can you identify whether the bifurcation buckling is exhibited or not? 13 Strength of the Thesis (of 16AE91R01): In this study, a reliable and accurate finite element model is formulated for linear and nonlinear composite shell element. Using the element bending and buckling analysis has done and compared with other references. An arc length method is applied to solve the postbuckling problems. A Navier type analytical methodology is also developed for linar bending and buckling analysis of panels to assess the accuracy and efficiency of the present shell model. The accuracy, performance, and applicability of the present FEM mode are examined through various bench problems. It has been shown that the present FEM model is efficient and can be conveniently implemented for both linear and nonlinear structural analysis of laminated and sandwich composite plates and panels. Weakness of the Thesis (of 16AE91R01): The buckling analysis is a linear system so that any program can be applied for it but the geometry nonlinear analysis problem is a nonlinear problem. Therefore Abaqus or geometry nonlinear support program should be used for the post buckling analysis. Questions to be asked during Thesis Defense (of 16AE91R01): In linear system, buckling analysis is formulated with eigenvalue problem. But the eigenvalue problem can not be applied for the nonlinear system. But in the thesis chapter 3.4.8 page 57, the nonlinear buckling analysis due to hygrothermal load equation is given by a linear system. 14