### Structural Analysis: In Theory and Practice

Free download.
Book file PDF easily for everyone and every device.
You can download and read online Structural Analysis: In Theory and Practice file PDF Book only if you are registered here.
And also you can download or read online all Book PDF file that related with Structural Analysis: In Theory and Practice book.
Happy reading Structural Analysis: In Theory and Practice Bookeveryone.
Download file Free Book PDF Structural Analysis: In Theory and Practice at Complete PDF Library.
This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats.
Here is The CompletePDF Book Library.
It's free to register here to get Book file PDF Structural Analysis: In Theory and Practice Pocket Guide.

Structural social work considers these environmental factors, and extends the analysis by consideration broader socioeconomic factors. Advocacy for client rights and adequate resources recognizes the importance of addressing issues of material resources. Conventional social work encourages social workers to emphasize personal emotions, skills, and interpersonal capacities and conveys the message that these are more important than advocacy for material resources. Conventional social workers maximize worker -- client power differences following the professional expert model.

While consultation may occur generally prescribed solutions are proposed which clients are expected to follow. This involves sharing opinions and encouraging feedback concerning how the clients situation links with the primary structures of oppression. Conventional social work methods tend to blame the victim and psychologizing problems. It may be used to obtain an expression for the entire elastic curve over the whole length of a beam.

Solution The expression for the bending moment at any point in the beam shown in Figure 4. The principle may be defined as follows: if a structure in equilibrium under a system of applied forces is subjected to a system of displacements compatible with the external restraints and the geometry of the structure, the total work done by the applied forces during these external displacements equals the work done by the internal forces, corresponding to the applied forces, during the internal deformations, corresponding to the external displacements.

To the cantilever shown in Figure 4. From the principle of conservation of energy and ignoring the effects of axial and shear forces, the external work done during the application of the loads must equal the internal energy stored in the beam. This results in a bending moment m in the element.

For a beam, moments produced by the virtual load or moment are considered positive, and moments produced by the applied loads, which are of opposite sense, are considered negative. A positive value for the displacement indicates that the displacement is in the same direction as the virtual force or moment. The deflection and slope at the free end of the cantilever shown in Figure 4. Taking the origin of coordinates at point 3, the expression for the bending moment due to the applied load is obtained from Figure 4.

The function m is always either constant along the length of the member or varies linearly. The function M may vary linearly for real concentrated loads or parabolically for real distributed loads. The volume of this solid is given by the area of cross-section multiplied by the height of the solid at the centroid of the cross-section. Example 4. Solution The functions M and m derived from Figure 4. Integration is carried out over Elastic deformations 85 all members of the frame and the values summed to provide the required displacement.

To determine the horizontal deflection of node 2 for the frame shown in Figure 4. These are shown at ii and iv. Member 34 has zero bending moment under both loading cases. The beam shown in Figure 4. The shear force diagram is shown at ii and is negative at end 1 and positive at end 2. The bending moment diagram is shown at iii and is negative as indicated. The elastic curve is shown at iv with deflections positive i.

An analogous beam known as the conjugate beam and of the same length as the real beam, as shown in Figure 4. The elastic load acts in a positive direction upward when the bending moment in the real beam is positive. The loading diagram is shown at v and is negative as indicated. The shear force diagram is shown at vi and is positive at end 1 and negative at end 2.

- Cosmopolitan Managers: Executive Development that Works?
- Trends in Structural Mechanics.
- Discrete and Combinatorial Mathematics: An Applied Introduction (5th Edition).
- Secondary Metabolism in Microorganisms, Plants and Animals;
- The Teaching of Structural Analysis.
- Submission Guidelines;
- Marxist Glossary.

Hence, as shown at iv , the slope of the elastic curve is positive at end 1 and negative at end 2. The bending moment diagram is shown at vii and is positive as indicated. Hence, as shown at iv , the deflection of the elastic curve is positive. The end slope and end deflection of the real beam are given by the end reaction and end moment of the conjugate beam. The maximum deflection in the real beam occurs at the position of zero shear in the conjugate beam. In the case of frames, the elastic load applied to the conjugate frame is positive i. Then, the displacement of the real frame at any section is perpendicular to the lever arm used to determine the moment in the conjugate frame and is outward when a positive bending moment occurs at the corresponding section of the conjugate frame.

The restraints of the conjugate structure must be consistent with the displacements of the real structure. Details of the necessary restraints in the conjugate structure are provided in Part 2, Chapter 4, Table 4. At a simple end support in a real structure, there is a rotation but no deflection. Thus, the corresponding restraints in the conjugate structure must be a shear force and a zero moment, which are produced by a simple end support in the conjugate structure.

At a fixed end in a real structure, there is neither a rotation nor a deflection. Thus, there must be no restraint at the corresponding point in the conjugate structure, which must be a free end. At a free end in a real structure, there is both a rotation and a deflection.

- Structural Analysis - In Theory and Practice.
- Hitler the Cat Goes West.
- STRUCTURAL ANALYSIS: IN THEORY & PRACTICE.
- Perfection of the Morning?
- Computer Forensics InfoSec Pro Guide!

Thus, the corresponding restraints in the conjugate structure are a shear force and a bending moment, which are produced at a fixed end. At an interior support in a real structure, there is no deflection and a smooth change in slope. Thus, there can be no moment and no reaction at the corresponding point in the conjugate structure, which must be an unsupported hinge. At an interior hinge in a real structure, there is a deflection and an abrupt change of slope. Thus, the corresponding restraints in the conjugate structure are a moment and a reaction, which are produced by an interior support.

The lateral load is applied to the frame at ii , which results in the bending moment diagram, drawn on the compression side of the members, shown at iv. To the pin-jointed frame shown in Figure 4. The external work done during the application of the loads must equal the internal energy stored in the structure, from the principle of conservation of energy. This results in a force u in any member.

Now, while the virtual load is still in position, the real loads W are gradually applied to the structure. For a pin-jointed frame, tensile forces are considered positive and compressive forces negative. Increase in the length of a member is considered positive and decrease in length negative. The unit virtual load is applied to the frame in the anticipated direction of the deflection. If the assumed direction is correct, the deflection obtained will have a positive value. The deflection obtained will be negative when the unit virtual load has been applied in the opposite direction to the actual deflection.

All members of the frame have a constant value for AE of , kips. Elastic deformations 95 4 2 14 6. The member forces P due to the applied loads and u due to the unit virtual load are tabulated in Table 4. Table 4. Determine the location of the maximum deflection in span 12 and the magnitude of the maximum deflection. Determine the deflection at node 3.

The static loads are applied to the structure as shown in Figure 5. However, a member such as a bridge girder or a crane gantry girder are subjected to moving loads, and the maximum design force in the member depends on the location of the moving load.

### Navigation menu

As shown in Figure 5. Alternatively, an influence line may be utilized to Structural Analysis: In Theory and Practice determine the location of the moving load that produces the maximum design force in a member. The displacement is applied in the same direction as the restraint. To obtain the influence line for the support reaction at end 1 of the simply supported beam shown in Figure 5.

This results in the elastic curve shown at ii. A unit load is applied to the beam at any point 3, as shown at iii , and the unit displacement applied to end 1. The influence line for bending moment at 3 is produced by cutting the beam at 3 and imposing a unit virtual rotation, as shown at iii. This produces the maximum positive shear at point 3 when the distributed load is located just to the right of 3. Example 5. Influence lines 5. This produces the maximum bending moment at a specific point 3 when a specified load is located at 3, as shown in ii , such that if the load is moved to the left of 3, the intensity of loading on section 13 is greater than on section 23, but if it moves to the right of point 3, the intensity of loading on section 23 is greater than on section The maximum moment usually occurs under one of the wheels adjacent to the centroid of the train.

The influence line for bending moment at the location of the maximum moment is shown at ii. Structural Analysis: In Theory and Practice 5. Hence, the maximum bending moment occurs under the second wheel load when it is located at a distance of 0. Diagrams indicating maximum values are known as envelope diagrams and are determined using influence lines at selected points along the member. Values of the moment are calculated in Table 5. This is shown in Figure 5.

The moving load is transferred from one panel point to the next as the load moves across the stringer. Hence, the influence line for axial force in a member is completed by connecting the influence line ordinates at the panel points on either side of a panel with a straight line. The influence lines for axial force in members 34, 35, and 24 are obtained by taking a section through these three members and considering the relevant free body diagrams. Positive sense of the influence line indicates tension in member Because of the effect of the stringers, the influence line between nodes 3 and 5 is obtained by connecting the ordinates at 3 and 5 with a straight line.

The influence line is positive, indicating tension in member The influence line for axial force in member 78 is identical in shape and of opposite sign. This influence line is identical in shape to the influence line for P57 and of opposite sign. The influence lines for axial force in members 89, , and 79 are obtained by taking a section through these three members and considering the relevant free body diagrams.

The influence line for P89 is shown at ii. The influence line for P is shown at iii. The influence line for P79 is shown at iv. Supplementary problems S5. Determine the maximum value of V4 due to a train of three wheel loads each of 3 kips at 2 ft on center. Construct the influence line for axial force in member Determine the maximum value of P24 due to a concentrated load of 5 kips. Determine the maximum value of P45 due to concentrated load of 10 kips. Determine the maximum value of P due to concentrated load of 20 kips. In some instances, however, it is necessary to consider the building as a whole and design it as a three-dimensional structure.

The sign convention shown in Figure 6. A space structure is illustrated in Figure 6. The plan view of the structure is in the xz plane, as shown in Figure 6. Displacements in a space structure may occur in six directions, a displacement in the x, y, and z directions and a rotation about the x-, y-, and z-axes. The sign convention for displacements is shown in Figure 6. A total of six displacement components define the restraint conditions at support 1 of the Structural Analysis: In Theory and Practice y x z Figure 6. The arrows indicate the positive directions of the displacement components, and, using the right-hand screw system, rotations are considered positive when acting clockwise as viewed from the origin.

Similarly, a total of six force components, as shown in Figure 6. Space frames 6. The cranked cantilever of Figure 6. Similarly, six member stresses may be determined at a section cut through the structure, at any point 4, as shown by the free-body diagram shown in Figure 6. Determine the magnitude of the reactions at support 1.

The supports consist of a fixed pin at node 1, providing three restraints as shown, and rollers at nodes 2, 3, and 4, providing only vertical restraint. Determine if the structure is statically determinate. Figure 6. The supports consist of fixed pins at nodes 1, 2, and 3, each providing three restraints as shown.

- [PDF] Structural Analysis in Theory and Practice By Alan Williams Book Free Download.
- Reading Guide to!
- Processed Apple Products.
- The Tango Briefing;
- Structural anthropology.
- Breathe: Everyone Has To Do It.
- See a Problem?!

Determine the member forces produced by the kip vertical load applied at node 4. The supports consist of fixed pins at nodes 1, 2, and 3, each providing three restraints. A statically determinate structure is one in which all member forces and external reactions may be determined by applying the equations of equilibrium. An indeterminate or redundant structure is one that possesses more unknown member forces and reactions than available equations of equilibrium.

To determine the member forces and reactions, additional equations must be obtained from conditions of geometrical compatibility. The number of unknowns, in excess of the available equations of equilibrium, is the degree of indeterminacy, and the unknown forces and reactions are the redundants. The redundants may be removed from the structure, leaving a stable, determinate structure, which is known as the cut-back structure. External redundants are Structural Analysis: In Theory and Practice redundants that exist among the external reactions. Several methods have been proposed1â€”5 for evaluating the indeterminacy of a structure.

The roller support provides only one degree of restraint in the vertical direction, and both horizontal and rotational displacements can occur. The hinge support provides two degrees of restraint in the vertical and horizontal directions, and only rotational displacement can occur. Thus, a structure is externally indeterminate when it possesses more than three external restraints and unstable when it possesses fewer than three.

The i ii Figure 1. M H V Figure 1. For the rigid frame shown in Figure 1. The degree of indeterminacy of the frames shown in Figure 1. M M Figure 1. In general, the introduction of hinges into i of the n members meeting at a rigid joint produces i releases. Figure S1. Matheson, J. Degree of redundancy of plane frameworks. Civil Eng. June Henderson, J. Statical indeterminacy of a structure. Aircraft Engineering. December Rockey, K. The degree of redundancy of structures.

Di Maggio, F. Statical indeterminacy and stability of structures. Rangasami, K. Degrees of freedom of plane and space frames. The Structural Engineer. March In conjunction with the principles of superposition and geometrical compatibility, the values of the redundants in indeterminate structures may then be evaluated. The principle may be defined as follows: if a structure in equilibrium under a system of applied forces is subjected to a system of displacements compatible with the external restraints and the geometry of the structure, the total work Structural Analysis: In Theory and Practice done by the applied forces during these external displacements equals the work done by the internal forces, corresponding to the applied forces, during the internal deformations, corresponding to the external displacements.

Thus, in Chapters 2 and 3 and Sections 6. A rigorous proof of the principle based on equations of equilibrium has been given by Di Maggio1. A derivation of the virtual-work expressions for linear structures is given in the following section. The external work done during the application of the loads must equal the internal energy stored in the structure from the principle of conservation of energy. This results in a force u, a bending moment m, and a shear force q in any element. For rigid frames, only the last two terms on the right-hand side of the expression are significant. For a rigid frame, moments produced by the virtual load or moment are considered positive, and moments produced by the applied loads, which are of opposite sense, are considered negative.

Table 2.

Solution Member forces u1 due to a vertical downward unit load at 4 have already been determined in Example 2. The moment of inertia has a constant value I over the length 23 and increases linearly from I at 2 to 2I at 1. The volume of this solid is given by the area of crosssection multiplied by the height of the solid at the centroid of the cross-section. Virtual work methods Table 2. Virtual work methods Alternatively, from Table 2. From Table 2. The principle of geometrical compatibility may be defined as follows: the displacement of any point in a structure due to a system of applied forces must be compatible with the deformations of the individual members.

The two above principles may be used to evaluate the redundants in indeterminate structures. The first stage in the analysis is to cut back the structure to a determinate condition and apply the external loads. The displacements corresponding to and at the point of application of the removed redundants may be determined by the virtual-work relations. To the unloaded cut-back structure, each redundant force is applied in turn and the displacements again determined.

The total displacement at each point is the sum of the displacements due to the applied loads and the redundants and must be compatible with the deformations of the individual members. Thus, a series of compatibility equations is obtained equal in number to the number of redundants. These equations are solved simultaneously to obtain the redundants and the remaining forces obtained from equations of equilibrium. The value of the thrust required to restore the arch to its original span is H1, and the value of the thrust required to reduce the deflection of 2 to zero is H2.

Determine the ratio of H2 to H1, neglecting the effects of axial and shearing forces. Determine the bending moment at 2, neglecting the effects of axial and shearing forces. Determine the resulting vertical deflection at panel points 2 and 3.

What additional vertical load W must be applied at panel point 3 to increase the deflection at panel point 2 by 50 percent? A horizontal force H is applied to the free end of the rib so that end 2 can deflect only vertically when the vertical load V is applied at 2. Determine the resulting vertical deflection at node 4. The cross-sectional area of the columns is A.

The modulus of elasticity of all members is E, and the modulus of rigidity is G. Determine the bending moments and shear forces in the members and calculate the horizontal deflection of node 3 due to the load of 4 kips. Di Maggio, I. Principle of virtual work in structural analysis. Ghose, S. The volume integration method of structural analysis. In the case of polygonal two-hinged arches, the preferred solution is by the method of column analogy and is dealt with in Section 6.

Indeterminate pin-jointed frames The horizontal deflection of joint 1 of the actual structure may be determined by considering a virtual unit load applied horizontally at 1 in systems i and ii. In general, to determine the deflection of an indeterminate structure, the unit virtual load may be applied to any cut-back structure that can support it. Initial inaccuracies in the lengths of members of an indeterminate frame produce forces in the members when they are forced into position. If the member 12 in Figure 3. The remaining member forces are given by the expression uR.

Determine a the forces in the members due to the applied load of 10 kips, b the horizontal deflection of point 2 due to the applied load of 10 kips, c the force in member 24 due to member 24 being 0. Indeterminate pin-jointed frames Table 3. All members have the same cross-sectional area and modulus of elasticity.

### Structural Analysis - In Theory and Practice

Solution The force R in member 13 is chosen as the redundant, and member forces P and u are obtained from i and ii and are tabulated in Table 3. Frames with more than two redundants are best solved by the flexibility matrix method given in Section Again, the two equations obtained may be solved for V1 and V2. An alternative method of solving the problem is to consider members 34 and 56 as internal redundants.

The values of the redundants may be obtained by the method of Section 3. In the particular structure shown in Figure 3. In system i , member 12 is subjected to a bending moment, and there are no axial forces in the members. The second moment of area of member 12 is in4, and the modulus of elasticity is constant for all members.

All members have the same cross-sectional area, modulus of elasticity, and second moment of area. Usually the arch axis closely follows the funicular polygon for the applied loads, and shear effects are small and may be neglected. In the case of reinforced-concrete arch ribs, the shrinkage of the concrete has a similar effect to a fall in temperature. The axial thrust, shear, and bending moment at any section 3 of the arch may be obtained from Figure 3. Neglecting the effects of axial and shear forces, determine the horizontal thrust at the supports and the bending moment at a point 20 ft from the left-hand support.

Neglecting the effects of axial and shear forces, determine the horizontal component of the axial forces in the arches. All members have the same cross-sectional area and the same modulus of elasticity. Table 3. Indeterminate pin-jointed frames Supplementary problems S3. Determine the resultant force in member 24 due to the lack of fit and the applied load of 20 kips. Determine the resultant force in members 24 and 26 due to the lack of fit. Determine the force in member 13 due to the applied load W.

The cross-sectional area and second moment of area of the beam are 86 in2 and in4. The cross-sectional area of the strut is 26 in2, and the ratio of the modulus of elasticity of steel and wood is Determine the force in the tie rod caused by the kip concentrated load at the crown. Neglecting the effects of axial and shear forces, determine the horizontal component of the axial force in the arches. Neglecting the effects of axial and shear forces, determine the horizontal thrust at the supports caused by the 10 kip load located as indicated.

The flexural rigidity of both columns also has a value of EIo. Determine the horizontal thrust at the supports caused by the concentrated load W at the crown. Neglect the effects of axial and shear force. The modulus of elasticity, cross-sectional areas, and second moments of area of the members are given in the Table.

## Structural Analysis - 1st Edition

The method may also be used to determine fixed-end moments and support reactions in continuous beams and frames, though generally the column analogy and moment distribution methods are preferred methods of solution. Structural Analysis: In Theory and Practice 4. An analogous beam, known as the conjugate beam, shown in Figure 4. Thus, the slope and deflection at any section in the real beam are given by the shear and bending moment at that section in the conjugate beam, and the elastic curve of the real beam is given by the bending moment diagram of the conjugate beam.

In the case of beams, the elastic load applied to the conjugate beam is positive i. The deflection of the real beam at any section is positive i. The slope of the real beam is positive when a positive shear force occurs at the corresponding section of the conjugate beam. The displacement of the real frame at any section is perpendicular to the lever arm used to determine the moment in the conjugate frame and outward when a positive bending moment occurs at the corresponding section of the conjugate frame.

At a simple end support in the real structure, shown in Table 4. At a free end in the real structure, shown in Table 4. Thus, the corresponding restraints in the conjugate structure are a shear force and a bending moment, which are produced at the fixed end. At an interior hinge in the real structure, shown in Table 4. Flange plates are added to the central 10 ft of the beam in order to limit the deflection to 1 of the span.

Determine the second moment of area required for the central portion if the second moment of area of the plain beam is in4 and the modulus of elasticity is 29, kips in2. All the members of the frame have the same second moment of area and the same modulus of elasticity. Solution The bending moment diagram, drawn on the compression side of the members, is shown at i , and the elastic load on the conjugate frame is shown at ii.

Flange plates are added to the 24 ft portion of the beam between the applied loads so as to double the second moment of area. Determine the distribution of bending moment in each member. Solution The deflections produced at the interconnections are indicated at i. The internal reactions at the interconnections may be considered as the redundants and are indicated at ii.

Along the diagonals, these reactions are zero due to the symmetry of the structure and applied loading. The applied loads and deflections of beams 11, 22, and 33 are indicated at iii , iv , and v. When the number of internal reactions exceeds three, the matrix method given by Stanek2 is a preferred method of solution. The change in slope of a pin-jointed frame is concentrated at the pins and is known as the angle change. Thus, the deflections at the panel points of a pinjointed frame are given by the bending moments at the corresponding points in a conjugate beam loaded with the total angle change at the panel points.

In the pin-jointed frame shown in Figure 4. Due to these changes in length, all angles in the frame change. All members of the frame have the same length, area, and modulus of elasticity. The loads on the conjugate beam are shown at i. The relative EI values are shown ringed. A uniformly distributed load w is applied over a length a from the support.

Conjugate beam methods 2a a 3 1 1 3 2 M Figure S4. Lee, S. The conjugate frame method and its application in the elastic and plastic theory of structures. Journal Franklin Institute. September Stanek, F. A matrix method for analyzing beam grillages. Developments in theoretical and applied mechanics. Plenum Press. New York. The bar-chain method of analysing truss deformations. May Influence lines for arches and multibay frames may be obtained by the methods given in Sections 6.

A comprehensive treatment of the determination of influence lines for indeterminate structures has been given by Larnach1. To obtain the influence line for reaction in the prop of the propped cantilever shown in Figure 5. Table 5. The beam is then supported at its center to form a two-span continuous beam. Obtain the influence line for bending moment for a point midway between one end support and the center support. The displacement is measured in the line of action of the applied force; the displacement corresponding to an applied moment is a rotation and to a force, a linear deflection.

Referring to Figure 5. Structural Analysis: In Theory and Practice A general procedure for obtaining the influence lines for any restraint in a structure is to apply a unit force to the structure in place of and corresponding to the restraint. The elastic curve produced is the influence line for displacement corresponding to the restraint. Dividing the ordinates of this elastic curve by the displacement occurring at the point of application of the unit force gives the influence line for the required restraint. From Figure 2. The second moment of area varies linearly from a value of I at end 1 to 2I at end 2.

The beam has a second moment of area three times that of each column. The moment required to produce the unit rotation is derived in Section 6. Thus, referring to Figure 5. For any other rotation, the elastic curve is obtained F at the as the product of the rotation and the influence line ordinate for M12 corresponding section. When a rotation also occurs at 2, the elastic curve is obtained as the algebraic sum of the two products.

The method may be readily extended to structures with nonâ€”prismatic members using stiffness factors, carry-over factors and fixed-end moments that have been tabulated4 for a large variety of non-prismatic members. The second moment of area of span 34 is twice that of spans 12 and Influence lines nl 1 4 3 2 l l 1 l Figure 5. These initial moments are distributed in Table 5. The second moment of area and modulus of elasticity is constant for all members. To determine the influence line for V1 for the two-span continuous beam shown in Figure 5.

The second moment of area is regarded as constant over the length of each segment, as indicated at i. The reaction V1 is replaced by a unit load acting vertically upward, and the bending moment diagram on the cut-back structure is shown at ii. The bending moment is regarded as constant over the length of each segment and has the values indicated.

The influence line for any other restraint may now be obtained by the direct application of statics. The beam is divided into short segments of length s, and the second moment of area is regarded as constant over the length of each segment, as indicated at i. A unit load is applied vertically upward at 2, as shown at ii , and the bending moment at the center of each segment is determined, as shown at iii.

The bending moment is regarded as constant over the length of each segment, and the elastic loads on the conjugate beam are considered to be concentrated at the center of each segment, as shown at iv. The ordinates of the elastic curve of the real beam, simply supported at 1 and 4 and with a unit load applied at 2, are given by the bending moments at the corresponding section in the conjugate beam and are shown at v. A correction must be applied to these ordinates to reduce the deflection at 3 to zero since, with V3 in position, there can be no deflection at 3.

A unit load acting vertically downward is applied at 3, as shown at vi. The ordinates of the elastic curve for this loading condition are obtained by the above procedure and are shown at vii. For a symmetrical structure, curve vii is the inverted mirror image of curve v.

## International Electronic Journal of Mathematics Education

Similarly, the influence line for V3 may be obtained and, for a symmetrical structure, is the mirror image of the influence line for V2. When the number of redundants exceeds 2, the flexibility matrix method, given in Section The relative EI values are indicated on the figure and may be assumed to be constant over the lengths indicated, as may also the elastic loads on the conjugate beam.

Solution The cut-back structure is produced by removing V1 and V4; a vertically upward load of 0. This produces the bending moment diagram shown at ii and the elastic loads on the conjugate beam at iii. The bending moment in the conjugate beam at the center of each segment is given in line 2 of Table 5.

Due to the symmetry of the structure, the ordinates of the elastic curve 1 2. This represents the ordinates of the elastic curve produced by a load of 0. The values of the influence line ordinates for M2 are given in line 7. Thus, all the information required to obtain the influence line ordinates for R may be obtained from one Williot-Mohr diagram constructed for system ii.

Thus, all the information required to obtain the influence lines for Rl and R2 may be obtained from the two Williot-Mohr diagrams constructed for systems ii and iii. For each panel point in turn, the required deflections are obtained from the diagrams and substituted in equations 1 and 2 , which are solved simultaneously to give the influence line ordinates for Rl and R2. When the internal redundants exceed two, the flexibility matrix method, given in Section The deflections of the panel points are most readily obtained using the method of angle changes. When the external redundants exceed two, the flexibility matrix method, given in Section All members of the frame have the same cross-sectional area, modulus of elasticity, and length.

The deflections at the bottom chord panel points are given by the bending moments at the corresponding section in the conjugate beam and are shown at iii. These deflections, when divided by , give the influence line ordinates for V3 as shown at iv. These values are shown at v. All members have the same cross-sectional area, modulus of elasticity, and length. The deflections of the lower panel points were obtained in Example 4. When unit load is applied vertically downward at 5, the relative deflections shown at ii are produced, and ii is the inverted mirror image of i due to the symmetry of the frame.

The ordinates at ii are multiplied by Determine the influence line ordinates, at intervals of 0. Determine the influence line ordinates for M2 at intervals of 0. Determine the influence line ordinates, at intervals of 20 ft, for horizontal thrust at the hinges as unit load moves from 5 to 7. Parallel submission of the same manuscript to more than one journal is considered as unethical publication behavior and is unacceptable. Proper recognition of the work of others must always be granted. The authors should also cite publications that influenced the inspiration and determined the nature of the submitted work.

Authorship should be limited to persons who have made a significant contribution to the concept, design, performance or interpretation of the results of the submitted manuscript. All those who have made a significant contribution should be listed as co-authors. If there are other people who have participated in certain aspects of the research project, they should be listed in the Acknowledgments section.

The author for correspondence should ensure that all the relevant co-authors as defined above are listed and no inappropriate co-authors are included in the list of authors of the manuscript. All authors should disclose in their manuscripts all financial or other conflicts of interest that can be interpreted as affecting the results or their interpretation in the manuscript. All sources of financial support for work should be indicated. When the author discovers a significant error or inaccuracy in his own published work, it is the responsibility of the author to immediately notify the editor or publisher of the journal and to cooperate to withdraw the article or to publish an appropriate errata.

Publisher confirmation In cases of alleged or proven scientific abuse, fraudulent publication or plagiarism, the publisher, in close cooperation with the editors, will take all appropriate measures to clarify the situation and change the article. This includes a quick errata publication or, in the most severe cases, complete withdrawal of the paper. Recent Articles Quarterly No.

Using composite coatings containing solid lubricants to minimize friction in piston combustion engines for aviation Andrzej Posmyk, Jerzy Myalski pages keywords: piston combustion engine, composite coatings, solid lubricant, loss of lubrication article version pdf 0. This lubrication simulates the operating conditions of the combustion engine components after lubrication loss.

The oil sorption is indicated by the increase in the mass of the sample with the coating. The contact with the reference coating seized after 8. The tests of the developed coating on the engine dynamometer as well as in the aircraft confirmed the usefulness of the developed coating. Abrasion resistance tests were carried out at room temperature in a ball-on-disc system. Balls with a diameter of 3. After the abrasion tests the sample surfaces were examined by scanning electron microscopy. The obtained results show that the tribological properties depend on the test conditions and content of the TiB 2 reinforcing phase.

Problems in examining thermal properties of insulating composite materials designed for technical means of transport Andrzej Posmyk, Krzysztof Hodor pages keywords: technical means of transport, thermal conductivity, insulating materials, measurement inaccuracy article version pdf 0.

To measure the thermal conductivity with measuring apparatuses available on the market, samples with a specific shape, dimensions and accuracy of workmanship are required. During the drying of water-soluble resins, bubbles and surface deformation develop as a result of water evaporation. Machining the samples to obtain flat and parallel surfaces is not recommended due to the possibility of damage to the microspheres.

Inaccuracy in producing the samples significantly affects the accuracy of the thermal conductivity measurement results by direct methods using the heat flow conducted by the test sample. The applied technological procedure included using the original element as a model to produce a mold, the production of the mold and the production of two elements from two alternatively reinforced laminates a chopped-strand mat and a plain-woven fabric , by the hand lay-up method.

On the basis of the observations made during the technological procedures and based on evaluation of the manufactured products, it was stated that the hand lay-up lamination technique with additional use of a vacuum bag is a very good and simple method of making single products. The use of vacuum prevents the occurrence of defects typical for classic manual lamination, such as delaminations, closed air bubbles, or especially lack of adhesion in low-radius curved areas.

It has no visible impact on the quality of the product and significantly facilitates demolding. An important issue during the application of vacuum bag assistance is proper selection of the amount of catalyst for the resin. This should be preceded by measuring the room temperature in which the process is carried out and the time of the hand lay-up lamination process - the number of layers should be selected for the predicted resin curing time so that a proper lay-up can be prepared and the vacuum process carried out before the laminate cures.

The manufactured elements require slight machining of the technological surplus, which is difficult to avoid when designing a technically simple form for a single or low- series product. Foaming extrusion of thermoplastic polyurethane modified by POSS nanofillers Piotr Stachak, Edyta Hebda, Krzysztof Pielichowski pages keywords: thermoplastic polyurethanes, chemical blowing agents, polyhedral oligosilsesquioxanes, POSS, extrusion article version pdf 1.

Combining extrusion with a foaming process leads to the fabrication of porous lightweight materials with novel properties. These properties can be further modified by applying additives, such as polyhedral oligomeric silsesquioxanes POSS. POSS are organic-inorganic hybrid nanofillers that could enhance polymer-based composites properties, e. In this work, the foaming extrusion process of TPU, utilizing azodicarboxamide ADC , sodium bicarbonate SC and citric acid monohydrate CA as blowing agents or their mixtures was described.