Theory of Structures (Stability and determinacy).pptx

Stability, determinacy and structural loads of structures

1. CEng 3203-Theory of Structures Chapter One Introduction to indeterminate structure 17-Oct-24 Stability & Determinacy of structures 1

2. Outlines Introduction Determinacy of structures for different types of beams, portal frames and truss Stability of structures Structural loads 17-Oct-24 2 Stability & Determinacy of structures

3. Learning outcomes of this chapter Describe the stability and determinacy. For analysis of indeterminate structures, adopt an appropriate structural analysis method. 17-Oct-24 3 Stability & Determinacy of structures

4. Introduction What is a structures? A structure refers to a system of connected parts used to support loads. The fundamental purpose of a structure is to transmit loads from the point of application to the point of support and through the foundations to the ground. 17-Oct-24 4 Stability & Determinacy of structures

5. Introduction Before going into the analysis of any structure , it is necessary to identify its statically type (classification ) i.e. whether it is determinate or indeterminate , stable or unstable . An unstable arrangement of supports and structural members should be avoided. 17-Oct-24 5 Stability & Determinacy of structures

6. Determinacy of structures What does mean determinacy of structures? The determinacy of structures refers to the degree to which the structural system is fixed. Why it is needed to fix structures? To determine the loads it must withstand, materials used to construct it. Indeterminacy is introduced in structures on account of functional requirements, limitations on types of framing and need for stiffness. 17-Oct-24 6 Stability & Determinacy of structures

7. Determinacy of structures Based on response of the structures to the external applied loads in the form of reactions or displacements, indeterminacy of the structures maybe classified into two general categories. Static indeterminacy due to support reactions Kinematic indeterminacy due to joints displacement 17-Oct-24 7 Stability & Determinacy of structures

8. Determinacy of structures Static indeterminacy Depending upon determinacy, structures maybe classified into two general categories. Statically determinate Statically indeterminate When all support reactions in a structure can be determined from equilibrium equations solely, the structure is referred to as statically determinate . 17-Oct-24 8 Stability & Determinacy of structures

9. Determinacy of structures When all support reactions in a structure cannot be determined from equilibrium equations solely (i.e. more unknown support reactions than available equilibrium equations), the structure is referred to as statically indeterminate . In addition to more unknown support reactions, a structure is statically indeterminate when it possesses more members than necessary for stability (and equilibrium). 17-Oct-24 9 Stability & Determinacy of structures

10. Determinacy of structures These excess members or support reactions are called redundant or degree of static indeterminacy . The degree of static indeterminacy is the number of unknown support reactions in excess of the available equilibrium equations . To analysis indeterminate structures, we need ways of establishing additional equations . 17-Oct-24 10 Stability & Determinacy of structures

11. Determinacy of structures These additional equations may be derived from compatibility of deformation or from conditions of symmetry and known as compatibility equations. Compatibility equations must be equal to the number of static degree indeterminacy. 17-Oct-24 11 Stability & Determinacy of structures

12. Determinacy of structures External and Internal Static Indeterminacy The static indeterminacy of a structure can be external (with respect to support reactions or restraints) or internal (with respect to member forces or internal forces). When all support reactions in a structure cannot be determined from equilibrium equations solely (i.e. more unknown support reactions than available equilibrium equations), the structure is referred to as statically indeterminate externally . 17-Oct-24 12 Stability & Determinacy of structures

13. Determinacy of structures A structure is s tatically indeterminate internally when it is not possible to determine all internal forces by using the equations of static equilibrium . For the great majority of structures, the question of whether or not they are statically indeterminate internally can be decided by inspection . For certain structures this is not so, and for these types rules have to be established. 17-Oct-24 13 Stability & Determinacy of structures

14. Determinacy of structures Total degree of static indeterminacy of structures can be given by the following equation: Degree of static indeterminacy of structures externally Degree of static indeterminacy of structures internally   17-Oct-24 14 Stability & Determinacy of structures

15. Determinacy of structures Where: is the number of internal forces for each member is the number of members is the number of unknown reactions is the number of equations of equilibrium at each joint is the number of joints of the structures is the number of internal releases (internal hinges and/or internal rollers)   17-Oct-24 15 Stability & Determinacy of structures

16. Determinacy of structures If a structure is supported by more than three reactions , then all the reactions cannot be determined from the three equations of equilibrium. Such structures are termed statically indeterminate externally ( ). The reactions in excess of those necessary for equilibrium are called external redundants , and the number of external redundants is referred to as the degree of external indeterminacy .   17-Oct-24 16 Stability & Determinacy of structures

17. Determinacy of structures If a structure is supported by fewer than three support reactions , the reactions are not sufficient to prevent all possible movements of the structure in its plane . Such a structure cannot remain in equilibrium under a general system of loads and is therefore, referred to as statically unstable externally ( ).   17-Oct-24 17 Stability & Determinacy of structures

18. Determinacy of Structures- Beams A beam is a structural element that is capable of withstanding load primarily by resisting bending . Beams generally carry vertical loadings (in plane moments and forces normal to the axis of the beam). Thus, for beams , , and the total degree of static indeterminacy for a beam can be calculated as:   17-Oct-24 18 Stability & Determinacy of structures

19. Determinacy of Structures- Beams Total degree of static indeterminacy for a beam is the degree of external static indeterminacy as no internal force in any member of a beam remains unknown, if beam is statically determinate externally. Thus for beams, and   17-Oct-24 19 Stability & Determinacy of structures

20. Determinacy of Structures- Beams Note : In the determination of , components of the support reactions in the direction parallel to the beam axis must be ignored since there is no internal axial force in any bean members. Stability of a beam is depends on external supports only and its determinacy relates on the number of available and conditional equations.   17-Oct-24 20 Stability & Determinacy of structures

21. Determinacy of Structures- Beams Let is the available number of reaction components , is the minimum number of reaction components required for stability usually , and n is the number of special/ conditional equation . If   17-Oct-24 21 Stability & Determinacy of structures

22. Determinacy of Structures- Beams Note: is not a sufficient condition for stability . F or a coplanar structure there are at most three equilibrium equations for each part, so that if there is a total of parts and force and moment reaction components, we have: , unstable , statically determinate , statically indeterminate   17-Oct-24 22 Stability & Determinacy of structures

23. Determinacy of Structures- Beams Examples (Determine total degree of static indeterminacy , degree of external and internal indeterminacy of the following beams) T otal degree of static indeterminacy for beam   17-Oct-24 23 Stability & Determinacy of structures    

24. Determinacy of Structures- Beams Where: is the number of members is the number of unknown reactions is the number of joints of the structures is the number of internal releases (internal hinges and/or shear releases)   17-Oct-24 24 Stability & Determinacy of structures

25. Determinacy of Structures- Beams Degree of external indeterminacy of a beam For beams Degree of internal indeterminacy of a beam For a beams A beam is s tatically indeterminate to the second degree . OR   17-Oct-24 25 Stability & Determinacy of structures

26. Determinacy of Structures- Beams is number force and moment reaction components is total number of parts Degree of static indeterminacy of a beam   17-Oct-24 26 Stability & Determinacy of structures

27. Determinacy of Structures- Beams is number force and moment reaction components is total number of parts Degree of static indeterminacy of a beam   17-Oct-24 27 Stability & Determinacy of structures    

28. Determinacy of Structures- Beams Degree of external indeterminacy of a beam Where: is the number of unknown reactions is the number of internal releases (internal hinges and/or internal rollers ) A beam is statically indeterminate to the first degree .   17-Oct-24 28 Stability & Determinacy of structures

29. Determinacy of Structures- Beams is number force and moment reaction components is total number of parts Degree of static indeterminacy of a beam   17-Oct-24 29 Stability & Determinacy of structures

30. Determinacy of Structures- Beams Degree of external indeterminacy of a beam Where: is the number of unknown reactions is the number of internal releases (internal hinges and/or internal rollers) A beam is statically determinate .   17-Oct-24 30 Stability & Determinacy of structures

31. Determinacy of Structures- Beams is number force and moment reaction components is total number of parts A beam is unstable .   17-Oct-24 31 Stability & Determinacy of structures    

32. Determinacy of Structures- Beams is number force and moment reaction components is total number of parts A beam is unstable .   17-Oct-24 32 Stability & Determinacy of structures    

33. Determinacy of Structures- Trusses A simple truss can be made by combining three bars to form a triangle . Stability depends partly on external supports and partly on the arrangement of members or bars . Three reaction components are required for external stability and determinacy of a plane truss without condition equations. 17-Oct-24 33 Stability & Determinacy of structures

34. Determinacy of Structures- Trusses For trusses (pin jointed frames, ) and thus total degree of static indeterminacy . External degree static of indeterminacy The external statically classification of the truss depends on the total number of reaction components, and their arrangement.   17-Oct-24 34 Stability & Determinacy of structures

35. Determinacy of Structures- Trusses where is the number of equilibrium equations and is the number of conditional equations. For trusses . Let is the available number of reaction components , is the minimum number of reaction components required for stability usually , and n is the number of special/ conditional equation .   17-Oct-24 35 Stability & Determinacy of structures

36. Determinacy of Structures- Trusses If The condition for is necessary but not sufficient conditions for statically classification because the arrangement of the reaction components may render the truss unstable.   17-Oct-24 36 Stability & Determinacy of structures

37. Determinacy of Structures- Trusses Internal degree static of indeterminacy . Let is the number of members required to form an internally statically determinate truss that connects joints and has reaction components required for external stability . If is the actual number of bar forces in the truss, then the following criteria hold true for internal classification:   17-Oct-24 37 Stability & Determinacy of structures

38. Determinacy of Structures- Trusses If Consider the trusses shown below. The truss shown in fig. (a) is stable whereas the truss shown in fig. (b) is unstable since the geometric arrangement of the members is not maintained.   17-Oct-24 38 Stability & Determinacy of structures

39. Determinacy of Structures- Trusses 17-Oct-24 39 Stability & Determinacy of structures

40. Determinacy of Structures- Frames A frame is a structure of continuous members being connected by rigidly joints. This type of structures ca resist both transverse and longitudinal loadings. For frames ( ) and thus total degree of static indeterminacy . Stability depends partly on external supports and partly on moment resisting joints. External degree static of indeterminacy where is the number of equilibrium equations and is the number of conditional equations.   17-Oct-24 40 Stability & Determinacy of structures

41. Determinacy of Structures- Frames The number of conditional equation introduced by a hinge joint is equal to the number of members at the joint minus one.   17-Oct-24 41 Stability & Determinacy of structures      

42. Determinacy of Structures- Frames Let is the available number of reaction components , is the minimum number of reaction components required for stability usually , and n is the number of special/ conditional equation . If   17-Oct-24 42 Stability & Determinacy of structures

43. Determinacy of Structures- Frames Internal degree static of indeterminacy . Let is the number of members required to form an internally statically determinate frame that connects joints and has reaction components required for external stability. If is the actual number of bar forces in the frame is the number of unknowns is the number of available equations   17-Oct-24 43 Stability & Determinacy of structures

44. Determinacy of Structures- Frames Then the following criteria hold true for internal classification : If   17-Oct-24 44 Stability & Determinacy of structures

45. Determinacy of Structures- Frames 17-Oct-24 45 Stability & Determinacy of structures

46. Stability of structures The resistance offered by a structure to undesirable movement like sliding, collapsing, overturning, etc. is called stability . A statically stable structure is a structure that can resist any actions without the development of the rigid body movement (mechanism) on the entire structure or within any parts of the structure. 17-Oct-24 46 Stability & Determinacy of structures

47. Stability of structures For the static stability of a structure , proper and sufficient number of constraints must be provided to the entire structure in terms of supports so as to prevent the rigid body movement (translation and rotation) in the structure. This means stability depends upon the support conditions and arrangements of members . 17-Oct-24 47 Stability & Determinacy of structures

48. Stability of structures In structural analysis, a structure is said to be stable when it can support any possible system of applied loads. To ensure the equilibrium of a structure or its members, it is not only necessary to satisfy the equations of equilibrium . But, the members must also be properly held or constrained by their supports . 17-Oct-24 48 Stability & Determinacy of structures

49. Stability of structures Based on this concept of static stability, a statically unstable structure is a structure that is not statically stable. In the other word, a structure that exhibits the rigid body movement (mechanism) for the entire structure or within any parts of the structure when subjected to a particular action. 17-Oct-24 49 Stability & Determinacy of structures

50. Stability of structures Stability can be divided into two as external and internal . External stability A structure in which there are insufficient number of constraints or improper constraints to prevent motion from taking place is called an unstable entire structure. This is external instability . What matters is not only the number of support reactions but also their arrangement . 17-Oct-24 50 Stability & Determinacy of structures

51. Stability of structures Structures for which the numbers of reaction components are greater than or equal to the number of available equilibrium equations but that are unstable due to arrangement of these reaction components are said to be geometrically unstable . When the reaction elements are three or more like supports that are either parallel or concurrent , they are not sufficient to maintain static equilibrium . 17-Oct-24 51 Stability & Determinacy of structures

52. Stability of structures For a structure to be externally stable, the following conditions must be met : Non-Parallel reactions The reactions at the supports should not be parallel to each other . If the reactions are parallel , even if there is sufficient numbers of reaction, it cannot be restrain rigid body movement which leading to external instability . 17-Oct-24 52 Stability & Determinacy of structures

53. Stability of structures Non-Concurrent reactions The reactions at the supports should not pass through the same point . If the reactions are concurrent, it indicates that the forces at the supports are acting in a single line of action and they cannot prevent rotation of the structure about their concurrent point which can result in external instability. By satisfying these conditions of external stability, a structure can maintain its overall stability and resist external loads and forces. 17-Oct-24 53 Stability & Determinacy of structures

54. Stability of structures 17-Oct-24 54 Stability & Determinacy of structures Parallel supports Beam Truss

55. Stability of structures For this case of parallel supports, even if we have sufficient number of supports, due to their improper arrangement, they will offer no resistance to horizontal movement, thus making the arrangement unstable. 17-Oct-24 55 Stability & Determinacy of structures

56. Stability of structures 17-Oct-24 56 Stability & Determinacy of structures Concurrent supports

57. Stability of structures Internal stability A structure is considered to be internally stable or rigid , if it maintains its shape and remains a rigid body when isolated from the supports . Conversely, a structure is termed internally unstable (or non-rigid) if it cannot maintain its shape and may undergo large displacements under small disturbances when not supported externally. 17-Oct-24 57 Stability & Determinacy of structures

58. Stability of structures This internal instability is caused by too many internal releases (i.e. internal hinges) or improper arrangement of members. 17-Oct-24 58 Stability & Determinacy of structures Examples of internally stable structures Note that each of the structures shown forms a rigid body and each can maintain its shape under loads.

59. Stability of structures Each structure is composed of two rigid parts, AB and BC connected by a hinged joint B which cannot prevent the rotation of one part with respect to the other. 17-Oct-24 59 Stability & Determinacy of structures Examples of internally unstable structures

60. Stability of structures 17-Oct-24 60 Stability & Determinacy of structures I nternally unstable truss due to improper arrangement of members Stable truss

61. Stability of structures 17-Oct-24 61 Stability & Determinacy of structures I nternally unstable structures due to improper arrangement of members

62. Stability of structures Generally, to investigate the static stability of a structure, first find degree of static indeterminacy . If the degree of static indeterminacy of a given structure is less than zero, i.e. , then the structure is statically unstable .   17-Oct-24 62 Stability & Determinacy of structures

63. Stability of structures For degree of static indeterminacy , investigate the structure for the development of rigid body motion (mechanism) within the structure under a particular action. If there is any mechanism , then the structure is statically unstable . If no mechanism , then the structure is statically stable .   17-Oct-24 63 Stability & Determinacy of structures

64. Stability of structures Mechanism of the entire structure is caused due to insufficient number of constraints or improper constraints (parallel and concurrent constraints). While the mechanism in any portion of the structure is caused by too many internal releases or improper arrangement of members. 17-Oct-24 64 Stability & Determinacy of structures

65. Stability of structures Investigate the static stability of the following structure Since , the structure is unstable internally.   17-Oct-24 65 Stability & Determinacy of structures    

66. Stability of structures However, by investigating all parts of this structure, there exists a pattern of rigid body motion (mechanism) as shown in the figure.   17-Oct-24 66 Stability & Determinacy of structures  

67. Stability of structures Therefore, the structure is statically unstable. Thus, static stability of the structure cannot be concluded from .   17-Oct-24 67 Stability & Determinacy of structures

68. Stability of structures Since , thus, static stability of the structure cannot be concluded from . However, by investigating all parts of this structure, there is no development of rigid body motion within any parts of the structure. Therefore, the structure is statically stable.   17-Oct-24 68 Stability & Determinacy of structures    

69. Structural loads What is a load ? A load is a weight or source of pressure due to materials or persons acting on a structure during their lifetime . Accurate estimation of the magnitudes of these loads is a very important aspect of the structural analysis process. 17-Oct-24 69 Stability & Determinacy of structures

70. Structural loads The loads that act on common civil engineering structures can be grouped according to their nature and source into three classes : D ead loads (Permanent loads) , Live loads (Variable loads) and Environmental loads 17-Oct-24 70 Stability & Determinacy of structures

71. Structural loads Dead loads Dead loads are those that act on the structure as a result of the weight of the structure itself and of the components of the system that are permanent fixtures . As a result, dead loads are characterized as having fixed magnitudes and positions . 17-Oct-24 71 Stability & Determinacy of structures

72. Structural loads Examples of dead loads are the weights of the structural members themselves such as beams and columns , the weights of roof surfaces , floor slabs , ceilings or permanent partitions walls and so on . How we can determine dead loads? Dead loads associated with the structure can be determined if the materials and sizes of the various components are known. 17-Oct-24 72 Stability & Determinacy of structures

73. Structural loads The determination of the characteristic values of self-weight and of the dimensions and densities shall be in accordance with ES EN 1990:2015 ‘General actions’ Part 1-1 section 4.1.2 . Nominal dimensions should be those as shown on the drawings . where A is members cross-section and is a unit weight of composed materials   17-Oct-24 73 Stability & Determinacy of structures

74. Structural loads 17-Oct-24 74 Stability & Determinacy of structures

75. Structural loads Live loads Live loads also known as imposed loads are those that may vary over time and often result from the occupancy of the structure . Typically live loads may include people, movable objects like furniture, vehicles and so on. Live loads are variable as they depend on usage and capacity. 17-Oct-24 75 Stability & Determinacy of structures

76. Structural loads How we can determine live loads ? The magnitude of live loads are varying on the structure and its impossible to obtain accurate values. Because of this, design codes such as our very nation’s ‘ ES EN 1991:2015 Part 1-1 ‘General Actions’ section 6.3 provide equivalent loads for various building categories and their specific usages. 17-Oct-24 76 Stability & Determinacy of structures

77. Structural loads 17-Oct-24 77 Stability & Determinacy of structures Imposed Loads on Buildings

78. Structural loads 17-Oct-24 78 Stability & Determinacy of structures

79. Structural loads The imposed loads are modelled by uniformly distributed loads, line loads or concentrated loads or combinations of these loads. 17-Oct-24 79 Stability & Determinacy of structures

80. Structural loads Values of imposed actions 17-Oct-24 80 Stability & Determinacy of structures

81. Structural loads 17-Oct-24 81 Stability & Determinacy of structures

82. Structural loads 17-Oct-24 82 Stability & Determinacy of structures

83. Structural loads Imposed Loads on Bridges Live loads on bridges consist primarily of vehicles, which vary in weight, size, and frequency of occurrence (cars, trucks, buses ). Live loads are often concentrated at specific points where vehicles are located on the bridge. 17-Oct-24 83 Stability & Determinacy of structures

84. Structural loads 17-Oct-24 84 Stability & Determinacy of structures Imposed Loads on Bridges

85. Structural loads 17-Oct-24 85 Stability & Determinacy of structures Weight shown are for each one axle = 2 wheels

86. Structural loads Differences between live loads on the buildings and live loads on the bridges The differences between live loads on buildings and live loads on bridges are primarily related to their functions, design considerations and usage patterns . Live loads in buildings vary based on the occupancy type (residential, commercial, etc.), furniture, equipment, and people. 17-Oct-24 86 Stability & Determinacy of structures

87. Structural loads Live loads are often assumed to be uniformly distributed across floors, although concentrated loads (like heavy furniture) may be considered . Live loads in buildings are typically considered to be sustained for longer periods , as buildings are generally static environments with continuous occupancy. 17-Oct-24 87 Stability & Determinacy of structures

88. Structural loads Live loads on buildings may not account for significant dynamic effects , as the movement of occupants is generally slower and less impactful compared to vehicles on a bridge . Live loads on bridges are often concentrated at specific points where vehicles are located on the bridge. Live loads on bridges are typically transient, as vehicles are not stationary; they pass over the structure relatively quickly . 17-Oct-24 88 Stability & Determinacy of structures

89. Structural loads Bridges must account for dynamic impact factors due to the acceleration and deceleration of vehicles, as well as potential collisions, which can increase the effective live load . In summary: Live loads on buildings are primarily influenced by occupancy and usage patterns, with a focus on static and sustained loads. Live loads on bridges are characterized by dynamic vehicle traffic, requiring considerations for concentrated loads and dynamic effects. 17-Oct-24 89 Stability & Determinacy of structures

90. Structural loads Environmental loads Environmental loads refer to the forces and stresses exerted on a structure due to natural environmental factors . These loads can significantly impact the design and performance of civil engineering structures, requiring careful consideration to ensure safety and functionality. 17-Oct-24 90 Stability & Determinacy of structures

91. Structural loads Environmental loads include : Wind Loads Seismic Loads Soil Loads Snow Loads Temperature Effects Rain Loads 17-Oct-24 91 Stability & Determinacy of structures

92. Structural loads Wind Loads Wind loads are the environmental load cases caused by a moving (blowing ) air mass against an obstructing structure and hence subsequently converting the kinetic energy of the wind into a potential energy of pressure . This wind pressures can be either positive or negative. 17-Oct-24 92 Stability & Determinacy of structures

93. Structural loads 17-Oct-24 93 Stability & Determinacy of structures

94. Structural loads 17-Oct-24 94 Stability & Determinacy of structures

95. Structural loads Positive wind pressures: are loads having pushing effects (towards the structure). Negative wind pressures (suctions): are those loads having pulling effects (away from the structure). These loadings caused by wind are expected to affect three perspective faces of a given structure. These are the windward , leeward and sides of a structure 17-Oct-24 95 Stability & Determinacy of structures

96. Structural loads W indward faces T hose in the upwind direction or from which the wind is coming. Windward faces are always loaded with positive wind pressures . 17-Oct-24 96 Stability & Determinacy of structures

97. Structural loads Leeward faces T hose which are away (downward or downwind) from the upcoming wind. Leeward faces are always loaded with negative wind pressures (suctions ) . 17-Oct-24 97 Stability & Determinacy of structures

98. Structural loads Side faces T hose faces parallel to the direction of the wind. Side walls can be loaded with positive or negative pressures depending on the geometric plan shape of the structure. 17-Oct-24 98 Stability & Determinacy of structures

99. Structural loads Wind Loads-Analysis Exposure extent of the structure is dependent on terrain roughness and topography of its location , expresses the vulnerability of the structure to wind actions. Because of this, design codes such as our very nation’s ‘ ES EN 1991:2015 ‘General Actions’ Part 1-4 section 5.2 provide characteristics wind loads value. 17-Oct-24 99 Stability & Determinacy of structures

100. Structural loads Earthquake Loads 17-Oct-24 100 Stability & Determinacy of structures

101. Structural loads An earthquake is a sudden motion in the earth’s upper crust (Lithosphere) caused by abrupt releases of accumulated strains energy in rocks below, volcanic activity, landslides and collapse of underground caves. The ground surface moves both in horizontal and vertical directions during an earthquake. The magnitude of the vertical ground motion is usually small and does not have a significant effect on most structures. 17-Oct-24 101 Stability & Determinacy of structures

102. Structural loads It is the horizontal component of ground motion that causes structural damage and must be considered in design of structures located in earthquake prone areas . The seismic base shear force for each horizontal direction in which the building is analyzed shall be determined by ES EN 1998:2015 section 4.3.3.2 .   17-Oct-24 102 Stability & Determinacy of structures

103. Quiz (5%) The preconditions for checking of the determinacy of a given structure is its stability. (True/False) (1 mark) Investigate the external and internal stability and determine the total degree of static indeterminacy, external and internal indeterminacy of the following framed structures. (3 marks) 17-Oct-24 103 Stability & Determinacy of structures

104. Quiz (5%) What are the causes of the external and internal instability of structures? ( 1 mark) 17-Oct-24 104 Stability & Determinacy of structures      

105. 17-Oct-24 105 END OF THE CHAPTER Stability & Determinacy of structures