Tos I - Chapter 1.pdf

Theory of Structure I Engr. Gabriel Gamana

Table of Contents

1.0 Types of Structures and Loads 2.0 Analysis of Statically Determinate Structures 3.0 Cables and Arches 4.0 Influence Lines for Statically Determinate Structures 5.0 Approximate Analysis of Statically indeterminate Structures 6.0 Deflections

2

1

1.0 Types of Structures and Loads

1.1 Introduction 1.2 Structural Elements 1.3 Types of Structures 1.4 Loads on Structure 1.5 Idealized Structure

3

1.1 Introduction

4

2

1.1 Introduction • A structure refers to a system of connected parts used to support a load. Important examples related to civil engineering include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are important. • When designing a structure to serve a specified function for public use, the engineer must account for its safety, esthetics, and serviceability, while taking into consideration economic and environmental constraints. • Often this requires several independent studies of different solutions before final judgment can be made as to which structural form is most appropriate. • This design process is both creative and technical and requires a fundamental knowledge of material properties and the laws of mechanics which govern material response. 5

1.1 Introduction • Once a preliminary design of a structure is proposed, the structure must then be analyzed to ensure that it has its required stiffness and strength. To analyze a structure properly, certain idealizations must be made as to how the members are supported and connected together. • The loadings are determined from codes and local specifications, and the forces in the members and their displacements are found using the theory of structural analysis. • The results of this analysis then can be used to redesign the structure, accounting for more accurate determination of the weight of the members and their size.

6

3

1.1 Introduction

7

1.2 Structural Elements

8

4

1.2 Structural Elements 1.2.1 Ties Those members that are subjected to axial tension forces only. Load is applied to ties only at the ends. Ties cannot resist flexural forces.

9

1.2 Structural Elements 1.2.2 Struts Those members that are subjected to axial compression forces only. Like ties, struts can be loaded only at their ends and cannot resist flexural forces.

10

5

1.2 Structural Elements 1.2.3 Brace A braced frame is a structural system designed to resist wind and earthquake forces. Members in a braced frame are not allowed to sway laterally (which can be done using shear wall or a diagonal steel sections, similar to a truss).

11

1.2 Structural Elements 1.2.4 Beams and Girders Those members that are primarily subjected to flexural forces. They usually are thought of as being horizontal members that are primarily subjected to gravity forces; but there are frequent exceptions (e.g., inclined rafters).

12

6

1.2 Structural Elements 1.2.5 Columns Those members that are primarily subjected to axial compression forces. A column may be subjected to flexural forces also. Columns usually are thought of as being vertical members, but they may be inclined.

13

1.2 Structural Elements 1.2.6 Diaphragms Structural components that are flat plates. Diaphragms generally have very high in-plane stiffness. They are commonly used for floors and shear resisting walls. Diaphragms usually span between beams or columns. They may be stiffened with ribs to better resist out-of-plane forces.

14

7

1.3 Types of Structures

15

1.3 Types of Structures 1.3.1 Trusses When the span of a structure is required to be large and its depth is not important criterion in design, a truss may be selected. Trusses consist of slender elements usually arranged in triangular fashion. Planar trusses are composed of members that lie in the same plane and are frequently used for bridge and roof support, whereas space trusses have members extending in three dimensions and are suitable for towers.

16

8

1.3 Types of Structures 1.3.2 Cables Cables are usually flexible and carry their loads in tension. Unlike tension ties, the external load is not applied along the axis of the cable and consequently the cable takes form that has defined as sag. Cables are commonly use when the span are greater than 46 m.

17

1.3 Types of Structures 1.3.3 Arches The arch achieves its strength in compression, since it has a reverse curvature to that of the cable. The arch must be rigid, however, in order to maintain its shape, and this results in secondary loadings involving shear and moment, which must be considered in its design.

18

9

1.3 Types of Structures 1.3.4 Frames Frames are often used in buildings and are composed of beams and columns that either pin or fixed. Like trusses, frames extend in two or three dimensions. The loading causes bending of its members and if it has rigid joints connections, this structure is generally “indeterminate” from a standpoint of analysis

19

1.3 Types of Structures 1.3.5 Surface Structure A surface structure is made from a material having a very small thickness compared to its other dimensions. Sometimes this material is very flexible and can take the form of a tent or airinflated structure. In both cases the material acts as a membrane that is subjected to pure tension.

20

10

1.4 Loads on Structure

21

1.4 Loads on Structure 1.4.1 Dead Load Dead loads are gravity loads of constant magnitudes and fixed positions that act permanently on the structure. Such loads consist of the weights of the structural system itself and of all other material and equipment permanently attached to the structural system. (NSCP 2010 - Section 204)

22

11

1.4 Loads on Structure Problem 1-1 The floor system of a building consists of a 15-cm-thick reinforced concrete slab resting on four steel floor beams, which in turn are supported by two steel girders, as shown in figure below. The cross-sectional areas of the floor beams and the girders are 94.8 cm2 and 337.4 cm2, respectively. Determine the axial load at column A.

Answer 𝑷𝑨 = 𝟖𝟏. 𝟐𝟓𝟎𝟔 𝒌𝑵

23

1.4 Loads on Structure 1.4.2 Live Load Loads of varying magnitudes and/or positions caused by the use of the structure. Sometimes, the term live loads is used to refer to all loads on the structure that are not dead loads. (NSCP 2010 - Section 205)

24

12

1.4 Loads on Structure 1.4.2.1 Impact Load When live loads are applied rapidly to a structure, they cause larger stresses. The dynamic effect of the load that causes this increase in stress in the structure is referred to as impact. To account for the increase in stress due to impact, the live loads expected to cause such a dynamic effect on structures are increased by certain impact percentages, or impact factors. (NSCP 2010 - Section 206)

25

1.4 Loads on Structure

26

13

1.4 Loads on Structure Problem 1-2 A two-story office building columns that are spaced directions. If the (flat) roof reduced live load supported at ground level.

shown in the figure has interior 7 m apart in two perpendicular loading is 1.0 kPa, determine the by a typical interior column located

Answer 𝑷 = 𝟏𝟏𝟔. 𝟕𝟖𝟖 𝒌𝑵

27

1.4 Loads on Structure 1.4.3 Environmental Loads 1.4.3.1 Wind Load Wind loads are produced by the flow of wind around the structure. The magnitudes of wind loads that may act on a structure depend on the geographical location of the structure, obstructions in its surrounding terrain, such as nearby buildings, and the geometry and the vibrational characteristics of the structure itself. (NSCP 2010 - Section 207)

28

14

1.4 Loads on Structure

29

1.4 Loads on Structure

30

15

1.4 Loads on Structure

31

1.4 Loads on Structure Problem 1-3 The enclosed building shown in the figure below is used for storage purposes and is located at Bulacan on open flat terrain. When the wind is directed as shown, determine the design wind pressure acting on the roof and sides of the building using the NSCP 2010 Specifications.

8m

23 m 46 m

23 m 32

16

1.4 Loads on Structure 1.4.3.2 Earthquake Load An earthquake is a sudden undulation of a portion of the earth’s surface. Although the ground surface moves in both horizontal and vertical directions during an earthquake, the magnitude of the vertical component of ground motion is usually small and does not have a significant effect on most structures. It is the horizontal component of ground motion that causes structural damage and that must be considered in designs of structures located in earthquake-prone areas.. (NSCP 2010 - Section 208)

33

1.4 Loads on Structure

34

17

1.4 Loads on Structure Problem 1-4 A four-storey building located at TUP-Manila has a storey height 5 m on the ground floor and 4 m on the other floors. The roof deck has a dead weight of 400 kN while the second to fourth floor level has a dead weight of 800 kN. The building is to be located at seismic zone 4 under special occupancy category. A steel moment resisting frames will be used. Determine the lateral force at roof deck level.

Answer 𝑭𝑿 = 𝟏𝟏𝟎. 𝟔𝟔𝟕 𝒌𝑵

35

1.4 Loads on Structure

36

18

1.4 Loads on Structure 1.4.3.3 Hydrostatic and soil Pressures • Structures used to retain water, such as dams and tanks, as well as coastal structures partially or fully submerged in water must be designed to resist hydrostatic pressure. Hydrostatic pressure acts normal to the submerged surface of the structure, with its magnitude varying linearly with height. • Underground structures, basement walls and floors, and retaining walls must be designed to resist soil pressure. The lateral soil pressure depends on the type of soil and is usually considerably smaller than the vertical pressure. For the portions of structures below the water table, the combined effect of hydrostatic pressure and soil pressure due to the weight of the soil, reduced for buoyancy, must be considered. (NSCP 2010 - Section 209 to 211) 37

1.4 Loads on Structure 1.4.4 Load Combinations As stated previously, once the magnitudes of the design loads for a structure have been estimated, an engineer must consider all loads that might act simultaneously on the structure at a given time. (NSCP 2010 - Section 203)

38

19

1.5 Idealized Structure

39

1.5 Idealized Structure 1.5.1 Support Connections Structural members are joined together in various ways depending on the intent of the designer. The three types of joints most often specified are the pin connection, the roller support, and the fixed joint. A pin-connected joint and a roller support allow some freedom for slight rotation, whereas a fixed joint allows no relative rotation between the connected members and is consequently more expensive to fabricate.

40

20

1.5 Idealized Structure In reality, however, all connections exhibit some stiffness toward joint rotations, owing to friction and material behavior. If the torsional spring constant 𝑘 = 0 the joint is a pin, and if 𝑘 = ∞ the joint is fixed.

41

1.5 Idealized Structure 1.5.2 Line Diagram The analytical model of the two or three-dimensional body selected for analysis is represented by a line diagram. On this diagram, each member of the structure is represented by a line coinciding with its centroidal axis. The dimensions of the members and the size of the connections are not shown on the diagram.

42

21

1.5 Idealized Structure 1.5.3 Plane VS Space Structure If all the members of a structure as well as the applied loads lie in a single plane, the structure is called a plane structure. The analysis of plane, or two-dimensional, structures is considerably simpler than the analysis of space, or three-dimensional, structures. Fortunately, many actual three-dimensional structures can be subdivided into plane structures for analysis.

43

1.5 Idealized Structure 1.5.4 Tributary Loadings When flat surfaces such as walls, floors, or roofs are supported by a structural frame, it is necessary to determine how the load on these surfaces is transmitted to the various structural elements used for their support. There are generally two ways in which this can be done. The choice depends on the geometry of the structural system, the material from which it is made, and the method of its construction.

44

22

1.5 Idealized Structure 1.5.4.1 One Way Slab A slab or deck that is supported such that it delivers its load to the supporting members by one-way action, is often referred to as a one-way slab

45

1.5 Idealized Structure 1.5.4.2 Two Way Slab According to the American Concrete Institute (ACI) 318 concrete code the support ratio is 𝐿 /𝐿 ≤ 2 the load is assumed to be delivered to the supporting beams and girders in two directions. When this is the case the slab is referred to as a two-way slab. 𝑚= 𝑤 = 𝑤 =

46

23

1.5 Idealized Structure Problem 1-5 Given a 3m x 4m concrete slab with a thickness of 175 mm for a 2nd floor plan of a two storey residential building. Determine the loading (Dead) to be carried by the beams.

Answer 𝒘𝟏 = 𝟑. 𝟓𝟏 𝒌𝑵/𝒎 𝒘𝟐 = 𝟒. 𝟐𝟖 𝒌𝑵/𝒎

47

1.5 Idealized Structure 1.5.5 Structural Analysis 1.5.5.1 Linear Static Analysis A linear static analysis is an analysis where a linear relation holds between applied forces and displacements. In practice, this is applicable to structural problems where stresses remain in the linear elastic range of the used material. In a linear static analysis the model’s stiffness matrix is constant, and the solving process is relatively short compared to a nonlinear analysis on the same model.

48

24

1.5 Idealized Structure 1.5.5.2 Nonlinear Analysis A nonlinear analysis is an analysis where a nonlinear relation holds between applied forces and displacements. Nonlinear effects can originate from geometrical nonlinearity’s (i.e. large deformations), material nonlinearity’s (i.e. elasto-plastic material), and contact. These effects result in a stiffness matrix which is not constant during the load application. As a result, a different solving strategy is required for the nonlinear analysis and therefore a different solver.

49

1.5 Idealized Structure 1.5.5.2.1 Geometric Nonlinearity In analyses involving geometric nonlinearity, changes in geometry as the structure deforms are considered in formulating the constitutive and equilibrium equations. Many engineering applications such as metal forming, tire analysis, and medical device analysis require the use of large deformation analysis based on geometric nonlinearity. Small deformation analysis based on geometric nonlinearity is required for some applications, like analysis involving cables, arches and shells.

50

25

1.5 Idealized Structure 1.5.5.2.2 Material Nonlinearity Material nonlinearity involves the nonlinear behavior of a material based on a current deformation, deformation history, rate of deformation, temperature, pressure, and so on. Examples of nonlinear material models are large strain (visco) elasto-plasticity and hyperelasticity (rubber and plastic materials). 1.5.5.2.3 Constraint and Contact Nonlinearity Constraint nonlinearity in a system can occur if kinematic constraints are present in the model. The kinematic degrees-offreedom of a model can be constrained by imposing restrictions on its movement. 51

1.5 Idealized Structure

52

26

1.5 Idealized Structure

53

27