AS101 Theory of Structures Lecture 1 PDF

Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University AS101 Theory of Structures PART I: STRUCTURAL ANALYSIS AND STATICALLY DETERMINATE BEAMS Introduction to Structural Analysis and L...

Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University AS101 Theory of Structures PART I: STRUCTURAL ANALYSIS AND STATICALLY DETERMINATE BEAMS Introduction to Structural Analysis and Loads Introduction to Structural Analysis Loads on Structures Analysis of Statically Determinate Beams Equilibrium and Support Reactions Determinacy and Stability of Structures ENGR. MICAH ELLA V. TAGLE INSTRUCTOR College of Engineering  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University THEORY OF STRUCTURES Deals with the principles and methods by which the direct stress, the shear and bending moment, and the deflection at any section of each constituent member in the structure may be calculated.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ENGINEERING STRUCTURES Something constructed or built Bridges, buildings, walls, dams, tower and shell structures Composed of one or more solid elements To design a structure involves many considerations among which are two major objectives that must be satisfied: - performance requirement - must carry loads safely  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ENGINEERING STRUCTURES The complete design of a structure is outlined in the following stages 1. Developing a general layout 2. Investigating the loads 3. Stress analysis 4. Selection of elements 5. Drawing and detailing  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ENGINEERING STRUCTURES Three major types of basic structures 1. Beam – is a straight member subjected only to transverse loads  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ENGINEERING STRUCTURES Three major types of basic structures 2. Truss – composed of members connected by frictionless hinges or pins  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ENGINEERING STRUCTURES Three major types of basic structures 3. Rigid Frame – built of members connected by rigid joints capable of resisting moments  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University LOAD CALCULATIONS Dead loads – includes the weight of the structure itself such as walls, plasters, ceilings, floors, beams, columns, and roofs Live loads – loadings to be carried by the structure. Live loads are moveable or temporarily attached to a structure. They include the loads on a building created by the storage of furniture and equipment, occupancy (people), and impact. Impact loads– dynamic effect of the application of live load. Impact loads are sudden or rapid loads applied on a structure over a relatively short period of time compared with other structural loads. Examples of impact loads are loads from moving vehicles, vibrating machinery, or dropped weights.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University LOAD CALCULATIONS Environmental loads 1. Rain loads - loads due to the accumulated mass of water on a rooftop during a rainstorm or major precipitation. 2. Wind loads - are pressures exacted on structures by wind flow. 3. Snow loads - the force exerted by accumulated snow and ice on buildings’ roofs 4. Seismic loads - loads exerted on a structure by the ground motion caused by seismic forces. 5. Hydrostatic and Earth Pressures - Hydrostatic pressure is the pressure that water exerts on a surface when it's standing still. Earth Pressure is the horizontal pressure applied by or to soil when it is against standing structures like basements (buildings) and retaining walls.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University LOAD CALCULATIONS  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University LOAD CALCULATIONS  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University EQUATIONS OF EQUILIBRIUM ෍ 𝐹𝑥 = 0 ෍ 𝐹𝑦 = 0 ෍ 𝑀𝑎 = 0 Where σ 𝐹𝑥 = summation of x component of each force in the system σ 𝐹𝑦 = summation of y component of each force in the system σ 𝑀𝑎 = summation of moment about any point a in the plane due to each force in the system  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 1. Link – one unknown, the reaction is a force that acts in the direction of link or cable  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 2. Smooth pin or hinged – two unknowns, consists of vertical and horizontal components  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 3. Rollers – one unknown, the reaction is always perpendicular to the surface at the point of contact  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 4. Slider – two unknowns, an axial force and a moment  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 5. Fixed connected collar – two unknowns, an axial and a moment  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 6. Fixed support – three unknowns, horizontal and vertical reaction plus a moment  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 7. Smooth pin-connected collar – one unknown, the reaction is acting perpendicular to the surface at the point of contact  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES 8. Smooth contact surface – one unknown, the reaction is acting perpendicular to the surface at the point of contact  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University TYPES OF SUPPORTS FOR PLANE STRUCTURES  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University LOADING IDEALIZATION CONCENTRATED LOADS Concentrated or Point Load is a load acting on a small elemental area. DISTRIBUTED LOADS Distributed loads are forces which are spread out over a length, area, or volume. i. Uniformly Distributed Load ii. Linearly Distributed Load  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University EQUIVALENT CONCENTRATED LOADS (ECL) NOTE: This conversion is only applicable when trying to simplify the equations of equilibrium.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ECL SAMPLE PROBLEM Determine the reactions on the beam below.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University ECL SAMPLE PROBLEM - SOLUTION Determine the reactions on the beam below.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University INCLINED LOADING Loads can be inclined as well. These type of loadings are defined by either an angle or a slope ratio. The easiest way to deal with these loading are to split them into their x and y components. This is done either by using trigonometric functions or the slope ratios directly.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University INCLINED LOADING 𝑜𝑝𝑝 𝑜𝑝𝑝 = 5𝑠𝑖𝑛60° 𝑜𝑝𝑝 = 5𝑠𝑖𝑛60° sin 60° = 5 𝑎𝑑𝑗 𝑎𝑑𝑗 = 5𝑐𝑜𝑠60° cos 60° = 𝑎𝑑𝑗 = 5𝑐𝑜𝑠60° 5  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University INCLINED LOADING c 𝑐 2 = 𝑎2 + 𝑏2 4 𝑐 2 = 42 + 32 𝑐 2 = 16 + 9 4 3 𝑐 2 = 25 (300) 5 𝑐=5 3 (300) 5  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University INCLINED LOADING – SAMPLE PROBLEM Determine the reactions on the beam shown below:  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University INCLINED LOADING – SAMPLE PROBLEM SOLUTION  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES Static Determinacy, Indeterminacy and Instability 1. Internally Stable – a structure is considered internally stable or rigid if it maintains its shape and remains in rigid body when detached from the supports 2. Internally Unstable – a structure is considered internally unstable or non rigid if it cannot maintain its shape and may undergo large displacements under small disturbances when not supported externally 3. Rigid Structures – a structure that offers significant resistance to its change of shape 4. Nonrigid Structures – a structure that offers negligible resistance to its change of shape when detached from the supports and would often collapse under its own weight when not supported externally 5. Statically Determinate Structures – an internally stable structure can be statically determinate externally if all its support reactions can be determined by solving the three static equations of equilibrium 6. Statically Indeterminate Structures – if a structure is supported by more than three reactions, then all reactions cannot be determined from the three static equations, such structures are termed as statically indeterminate externally 7. External Redundant – the excess reactions of those necessary for equilibrium 8. Degree of Indeterminacy – this is the number of external redundant.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES Determinacy of Beams 1. If 𝑟 < 3n The beam is unstable 2. If 𝑟 = 3n The beam is statically determinate provided that no geometric instability (internal and external) is involved 3. If 𝑟 > 3n The beam is statically indeterminate 4. If the 3 reactions will meet at a common point, the beam is unstable 5. If the 3 reactions are parallel to each other, the beam is unstable r – number of reactions n – number of parts or sections r-3n = Degree of Indeterminacy  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES Determinacy of Beams r – number of reactions n – number of parts or sections r-3n = Degree of Indeterminacy A B Internal Hinge P P B A V V  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES Determinacy of Beams r – number of reactions n – number of parts or sections r-3n = Degree of Indeterminacy A B Internal Roller B A V V  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES Determinacy of Trusses 1. If 𝑏 + 𝑟 < 2j The system is unstable 2. If 𝑏 + 𝑟 = 2j The beam is statically determinate provided that it is also stable 3. If 𝑏 + 𝑟 > 2j The beam is statically indeterminate r – number of reactions b – number of bars j – joints Degree of Indeterminacy= 𝑏 + 𝑟 − 2j  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES Determinacy of Rigid Frames 1. If 3𝑏 + 𝑟 < 3j + c The frame is unstable 2. If 3𝑏 + 𝑟 = 3j + c The frame is statically determinate provided that it is also stable 3. If 3𝑏 + 𝑟 > 3j + c The frame is statically indeterminate r – number of reactions b – number of bars j – rigid joints h – hinges c – number of equations of conditions ( c = 1 for hinge; c = 2 for roller; c = 0 for a beam without internal connection) Degree of Indeterminacy= 3𝑏 + 𝑟 − 3j + c  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 1. Indicate from the given structure if it is: a. Unstable b. Stable and determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 1. Indicate from the given structure if it is: a. Unstable b. Stable and determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree The beam is unstable because the 3 reactions will meet at a common point.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 2. Indicate from the given structure if it is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 2. Indicate from the given structure if it is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree The frame has only 2 reactions which is less than the 3 static equations, therefore, it is unstable.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 3. Indicate if the structure shown is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 3. Indicate if the structure shown is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree The structure is unstable since the 3 reactions are concurrent at B.  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 4. Indicate if the structure shown is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 4. Indicate if the structure shown is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree r=3 r _ 3n 3 _ 3(1) 3 = 3 (statically determinate)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 5. Indicate if the structure shown is: a. Determinate b. Indeterminate to the 1st degree c. Indeterminate to the 2nd degree d. Indeterminate to the 3rd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 5. Indicate if the structure shown is: a. Determinate b. Indeterminate to the 1st degree c. Indeterminate to the 2nd degree d. Indeterminate to the 3rd degree r=6 r _ 3n 6 _ 3(1) 6 > 3 (statically indeterminate to the 3rd degree)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 6. A beam is fixed at both ends and has an internal hinge along the span. All loads are acting in any direction of the beam. Indicate if it is: a. Unstable b. Determinate c. Indeterminate to the 2nd degree d. Indeterminate to the 3rd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 6. A beam is fixed at both ends and has an internal hinge along the span. All loads are acting in any direction of the beam. Indicate if it is: a. Unstable b. Determinate c. Indeterminate to the 2nd degree d. Indeterminate to the 3rd degree r=8 r _ 3n 8 _ 3(2) 8 ) 6 (statically indeterminate to the 2nd degree)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 7. Classify the structure whether it is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 7. Classify the structure whether it is: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree r=5 r _ 3n 5 _ 3(1) 5 ) 3 (statically indeterminate to the 2nd degree)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 8. Determine the determinacy of the given trusses: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 8. Determine the determinacy of the given trusses: a. Unstable b. Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree b = 10 r=6 j=8 b + r _ 2j 10 + 6 _ 2(8) 16 = 16 (statically determinate)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 9. Determine the given trusses whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 9. Determine the given trusses whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree b=9 r=6 j=7 b + r _ 2j 9 + 6 _ 2(7) 15 ) 14 (statically indeterminate to the 1st degree)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 10. Determine the given trusses whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 10. Determine the given trusses whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree b = 15 r = 6 j = 10 b + r _ 2j 15 + 6 _ 2(10) 21 ) 20 (statically indeterminate to the 1st degree)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 11. Determine the given frame whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 11. Determine the given frame whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree b=3 r=3 j=4 c=0 3b + r _ 3j + c 3(3) + 3 _ 3(4) + 0 12 = 12 (statically determinate)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 12. Determine the given frame whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 12. Determine the given frame whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 1st degree d. Indeterminate to the 2nd degree b=3 r=6 j=4 c=1 3b + r _ 3j + c 3(3) + 6 _ 3(4) + 1 15 ) 13 (statically indeterminate to the 2nd degree)  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 13. Determine the given frame whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 6th degree d. Indeterminate to the 7th degree  Manuel S. Enverga University Foundation, Lucena City College of Engineering An Autonomous University STABILITY AND DETERMINACY OF STRUCTURES 13. Determine the given frame whether it is: a. Unstable b. Statically Determinate c. Indeterminate to the 6th degree d. Indeterminate to the 7th degree b=9 r=4 j=8 c=0 3b + r _ 3j + c 3(9) + 4 _ 3(8) + 0 31 ) 24 (statically indeterminate to the 7th degree