13 Principle of Superposition Total disp. (or internal loadings, stress) at a point in a structure subjected to several external loadings can be determined by adding together the displacements (or internal loadings, stress) caused by each of the external loads acting separatelyLinear relationship exists among loads, stresses & displacements2 requirements for the principle to apply:Material must behave in a linear-elastic manner, Hooke’s Law is validThe geometry of the structure must not undergo significant change when the loads are applied, small displacement theory
16 Static EquilibriumThe state of an object when it is at rest or moving with a constant velocity.There may be several forces acting on the object.If they are canceling each other out and the object is not accelerating, then it is in a state of static equilibrium.
17 Equations of Equilibrium A structure or one of its members in equilibrium is called statics member when its balance of force and moment.In general this requires that force and moment in three independent axes, namely
18 Equations of Equilibrium In a single plane or 2D structures, we consider
19 Static Equilibrium * The assumed positive direction is as indicated. Since the externally applied force system is in equilibrium, the three equations of static equilibrium must be satisfied, i.e.+ve ↑ ΣFy = 0 The sum of the vertical forces must equal zero.+ve ΣM = 0 The sum of the moments of all forces about any point on the plane of the forces must equal zero.+ve → ΣFx = 0 The sum of the horizontal forces must equal zero.* The assumed positive direction is as indicated.
21 DeterminacyIf the reaction forces can be determined solely from the equilibrium EQs STATICALLY DETERMINATE STRUCTURENo. of unknown forces > equilibrium EQs STATICALLY INDETERMINATE STRUCTURECan be viewed globally or locally (via free body diagram)
22 Determinacy Determinacy and Indeterminacy For a 2D structure The additional EQs needed to solve for the unknown forces are referred to as compatibility EQsNo. of componentsNo. of reaction support
23 Reaction on a support connection For rolled support (r = 1)For pin support (r = 2)For fixed support (r=3)FyFyFxFyFxM
28 StabilityStability - Structures must be properly held or constrained by their supportsIn general, a structure is geometrically unstable if there are fewer reactive forces then equations of equilibrium.An unstable structure must be avoided in practice regardless of determinacy.
29 Stability Partial constraints Fewer reactive forces than equilibrium EQsSome equilibrium EQs can not be satisfiedStructure or Member is unstable
30 Stability Improper constraints In some cases, unknown forces may equal equilibrium EQsHowever, instability or movement of structure could still occur if support reactions are concurrent at a point
33 Application of the Equations of Equilibrium Free–Body Diagram - disassemble the structure and draw a free–body diagram of each member.Equations of Equilibrium - The total number of unknowns should be equal to the number of equilibrium equations
34 Determine the reactions on the beam as shown. Example 2-7Determine the reactions on the beam as shown.135 kN60.4 kN173.4 kN50.7 kNIgnore thickness183.1 kNIgnore thicknessDraw Free Body DiagramUse Eq of Equilibrium