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 separately Linear relationship exists among loads, stresses & displacements 2 requirements for the principle to apply: o Material must behave in a linear-elastic manner, Hooke’s Law is valid o The geometry of the structure must not undergo significant change when the loads are applied, small displacement theory
The 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.
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
In a single plane or 2D structures, we consider
Since the externally applied force system is in equilibrium, the three equations of static equilibrium must be satisfied, i.e. +ve ↑ Σ Fy = 0 The sum of the vertical forces must equal zero. +ve Σ M = 0 The sum of the moments of all forces about any point on the plane of the forces must equal zero. +ve → Σ Fx = 0 The sum of the horizontal forces must equal zero. * The assumed positive direction is as indicated.
If the reaction forces can be determined solely from the equilibrium EQs STATICALLY DETERMINATE STRUCTURE No. of unknown forces > equilibrium EQs STATICALLY INDETERMINATE STRUCTURE Can be viewed globally or locally (via free body diagram)
Determinacy and Indeterminacy o For a 2D structure o The additional EQs needed to solve for the unknown forces are referred to as compatibility EQs No. of reaction support No. of components
For rolled support (r = 1) For pin support (r = 2) For fixed support (r=3) FyFy FyFy FxFx FyFy FxFx M
Stability - Structures must be properly held or constrained by their supports In 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.
Partial constraints o Fewer reactive forces than equilibrium EQs o Some equilibrium EQs can not be satisfied o Structure or Member is unstable
Improper constraints o In some cases, unknown forces may equal equilibrium EQs o However, instability or movement of structure could still occur if support reactions are concurrent at a point
Improper constraints Parallel Concurrent
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
Determine the reactions on the beam as shown. 135 kN 60.4 kN kN 50.7 kN Ignore thickness kN Ignore thickness 1.Draw Free Body Diagram 2.Use Eq of Equilibrium