Deflection and Slope by IntegrationEIw" = -M(x)EIw' =-/ M(x)dx +CElw = -{ [ M(x)dxdx +Cx + D: Conventionally assuming constant flexural rigidity (E) Integration constants C and D can be determined from boundaryconditions, symmetry conditions, and continuity conditions6
EIw M(x) EIw M x x C ( )d EIw M(x)dxdx Cx D • Integration constants C and D can be determined from boundary conditions, symmetry conditions, and continuity conditions. Deflection and Slope by Integration • Conventionally assuming constant flexural rigidity (EI) 6
Boundary Conditions - Simple Beams30kN/m30kN/m30kN·mBCA2m2m11. Deflections are restrained at the hinged/rolled supports=W^=0; W=01
0; 0 w w A B Boundary Conditions – Simple Beams • Deflections are restrained at the hinged/rolled supports 7
Boundary Conditions- Cantilever BeamsmAT. Both the deflection and rotation are restrained at theclamped end=W=0; =08
0; 0 wA A 8 Boundary Conditions- Cantilever Beams • Both the deflection and rotation are restrained at the clamped end
Symmetry Conditions? Both the geometry andloads are symmetric aboutBthe mid-section (x = L/2)L=→=02L2l2qM123EBDCAB219
Symmetry Conditions • Both the geometry and loads are symmetric about the mid-section (x = L/2) 0 C 9
Continuity ConditionswPAB1aX2LL0≤ ≤a,α≤x≤L,O≤≤L-αw(x =a)=w(x2 =a); Φ(x =a)=0(x2 =a)w(x =a)=w(x =L-a); (xi =α)=-0(x =L-α10
Continuity Conditions 1 2 3 1 2 1 2 1 3 1 3 0 , , 0 ; ; x a a x L x L a w x a w x a x a x a w x a w x L a x a x L a 10 P