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A cantilever beam AB, fixed at the end A and carrying a load W at the force end B, is found to deflect by δ at the midpoint of AB. The deflection of B due to load W/2 at the midpoint will be
Using Maxwell Reciprocal theorem
Virtual Work Done = Constant
(Load at Mid Point) (Deflection at Mid-Point due to load at B) = (Load at B) (Deflection at B due to load at Mid Point)
(0.5 W) (δ) = ( W) (Deflection at B)
Deflection at B = (δ)/2
Due to some point load anywhere one fixed beam, the maximum free bending moment is M. The sum of fixed end moments is?
Here we will apply the principle of superposition. For a simply supported beam, a concentrated load will form a triangle. The peak of triangle will be M as given in the question,
Area of free bending moment diagram = Area of triangle = 0.5 x M x L
Now, superposing the fixed end moments of supports say M1 and M2. The BMD by these fixed end moments will be a trapezoid with parallel sides having magnitude M1 and M2.
Area of fixed bending moment diagram = Area of trapezoid = - 0.5 x (M1 + M2) x L
As the slopes at the ends of a fixed beam is zero. So, the area of total B.M.D. )fixed end moment + free moment) will be zero.
-0.5 x (M1 + M2) x L = 0.5 x M x L
(M1 + M2) = M
The bending moment diagram for an overhanging beam is shown in the below figure
The points of contra-flexure for the above beam are
Point of contraflexure is the point where bending moment changes its sign. As it can be seen at point C it becomes negative to positive and at point D it becomes positive to negative, therefore, point C and D are points of contraflexure.
A fixed beam of uniform section is carrying a point load at its midspan. When the moment of inertia of the middle half length is reduced to half its original value, then the fixed and moments
Fixed end moment due to concentrated load P is given by PL/8.
Since, the fixed end moment does not depend on the cross-sectional properties, hence, it will remain same.
In case of a cantilever beam carrying uniformly varying load, the ratio of the maximum bending moment at free end for conditions, that is, when the load increase from zero at free end to w at fixed end and to that when the load increase from zero at fixed end is
A simply supported beam is subjected to an eccentric concentrated load. Where does the maximum deflection of the beam due to the applied load occur?
The deflected shape of beam is shown in the figure below. It can be seen that the maximum deflection occurs between the load point and center of the beam.
A simply supported beam AB of span 1 has a uniform cross section throughout. It carries a central concentrated load W and another load that is uniformly distributed over the entire span, its total magnitude being W. The minimum deflection in the beam is
For the beam shown below, the collapse load P is given by
The shear force diagram for the portal frame loaded as shown below is
The influence line diagram for the force in members of the truss shown below is
What is the carry-over factor from A to B while using moment distribution for analyzing the given beam?
Carry-over factor means the amount by which moment applied on one support is developed on another support. For a fixed adjacent support in a uniform beam carry over factor is ½. But here, since the cross-section is larger on side A, thus, moment of Inertia will be higher leading to higher flexural rigidity. Hence, support A can bear larger moment in comparison to B.
Therefore, carry-over factor from A to B will be less than 0.5 while using moment distribution method.
The influence line for force in member BC is
Member BC will act as a zero force member unless the unit load is between A and D. This is because member AC and CD are collinear horizontal members and any vertical force, developed in member BC cannot exist due to violation of Fy =0 equilibrium equation Moreover when the unit load is at C entire unit load will be balanced by member BC.
Therefore, the option D matches with the above description.
What is the value of flexibility coefficient f12 for the continuous beam shown below?
What is the ordinate of influence line at B for reaction RD in the figure below?
What is the value of vertical reaction at A for the frame shown below?