Shear & Moment in Beams
DEFINITION OF A BEAM
A beam is a bar subject to forces or couples that lie in a plane containing the
longitudinal of the bar. According to determinacy, a beam may be determinate or
indeterminate.
STATICALLY DETERMINATE BEAMS
Statically determinate beams are those beams in which the reactions of the supports
may be determined by the use of the equations of static equilibrium. The beams shown
below are examples of statically determinate beams.
STATICALLY INDETERMINATE BEAMS
If the number of reactions exerted upon a beam exceeds the number of equations in
static equilibrium, the beam is said to be statically indeterminate. In order to solve the
reactions of the beam, the static equations must be supplemented by equations based
upon the elastic deformations of the beam.
The degree of indeterminacy is taken as the difference between the umber of reactions
to the number of equations in static equilibrium that can be applied. In the case of the
propped beam shown, there are three reactions R1, R2, and M and only two equations
(∑M = 0 and sum;Fv = 0) can be applied, thus the beam is indeterminate to the first
degree (3 – 2 = 1).
,TYPES OF LOADING
Loads applied to the beam may consist of a concentrated load (load applied at a point),
uniform load, uniformly varying load, or an applied couple or moment. These loads are
shown in the following figures.
, Shear and Moment Diagrams
Consider a simple beam shown of length L that
carries a uniform load of w (N/m) throughout its
length and is held in equilibrium by reactions R1
and R2. Assume that the beam is cut at point C a
distance of x from he left support and the portion of
the beam to the right of C be removed. The portion
removed must then be replaced by vertical
shearing force V together with a couple M to hold
the left portion of the bar in equilibrium under the
action of R1 and wx. The couple M is called the resisting moment or moment and the
force V is called the resisting shear or shear. The sign of V and M are taken to be
positive if they have the senses indicated above.
Solved Problems in Shear and Moment Diagrams
INSTRUCTION
Write shear and moment equations for the beams in the following problems. In each
problem, let x be the distance measured from left end of the beam. Also, draw shear
and moment diagrams, specifying values at all change of loading positions and at points
of zero shear. Neglect the mass of the beam in each problem.
Problem 403
Beam loaded as shown in Fig. P-403.
DEFINITION OF A BEAM
A beam is a bar subject to forces or couples that lie in a plane containing the
longitudinal of the bar. According to determinacy, a beam may be determinate or
indeterminate.
STATICALLY DETERMINATE BEAMS
Statically determinate beams are those beams in which the reactions of the supports
may be determined by the use of the equations of static equilibrium. The beams shown
below are examples of statically determinate beams.
STATICALLY INDETERMINATE BEAMS
If the number of reactions exerted upon a beam exceeds the number of equations in
static equilibrium, the beam is said to be statically indeterminate. In order to solve the
reactions of the beam, the static equations must be supplemented by equations based
upon the elastic deformations of the beam.
The degree of indeterminacy is taken as the difference between the umber of reactions
to the number of equations in static equilibrium that can be applied. In the case of the
propped beam shown, there are three reactions R1, R2, and M and only two equations
(∑M = 0 and sum;Fv = 0) can be applied, thus the beam is indeterminate to the first
degree (3 – 2 = 1).
,TYPES OF LOADING
Loads applied to the beam may consist of a concentrated load (load applied at a point),
uniform load, uniformly varying load, or an applied couple or moment. These loads are
shown in the following figures.
, Shear and Moment Diagrams
Consider a simple beam shown of length L that
carries a uniform load of w (N/m) throughout its
length and is held in equilibrium by reactions R1
and R2. Assume that the beam is cut at point C a
distance of x from he left support and the portion of
the beam to the right of C be removed. The portion
removed must then be replaced by vertical
shearing force V together with a couple M to hold
the left portion of the bar in equilibrium under the
action of R1 and wx. The couple M is called the resisting moment or moment and the
force V is called the resisting shear or shear. The sign of V and M are taken to be
positive if they have the senses indicated above.
Solved Problems in Shear and Moment Diagrams
INSTRUCTION
Write shear and moment equations for the beams in the following problems. In each
problem, let x be the distance measured from left end of the beam. Also, draw shear
and moment diagrams, specifying values at all change of loading positions and at points
of zero shear. Neglect the mass of the beam in each problem.
Problem 403
Beam loaded as shown in Fig. P-403.
