4.6 through 4.28 Determine the force in each member of the truss shown by the method of joints. FIG P4.6

Find the forces in the members of the truss by the method of joints.

Given information:

Apply the sign conventions for calculating reactions, forces and moments using the three equations of equilibrium as shown below.

• For summation of forces along x-direction is equal to zero (∑Fx=0), consider the forces acting towards right side as positive (→+) and the forces acting towards left side as negative (←−).
• For summation of forces along y-direction is equal to zero (∑Fy=0), consider the upward force as positive (↑+) and the downward force as negative (↓−).
• For summation of moment about a point is equal to zero (∑Mat a point=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Method of joints:

The negative value of force in any member indicates compression (C) and the positive value of force in any member indicates Tension (T).

Consider the horizontal and vertical reactions at D are Dx and Dy.

Consider the horizontal reaction at A are Ax.

Calculation:

Show the free body diagram of the truss as shown in Figure 1.

Refer Figure 1.

For Equilibrium of forces,

Take the sum of all horizontal forces in x direction as zero.

∑Fx=0Dx+Ax=0Dx=−Ax (1)

Take the sum of all vertical forces in y direction as zero.

∑Fy=0−Dy−5−10−10=0Dy=−25 k

Take sum of moments about the support A as zero.

∑MA=0−Dx×10−10×12−10×24=0−10Dx=360Dx=−36 k

Substitute −36 k for Dx in Equation (1).

Ax=36 k

Consider the triangle DCA.

Consider the angle DCA is θ. Then,

tanθ=1024θ=tan−1(1024)θ=22.61°

Show the joint C as shown in Figure 2.

Refer Figure 2.

Take the sum of the forces acting in the vertical direction as zero. Then,

FECsinθ=10FECsin(22.61°)=10FEC=10sin(22.61°)FEC=26 k(T)

Take the sum of the forces acting in the horizontal direction as zero. Then,

−FBC−FECcosθ=0FBC=−FECcos(22.61°)FBC=−26cos(22.61°)FBC=−24 kFBC=24 k(C)

Show the joint B as shown in Figure 3.

Refer Figure 3.

Take the sum of the forces acting in the vertical direction as zero. Then,

FBE=10 k(T)

Take the sum of the forces acting in the horizontal direction as zero