Solving Indeterminate beam by slope-deflection equations
Use slope deflection equations to find the resultant end moments and draw resultant bending moment diagram for the continuous beam shown in figure 7-4(a). Take EI as constant for the beam.
The given beam in figure 7-4(a) is statically indeterminate of degree 4. In this case two spans (AC and CD) of the beam are to be considered.
Step1: Calculate Fixed end moments
(FEM)AC = -PL/8 = 50*4/8 = -25 kNm
(FEM)CA = PL/8 = 25 kNm
(FEM)CD = -wL2 /8 = -20*3*3/8 = -15 kNm
(FEM)DC = 15 kNm
Step 2: Applying slope-deflection equations for AC;
as support A is fixed there is no possibility of rotation at A, therefore θA = 0; and also there is no settlement of support because the supports are rigid, so δ = 0;
Now substitute all the values in the above equations for span AC; we get
MAC =(EIθC )/2 -25 Eq. (1)
MCA =(EIθC ) +25 Eq. (2)
Step 3: Apply slope-deflection equations for span CD,
the support D is fixed therefore θD = 0; and also there is no settlement of support because the supports are rigid, so δ = 0;
Now substitute all the values in the above equations for span CD; we get
MCD =(4EIθC )/3 -15 Eq. (3)
MDC =(2EIθC )/3 +25 Eq. (4)
Now we have 4 equations (Eq. 1, 2 , 3 & 4) with 5 unknowns. The additional equation required is obtained by applying the moment equilibrium at support C;
Step 4: Σ MC = 0;
MCA + MCD = 0; therefore,
(EIθC ) +25 + (4EIθC )/3 -15 = 0;
yields θC = (-30/7)/EI
substituting the value of θC in eq. 1, 2, 3 & 4 we get the values for end moments
MAC = -27.14 kNm;
MCA = 20.71 kNm;
MCD = - 20.71 kNm;
MDC = 12.14 kNm;
Sign covention for Fixed end moment:
For left end the negative value is hogging whereas positive value is sagging;
For right end the positve value is hogging whereas negative value is sagging
This way all the above moment calculated at the fixed ends are hogging in nature.
To determine the sagging moments, each span is considered as simply supported and the bending moments are calculated.
The Sagging moment diagram for AC will be triangle with maximum value = PL/4 = 50 x 4/4 = 50 kNm at the position of point load.
The sagging moment diagram for CD will be parabolic with maximum at the center = wL2/8 = 20 x 32/8 = 22.5 kNm
You can also use our bending moment calculator to determine the values of sagging moment for AC and CD.
The Resultant bending moment diagram is shown in Figure 7-4(b) and is drawn by considering sagging moments as positive (black in colour) and hogging moments as negative (red in colour).
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